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b(class)30 b(SOMAdaptionNetGNG)82 b(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(63)727 5337 y(5.2.7)106 b(class)30 b(SOMAdaptionGCells)f(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(64)p eop %%Page: 5 5 5 4 bop 382 249 a Fs(CONTENTS)2423 b Fr(v)727 548 y(5.2.8)106 b(class)30 b(SOMAdaptionNetGCells)d(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)f(.)h(.)93 b(64)382 752 y Fp(6)84 b(Examples)2309 b(66)518 865 y Fr(6.1)94 b(A)31 b(simple)d(sample)82 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.) f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(66)727 978 y(6.1.1)106 b(Kohonen)30 b(Grid)91 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(66)727 1090 y(6.1.2)106 b(Neural)30 b(Gas)70 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(67)727 1203 y(6.1.3)106 b(Gro)m(wing)30 b(Neural)g(Gas)56 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) f(.)h(.)93 b(68)518 1316 y(6.2)h(Animals)57 b(.)45 b(.)h(.)g(.)g(.)f(.) h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(68)727 1429 y(6.2.1)106 b(Kohonen)30 b(Grid)91 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(70)727 1542 y(6.2.2)106 b(Neural)30 b(Gas)70 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(71)518 1655 y(6.3)h(Iris)29 b(\015o)m(w)m(ers)e(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(72)727 1768 y(6.3.1)106 b(Neural)30 b(Gas)70 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(74)727 1881 y(6.3.2)106 b(Gro)m(wing)30 b(Neural)g(Gas)56 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) f(.)h(.)93 b(75)382 2085 y Fp(7)84 b(Outlo)s(ok)2379 b(77)518 2198 y Fr(7.1)94 b(Extensions)25 b(.)46 b(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.) f(.)h(.)g(.)f(.)h(.)93 b(77)727 2311 y(7.1.1)106 b(New)31 b(algorithms)i(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(77)727 2423 y(7.1.2)106 b(New)31 b(comp)s(onen)m(ts)55 b(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f (.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(78)727 2536 y(7.1.3)106 b(Sp)s(ecial)29 b(Applications)60 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h (.)93 b(79)518 2649 y(7.2)h(Conclusion)89 b(.)46 b(.)g(.)f(.)h(.)g(.)f (.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.) h(.)g(.)f(.)h(.)93 b(80)382 2853 y Fp(A)57 b(Input)35 b(\014les)2282 b(82)518 2966 y Fr(A.1)71 b(Map)31 b(data)69 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.) f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(82)518 3079 y(A.2)71 b(Map)31 b(op)s(erators)86 b(.)45 b(.)h(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) f(.)h(.)93 b(86)518 3192 y(A.3)71 b(Basic)31 b(utilities)54 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(89)518 3305 y(A.4)71 b(Neural)30 b(Gas)78 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f (.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.) h(.)93 b(92)518 3418 y(A.5)71 b(Gro)m(wing)30 b(Neural)g(Gas)65 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)f(.)h(.)93 b(93)382 3621 y Fp(B)62 b(T)-9 b(est)35 b(\014les)2343 b(95)518 3734 y Fr(B.1)75 b(A)31 b(simple)d(sample)82 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(95)727 3847 y(B.1.1)87 b(Kohonen)30 b(Grid)91 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)f(.)h(.)93 b(95)727 3960 y(B.1.2)87 b(Neural)30 b(Gas)70 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)93 b(99)727 4073 y(B.1.3)87 b(Gro)m(wing)30 b(Neural)g(Gas)56 b(.)46 b(.)g(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)h(102)518 4186 y(B.2)75 b(Animals)57 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) f(.)h(.)h(106)727 4299 y(B.2.1)87 b(Animals)29 b(Data)j(set)k(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)f(.)h(.)h(106)727 4412 y(B.2.2)87 b(Kohonen)30 b(Grid)91 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.) f(.)h(.)g(.)f(.)h(.)h(109)727 4525 y(B.2.3)87 b(Neural)30 b(Gas)70 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)h(112)518 4638 y(B.3)75 b(Iris)42 b(.)j(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)f(.)h(.)h(115)727 4751 y(B.3.1)87 b(Iris)29 b(Data)j(set)92 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)f(.)h(.)h(115)727 4863 y(B.3.2)87 b(Neural)30 b(Gas)70 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)h(122)727 4976 y(B.3.3)87 b(Gro)m(wing)30 b(Neural)g(Gas)56 b(.)46 b(.)g(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)h(125)382 5180 y Fp(C)60 b(Notation)2290 b(128)p eop %%Page: 1 6 1 5 bop 1676 920 a Fp(Abstract)523 1132 y Fr(T)-8 b(o)38 b(mo)s(del)e(a)i(ph)m(ysical)e(system)i(on)f(the)g(computer,)j(one)e (tries)e(to)i(understand)382 1245 y(what)c(parameters)h(in\015uence)f (the)g(system)h(and)f(ho)m(w)h(they)g(do.)53 b(This)33 b(information)382 1358 y(can)g(b)s(e)e(gained)h(using)f Fo(Data)k(A)n(nalysis)p Fr(.)47 b(Analysis)30 b(ma)m(y)j(b)s(e)f(done)g (using)f(statistical)382 1471 y(metho)s(ds)j(\(e.g)i(regression)e (analysis\))f(or)i Fo(Machine)h(L)-5 b(e)g(arning)44 b Fr(metho)s(ds)34 b(\(ML\))h(lik)m(e)382 1584 y(decision)f(trees,)k (cluster)d(analysis)f(or)i(Arti\014cial)e(Neural)h(Nets.)57 b(Though)35 b(all)f(these)382 1697 y(ML)22 b(metho)s(ds)f(di\013er)f (in)h(structural)f(and)h(algorithmic)g(form)g(and)g(in)f(ho)m(w)i(they) g(searc)m(h)382 1810 y(for)h(a)g(solution)f(\(in)g(some)i(kind)d(an)i (optimization)f(pro)s(cess\),)j(they)e(all)f(try)h(to)h(abstract)382 1923 y(general)30 b(concepts)h(using)e(sp)s(eci\014c)h(examples)f(\()p Fo(inductive)k(le)-5 b(arning)8 b Fr(\).)523 2036 y(Neural)22 b(Nets)i(are)f(inspired)d(b)m(y)i(the)h(h)m(uman)f(brain)g(as)h(they)g (use)f(small)g(but)g(highly)382 2149 y(connected)32 b(comp)s(onen)m(ts) f(\(neurons\))f(to)i(compute)f(the)g(result.)42 b(Information)29 b(is)h(not)382 2262 y(stored)g(at)h(a)g(single)e(lo)s(cation)h(but)g (spread)g(o)m(v)m(er)h(the)g(whole)e(net)m(w)m(ork.)523 2374 y(Self)39 b(organizing)g(maps)g(\(a)h(sp)s(ecial)f(kind)e(of)j (arti\014cial)e(neural)h(nets\))h(are)g(able)382 2487 y(to)35 b(map)f(a)h(data)g(set)g(on)m(to)g(a)g(prede\014ned)d(or)j (dynamicly)d(created)j(\(discrete\))g(top)s(o-)382 2600 y(logical)c(structure.)43 b(This)30 b(abilit)m(y)g(can)i(b)s(e)f(used)f (for)i(generalization)f(purp)s(ose)f(\(\014nd-)382 2713 y(ing)g(clusters)f(and)h(neigh)m(b)s(orho)s(o)s(d)e(relations\))i(as)h (w)m(ell)f(as)g(for)h(appro)m(ximation)e(tasks)382 2826 y(\(e.g.)39 b(appro)m(ximating)20 b(a)i(more)g(dimensional)c(function)i (with)g(a)i Fn(n)p Fr(-dimensional)c(grid\).)382 2939 y(Though)32 b(the)g(original)f(mo)s(del)g(dev)m(elop)s(ed)h(b)m(y)h (Kohonen)f(uses)g(a)h(\014x)f(net)h(structure,)382 3052 y(sev)m(eral)28 b(extensions)f(ha)m(v)m(e)i(b)s(een)e(dev)m(elop)s(ed)g (recen)m(tly)-8 b(,)29 b(whic)m(h)e(create)i(or)f(mo)s(dify)d(the)382 3165 y(size)30 b(of)h(the)f(net)h(on)f(demand)f(\()p 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y(system)f(is)g(Kohonens)f(original)g(algorithm)g(and)h(a)h (net,)j(based)c(up)s(on)f(a)i(general)382 4294 y(graph.)74 b(Using)41 b(this,)j(some)e(extensions)f(are)h(describ)s(ed:)62 b(Neural)41 b(Gas,)k(Gro)m(wing)382 4407 y(Neural)38 b(Gas,)k(Gro)m(wing)d(Cells,)h(Gro)m(wing)f(Grid)f(and)g(GSOMs.)67 b(The)38 b(general)h(net)382 4520 y(structure)33 b(is)g(also)h (extended)g(to)h(allo)m(w)e(the)h(setup)g(of)g(a)g Fn(n)p Fr(-dimensional)d(grid.)50 b(The)382 4633 y(framew)m(ork)30 b(w)m(as)h(implemen)m(ted)e(within)f(the)i Fo(A)n(nalysis++)k Fr(pac)m(k)-5 b(age)32 b(using)d Fo(C++)p Fr(.)523 4746 y(T)-8 b(o)39 b(giv)m(e)f(an)g(o)m(v)m(erview)h(of)g(the)f(abilities)e (of)i(the)g(metho)s(ds)g(and)f(to)i(sho)m(w,)i(ho)m(w)382 4858 y(di\013eren)m(t)34 b(algorithms)f(deal)g(with)g(the)h(same)h (data-set,)i(three)d(examples)g(are)h(giv)m(en)382 4971 y(at)c(the)g(end.)p eop %%Page: 1 7 1 6 bop 382 1171 a Fm(Chapter)65 b(1)382 1586 y Fq(In)-6 b(tro)6 b(duction)382 2031 y Fr(In)24 b(the)g(past,)i(data)g(w)m(as)e (usually)f(\\pro)s(cessed")h(b)m(y)h(a)g(giv)m(en)f(algorithm)g(or)g (form)m(ula)g(to)382 2144 y(gain)31 b(a)i(sp)s(eci\014c)e(result.)44 b(This)30 b(required)g(detailed)h(kno)m(wledge)h(ab)s(out)g(the)g(con)m (ten)m(ts)382 2257 y(and)20 b(structure)g(of)g(the)h(giv)m(en)f(data)h (set.)38 b(As)20 b(this)g(kno)m(wledge)g(is)f(not)i(alw)m(a)m(ys)g(a)m (v)-5 b(ailable,)382 2369 y(a)41 b(new)f(branc)m(h)g(of)h(metho)s(ds)f (w)m(as)h(dev)m(elop)s(ed,)i(that)e(should)e(allo)m(w)h(to)h (\\analyze")382 2482 y(new)g(\(and)g(prop)s(erly)e(unkno)m(wn\))h(data) i(in)e(some)i(w)m(a)m(y)-8 b(.)74 b(In)41 b(the)g(b)s(eginning,)h Fo(Data)382 2595 y(A)n(nalysis)48 b Fr(w)m(as)41 b(usually)d(done)i (using)f(statistical)i(metho)s(ds.)70 b(But)40 b(these)h(metho)s(ds)382 2708 y(also)d(required)e(a)i(lot)g(of)h(kno)m(wledge,)h(an)e(in-sigh)m (t)e(of)j(the)f(giv)m(en)g(data.)64 b(With)37 b(the)382 2821 y(arising)31 b(of)j(computers,)g(and)e(their)g(abilit)m(y)g(to)i (deal)f(with)f(great)i(amoun)m(ts)f(of)h(data,)382 2934 y(it)c(w)m(as)h(no)m(w)f(p)s(ossible)e(to)j(do)f(additional)f (analysis.)523 3047 y(One)48 b(of)h(these)g(new)g(dev)m(elop)s(ed)f (branc)m(hes)g(w)m(as)h Fo(Machine)g(L)-5 b(e)g(arning)p Fr(.)97 b(This)382 3160 y(means,)32 b(that)g(a)g Fo(machine)39 b Fr(\(the)32 b(computer\),)g(tries)f(to)h(\014nd)e(similar)f(parts)i (\(general-)382 3273 y(ization\))g(and)g(put)f(di\013eren)m(t)h(parts)g (in)m(to)g(relation)f(to)i(eac)m(h)h(other)e(\()p Fo(le)-5 b(arning)8 b Fr(\).)45 b(Ho)m(w)382 3386 y(learning)36 b(is)h(done,)j(di\013ers)d(a)h(lot)g(b)s(et)m(w)m(een)h(the)f (di\013eren)m(t)f(metho)s(ds,)j(but)d(one)h(can)382 3499 y(distinguish)33 b(t)m(w)m(o)38 b(approac)m(hes:)55 b Fo(Sup)-5 b(ervise)g(d)39 b(L)-5 b(e)g(arning)p Fr(,)40 b(where)d(learning)e(pro)s(ceeds)382 3611 y(b)m(y)c(presen)m(ting)f (not)h(only)f(the)h(input)e(data-set,)k(but)d(also)h(the)g(desired)e (output)i(\(e.g.)382 3724 y(if)41 b(animals)f(are)j(describ)s(ed,)g (the)f(sp)s(ecies\),)j(and)c Fo(Unsup)-5 b(ervise)g(d)45 b(L)-5 b(e)g(arning)p Fr(,)46 b(where)382 3837 y(learning)29 b(pro)s(ceeds)h(without)f(this)g(additional)g(information)f([5)q(,)i(6) q(].)523 3950 y(A)23 b(sp)s(ecial)f(branc)m(h)h(of)h(Mac)m(hine)f (Learning)f(metho)s(ds)h(are)h(Arti\014cial)d(Neural)i(Nets)382 4063 y(\(ANN\).)34 b(These)f(metho)s(ds)f(try)h(to)h(recreate)h(the)e (features)g(of)g(the)g(h)m(uman)f(brain)g([1,)382 4176 y(2,)f(3,)g(4].)41 b(Sev)m(eral)30 b(di\013eren)m(t)g(v)-5 b(arian)m(ts)30 b(exist,)g(sup)s(ervised)e(as)i(w)m(ell)g(as)g(unsup)s (ervised.)382 4289 y(One)g(of)g(the)h(most)g(\015exible)d(ANN)j(metho)s (ds)f(are)h Fo(Neur)-5 b(al)33 b(Maps)p Fr(.)523 4402 y(In)43 b(this)g(pap)s(er)g(I)h(will)d(presen)m(t)j(an)g(ob)5 b(ject)45 b(orien)m(ted)e(framew)m(ork)h(for)g Fo(Neur)-5 b(al)382 4515 y(Maps)p Fr(.)54 b(Neural)33 b(Maps)i(are)g(able,)g(to)g (appro)m(ximate)g(the)g(distribution)30 b(of)35 b(unkno)m(wn)382 4628 y(input)c(data)i(and)g(to)g(map)g(it)f(on)m(to)i(a)f(new)g(top)s (ological)f(structure.)48 b(As)33 b(opp)s(osed)f(to)382 4741 y(most)40 b(Neural)f(Net)h(metho)s(ds,)h(where)e(learning)f(is)h (done)g(b)m(y)g(sup)s(ervised)e(learning)382 4853 y(metho)s(ds,)30 b(the)g(algorithms)g(used)f(here)h(supp)s(ort)f(unsup)s(ervised)e (learning.)523 4966 y(The)43 b(original)f(mo)s(del)g(w)m(as)i(dev)m (elop)s(ed)e(b)m(y)i(Kohonen)f(and)g(is)f(kno)m(wn)h(as)h Fo(Self)382 5079 y(Or)-5 b(ganizing)32 b(Map)k Fr(or)29 b(SOM)g([13)q(,)h(16)q(].)40 b(Though)29 b(the)g(framew)m(ork)h(is)e (\015exible)g(enough)382 5192 y(to)41 b(hold)e(most)i(deriv)-5 b(ations)39 b(of)i(the)g(basic)e(mo)s(del,)k(the)d(fo)s(cus)g(in)f (this)h(pap)s(er)f(w)m(as)382 5305 y(set)e(on)f(gro)m(wing)g(nets.)59 b(This)35 b(group)h(of)g(metho)s(ds)g(allo)m(ws)g(not)g(only)g(the)g (mapping)1854 5652 y(1)p eop %%Page: 2 8 2 7 bop 382 249 a Fs(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)1570 b Fr(2)382 548 y(of)34 b(the)f(input)f(data)i(on)m(to)h(a)f(giv)m(en)g (top)s(ological)f(structure,)h(but)f(also)h(creating)g(this)382 661 y(structure)c(on)g(demand.)523 774 y(The)c(goal)h(of)g(this)e (thesis)g(is)h(to)h(generalize)g(the)f(basic)g(algorithm.)39 b(F)-8 b(or)27 b(that)g(pur-)382 887 y(p)s(ose)22 b(the)h(main)f(comp)s (onen)m(ts)h(are)g(isolated)f(and)g(put)g(in)m(to)h(relation)f(to)i (the)e(the)h(whole)382 1000 y(algorithm.)53 b(This)33 b(generalization)h(is)g(then)h(used,)g(to)h(describ)s(e)d(sev)m(eral)i (extensions)382 1112 y(in)29 b(their)g(relation)h(to)h(the)g(original)d (algorithm)i(and)f(to)i(eac)m(h)h(other.)523 1225 y(The)d(metho)s(ds)g (w)m(ere)h(implemen)m(ted)e(as)h(part)h(of)f(the)h(Analysis++)d(pac)m (k)-5 b(age,)32 b(us-)382 1338 y(ing)20 b(an)h(Ob)5 b(ject)21 b(Orien)m(ted)f(approac)m(h.)38 b(This)19 b(enables)i(users)f(of)h(the) g(metho)s(ds)g(to)g(easily)382 1451 y(in)m(terc)m(hange)28 b(parts)f(of)h(the)g(algorithms)e(and)h(to)h(compare)g(the)g(results.) 38 b(While)26 b(most)382 1564 y(previous)i(implemen)m(tations)g(of)i (these)h(algorithms)d(only)h(supp)s(orted)f(a)i(sp)s(ecial)f(case)382 1677 y(of)36 b(eac)m(h)h(algorithm,)g(this)e(framew)m(ork)h(is)f (completely)h(mo)s(di\014able.)55 b(Eac)m(h)37 b(metho)s(d)382 1790 y(can)j(b)s(e)g(replaced)g(to)h(test)g(the)f(results)f(with)g (di\013eren)m(t)h(setups.)70 b(This)38 b(is)i(a)g(great)382 1903 y(b)s(ene\014t,)27 b(b)s(ecause)g(one)g(program)g(can)g(b)s(e)g (used)f(to)i(do)e(sev)m(eral)i(tests)g(using)d(the)i(same)382 2016 y(input)d(data.)40 b(Also)25 b(dev)m(eloping)h(new)f(metho)s(ds)g (is)g(enormous)h(simpli\014ed,)d(as)j(an)m(y)g(old)382 2129 y(metho)s(d)31 b(can)g(b)s(e)g(in)m(terc)m(hanged)h(within)d(the)i (algorithm)g(while)e(the)j(other)f(metho)s(ds)382 2242 y(are)g(reused,)f(so)g(that)h(one)g(can)f(concen)m(trate)j(on)d(the)h (new)f(parts.)523 2355 y(The)22 b(application)e(of)i(these)g(metho)s (ds)g(range)g(from)g(dimension)d(reduction)i(of)h(high)382 2467 y(dimensional)k(data)k(o)m(v)m(er)g(v)m(ector)h(quan)m (ti\014cation)e(to)g(clustering.)39 b(A)m(t)30 b(the)g(end)e(of)h(the) 382 2580 y(pap)s(er)g(some)i(examples)f(are)h(giv)m(en)f(to)h (illustrate)e(these)i(abilities.)523 2693 y(The)g(program)g(w)m(as)g (completely)g(written)f(in)g(C++)g(and)h(is)f(part)h(of)h(the)f(Analy-) 382 2806 y(sis++)e(pro)5 b(ject.)382 3093 y Fl(1.1)135 b(Motiv)-7 b(ation)382 3296 y Fr(Arti\014cial)43 b(Neural)i(Nets)h (\(ANN\))g(try)f(to)h(recreate)h(the)e(features)h(of)f(the)g(h)m(uman) 382 3408 y(brain.)84 b(Lik)m(e)46 b(our)f(brain,)j(eac)m(h)e(net)g (consists)f(of)h(a)g(great)h(n)m(um)m(b)s(er)d(of)i(neurons,)382 3521 y(whic)m(h)36 b(are)j(connected)f(with)f(eac)m(h)i(other.)63 b(Information)37 b(pro)s(cessing)g(is)g(not)h(done)382 3634 y(b)m(y)i(taking)h(the)f(input)f(data)i(and)f(p)s(erforming)e(a)j (single)e(computation)i(to)g(obtain)382 3747 y(the)30 b(\014nal)e(result,)h(but)g(rather)g(b)m(y)h(spreading)e(the)i (information)e(o)m(v)m(er)j(the)f(whole)e(net)382 3860 y(and)g(m)m(ultiple)f(small)g(computations)i(in)e(eac)m(h)k(neuron.)39 b(The)28 b(neurons,)g(that)i(receiv)m(e)382 3973 y(input)j(information) g(form)i(the)g(input)e(la)m(y)m(er,)38 b(the)d(neurons)f(that)h(con)m (tain)h(the)f(\014nal)382 4086 y(result,)26 b(form)g(the)g(output)g(la) m(y)m(er)h(\(see)g(\014gure)f(1.1\).)41 b(Some)26 b(nets)g(use)g (neurons)f(that)i(lie)382 4199 y(b)s(et)m(w)m(een)g(this)e(t)m(w)m(o)i (la)m(y)m(ers.)40 b(They)26 b(build)d(the)k(so)f(called)f(hidden)f(la)m (y)m(er.)40 b(The)26 b(neurons)382 4312 y(are)k(connected)h(b)m(y)f (directed)f(w)m(eigh)m(ted)i(edges)f(so)g(that)h(the)f(output)g(of)g(a) g(neuron)f(in)382 4425 y(one)e(la)m(y)m(er)g(is)f(used)g(as)h(the)g (input)e(of)i(a)g(neuron)f(in)f(the)i(next)g(la)m(y)m(er.)40 b(An)27 b(in)m(tro)s(duction)382 4538 y(and)j(o)m(v)m(erview)h(of)f (ANNs)h(can)g(b)s(e)e(found)g(in)g([1)q(,)i(2,)g(3,)g(4].)523 4650 y(As)40 b(opp)s(osed)f(to)i(most)g(classical)e(metho)s(ds,)j(the)e (b)s(eha)m(vior)f(of)i(the)f(net)g(is)f(not)382 4763 y(prede\014ned,)31 b(but)g(mo)s(di\014ed)e(as)k(part)e(of)h(the)g (learning)e(pro)s(cess.)45 b(F)-8 b(or)33 b(that)f(purp)s(ose,)382 4876 y(sample)d(data)h(is)f(presen)m(ted)h(to)g(the)g(net)g(and)f(then) h(the)g(net)g(is)e(adapted)i(to)h(impro)m(v)m(e)382 4989 y(the)38 b(results.)62 b(This)35 b(ma)m(y)k(b)s(e)e(done)h(b)m(y)f (comparing)g(the)h(results)f(with)f(the)i(desired)382 5102 y(output)30 b(data)h(or)f(b)m(y)h(self-organizing)e(metho)s(ds.) 523 5215 y(As)39 b(the)g(information)e(is)h(spread)g(o)m(v)m(er)i(the)g (whole)e(net)h(and)f(eac)m(h)i(neuron)e(has)382 5328 y(only)29 b(a)h(limited)e(in\015uence)g(to)j(the)f(\014nal)e(result,)h (ANN's)i(\(an)f(in)e(particular)h(SOM's\))p eop %%Page: 3 9 3 8 bop 382 249 a Fs(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)1570 b Fr(3)1129 1400 y currentpoint currentpoint translate 0.47949 0.47949 scale neg exch neg exch translate 1129 1400 a 1129 1400 a gsave currentpoint currentpoint translate 0 neg rotate neg exch neg exch translate 1129 1400 a @beginspecial 27 @llx 587 @lly 401 @urx 823 @ury 3740 @rwi @setspecial %%BeginDocument: figures/ann.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 27 587 401 823 %%Creator: CorelDRAW 7 %%Title: %%CreationDate: Sun Oct 24 22:01:02 1999 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness true def /AutoSteps 0 def /CMYKMarks false def /UseLevel2 false def %Color profile: Deaktiviert %%BeginResource: procset wCorel7Dict 7.0 0 /wCorel7Dict 300 dict def wCorel7Dict begin % Copyright (c)1992-96 Corel Corporation % All rights reserved. v7.0 r1.0 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/setscreen{pop pop pop}bd/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd /SepMode_5 where{pop}{/SepMode_5 0 def}ifelse/CurrentInkName_5 where{pop} {/CurrentInkName_5(Composite)def}ifelse/$ink_5 where{pop}{/$ink_5 -1 def} ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def/$t 1 def/$n _ def/$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def/$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def/s1t 0 def/s1n _ def/$bkg false def /SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd /$ptm matrix def/$ttm matrix def/$stm matrix def/$ffpnt true def /CorelDrawReencodeVect[16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /L2?/languagelevel where{pop languagelevel 2 ge}{false}ifelse def /@BeginSysCorelDict{systemdict/Corel30Dict known{systemdict/Corel30Dict get exec}if}bd/@EndSysCorelDict{systemdict/Corel30Dict known{end}if}bd AutoFlatness {/@ifl{dup currentflat exch sub 10 gt{ ([Error: PathTooComplex; OffendingCommand: AnyPaintingOperator]\n)print flush @np exit}{currentflat 2 add setflat}ifelse}bd/@fill/fill ld/fill{currentflat{ {@fill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@eofill/eofill ld/eofill {currentflat{{@eofill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@clip /clip ld/clip{currentflat{{@clip}stopped{@ifl}{exit}ifelse}bind loop setflat} bd/@eoclip/eoclip ld/eoclip{currentflat{{@eoclip}stopped{@ifl}{exit}ifelse} bind loop setflat}bd/@stroke/stroke ld/stroke{currentflat{{@stroke}stopped {@ifl}{exit}ifelse}bind loop setflat}bd}if L2?{/@ssa{true setstrokeadjust}bd}{ /@ssa{}bd}ifelse/d/setdash ld/j/setlinejoin ld/J/setlinecap ld/M/setmiterlimit ld/w/setlinewidth ld/O{/$o xd}bd/R{/$O xd}bd/W/eoclip ld/c/curveto ld/C/c ld/l /lineto ld/L/l ld/rl/rlineto ld/m/moveto ld/n/newpath ld/N/newpath ld/P{11 rp} bd/u{}bd/U{}bd/A{pop}bd/q/@gs ld/Q/@gr ld/&{}bd/@j{@sv @np}bd/@J{@rs}bd/g{1 exch sub/$k xd/$c 0 def/$m 0 def/$y 0 def/$t 1 def/$n _ def/$fil 0 def}bd/G{1 sub neg/$K xd _ 1 0 0 0/$C xd/$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type /stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd /$fil 0 def}bd/K{1 index type/stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def} ifelse/$K xd/$Y xd/$M xd/$C xd}bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{ /s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{ dup 0 ne{setflat}{pop}ifelse}bd/v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2 copy c}bd/Y/y ld/@w{matrix rotate/$ptm xd matrix scale $ptm dup concatmatrix /$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd {/$scp xd/$sca xd/$scf xd}if}bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd} if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch load SetScr/$dsp xd/$dsa xd/$dsf xd}bd/$ngx{$SDF{$SCF SepMode_5 0 eq{$SCA} {$dsa}ifelse $SCP @ss}if}bd/p{/$pm xd 7 rp/$pyf xd/$pxf xd/$pn xd/$fil 1 def} bd/@MN{2 copy le{pop}{exch pop}ifelse}bd/@MX{2 copy ge{pop}{exch pop}ifelse}bd /InRange{3 -1 roll @MN @MX}bd/@sqr{dup 0 rl dup 0 exch rl neg 0 rl @cp}bd /currentscale{1 0 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt 0 1 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt}bd/@unscale{}bd/wDstChck{2 1 roll dup 3 -1 roll eq{1 add}if}bd /@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1 exch sub}bd /cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop}bd /rgb2cmyk{3{1 exch sub 3 1 roll}repeat 3 copy @MN @MN 3{dup 5 -1 roll sub neg exch}repeat}bd/rgb2g{2 index .299 mul 2 index .587 mul add 1 index .114 mul add 4 1 roll pop 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(defect)609 1563 y(of)37 b(one)h(neuron)e(has)g(also)h(only)g(limited)d (e\013ect.)63 b(This)35 b(is)h(esp)s(ecially)f(imp)s(or-)609 1676 y(tan)m(t)c(if)e(computation)h(is)e(done)i(in)f(a)m(wkw)m(ard)h (en)m(vironmen)m(ts)f(\(e.g.)42 b(b)s(eneath)29 b(a)609 1789 y(furnace\).)382 2033 y Fj(1.1.1)112 b(Self)38 b(Organizing)e (Maps)382 2204 y Fr(Self)j(Organizing)f(Maps)j(w)m(ere)f(in)m(tro)s (duced)f(in)g(1982)i(b)m(y)f(T.)g(Kohonen)g([12)q(].)70 b(The)382 2317 y(motiv)-5 b(ation)39 b(w)m(as)h(to)h(\014nd)d(an)i (algorithm,)h(\\that)g(w)m(ould)d(e\013ectiv)m(ely)j(map)e(similar)382 2430 y(patterns)28 b(\(pattern)h(v)m(ectors)h(close)e(to)h(eac)m(h)h (other)e(in)f(the)i(input)d(signal)h(space\))i(on)m(to)382 2543 y(con)m(tin)m(uous)c(lo)s(cations)f(in)g(the)i(output)f(space".)40 b(T)-8 b(o)25 b(solv)m(e)h(this)e(problem,)h(he)g(in)m(v)m(esti-)382 2656 y(gated)j(the)f(SOM)f(algorithm.)38 b(The)26 b(algorithm)g (creates)i(a)f(mapping)e(from)i(the)g(input)382 2769 y(space)e(on)m(to)h(a)f(discrete)f(top)s(ological)g(structure)g(\(e.g.) 41 b(a)24 b(grid\),)i(where)e(the)g(ab)s(o)m(v)m(e)i(cri-)382 2882 y(teria)e(is)g(ful\014lled.)35 b(This)22 b(mapping)h(is)h(mo)s (di\014ed)e(in)h(eac)m(h)i(learning)e(step,)j(to)f(minimize)382 2995 y(the)36 b(o)m(v)m(erall)h(error.)58 b(T)-8 b(o)37 b(ac)m(hiev)m(e)g(a)g(con)m(tin)m(uous)f(mapping,)g(not)h(only)e(one)h (neuron,)382 3107 y(but)j(also)g(its)g(neigh)m(b)s(ors)g(in)f(the)i (output)f(space)h(are)g(mo)s(di\014ed.)67 b(This)37 b(metho)s(d)i(of) 382 3220 y(lateral)31 b(in)m(teraction)g(w)m(as)h(used)f(b)s(efore)g (for)g(asso)s(ciativ)m(e)h(memory)g([7,)g(8)q(,)f(9)q(,)g(10)q(],)h (but)382 3333 y(opp)s(osed)d(to)j(Kohonen's)d(mo)s(del,)h(a)h(con)m (tin)m(uous)f(output)g(space)h(w)m(as)f(used.)523 3446 y(While)c(this)g(algorithm)h(w)m(orks)g(unsup)s(ervised)d(\(that)k (means)f(that)h(learning)e(pre-)382 3559 y(cedes)41 b(b)m(y)f(presen)m (ting)g(new)g(input)f(samples)g(but)h(not)h(the)f(desired)f(output\),) 44 b(also)382 3672 y(sup)s(ervised)38 b(learning)i(algorithms)g(w)m (ere)h(dev)m(elop)s(ed,)i(but)e(will)d(not)j(b)s(e)g(discussed)382 3785 y(here.)f(Nev)m(ertheless)31 b(it)f(is)f(p)s(ossible)f(to)j (extend)f(the)g(giv)m(en)g(framew)m(ork)h(to)f(in)m(tegrate)382 3898 y(these)i(metho)s(ds.)45 b(The)32 b(most)g(p)s(opular)e(extension) i(is)f(L)-10 b(V)m(Q)32 b(\(learning)e(v)m(ector)k(quan-)382 4011 y(tization\))27 b(dev)m(elop)s(ed)f(from)g(Kohonen)g([13)q(])h (where)f(sup)s(ervised)e(learning)h(is)h(used)g(for)382 4124 y(clustering.)39 b(A)31 b(go)s(o)s(d)f(o)m(v)m(erview)h(of)f(p)s (ossible)e(deriv)-5 b(ations)29 b(can)i(b)s(e)f(found)f(in)g([16)q(].) 523 4237 y(A)d(generalization)g(of)g(Kohonen's)g(SOMs)f(are)i(gro)m (wing)f(maps.)38 b(While)25 b(Kohonen)382 4349 y(used)i(a)h(\014x)f (top)s(ological)h(structure)f(in)g(the)h(output)f(space,)i(Gro)m(wing)f (Maps)g(are)g(able)382 4462 y(to)37 b(create)i(this)d(structure)g(on)h (their)f(o)m(wn.)60 b(Some)37 b(metho)s(ds)f(are)h(able)f(to)i(create)g (a)382 4575 y(complete)27 b(new)g(structure,)g(while)e(other)i(only)f (\\expand")i(a)f(giv)m(en)g(one)g(\(e.g.)41 b(a)28 b(grid\))382 4688 y(to)i(the)f(needed)g(size.)40 b(This)28 b(allo)m(ws)g(an)i (easier)f(analysis)f(of)h(unkno)m(wn)f(and)g(/)i(or)f(v)m(ery)382 4801 y(unequal)g(distributed)f(input)h(data.)43 b(It)31 b(is)f(also)h(p)s(ossible)d(to)k(use)e(these)i(metho)s(ds)e(for)382 4914 y(simple)e(cluster)i(analysis.)p eop %%Page: 5 11 5 10 bop 382 249 a Fs(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)1570 b Fr(5)382 548 y Fl(1.2)135 b(State)46 b(of)f(Approac)l(h)382 751 y Fr(This)27 b(thesis)g(will)f(put)i(Kohonen)g(maps)g(and)f(some)i (gro)m(wing)g(v)-5 b(arian)m(ts)28 b(in)f(one)i(struc-)382 864 y(tural)38 b(framew)m(ork.)68 b(After)39 b(a)h(detailed)e (description)f(of)i(the)h(di\013eren)m(t)e(algorithms,)382 977 y(they)33 b(are)g(analyzed)f(to)i(\014nd)d(similarities)e(and)j (di\013erences.)47 b(This)31 b(analysis)g(is)g(used)382 1090 y(to)39 b(create)h(a)f(common)f(class)g(structure)g(for)h(all)e (algorithms)g(and)h(p)s(ossible)e(exten-)382 1203 y(sions.)i(Afterw)m (ards)27 b(the)h(implemen)m(tation)e(of)h(these)h(classes)f(is)f (describ)s(ed.)38 b(T)-8 b(o)28 b(sho)m(w)382 1315 y(the)i(abilities)e (of)j(these)g(metho)s(ds,)f(some)g(examples)g(are)h(giv)m(en)f(in)g (the)g(end.)382 1555 y Fj(1.2.1)112 b(State)38 b(of)f(the)h(art)382 1727 y Fr(Starting)27 b(with)f(Kohonen's)h(SOM)g(algorithm)g(sev)m (eral)g(extensions)g(are)h(discussed)e(in)382 1840 y(c)m(hapter)31 b(2.)518 2030 y Fk(\017)46 b Fr(Kohonen's)30 b(Basic)h(Algorithm)609 2177 y(Kohonen's)36 b(algorithm)f(is)h(the)g(basic)g(part)g(of)g(all)f (other)i(Neural)e(Map)i(algo-)609 2290 y(rithms.)i(Nev)m(ertheless)30 b(there)g(are)h(a)f(lot)g(of)g(p)s(ossible)d(mo)s(di\014cations,)h(lik) m(e)h(the)609 2402 y(de\014nition)35 b(of)j(\\similar")e(or)h(the)h(up) s(date)f(function.)61 b(It)38 b(is)e(one)i(goal)g(of)g(this)609 2515 y(pap)s(er)e(to)h(isolate)g(these)g(comp)s(onen)m(ts,)i(to)e(allo) m(w)f(easy)h(in)m(terc)m(hange)h(and)e(to)609 2628 y(enable)30 b(the)h(user)e(to)i(test)h(di\013eren)m(t)d(setups)h(for)g(a)h(giv)m (en)g(dataset.)518 2808 y Fk(\017)46 b Fr(Neural)30 b(Gas)h(plus)d (Comp)s(etitiv)m(e)i(Hebbian)f(Learning)609 2955 y(The)35 b Fo(Neur)-5 b(al)37 b(Gas)43 b Fr(algorithm)34 b(is)g(an)h(extension)g (of)g(classical)f(v)m(ector)j(quan)m(ti-)609 3068 y(zation)32 b(metho)s(ds,)f(that)g(try)h(to)f(appro)m(ximate)h(the)f(input)e(data)j (with)e(a)i(set)f(of)609 3181 y(reference)24 b(v)m(ectors.)40 b(This)22 b(concept)j(is)e(enhanced)g(b)m(y)h(a)g(lateral)f(in)m (teraction)h(b)s(e-)609 3294 y(t)m(w)m(een)32 b(these)e(v)m(ectors)i (\(nearb)m(y)e(v)m(ectors)i(in\015uence)d(eac)m(h)i(other\),)g(so)g (that)f(not)609 3407 y(only)c(the)g(distribution)d(of)k(the)f(input)f (data,)j(but)d(additionally)f(neigh)m(b)s(orho)s(o)s(d)609 3520 y(relations)30 b(are)g(learned.)518 3700 y Fk(\017)46 b Fr(Gro)m(wing)30 b(Neural)g(Gas)609 3846 y(This)23 b(mo)s(del)g(uses)g(a)i(\015exible)d(n)m(um)m(b)s(er)h(of)i(neurons)e (to)i(de\014ne)e(the)i(map.)38 b(These)609 3959 y(neurons)30 b(are)g(connected,)i(using)d Fo(Comp)-5 b(etitive)34 b(Hebbian)f(L)-5 b(e)g(arning)39 b Fr(as)31 b(in)e(the)609 4072 y(Neural)24 b(Gas)i(\(often)g(used)e(connections)h(are)g (strengthened)g(while)e(rarely)h(used)609 4185 y(ones)j(are)h(remo)m(v) m(ed\).)41 b(A)27 b(lo)s(cal)f(error)h(measure)g(is)f(used,)h(to)h (determine,)f(where)609 4298 y(to)k(insert)f(new)f(neurons.)518 4478 y Fk(\017)46 b Fr(Gro)m(wing)30 b(Cell)f(Structures)609 4625 y(This)i(mo)s(del)h(is)g(closely)h(related)f(to)i(the)f Fo(gr)-5 b(owing)36 b(neur)-5 b(al)36 b(gas)41 b Fr(mo)s(del,)33 b(but)f(it)609 4738 y(uses)e(a)g(\014x)f(net)m(w)m(ork)i(dimension.)38 b(This)28 b(is)h(done,)h(b)m(y)f(not)h(only)f(adding)g(a)h(new)609 4851 y(connection,)h(but)f(a)g(complete)h(simplex)e(to)i(the)f(net.)518 5031 y Fk(\017)46 b Fr(Gro)m(wing)30 b(Grid)609 5177 y(This)35 b(mo)s(del)h(is)h(a)g(com)m(bination)g(of)g(ideas)f(from)h (the)g(gro)m(wing-cells)g(using)e(a)609 5290 y(classic)g(grid)f(as)h (underlying)d(structure.)55 b(Gro)m(wing)35 b(is)f(done,)j(b)m(y)e (inserting)e(a)609 5403 y(whole)24 b(ro)m(w)h(or)f(column)g(at)h(the)g (p)s(oin)m(t)e(with)h(the)h(highest)e(lo)s(cal)h(error)g(measure.)p eop %%Page: 6 12 6 11 bop 382 249 a Fs(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)1570 b Fr(6)518 548 y Fk(\017)46 b Fr(Gro)m(wing)30 b(Hyp)s(ercubial)e (Output)h(Space)i(\(GSOM\))609 698 y(The)k(GSOM)g(mo)s(del)f(is)g(in)g (its)h(basic)g(idea)f(also)i(a)f(gro)m(wing)g(grid)f(algorithm.)609 811 y(As)27 b(opp)s(osed)f(to)h(the)g(Gro)m(wing)g(Grid,)f(the)h(GSOM)g (can)g(also)g(gro)m(w)g(in)m(to)g(a)g(new)609 924 y(dimension,)h(so)i (that)h(the)f(net)m(w)m(ork)h(dimensionalit)m(y)c(has)j(not)g(to)h(b)s (e)f(kno)m(wn)f(a)609 1037 y(priori.)382 1280 y Fj(1.2.2)112 b(Comp)s(onen)m(t)38 b(analysis)382 1452 y Fr(The)k(goal)h(of)g(this)e 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b(a)g(new)f(class)h(is)f(created,)44 b(it)39 b(is)g(p)s(ossible)f(to)i (reuse)g(previously)e(designed)609 4184 y(classes)33 b(to)g(minimize)e(co)s(ding.)47 b(This)30 b(is)i(done)h(via)f(heredit)m (y)g(of)h(comp)s(onen)m(ts.)609 4297 y(A)44 b(grid)e(for)h(example,)j (has)e(the)f(same)h(in)m(terface)g(as)f(a)h(graph,)i(but)d(di\013ers) 609 4410 y(in)38 b(some)h(metho)s(ds,)h(lik)m(e)e(adding)f(a)i(neuron.) 65 b(These)38 b(metho)s(ds)g(need)h(to)g(b)s(e)609 4523 y(rewritten,)30 b(while)e(the)j(rest)f(of)h(the)f(graph-class)g(can)h (b)s(e)f(reused.)523 4736 y(Esp)s(ecially)38 b(when)h(testing)h (di\013eren)m(t)f(metho)s(ds)g(and)h(creating)g(new)g(ones,)i(the)382 4848 y(ab)s(o)m(v)m(e)23 b(prop)s(erties)e(of)h(OO-Design)g(are)h(v)m (ery)f(useful.)36 b(They)22 b(enable)g(the)g(programmer)382 4961 y(to)34 b(minimize)d(new)i(co)s(de,)i(when)d(creating)i(new)f (metho)s(ds)g(and)g(enable)f(the)i(user)f(to)382 5074 y(easily)26 b(in)m(terc)m(hange)h(metho)s(ds)f(and)g(com)m(bine)h(them) f(to)i(new)e(algorithms)g(as)h(needed.)382 5187 y(As)c(this)f(can)h(b)s (e)g(con)m(trolled)g(via)f(the)i(input)d(\014les,)j(results)e(can)h(b)s (e)f(compared)i(without)382 5300 y(creating)31 b(new)f(programs.)p eop %%Page: 7 13 7 12 bop 382 249 a Fs(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)1570 b Fr(7)523 548 y(In)27 b(c)m(hapter)g(3,)i(a)f(generalized)f(algorithm) f(will)e(b)s(e)j(dev)m(elop)s(ed,)h(that)f(allo)m(ws)g(easy)382 661 y(mo)s(di\014cations)g(of)i(the)g(basic)f(steps,)i(so)f(that)g(ev)m (ery)h(v)-5 b(arian)m(t)29 b(of)g(SOM)f(algorithms)g(is)382 774 y(p)s(ossible)g(to)j(\014t)f(in)f(it.)523 887 y(This)k(w)m(as)i (done,)g(b)m(y)f(dividing)e(Kohonen's)i(algorithm)f(in)g(its)h(basic)g (parts)g(and)382 1000 y(then)43 b(mo)s(difying)f(and)h(extending)g (these)h(parts)f(to)i(\014t)f(also)f(for)h(the)g(other)g(algo-)382 1112 y(rithms.)k(This)32 b(horizon)m(tal)h(view)f(allo)m(ws)h(easy)h (comparison)f(b)s(et)m(w)m(een)h(the)f(di\013eren)m(t)382 1225 y(algorithms.)65 b(It)39 b(is)f(sho)m(wn,)j(that)e(only)f(small)g (parts)h(di\013er)e(in)h(eac)m(h)i(metho)s(d)e(and)382 1338 y(that)31 b(great)g(parts)g(are)f(in)m(terc)m(hangeable.)382 1582 y Fj(1.2.3)112 b(Implemen)m(tation)382 1753 y Fr(In)43 b(c)m(hapter)h(4)g(the)g(theoretical)g(results)f(from)g(c)m(hapter)h(3) g(are)g(tak)m(en)h(to)g(build)40 b(a)382 1866 y(concrete)29 b(class)d(structure.)40 b(First)26 b(the)i(general)f(framew)m(ork)g(is) f(build)e(up.)39 b(Eac)m(h)28 b(part)382 1979 y(of)36 b(the)h(algorithm)e(is)g(isolated)h(in)f(a)i(o)m(wn)f(class,)i(con)m (taining)d(all)g(needed)h(metho)s(ds)382 2092 y(and)31 b(data.)46 b(Dynamic)32 b(references)g(are)g(used)f(to)i(link)d(these)i (ob)5 b(jects)32 b(together.)47 b(The)382 2205 y(class)34 b(dep)s(endencies)e(are)j(describ)s(ed)d(using)h(UML)h(\(Uni\014ed)f (Mo)s(deling)g(Language\))382 2318 y(diagrams.)81 b(Then)43 b(the)h(most)h(imp)s(ortan)m(t)e(parts)h(are)g(describ)s(ed)e(in)h (detail)g(using)382 2431 y(pseudo-co)s(de)30 b(algorithms.)523 2544 y(Implemen)m(tation)i(w)m(as)h(done)g(within)d(the)j Fo(A)n(nalysis++)j Fr(pac)m(k)-5 b(age,)36 b(whic)m(h)31 b(o\013ers)382 2657 y(a)40 b(C++)f(framekit)h(for)f(Mac)m(hine)h (Learning.)69 b(Here)40 b(the)g(main)f(comp)s(onen)m(ts)h(w)m(ere)382 2770 y(isolated)34 b(and)g(implemen)m(ted)f(as)i(C++)f(ob)5 b(jects.)54 b(This)33 b(allo)m(ws)h(the)h(in)m(terc)m(hange)g(of)382 2882 y(new)30 b(mo)s(dules)e(with)i(previously)e(dev)m(elop)s(ed)h (comp)s(onen)m(ts.)41 b(This)29 b(enables)h(the)g(user)382 2995 y(to)22 b(compare)g(the)f(results)f(of)h(the)h(metho)s(ds)e(with)g (the)i(results)e(of)h(classical)f(algorithms.)382 3108 y(As)38 b(all)g(new)g(metho)s(ds)g(w)m(ere)h(\014t)f(in)m(to)h(the)f (framew)m(ork,)j(it)d(is)g(p)s(ossible)e(to)j(use)f(the)382 3221 y(datat)m(yp)s(es)c(and)e(data)i(manipulation)d(functions)g (de\014ned)h(in)g(the)h(pac)m(k)-5 b(age.)51 b(So)33 b(not)382 3334 y(only)h(con)m(tin)m(uous)g(data)h(lik)m(e)f(\015oats)i (can)f(b)s(e)f(used)f(as)i(an)g(input,)f(but)g(also)h(\(fuzzy-\))382 3447 y(logical)30 b(v)-5 b(alues.)523 3560 y(On)38 b(the)h(other)g (hand,)h(the)e(results)g(of)h(the)g(new)f(metho)s(ds)g(ma)m(y)h(b)s(e)f (reused)g(in)382 3673 y(the)g(pac)m(k)-5 b(age.)64 b(As)37 b(Neural)g(Maps)h(are)f(able)g(to)i(do)e(dimension-reduction)d(of)k (high)382 3786 y(dimensional)20 b(input)i(data,)j(they)f(ma)m(y)g(b)s (e)f(used)f(to)i(prepro)s(cess)f(suc)m(h)g(datasets)h(b)s(efore)382 3899 y(using)31 b(them)h(in)f(other)h(metho)s(ds.)46 b(This)30 b(is)h(esp)s(ecially)g(useful,)g(if)g(the)h(input)f(data)i (is)382 4012 y(v)m(ery)d(complex)f(or)h(the)f(data)i(is)d(c)m(hanging)i (in)e(time.)40 b(In)29 b(b)s(oth)g(cases,)i(the)f(net)f(is)g(able)382 4124 y(to)i(act)h(as)e(a)h(bu\013er,)f(that)g(\\normalizes")h(the)f (input)f(data.)523 4237 y(The)i(basic)g(comp)s(onen)m(ts)h(are)g (implemen)m(ted)e(in)g(t)m(w)m(o)j(basic)e(mo)s(dules,)f(one)i(with)382 4350 y(the)d(k)m(ernel)f(comp)s(onen)m(ts)h(and)f(virtual)f(classes)h (needed)h(for)f(all)f(Neural)h(Map)h(imple-)382 4463 y(men)m(tations)f(and)g(a)g(second)g(mo)s(dule)f(with)f(some)j(sp)s (ecial)d(utilities)g(\(e.g.)41 b(a)28 b(grid)f(and)382 4576 y(a)k(n)m(umeric)e(v)m(ector)j(class\).)523 4689 y(T)-8 b(o)25 b(gain)e(maximal)g(\015exibilit)m(y)e(the)j(map)g(data)h (is)e(stored)h(as)g(a)h(undirected)d(graph.)382 4802 y(Eac)m(h)34 b(comp)s(onen)m(t)f(has)g(a)h(prede\014ned)e(in)m(terface) h(and)g(is)f(dynamically)f(link)m(ed)h(with)382 4915 y(the)39 b(other)g(mo)s(dules.)65 b(This)38 b(enables)g(the)h(user)g (to)g(in)m(terc)m(hange)h(the)f(comp)s(onen)m(ts)382 5028 y(ev)m(en)30 b(at)g(run-time.)40 b(The)29 b(algorithms)f(can)i(b)s (e)f(con)m(trolled)g(through)g(the)h(input)d(\014les.)382 5141 y(The)j(input)e(\014les)i(can)g(b)s(e)g(found)f(in)g(the)i(App)s (endix.)523 5254 y(In)24 b(c)m(hapter)h(5)g(this)e(set)i(of)g(metho)s (ds)f(is)f(expanded)h(b)m(y)h(some)f(gro)m(wing)h(algorithms)382 5366 y(in)35 b(a)h(separate)h(mo)s(dule.)56 b(Using)35 b(OO-design)g(the)h(new)g(parts)f(w)m(ere)i(easily)e(\014t)h(in)m(to)p eop %%Page: 8 14 8 13 bop 382 249 a Fs(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)1570 b Fr(8)382 548 y(the)43 b(basic)f(mo)s(del.)76 b(As)43 b(some)g(mo)s(dels)e(are)i(extensions)f(of)h(other)g(ones,)j(a)d(lot)f (of)382 661 y(comp)s(onen)m(ts)30 b(could)e(b)s(e)h(reused.)40 b(This)28 b(ensures,)h(that)h(the)g(in)m(terface)g(is)e(similar)f(and) 382 774 y(co)s(ding)i(is)h(minimized.)523 887 y(The)k(ANAL)-8 b(YSIS++)34 b(program)g(is)g(a)h(ob)5 b(ject-orien)m(ted)36 b(framew)m(ork)e(written)g(in)382 1000 y(C++)20 b(for)h(the)g (manipulation)e(and)h(dev)m(elopmen)m(t)i(of)f(m)m(ulti-strategy)g(mac) m(hine)g(learn-)382 1112 y(ing)35 b(metho)s(ds)g(and)g(data)h(ob)5 b(jects.)58 b(The)35 b(pro)5 b(jects)36 b(is)f(b)s(eing)f(dev)m(elop)s (ed)i(within)d(an)382 1225 y(EC)h(pro)5 b(ject,)37 b(REF)m(ORM)f(in)d (close)j(co)s(op)s(eration)f(with)e(UNI-SOFTW)-10 b(ARE)35 b(PLUS)382 1338 y(\(USP\))e(and)f(the)h(F)-8 b(uzzy)34 b(Logic)f(Lab)s(oratory)g(Linz)e(\(FLLL\).)j(The)e(emphasis)f(of)i(the) 382 1451 y(pro)5 b(ject)21 b(is)f(to)h(pro)m(vide)f(a)h(platform)f(in)f (whic)m(h)g(man)m(y)i(mac)m(hine)f(learning)f(metho)s(ds)h(can)382 1564 y(b)s(e)33 b(used)h(in)f(harmon)m(y)h(together,)j(i.e.)52 b(the)34 b(m)m(ulti-strategy)h(approac)m(h.)52 b(Tw)m(o)35 b(main)382 1677 y(concepts)27 b(to)g(this)e(end)h(are)h(the)g(in)m (telligen)m(t)e(use)h(of)g(in)m(terc)m(hangeable)h(datat)m(yp)s(es)g (and)382 1790 y(a)k(mec)m(hanism)g(to)h(store)g(the)f(history)f(of)h (the)h(data)g(analyses)f(p)s(erformed)e(on)i(a)h(data)382 1903 y(set.)53 b(Another)34 b(branc)m(h)f(of)i(the)f(pro)5 b(ject)35 b(is)e(concerned)h(with)f(the)i(generalization)e(of)382 2016 y(traditional)h(metho)s(ds)h(in)m(to)h(an)g(ob)5 b(ject-orien)m(ted)37 b(framew)m(ork.)57 b(Another)35 b(asp)s(ect)i(is)382 2129 y(the)28 b(generalization)f(of)h(mac)m(hine)f (learning)f(metho)s(ds)h(using)g(simple)e(true-false)j(logic)382 2242 y(to)34 b(fuzzy)f(logics)g(\(not)g(only)f(as)i(a)f(p)s(ost-pro)s (cessing)f(optimization,)h(but)g(also)g(in)e(the)382 2355 y(direct)f(creation)g(in)g(a)g(fuzzy)h(logic)f(domain\).)382 2598 y Fj(1.2.4)112 b(Examples)382 2770 y Fr(T)-8 b(o)32 b(illustrate)e(the)h(p)s(o)m(w)m(er)h(and)f(\015exibilit)m(y)d(of)k (the)g(implemen)m(ted)e(metho)s(ds,)h(sev)m(eral)382 2882 y(examples)e(are)h(presen)m(ted)g(in)f(c)m(hapter)h(6.)41 b(F)-8 b(or)31 b(that)f(purp)s(ose)e(three)i(di\013eren)m(t)g(data-)382 2995 y(sets)h(are)f(analyzed)h(using)e(di\013eren)m(t)g(metho)s(ds.)518 3208 y Fk(\017)46 b Fr(Simple)28 b(sample)609 3358 y(As)h(an)g(in)m (tro)s(ducing)d(example,)j(a)g(simple)e(square)i(is)f(presen)m(ted)g (to)i(eac)m(h)g(algo-)609 3471 y(rithm.)36 b(The)21 b(goal)g(is)f(to)i (distribute)c(the)j(net)g(homogeneously)g(o)m(v)m(er)i(the)e(square.) 609 3584 y(This)36 b(sample)h(w)m(as)h(run)e(with)g(Kohonen's)h (original)f(algorithm,)j(the)e(Neural)609 3697 y(Gas)30 b(and)f(the)h(Gro)m(wing)g(Neural)e(Gas)j(algorithm.)39 b(This)28 b(sample)h(sho)m(ws,)h(ho)m(w)609 3810 y(the)38 b(net)f(self-organizes)g(to)h(appro)m(ximate)f(the)g(input)f(data)i (and)e(to)i(prev)m(en)m(t)609 3923 y(neigh)m(b)s(orho)s(o)s(d)28 b(relations.)518 4110 y Fk(\017)46 b Fr(Iris)29 b(\015o)m(w)m(ers)609 4261 y(This)23 b(is)g(a)i(classic)f(AI)g(sample)g(data-set,)j(where)d (the)h(goal)g(is)e(to)i(separate)g(three)609 4374 y(sp)s(ecies)g(of)g (iris)e(\015o)m(w)m(ers,)k(whic)m(h)d(are)i(describ)s(ed)d(b)m(y)i (four)g(discrete)g(parameters.)609 4486 y(The)33 b(result)g(is)f (comparable)i(to)g(that,)h(obtained)e(b)m(y)h(classical)e(metho)s(ds,)i (but)609 4599 y(without)40 b(an)m(y)i(class)f(information)e(in)h(the)h (learning)f(pro)s(cess.)72 b(This)40 b(sho)m(ws,)609 4712 y(ho)m(w)h(the)f(net)h(is)e(able)h(to)h(\014nd)e(structures)h(in)f (the)i(input)d(data)j(and)f(create)609 4825 y(according)31 b(nets.)518 5013 y Fk(\017)46 b Fr(Animal)29 b(sample)609 5163 y(Here)46 b(di\013eren)m(t)f(animals)f(are)i(describ)s(ed)e(b)m(y) h(sev)m(eral)h(b)s(o)s(olean)f(attributes.)609 5276 y(Goal)33 b(of)g(the)g(learning)f(pro)s(cess)g(is,)h(to)g(\014nd)f(relations)g(b) s(et)m(w)m(een)h(the)g(animals)p eop %%Page: 9 15 9 14 bop 382 249 a Fs(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)1570 b Fr(9)609 548 y(and)37 b(to)g(group)g(similar)d(ones)k(together.)62 b(The)36 b(data)i(w)m(as)f(presen)m(ted)g(to)h(Ko-)609 661 y(honen's)c(original)e(algorithm)h(and)h(to)h(the)f(Neural)g(Gas.) 52 b(Both)35 b(create)h(go)s(o)s(d)609 774 y(mappings)j(of)h(the)h (relations)e(b)s(et)m(w)m(een)i(the)f(animals.)69 b(The)40 b(Neural)g(Gas)h(is)609 887 y(furthermore)29 b(able)h(to)h(create)h (clusters)e(of)g(assimiable)f(animals.)523 1099 y(These)d(samples)f (giv)m(e)h(a)h(surv)m(ey)e(of)h(the)h(p)s(ossible)c(applications)h(of)i (the)h(presen)m(ted)382 1212 y(algorithms.)39 b(As)28 b(the)h(input)d(data)j(is)f(handled)e(in)h(a)i(v)m(ery)g(\015exible)d (w)m(a)m(y)-8 b(,)31 b(and)c(a)i(lot)f(of)382 1325 y(di\013eren)m(t)36 b(information)e(ma)m(y)j(b)s(e)f(receiv)m(ed)h(form)e(the)i(results,)g (these)g(metho)s(ds)e(are)382 1438 y(ideal)22 b(for)g(dealing)g(with)g (unkno)m(wn)f(and)i(complex)g(data.)39 b(Other)22 b(classic)g (applications)382 1551 y(for)j(SOMs)g(are)h(sp)s(eec)m(h)g(recognition) f([14)q(])h(or)f(the)h(tra)m(v)m(eling)g(salesman)f(problem)f([15)q(].) 523 1664 y(By)g(using)e(one)h(dataset)i(with)d(sev)m(eral)i(algorithms) e(it)h(is)f(also)i(sho)m(wn,)g(ho)m(w)g(simple)382 1777 y(the)g(b)s(eha)m(vior)f(of)i(the)f(net)g(can)g(b)s(e)g(c)m(hanged.)39 b(This)22 b(enables)i(the)g(user)g(to)g(\\pla)m(y")h(with)382 1890 y(di\013eren)m(t)36 b(metho)s(ds,)i(compare)f(the)g(results)e(and) h(select)i(the)e(b)s(est)h(metho)s(d)f(for)g(his)382 2002 y(needs.)382 2246 y Fj(1.2.5)112 b(Outlo)s(ok)382 2418 y Fr(The)21 b(pap)s(er)g(is)f(closed)i(with)e(an)i(short)f(outlo)s (ok)h(on)f(p)s(ossible)f(extensions)h(and)g(v)-5 b(arian)m(ts)382 2530 y(of)36 b(the)g(metho)s(ds)f(implemen)m(ted)g(so)h(far.)57 b(In)36 b(c)m(hapter)g(7)g(the)h(dev)m(elopmen)m(t)f(of)g(new)382 2643 y(algorithm)24 b(is)h(discussed)f(as)h(w)m(ell)g(as)h(the)g (design)e(of)i(new)f(comp)s(onen)m(ts.)39 b(Also)25 b(sev)m(eral)382 2756 y(mo)s(di\014cations)j(that)j(\014t)g(ANNs)f(esp)s(ecially)f(for)h (cluster)g(analysis)f(are)h(describ)s(ed.)523 2869 y(In)36 b(the)h(future,)g(the)g(framew)m(ork)g(presen)m(ted)f(in)g(this)f(pap)s (er)h(will)e(hop)s(efully)f(b)s(e)382 2982 y(extended)d(with)f(a)i(lot) f(of)h(new)f(v)-5 b(arian)m(ts)30 b(and)g(algorithms.)p eop %%Page: 10 16 10 15 bop 382 1171 a Fm(Chapter)65 b(2)382 1586 y Fq(State)78 b(of)g(the)g(art)382 2031 y Fr(In)41 b(this)f(c)m(hapter)j(I)e(will)e (giv)m(e)j(a)g(brief)f(in)m(tro)s(duction)e(to)k(the)f(main)e(ideas)h (of)h(self-)382 2144 y(organizing)26 b(maps.)39 b(The)27 b(basic)g(concepts)h(are)f(explained)e(starting)i(with)f(Kohonen's)382 2257 y(mo)s(del,)35 b(whic)m(h)f(is)g(presen)m(ted)g(from)h(a)g (functional)e(p)s(oin)m(t)h(of)h(view)g(and)f(as)h(an)g(algo-)382 2369 y(rithm.)k(The)30 b(algorithm)f(is)h(analyzed)g(and)g(divided)d (in)m(to)k(its)f(basic)f(comp)s(onen)m(ts.)523 2482 y(Though)34 b(SOMs)g(can)i(b)s(e)e(describ)s(ed)f(as)i(t)m(w)m(o-la)m(y)m(er)i (ANNs,)g(they)e(also)g(de\014ne)f(a)382 2595 y(mapping)g(from)h(the)h (input)e(on)m(to)j(the)f(output)g(space.)57 b(The)35 b(output)h(space)g(can)g(b)s(e)382 2708 y(an)m(y)c(discrete)f(top)s (ological)g(structure.)43 b(Lateral)31 b(in)m(teraction)h(is)e(used)h (to)h(adjust)e(the)382 2821 y(net.)38 b(This)21 b(pro)m(vides)h(a)h (smo)s(oth)g(global)f(ordering)g(of)h(the)g(net,)i(so)e(that)g(similar) e(inputs)382 2934 y(are)31 b(mapp)s(ed)e(on)m(to)i(top)s(ological)f (close)h(regions)f(on)g(the)h(net)f([12)q(].)523 3047 y(Kohonen)g(used)f(a)i(prede\014ned)d(output)i(structure,)g(usually)e (a)j(t)m(w)m(o-dimensional)382 3160 y(grid.)38 b(F)-8 b(or)26 b(some)f(applications,)g(this)f(de\014nition)f(is)h(to)i (restrictiv)m(e.)39 b(E.g.)h(if)24 b(the)i(input)382 3273 y(data)i(is)e(v)m(ery)h(high)e(dimensional)g(or)i(has)f(a)i(lot)e (of)i(distinct)d(clusters,)i(a)g(grid)f(will)e(not)382 3386 y(su\016ce.)50 b(Therefore)34 b(others)f(ha)m(v)m(e)i(dev)m(elop)s (ed)e(sev)m(eral)h(extensions)f(that)h(create)h(the)382 3499 y(output)30 b(structure)g(on)g(demand.)40 b(Some)30 b(of)h(them)f(will)e(b)s(e)i(describ)s(ed)e(here:)382 3679 y Fp(Neural)34 b(Gas)i(plus)f(Comp)s(etitiv)m(e)e(Hebbian)i (Learning:)45 b Fr(The)20 b(Neural)g(Gas)h(al-)609 3792 y(gorithm)33 b(starts)i(with)d(a)i(set)h(of)f(unconnected)g(neurons.)50 b(Then)33 b(connections)609 3905 y(are)e(created)g(as)g(an)f(part)g(of) h(the)g(learning)d(pro)s(cess)i([22)q(,)h(24)q(].)382 4082 y Fp(Gro)m(wing)k(Neural)g(Gas:)46 b Fr(This)18 b(mo)s(del)h(extends)h(the)h(Neural)e(Gas)i(algorithm.)37 b(Only)609 4195 y(t)m(w)m(o)32 b(neurons)d(are)i(created)h(at)f (startup,)f(while)f(additional)f(neurons)h(are)i(gen-)609 4308 y(erated)g(in)e(the)i(learning)e(pro)s(cess)g([25)r(].)382 4485 y Fp(Gro)m(wing)35 b(Cell)g(Structures:)45 b Fr(T)-8 b(o)38 b(create)h(a)f(map)f(with)g(a)g(homogeneous)i(struc-)609 4598 y(ture,)f(this)d(algorithm)h(not)g(just)g(adds)g(single)f (neurons,)i(but)f(complete)h(sim-)609 4710 y(plices)29 b([23)q(].)382 4887 y Fp(Gro)m(wing)35 b(Grid:)46 b Fr(In)37 b(some)i(applications)d(one)i(ma)m(y)g(need)g(the)g(strong)g(structure) 609 5000 y(of)d(a)f(grid)f(as)i(underlying)c(structure.)52 b(F)-8 b(or)35 b(that)g(purp)s(ose)e(this)g(algorithm)h(is)609 5113 y(able)c(to)h(insert)e(new)h(ro)m(ws)h(to)g(the)f(grid)g([26)q(].) 382 5290 y Fp(Gro)m(wing)35 b(Hyp)s(ercubial)h(Output)e(Space)h (\(GSOM\):)45 b Fr(The)29 b(GSOM)g(Mo)s(del)h(is)609 5403 y(in)40 b(its)g(basic)h(idea)f(also)h(a)h(gro)m(wing)f(grid)f (algorithm.)71 b(As)41 b(opp)s(osed)g(to)g(the)1831 5652 y(10)p eop %%Page: 11 17 11 16 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(11)609 548 y(Gro)m(wing)38 b(Grid,)h(the)f(GSOM)g(is)f(able)g(to)i(gro)m(w)g(in)m(to)f(a)g(new)g (dimension,)f(so)609 661 y(that)31 b(the)g(net)m(w)m(ork)g (dimensionalit)m(y)c(has)k(not)f(to)h(b)s(e)f(kno)m(wn)g(a)h(priori)d ([28)q(].)382 947 y Fl(2.1)135 b(Kohonen's)46 b(mo)t(del)382 1150 y Fr(Kohonen)38 b(in)m(v)m(estigated)i(the)f(Self)e(Organizing)g (Maps)i(\(SOM\))g(in)f(1982)i([12)q(].)66 b(His)382 1263 y(idea)45 b(w)m(as)g(to)h(create)h(a)f(neural)e(net)i(to)g(represen)m (t)f(the)g(input)f(space,)50 b(using)43 b(the)382 1376 y(top)s(ological)30 b(structure)h(of)g(a)g(grid)e(to)j(store)f(neigh)m (b)s(orho)s(o)s(d)e(relations.)41 b(As)30 b(opp)s(osed)382 1489 y(to)24 b(most)f(Neural)g(Net)g(metho)s(ds,)i(where)d(the)h (desired)f(result)g(is)g(used)g(to)i(compute)f(the)382 1602 y(w)m(eigh)m(ts)i(of)g(the)g(net,)h(SOM's)e(need)h(no)f(reference) h(output)g(to)g(tune)g(their)e(parameters)382 1715 y(but)30 b(are)g(able)g(to)h(self-organize)g(themself.)523 1828 y(A)c(neural)e(net)i Fo(self-or)-5 b(ganizes)35 b Fr(if)25 b(learning)g(pro)s(ceeds)h(without)g(an)m(y)h(sup)s(ervision.)382 1941 y(The)k(input)e(data)j(is)e(presen)m(ted)h(to)h(the)f(net,)h(and)e (it)h(tries)g(to)h(\014nd)d(a)j(corresp)s(onding)382 2054 y(represen)m(tation)h(of)h(the)g(neigh)m(b)s(orho)s(o)s(d)c (relations)j(of)h(the)f(input)f(space)i(itself.)48 b(The)382 2166 y(goal)32 b(of)g(the)g(learning)e(pro)s(cess)h(is)g(to)h(minimize) d(a)j(global)f(error)g(function)g(\(e.g.)45 b(the)382 2279 y(a)m(v)m(erage)g(prediction)c(error\).)78 b(F)-8 b(or)43 b(eac)m(h)h(new)e(sample,)j(the)e(net)g(parameters)g(are)382 2392 y(mo)s(di\014ed)28 b(to)j(optimize)f(this)f(error)h(function.)382 2505 y(Kohonen's)g(algorithm)f(has)h(t)m(w)m(o)i(ma)5 b(jor)30 b(goals:)382 2718 y Fp(T)-9 b(op)s(ology-preserving)47 b Fr(Similar)30 b(input)g(v)m(ectors)j(should)e(b)s(e)g(mapp)s(ed)g(to) i(nearb)m(y)609 2831 y(neurons)45 b(of)i(the)f(net,)51 b(and)46 b(adjacen)m(t)i(neurons)d(should)f(ha)m(v)m(e)k(comparable)609 2944 y(reference-v)m(ectors.)382 3131 y Fp(Distribution-preserving)e Fr(The)d(distribution)c(of)k(the)h(w)m(eigh)m(t)f(v)m(ectors)i(should) 609 3244 y(appro)m(ximate)31 b(the)f(densit)m(y)g(of)g(the)h(input)d (data)j Fn(P)13 b Fr(\()p Fn(x)p Fr(\).)523 3457 y(As)25 b(learning)f(is)h(done)g(lo)s(cally)f(but)h(with)f(limited)f(in)m (teraction,)k(a)f(detailed)e(math-)382 3569 y(ematical)34 b(analysis)f(of)h(ho)m(w)g(learning)e(is)h(done,)i(is)e(di\016cult)f (and)h(still)f(missing.)49 b(An)382 3682 y(analysis)24 b(ho)m(w)h(the)h(neurons)e(are)i(distributed)d(after)j(the)f(learning)f (pro)s(cess)h(w)m(as)h(done)382 3795 y(b)m(y)k(Kohonen)g([12)q(])h(and) e(Ritter)i([17)q(].)523 3908 y(This)36 b(mo)s(del)h(is)g(in)f(some)j(w) m(a)m(y)f(a)h(clustering)d(metho)s(d)h(as)h(it)g(divide)e(the)i(input) 382 4021 y(space)26 b(in)f(sev)m(eral)h(partitions.)37 b(It)26 b(is)f(closely)g(related)h(to)g(the)g Fo(k-me)-5 b(ans)29 b(cluster)g(analy-)382 4134 y(sis)p Fr(,)e(whic)m(h)e(w)m(as)i (dev)m(elop)s(ed)e(from)h(Andersb)s(erg)f(in)g(1973)j([11)q(].)39 b(The)26 b(main)f(di\013erence)382 4247 y(is)43 b(the)h Fo(top)-5 b(olo)g(gic)g(al)49 b(structur)-5 b(e)51 b Fr(that)45 b(stores)f(the)g(neigh)m(b)s(orho)s(o)s(d)e(relations.)80 b(This)382 4360 y(enables)30 b(ANNs)g(to)h(b)s(e)f(uses)g(also)g(for)h (other)f(applications)f(then)h(cluster)f(analysis.)382 4603 y Fj(2.1.1)112 b(Mathematical)37 b(mo)s(del)382 4775 y Fr(Self)h(Organizing)f(Maps)i(b)s(elong)f(to)h(the)g(class)g(of) g(Neural)f(Nets.)66 b(Though)38 b(SOMs)382 4888 y(are)26 b(usually)d(describ)s(ed)h(as)i(an)f(indep)s(enden)m(t)e(dev)m(elopmen) m(t,)28 b(I)d(will)e(try)i(to)i(in)m(tro)s(duce)382 5001 y(them)j(as)h(a)g(classic)e(neural)g(net)i(\014rst.)523 5114 y(SOMs)f(can)h(b)s(e)f(describ)s(ed)e(as)j(simple)e(t)m(w)m(o-la)m (y)m(er)j(net)m(w)m(orks.)42 b(This)29 b(means,)i(that)382 5227 y(they)g(consist)f(of)h(an)g(input)e(la)m(y)m(er,)j(where)e(the)h (input)e(data)i(is)f(presen)m(ted)h(to)g(the)g(net)382 5339 y(and)h(an)h(output)g(la)m(y)m(er,)h(where)f(the)g(results)f(are)h (displa)m(y)m(ed.)48 b(These)32 b(t)m(w)m(o)j(la)m(y)m(ers)e(are)p eop %%Page: 12 18 12 17 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(12)382 548 y(connected)34 b(b)m(y)f(w)m(eigh)m(ted)h(edges.)49 b(The)33 b(input)e(space)j(has)f (to)h(b)s(e)f(a)g(v)m(ector-space)j(to)382 661 y(allo)m(w)c (computation)g(with)f(the)i(input)d(data)k(inside)c(the)i(net)m(w)m (ork.)48 b(F)-8 b(or)33 b(the)g(output)382 774 y(space,)e(an)m(y)g (graph)f(structure)g(su\016ces.)523 887 y(As)35 b(this)e(represen)m (tation)i(is)f(v)m(ery)h(limited,)f(an)h(alternativ)m(e)g(mo)s(del)e (will)f(b)s(e)i(de-)382 1000 y(v)m(elop)s(ed)39 b(afterw)m(ards,)j (that)e(describ)s(es)e(the)h(map)g(from)g(a)h(more)g(general)f(p)s(oin) m(t)f(of)382 1112 y(view.)382 1352 y Fp(La)m(y)m(er)d(represen)m (tation)382 1524 y Fr(The)30 b(net)g(consists)f(of)i(t)m(w)m(o)g(la)m (y)m(ers.)41 b(The)30 b(input)e(or)i(source)g(la)m(y)m(er)h Fk(R)f Fr(and)g(the)g(output)382 1636 y(or)35 b(target)i(la)m(y)m(er)f Fk(C)5 b Fr(.)55 b(Kohonen)34 b(used)h(a)g(grid)f(as)i(output)e (structure,)j(so)e(that)h(ev)m(ery)382 1749 y(neuron)f(has)h(exactly)h (four)f(neigh)m(b)s(ors)f(\(except)i(the)g(neurons)e(on)h(the)h (edge\).)59 b(The)382 1862 y(input)39 b(la)m(y)m(er)i(comprises)f Fn(N)1364 1876 y Fi(R)1462 1862 y Fr(\(this)g(is)g(the)h(dimension)d (of)j(the)g(input)e(space\),)44 b(the)382 1975 y(output)27 b(la)m(y)m(er)h Fn(N)967 1989 y Fi(C)1054 1975 y Fr(units)e(\(the)j(n)m (um)m(b)s(er)d(of)i(di\013eren)m(t)f(output)g(states\).)42 b(The)27 b(neurons)382 2088 y(of)f(the)g(input)f(la)m(y)m(er)h(are)h (fully)c(in)m(terconnected)k(to)g(the)f(neurons)f(of)h(the)g(output)g (la)m(y)m(er)382 2201 y(b)m(y)h(an)h(edge)g(with)e(w)m(eigh)m(t)j Fn(w)1388 2215 y Fi(is)1476 2201 y Fr(\(see)g(\014gure)e(2.1\).)41 b(Both,)29 b(the)f(input)d(v)m(ectors)k(and)e(the)382 2314 y(w)m(eigh)m(t)j(v)m(ectors)h(ha)m(v)m(e)f(to)g(b)s(e)f (normalized.)39 b(This)28 b(is)g(necessary)i(to)g(prev)m(en)m(t)g(fa)m (v)m(oring)382 2427 y(of)g(v)m(ectors)i(with)d(great)j(v)-5 b(alues.)523 2540 y(If)43 b(a)g(input)e(v)m(ector)k Fn(x)h Fr(=)h([)p Fn(\030)1537 2554 y Fh(1)1576 2540 y Fn(;)15 b(\030)1656 2554 y Fh(2)1696 2540 y Fn(;)g(:)g(:)g(:)h(;)f(\030)1937 2554 y Fi(N)1993 2565 y Fg(R)2048 2540 y Fr(])2073 2507 y Fi(T)2175 2540 y Fk(2)46 b(R)d Fr(is)f(presen)m(ted)h(to)h(the)f (net,)382 2653 y(for)e(ev)m(ery)h(neuron)f(on)g(the)h(output)f(la)m(y)m (er)h Fn(j)49 b Fk(2)44 b(C)i Fr(the)c Fo(ener)-5 b(gy)43 b(function)49 b Fn(e)3109 2667 y Fi(j)3146 2653 y Fr(\()p Fn(x)p Fr(\))42 b(is)382 2766 y(computed:)1458 2878 y Fn(e)1500 2892 y Fi(j)1537 2878 y Fr(\()p Fn(x)p Fr(\))26 b(=)1861 2798 y Ff(X)1781 2981 y Fi(i)p Fh(=1)p Fi(:::)o(N)2010 2992 y Fg(R)2075 2878 y Fn(\030)2115 2892 y Fi(i)2143 2878 y Fn(w)2208 2892 y Fi(ij)2269 2878 y Fn(:)890 b Fr(\(2.1\))382 3124 y(The)30 b(most)g(stim)m(ulated)g(neuron)f Fn(s)c Fk(2)g(C)1410 3324 y Fn(s)g Fr(=)f(arg)r(max)1870 3338 y Fi(j)t Fe(2C)2025 3324 y Fk(f)p Fn(e)2112 3338 y Fi(j)2150 3324 y Fr(\()p Fn(x)p Fr(\))p Fk(g)p Fn(:)842 b Fr(\(2.2\))382 3524 y(is)26 b(de\014ned)g(as)h(the)g Fo(b)-5 b(est)30 b(c)-5 b(orr)g(esp)g(onding)33 b(neur)-5 b(on)35 b Fr(and)26 b(its)h(w)m(eigh)m(ts)g Fn(w)2777 3538 y Fi(is)2838 3524 y Fn(;)43 b(i)25 b Fr(=)g(1)15 b Fn(:)g(:)g(:)i(N)3313 3538 y Fi(R)382 3637 y Fr(are)31 b(up)s(dated.)42 b(Adjacen)m(t)32 b(output)f(neurons)f(are)h(also)h(mo) s(di\014ed)d(according)i(to)h(their)382 3750 y(distance)40 b(to)h Fn(s)p Fr(.)70 b(Usually)38 b(the)j(minimal)c(path)j(distance)g (is)f(used,)j(but)e(an)m(y)g(other)382 3863 y(metric)30 b(\(e.g.)42 b(the)31 b(cit)m(y-blo)s(c)m(k)f(metric\))h(w)m(ould)e (su\016ce)h(to)s(o.)523 3976 y(F)-8 b(or)45 b(eac)m(h)g(input)d 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neg exch neg exch translate 980 1647 a @beginspecial 142 @llx 468 @lly 472 @urx 687 @ury 3300 @rwi @setspecial %%BeginDocument: figures/layers.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 142 468 472 687 %%Creator: CorelDRAW 7 %%Title: %%CreationDate: Mon Mar 22 10:57:36 1999 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness true def /AutoSteps 0 def /CMYKMarks false def /UseLevel2 false def %Color profile: %%BeginResource: procset wCorel7Dict 7.0 0 /wCorel7Dict 300 dict def wCorel7Dict begin % Copyright (c)1992-96 Corel Corporation % All rights reserved. v7.0 r1.0 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/setscreen{pop 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%%PageTrailer @rs @rs %%Trailer @EndSysCorelDict end %%DocumentSuppliedResources: procset wCorel7Dict %%EOF %%EndDocument @endspecial 3730 1647 a currentpoint grestore moveto 3730 1647 a 980 1647 a currentpoint currentpoint translate 1 0.6521 div 1 0.6521 div scale neg exch neg exch translate 980 1647 a 951 1942 a Fr(Figure)30 b(2.1:)42 b(Kohonen)30 b(net,)h(La)m(y)m(er)g (represen)m(tation)980 3472 y currentpoint currentpoint translate 0.56334 0.56334 scale neg exch neg exch translate 980 3472 a 980 3472 a gsave currentpoint currentpoint translate 0 neg rotate neg exch neg exch translate 980 3472 a @beginspecial 35 @llx 503 @lly 417 @urx 804 @ury 3820 @rwi @setspecial %%BeginDocument: figures/kohonen1.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 35 503 417 804 %%Creator: CorelDRAW 7 %%Title: %%CreationDate: Mon Mar 22 10:50:30 1999 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness true def /AutoSteps 0 def /CMYKMarks false def /UseLevel2 false def %Color profile: %%BeginResource: procset wCorel7Dict 7.0 0 /wCorel7Dict 300 dict def wCorel7Dict begin % Copyright (c)1992-96 Corel Corporation % All rights reserved. v7.0 r1.0 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/setscreen{pop pop pop}bd/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd /SepMode_5 where{pop}{/SepMode_5 0 def}ifelse/CurrentInkName_5 where{pop} {/CurrentInkName_5(Composite)def}ifelse/$ink_5 where{pop}{/$ink_5 -1 def} ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def/$t 1 def/$n _ def/$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def/$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def/s1t 0 def/s1n _ def/$bkg false def /SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd /$ptm matrix def/$ttm matrix def/$stm matrix def/$ffpnt true def /CorelDrawReencodeVect[16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /L2?/languagelevel where{pop languagelevel 2 ge}{false}ifelse def /@BeginSysCorelDict{systemdict/Corel30Dict known{systemdict/Corel30Dict get exec}if}bd/@EndSysCorelDict{systemdict/Corel30Dict known{end}if}bd AutoFlatness {/@ifl{dup currentflat exch sub 10 gt{ ([Error: PathTooComplex; OffendingCommand: AnyPaintingOperator]\n)print flush @np exit}{currentflat 2 add setflat}ifelse}bd/@fill/fill ld/fill{currentflat{ {@fill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@eofill/eofill ld/eofill {currentflat{{@eofill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@clip /clip ld/clip{currentflat{{@clip}stopped{@ifl}{exit}ifelse}bind loop setflat} bd/@eoclip/eoclip ld/eoclip{currentflat{{@eoclip}stopped{@ifl}{exit}ifelse} bind loop setflat}bd/@stroke/stroke ld/stroke{currentflat{{@stroke}stopped {@ifl}{exit}ifelse}bind loop setflat}bd}if L2?{/@ssa{true setstrokeadjust}bd}{ /@ssa{}bd}ifelse/d/setdash ld/j/setlinejoin ld/J/setlinecap ld/M/setmiterlimit ld/w/setlinewidth ld/O{/$o xd}bd/R{/$O xd}bd/W/eoclip ld/c/curveto ld/C/c ld/l /lineto ld/L/l ld/rl/rlineto ld/m/moveto ld/n/newpath ld/N/newpath ld/P{11 rp} bd/u{}bd/U{}bd/A{pop}bd/q/@gs ld/Q/@gr ld/&{}bd/@j{@sv @np}bd/@J{@rs}bd/g{1 exch sub/$k xd/$c 0 def/$m 0 def/$y 0 def/$t 1 def/$n _ def/$fil 0 def}bd/G{1 sub neg/$K xd _ 1 0 0 0/$C xd/$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type /stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd /$fil 0 def}bd/K{1 index type/stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def} ifelse/$K xd/$Y xd/$M xd/$C xd}bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{ /s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{ dup 0 ne{setflat}{pop}ifelse}bd/v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2 copy c}bd/Y/y ld/@w{matrix rotate/$ptm xd matrix scale $ptm dup concatmatrix /$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd {/$scp xd/$sca xd/$scf xd}if}bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd} if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch load SetScr/$dsp xd/$dsa xd/$dsf xd}bd/$ngx{$SDF{$SCF SepMode_5 0 eq{$SCA} {$dsa}ifelse $SCP @ss}if}bd/p{/$pm xd 7 rp/$pyf xd/$pxf xd/$pn xd/$fil 1 def} bd/@MN{2 copy le{pop}{exch pop}ifelse}bd/@MX{2 copy ge{pop}{exch pop}ifelse}bd /InRange{3 -1 roll @MN @MX}bd/@sqr{dup 0 rl dup 0 exch rl neg 0 rl @cp}bd /currentscale{1 0 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt 0 1 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt}bd/@unscale{}bd/wDstChck{2 1 roll dup 3 -1 roll eq{1 add}if}bd /@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1 exch sub}bd /cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop}bd /rgb2cmyk{3{1 exch sub 3 1 roll}repeat 3 copy @MN @MN 3{dup 5 -1 roll sub neg exch}repeat}bd/rgb2g{2 index .299 mul 2 index .587 mul add 1 index .114 mul add 4 1 roll pop pop pop}bd/WaldoColor_5 where{pop}{/SetRgb/setrgbcolor ld/GetRgb /currentrgbcolor ld/SetGry/setgray ld/GetGry/currentgray ld/SetRgb2 systemdict /setrgbcolor get def/GetRgb2 systemdict/currentrgbcolor get def/SetHsb systemdict/sethsbcolor get def/GetHsb systemdict/currenthsbcolor get def /rgb2hsb{SetRgb2 GetHsb}bd/hsb2rgb{3 -1 roll dup floor sub 3 1 roll SetHsb GetRgb2}bd/setcmykcolor where{pop/SetCmyk_5/setcmykcolor ld}{/SetCmyk_5{ cmyk2rgb SetRgb}bd}ifelse/currentcmykcolor where{pop/GetCmyk/currentcmykcolor ld}{/GetCmyk{GetRgb rgb2cmyk}bd}ifelse/setoverprint where{pop}{/setoverprint{ /$op xd}bd}ifelse/currentoverprint where{pop}{/currentoverprint{$op}bd}ifelse /@tc_5{5 -1 roll dup 1 ge{pop}{4{dup 6 -1 roll mul exch}repeat pop}ifelse}bd /@trp{exch pop 5 1 roll @tc_5}bd/setprocesscolor_5{SepMode_5 0 eq{SetCmyk_5}{0 4 $ink_5 sub index exch pop 5 1 roll pop pop pop pop SepsColor true eq{$ink_5 3 gt{1 sub neg SetGry}{0 0 0 4 $ink_5 roll SetCmyk_5}ifelse}{1 sub neg SetGry} ifelse}ifelse}bd/findcmykcustomcolor where{pop}{/findcmykcustomcolor{5 array astore}bd}ifelse/setcustomcolor where{pop}{/setcustomcolor{exch aload pop SepMode_5 0 eq{pop @tc_5 setprocesscolor_5}{CurrentInkName_5 eq{4 index}{0} ifelse 6 1 roll 5 rp 1 sub neg SetGry}ifelse}bd}ifelse/@scc_5{dup type /booleantype eq{setoverprint}{1 eq setoverprint}ifelse dup _ eq{pop setprocesscolor_5 pop}{findcmykcustomcolor exch setcustomcolor}ifelse SepMode_5 0 eq{true}{GetGry 1 eq currentoverprint and 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h(b)m(y)f(Yin)f(and)g(Allision)e([20)q(].)46 b(T)-8 b(o)609 4599 y(guaran)m(tee)32 b(con)m(v)m(ergence,)h(t)m(w)m(o)f(additional)c (conditions)h(ha)m(v)m(e)j(to)f(b)s(e)e(ful\014lled:)1420 4728 y Fe(1)1395 4753 y Ff(X)1397 4934 y Fi(t)p Fh(=0)1530 4834 y Fn(\013)p Fr(\()p Fn(t)p Fr(\))e(=)e Fk(1)p Fn(;)2061 4728 y Fe(1)2036 4753 y Ff(X)2038 4934 y Fi(t)p Fh(=0)2171 4834 y Fn(\013)p Fr(\()p Fn(t)p Fr(\))2332 4796 y Fh(2)2398 4834 y Fn(<)g Fk(1)553 b Fr(\(2.15\))382 5119 y Fj(2.1.3)112 b(Analysis)382 5290 y Fr(The)38 b(b)s(ene\014t)g(of)h(Kohonen's)f(SOM)g (algorithm)g(is)g(its)g(abilit)m(y)f(to)i(\014t)g(the)g(net)g(in)m(to) 382 5403 y(di\013eren)m(t)f(input)e(data)j(spaces,)i(appro)m(ximating)d (the)g(densit)m(y)g(distribution)d(of)j(the)p eop %%Page: 18 24 18 23 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(18)382 548 y(data)31 b Fn(P)13 b Fr(\()p Fn(x)p Fr(\))31 b(in)e(some)i(w)m(a)m(y)-8 b(.)42 b(An)30 b(imp)s(ortan)m(t)f(prop)s(ert)m(y)h(is)g(the)g(pro)s (of)g(of)g(con)m(v)m(ergence)382 661 y(under)d(certain)h(conditions.)39 b(Also)28 b(non-trivial)e(mappings)g(are)j(found,)f(without)f(an)m(y) 382 774 y(a-priori)i(kno)m(wledge)h(of)h(the)f(input-distribution.)523 887 y(As)38 b(a)g(dra)m(wbac)m(k,)i(one)e(has)g(to)g(kno)m(w)g(the)g (exp)s(ected)g(dimension)e(of)i(the)g(input)382 1000 y(data,)30 b(b)s(ecause)f(the)h(dimension)c(and)j(size)g(of)g(the)g (net)h(is)e(de\014ned)g(at)i(the)f(b)s(eginning.)382 1112 y(The)24 b(dimension)d(has)j(also)g(to)h(b)s(e)f(homogeneous)h(o)m (v)m(er)g(the)g(whole)e(input)f(space.)39 b(This)382 1225 y(is)21 b(esp)s(ecially)f(a)i(problem,)g(if)f(the)h(input)e(data,) k(breaks)e(up)f(in)m(to)h(clusters,)h(or)e(has)h(areas)382 1338 y(of)30 b(di\013eren)m(t)g(densit)m(y)-8 b(.)523 1451 y(It)29 b(is)g(imp)s(ortan)m(t)f(to)i(mak)m(e)h(a)f(di\013erence)e (b)s(et)m(w)m(een)i(the)g(dimension)d(of)i(the)h(input)382 1564 y(data,)42 b(and)d(the)g(dimension)d(of)j(the)h(input)d(space.)67 b(The)39 b(dimension)d(of)j(the)h(input)382 1677 y(space)45 b(has)f(no)g(in\015uence)e(at)j(all,)i(as)e(the)f(net)g(tries)g(to)h (appro)m(ximate)f(the)h(input)382 1790 y(manifold.)d(If)31 b(the)h(input)d(space)j(is)f(v)m(ery)h(high)e(dimensional)e(but)j(the)h (input)d(data)k(is)382 1903 y(not,)i(the)e(map)h(can)g(b)s(e)e(used)h (to)i(eliminate)d(the)h(unnecessary)g(dimensions.)48 b(But)34 b(if)382 2016 y(the)c(dimension)e(of)i(the)h(net)f(is)f(lo)m (w)m(er)i(than)e(the)i(dimension)c(of)k(the)f(input)e(data,)j(the)382 2129 y(net)k(ma)m(y)h(corrugate.)56 b(E.g.)g(using)34 b(a)h(one)h(dimensional)c(net,)37 b(to)f(represen)m(t)f(a)h(ev)m(en)382 2242 y(distributed)25 b(t)m(w)m(o-dimensional)i(input)g(data,)j (results)d(in)g(a)h(net,)i(that)f(is)e(comparable)382 2355 y(to)i(a)h(p)s(eano-curv)m(e)f(\(see)h(\014gure)e(2.3\).)41 b(This)27 b(b)s(ecomes)j(clear,)f(if)f(one)h(sees)g(Kohonen's)382 2467 y(algorithm)g(as)i(a)f(pro)s(cess)g(of)h(minimization)c(of)j(the)h (the)f(a)m(v)m(erage)j(distance)d(b)s(et)m(w)m(een)382 2580 y(the)g(sample)f(p)s(oin)m(ts)g(and)g(their)g(nearest)i (reference-v)m(ector.)43 b(F)-8 b(or)31 b(this)d(minimization)382 2693 y(problem)h(a)h(p)s(eano)h(curv)m(e)f(is)g(the)g(optimal)g (solution.)1129 4378 y currentpoint currentpoint translate 0.48863 0.48863 scale neg exch neg exch translate 1129 4378 a 1129 4378 a gsave currentpoint currentpoint translate 0 neg rotate neg exch neg exch translate 1129 4378 a @beginspecial 117 @llx 195 @lly 484 @urx 580 @ury 3670 @rwi @setspecial %%BeginDocument: figures/peano.ps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: GNU libplot drawing library 1.6 %%Title: PostScript plot %%CreationDate: Tue Oct 5 15:09:12 1999 %%DocumentData: Clean7Bit %%LanguageLevel: 1 %%Pages: 1 %%PageOrder: Ascend %%Orientation: Portrait %%BoundingBox: 117 195 484 580 %%DocumentNeededResources: font Helvetica %%DocumentSuppliedResources: procset GNU_libplot 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/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis /yacute/thorn/ydieresis ] def /reencodeISO { dup dup findfont dup length dict begin { 1 index /FID ne { def }{ pop pop } ifelse } forall /Encoding ISOLatin1Encoding def currentdict end definefont } def /Helvetica reencodeISO def %%BeginResource procset GNU_libplot 1.0 0 /none null def /numGraphicParameters 17 def /stringLimit 65535 def /arrowHeight 8 def /arrowWidth 4 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc closepath patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 1 printSize sub descender sub def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def %%EndResource %%EndSetup %%Page: 1 1 %%PageResources: font Helvetica %%PageBoundingBox: 117 195 484 580 %%BeginPageSetup %I Idraw 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 0 0 ] concat /originalCTM matrix currentmatrix def /trueoriginalCTM matrix currentmatrix def %%EndPageSetup Begin %I Poly [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4 2304 2304 2304 9216 9216 9216 9216 2304 4 Poly End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 128.156 213.1332 ] concat %I [ (0) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 9216 2304 9078 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 2304 2304 2442 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 197.276 213.1332 ] concat %I [ (1) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3686 9216 3686 9078 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3686 2304 3686 2442 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 266.396 213.1332 ] concat %I [ (2) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5069 9216 5069 9078 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5069 2304 5069 2442 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 335.516 213.1332 ] concat %I [ (3) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6451 9216 6451 9078 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6451 2304 6451 2442 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 404.636 213.1332 ] concat %I [ (4) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7834 9216 7834 9078 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7834 2304 7834 2442 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 473.756 213.1332 ] concat %I [ (5) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 9216 9216 9078 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 2304 9216 2442 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 9216 2304 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 2304 2304 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2580 9216 2580 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2580 2304 2580 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2857 9216 2857 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2857 2304 2857 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3133 9216 3133 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3133 2304 3133 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3410 9216 3410 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3410 2304 3410 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3686 9216 3686 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3686 2304 3686 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3963 9216 3963 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3963 2304 3963 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4239 9216 4239 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4239 2304 4239 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4516 9216 4516 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4516 2304 4516 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4792 9216 4792 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4792 2304 4792 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5069 9216 5069 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5069 2304 5069 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5345 9216 5345 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5345 2304 5345 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5622 9216 5622 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5622 2304 5622 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5898 9216 5898 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5898 2304 5898 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6175 9216 6175 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6175 2304 6175 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6451 9216 6451 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6451 2304 6451 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6728 9216 6728 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6728 2304 6728 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7004 9216 7004 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7004 2304 7004 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7281 9216 7281 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7281 2304 7281 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7557 9216 7557 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7557 2304 7557 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7834 9216 7834 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7834 2304 7834 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8110 9216 8110 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8110 2304 8110 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8387 9216 8387 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8387 2304 8387 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8663 9216 8663 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8663 2304 8663 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8940 9216 8940 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8940 2304 8940 2359 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 9216 9216 9161 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 2304 9216 2359 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 117.9279 229.3306 ] concat %I [ (0) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 2304 9078 2304 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 2304 2442 2304 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 117.9279 298.4506 ] concat %I [ (1) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 3686 9078 3686 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 3686 2442 3686 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 117.9279 367.5706 ] concat %I [ (2) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 5069 9078 5069 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 5069 2442 5069 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 117.9279 436.6906 ] concat %I [ (3) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 6451 9078 6451 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 6451 2442 6451 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 117.9279 505.8106 ] concat %I [ (4) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 7834 9078 7834 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 7834 2442 7834 2 MLine End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-18-*-*-*-*-*-*-* /Helvetica 18.144000 SetF %I t [ 1 0 0 1 117.9279 574.9306 ] concat %I [ (5) ] Text End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 9216 9078 9216 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 9216 2442 9216 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 2304 9161 2304 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 2304 2359 2304 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 2580 9161 2580 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 2580 2359 2580 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 2857 9161 2857 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 2857 2359 2857 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 3133 9161 3133 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 3133 2359 3133 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 3410 9161 3410 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 3410 2359 3410 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 3686 9161 3686 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 3686 2359 3686 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 3963 9161 3963 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 3963 2359 3963 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 4239 9161 4239 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 4239 2359 4239 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 4516 9161 4516 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 4516 2359 4516 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 4792 9161 4792 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 4792 2359 4792 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 5069 9161 5069 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 5069 2359 5069 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 5345 9161 5345 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 5345 2359 5345 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 5622 9161 5622 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 5622 2359 5622 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 5898 9161 5898 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 5898 2359 5898 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 6175 9161 6175 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 6175 2359 6175 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 6451 9161 6451 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 6451 2359 6451 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 6728 9161 6728 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 6728 2359 6728 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 7004 9161 7004 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 7004 2359 7004 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 7281 9161 7281 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 7281 2359 7281 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 7557 9161 7557 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 7557 2359 7557 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 7834 9161 7834 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 7834 2359 7834 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 8110 9161 8110 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 8110 2359 8110 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 8387 9161 8387 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 8387 2359 8387 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 8663 9161 8663 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 8663 2359 8663 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 8940 9161 8940 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 8940 2359 8940 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 9216 9216 9161 9216 2 MLine End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2304 9216 2359 9216 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 6127 6207 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5756 6543 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6127 6207 5756 6543 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5756 6543 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5400 7363 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5756 6543 5400 7363 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5400 7363 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5422 7901 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5400 7363 5422 7901 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5422 7901 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5275 8553 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5422 7901 5275 8553 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5275 8553 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4321 8580 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5275 8553 4321 8580 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4321 8580 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3525 8673 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4321 8580 3525 8673 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3525 8673 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3057 8392 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3525 8673 3057 8392 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3057 8392 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3010 7566 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3057 8392 3010 7566 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3010 7566 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3477 6824 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3010 7566 3477 6824 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3477 6824 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3201 6289 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3477 6824 3201 6289 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3201 6289 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3181 5788 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3201 6289 3181 5788 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3181 5788 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3199 5193 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3181 5788 3199 5193 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3199 5193 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3088 4605 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3199 5193 3088 4605 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3088 4605 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2906 4039 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3088 4605 2906 4039 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2906 4039 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2847 3350 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2906 4039 2847 3350 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2847 3350 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3389 3018 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 2847 3350 3389 3018 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 3389 3018 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4193 2930 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 3389 3018 4193 2930 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4193 2930 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4862 3288 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4193 2930 4862 3288 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4862 3288 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4991 3837 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4862 3288 4991 3837 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4991 3837 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4470 4413 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4991 3837 4470 4413 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4470 4413 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4741 5284 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4470 4413 4741 5284 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 4741 5284 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5439 5296 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 4741 5284 5439 5296 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 5439 5296 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 6129 4735 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 5439 5296 6129 4735 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 6129 4735 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 6622 4038 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6129 4735 6622 4038 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 6622 4038 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 6570 3055 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6622 4038 6570 3055 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 6570 3055 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7489 2846 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 6570 3055 7489 2846 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7489 2846 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8227 2822 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7489 2846 8227 2822 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8227 2822 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8714 3050 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8227 2822 8714 3050 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8714 3050 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8582 3710 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8714 3050 8582 3710 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8582 3710 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8298 4439 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8582 3710 8298 4439 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8298 4439 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7945 5115 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8298 4439 7945 5115 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7945 5115 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7441 5464 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7945 5115 7441 5464 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7441 5464 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7553 6361 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7441 5464 7553 6361 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7553 6361 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8346 6527 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7553 6361 8346 6527 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8346 6527 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8407 7038 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8346 6527 8407 7038 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8407 7038 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8551 7729 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8407 7038 8551 7729 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8551 7729 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8046 8575 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8551 7729 8046 8575 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 8046 8575 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7133 8353 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 8046 8575 7133 8353 2 MLine End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 7133 8353 65 Circ End Begin %I Circ [0.324 0 0 0.324 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 6661 8137 65 Circ End Begin %I MLine [0.6776471 0 0 0.6776471 0 0 ] trueoriginalCTM originalCTM concatmatrix pop 0 setlinecap 0 setlinejoin %I b 65535 1 0 0 [ ] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p none SetP %I t [0.05 0 0 0.05 18 108 ] concat %I 2 7133 8353 6661 8137 2 MLine End %%PageTrailer End %I eop showpage %%Trailer end %%EOF %%EndDocument @endspecial 4187 4378 a currentpoint grestore moveto 4187 4378 a 1129 4378 a currentpoint currentpoint translate 1 0.48863 div 1 0.48863 div scale neg exch neg exch translate 1129 4378 a 492 4674 a Fr(Figure)g(2.3:)42 b(A)30 b(one)h(dimensional)c (net,)k(learning)e(a)i(t)m(w)m(o)g(dimensional)d(square)523 4886 y(This)35 b(ma)m(y)i(b)s(ecome)g(a)g(problem,)g(as)g(neigh)m(b)s (orho)s(o)s(d)d(relations)i(are)h(not)g(repre-)382 4999 y(sen)m(ted)28 b(in)f(a)h(correct)h(w)m(a)m(y)-8 b(.)41 b(Nearb)m(y)28 b(neurons)f(in)f(the)i(input)e(space)j(can)f(b)s(e)f (far)g(apart)382 5112 y(in)e(the)i(output)g(space.)40 b(T)-8 b(o)27 b(a)m(v)m(oid)g(this,)g(the)g(righ)m(t)f(dimension)e(for) j(the)f(net)h(has)g(to)g(b)s(e)382 5225 y(c)m(hosen.)41 b(In)27 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figures/gngas.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 27 670 185 823 %%Creator: CorelDRAW 7 %%Title: %%CreationDate: Sat May 15 17:18:29 1999 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness true def /AutoSteps 0 def /CMYKMarks false def /UseLevel2 false def %Color profile: %%BeginResource: procset wCorel7Dict 7.0 0 /wCorel7Dict 300 dict def wCorel7Dict begin % Copyright (c)1992-96 Corel Corporation % All rights reserved. v7.0 r1.0 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/setscreen{pop pop pop}bd/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd /SepMode_5 where{pop}{/SepMode_5 0 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[0.00028346 0.00000000 0.00000000 0.00028346 145.26821516 787.82964047] @tm 0 O 0 @g 0.00 0.00 0.00 1.00 k e % FontChange:/_TimesNewRoman 28222.00000 z %CHAR: 76771 0 (1) @t 80080 16852 m 84627 19071 L 85079 19071 L 85079 3295 l 85079 2247 85125 1595 85213 1337 c 85298 1079 85481 882 85756 744 c 86031 607 86592 529 87435 512 C 87435 0 L 80408 0 L 80408 512 L 81290 529 81862 603 82119 737 c 82377 871 82553 1051 82655 1274 c 82758 1499 82807 2173 82807 3295 c 82807 13381 l 82807 14739 82761 15614 82669 15998 c 82606 16291 82486 16510 82317 16647 c 82147 16785 81943 16852 81703 16852 c 81364 16852 80891 16711 80285 16425 C 80080 16852 L @c F T %%PageTrailer @rs @rs %%Trailer @EndSysCorelDict end %%DocumentSuppliedResources: procset wCorel7Dict %%EOF %%EndDocument @endspecial 2745 4345 a currentpoint grestore moveto 2745 4345 a 1428 4345 a currentpoint currentpoint translate 1 0.68102 div 1 0.68102 div scale neg exch neg exch translate 1428 4345 a 707 4640 a Fr(Figure)f(2.4:)42 b(Gro)m(wing)30 b(Neural)f(Net,)j(Adding)d(a)h(new)g(connection)518 4880 y Fk(\017)46 b Fr(Up)s(date)30 b(W)-8 b(eigh)m(ts)711 5083 y Fp({)46 b Fr(The)39 b(squared)g(distance)g(b)s(et)m(w)m(een)g (the)h(input)d(data)j(and)f(the)h(winning)809 5196 y(w)m(eigh)m(t)31 b(v)m(ector)h(is)d(computed)i(and)e(stored)i(in)e(the)i(lo)s(cal)e (error)h(v)-5 b(ariable:)1426 5380 y Fn(E)1493 5394 y Fi(s)1526 5403 y Fd(1)1564 5380 y Fr(\()p Fn(t)21 b Fr(+)f(1\))26 b(=)f Fn(E)2013 5394 y Fi(s)2046 5403 y Fd(1)2084 5380 y Fr(\()p Fn(t)p Fr(\))c(+)f Fk(k)p Fn(x)g Fk(\000)g Fn(w)2572 5394 y Fi(s)2605 5403 y Fd(1)2644 5380 y Fk(k)2689 5343 y Fh(2)2729 5380 y Fn(:)384 b Fr(\(2.24\))p eop %%Page: 22 28 22 27 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(22)711 548 y Fp({)46 b Fr(Up)s(date)35 b(the)g(b)s(est)g(matc)m(hing)g(w)m(eigh)m(t)g(v)m (ector)i Fn(s)2512 562 y Fh(1)2586 548 y Fr(and)d(its)h(direct)f(top)s (o-)809 661 y(logical)c(neigh)m(b)s(ors)f Fk(N)1577 675 y Fi(s)1610 684 y Fd(1)1678 661 y Fr(using)g(\014x)h(learning)f(rates)i Fn(\013)2675 676 y Fi(b)2739 661 y Fr(and)f Fn(\013)2974 675 y Fi(n)3021 661 y Fr(:)1131 865 y Fn(w)1196 879 y Fi(c)1227 888 y Fd(1)1266 865 y Fr(\()p Fn(t)20 b Fr(+)g(1\))84 b(=)f Fn(w)1828 879 y Fi(s)1861 888 y Fd(1)1899 865 y Fr(\()p Fn(t)p Fr(\))21 b(+)f Fn(\013)2172 880 y Fi(b)2206 865 y Fr(\()p Fn(x)h Fk(\000)f Fn(w)2470 879 y Fi(s)2503 888 y Fd(1)2541 865 y Fr(\))1172 1003 y Fn(w)1237 1017 y Fi(i)1266 1003 y Fr(\()p Fn(t)g Fr(+)g(1\))84 b(=)f Fn(w)1828 1017 y Fi(i)1856 1003 y Fr(\()p Fn(t)p Fr(\))21 b(+)e Fn(\013)2128 1017 y Fi(n)2176 1003 y Fr(\()p Fn(x)h Fk(\000)g Fn(w)2439 1017 y Fi(i)2467 1003 y Fr(\))182 b Fk(8)p Fn(i)25 b Fk(2)g(N)2952 1017 y Fi(s)2985 1026 y Fd(1)3023 1003 y Fn(:)90 b Fr(\(2.25\))711 1224 y Fp({)46 b Fr(Incremen)m(t)34 b(the)g(time)f(stamp)g(of)h(all)e(connections,)j (and)e(remo)m(v)m(e)i(those,)809 1337 y(that)30 b(exceed)g(the)f(maxim) m(um)f(age)i Fn(L)p Fr(.)40 b(If)29 b(a)g(neuron)f(has)h(no)g (connections)809 1450 y(left,)i(remo)m(v)m(e)g(it)f(as)h(w)m(ell.)518 1662 y Fk(\017)46 b Fr(Up)s(date)30 b(net)h(structure)f(I)s(I)609 1812 y(If)j(the)g(n)m(um)m(b)s(er)f(of)h(input)f(signals)f(generated)k (so)e(far)g(is)f(an)h(in)m(teger)h(m)m(ultiple)609 1925 y(of)d(a)f(prede\014ned)f(parameter,)i(insert)e(a)i(new)f(neuron)f(as)i (follo)m(ws:)711 2138 y Fp({)46 b Fr(Determine)31 b(the)f(neuron)f Fn(q)34 b Fr(with)29 b(the)h(maxim)m(um)f(accum)m(ulated)i(error:)1747 2342 y Fn(q)d Fr(=)d(arg)r(max)2209 2356 y Fi(c)p Fe(2C)2331 2342 y Fn(E)2398 2356 y Fi(c)3138 2342 y Fr(\(2.26\))711 2563 y Fp({)46 b Fr(Determine)38 b(the)g(direct)f(neigh)m(b)s(or)f Fn(f)47 b Fr(of)38 b Fn(q)i Fr(with)c(the)i(maxim)m(um)f(accu-)809 2676 y(m)m(ulated)30 b(error:)1716 2789 y Fn(f)k Fr(=)25 b(arg)r(max)2188 2803 y Fi(c)p Fe(2N)2324 2811 y Fg(q)2362 2789 y Fn(E)2429 2803 y Fi(c)3138 2789 y Fr(\(2.27\))711 2972 y Fp({)46 b Fr(Add)32 b(a)h(new)f(neuron)f Fn(r)k Fr(and)d(de\014ne)g(its)g(reference)h(v)m(ector)h(b)s(et)m(w)m(een)f (the)809 3085 y(t)m(w)m(o)f(neigh)m(b)s(ors:)1379 3289 y Fk(C)f Fr(=)25 b Fk(C)g([)20 b(f)p Fn(r)s Fk(g)p Fn(;)198 b(w)2130 3303 y Fi(r)2193 3289 y Fr(=)25 b(\()p Fn(w)2389 3303 y Fi(q)2448 3289 y Fr(+)19 b Fn(w)2603 3304 y Fi(f)2649 3289 y Fr(\))p Fn(=)p Fr(2)p Fn(:)339 b Fr(\(2.28\))711 3510 y Fp({)46 b Fr(Connect)33 b(this)e(new)h(neuron)f Fn(r)k Fr(to)f Fn(q)h Fr(and)d Fn(f)10 b Fr(,)32 b(and)g(remo)m(v)m(e)i (the)e(old)g(con-)809 3623 y(nection)e(b)s(et)m(w)m(een)h(them:)1443 3827 y Fk(E)i Fr(=)25 b Fk(E)j([)20 b(f)p Fr(\()p Fn(r)m(;)15 b(q)s Fr(\))p Fn(;)g Fr(\()p Fn(r)m(;)g(f)10 b Fr(\))p Fk(g)34 b Fn(=)d Fk(f)p Fr(\()p Fn(q)s(;)15 b(f)10 b Fr(\))p Fk(g)p Fn(:)402 b Fr(\(2.29\))711 4048 y Fp({)46 b Fr(Mo)s(dify)29 b(the)i(error)f(v)-5 b(ariables)29 b(of)h Fn(q)s Fr(,)h Fn(f)39 b Fr(and)30 b Fn(r)s Fr(:)1275 4252 y Fn(E)1342 4266 y Fi(q)1380 4252 y Fr(\()p Fn(t)20 b Fr(+)g(1\))84 b(=)f Fn(E)1944 4266 y Fi(q)1982 4252 y Fr(\()p Fn(t)p Fr(\))21 b Fk(\000)e Fn(\017E)2300 4266 y Fi(q)2339 4252 y Fr(\()p Fn(t)p Fr(\))696 b(\(2.30\))1267 4390 y Fn(E)1334 4405 y Fi(f)1380 4390 y Fr(\()p Fn(t)20 b Fr(+)g(1\))84 b(=)f Fn(E)1944 4405 y Fi(f)1989 4390 y Fr(\()p Fn(t)p Fr(\))21 b Fk(\000)f Fn(\017E)2308 4405 y Fi(f)2353 4390 y Fr(\()p Fn(t)p Fr(\))682 b(\(2.31\))1275 4528 y Fn(E)1342 4542 y Fi(r)1380 4528 y Fr(\()p Fn(t)20 b Fr(+)g(1\))84 b(=)f(\()p Fn(E)1979 4542 y Fi(q)2017 4528 y Fr(\()p Fn(t)21 b Fr(+)e(1\))j(+)d Fn(E)2455 4543 y Fi(f)2501 4528 y Fr(\()p Fn(t)h Fr(+)g(1\)\))p Fn(=)p Fr(2)p Fn(:)228 b Fr(\(2.32\))518 4786 y Fk(\017)46 b Fr(Up)s(date)30 b(error)g(v)-5 b(ariables)609 4936 y(Decrease)32 b(all)e(error)g(v)-5 b(ariables)29 b(b)m(y)h(a)h(\014x)f(amoun)m(t:) 1229 5141 y Fn(E)1296 5155 y Fi(c)1331 5141 y Fr(\()p Fn(t)20 b Fr(+)g(1\))26 b(=)f Fn(E)1779 5155 y Fi(c)1814 5141 y Fr(\()p Fn(t)p Fr(\))20 b Fk(\000)g Fn(\014)5 b(E)2151 5155 y Fi(c)2186 5141 y Fr(\()p Fn(t)p Fr(\))183 b Fk(8)p Fn(c)25 b Fk(2)g(C)5 b Fn(:)387 b Fr(\(2.33\))p eop %%Page: 23 29 23 28 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(23)980 1475 y currentpoint currentpoint translate 0.93977 0.93977 scale neg exch neg exch translate 980 1475 a 980 1475 a gsave currentpoint currentpoint translate 0 neg rotate neg exch neg exch translate 980 1475 a @beginspecial 27 @llx 693 @lly 256 @urx 823 @ury 2290 @rwi @setspecial %%BeginDocument: figures/gngas2.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 27 693 256 823 %%Creator: CorelDRAW 7 %%Title: %%CreationDate: Sat May 15 17:30:39 1999 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness true def /AutoSteps 0 def /CMYKMarks false def /UseLevel2 false def %Color profile: %%BeginResource: procset wCorel7Dict 7.0 0 /wCorel7Dict 300 dict def wCorel7Dict begin % Copyright (c)1992-96 Corel Corporation % All rights reserved. v7.0 r1.0 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/setscreen{pop pop pop}bd/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd /SepMode_5 where{pop}{/SepMode_5 0 def}ifelse/CurrentInkName_5 where{pop} {/CurrentInkName_5(Composite)def}ifelse/$ink_5 where{pop}{/$ink_5 -1 def} ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def/$t 1 def/$n _ def/$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def/$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def/s1t 0 def/s1n _ def/$bkg false def /SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd /$ptm matrix def/$ttm matrix def/$stm matrix def/$ffpnt true def /CorelDrawReencodeVect[16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /L2?/languagelevel where{pop languagelevel 2 ge}{false}ifelse def /@BeginSysCorelDict{systemdict/Corel30Dict known{systemdict/Corel30Dict get exec}if}bd/@EndSysCorelDict{systemdict/Corel30Dict known{end}if}bd AutoFlatness {/@ifl{dup currentflat exch sub 10 gt{ ([Error: PathTooComplex; OffendingCommand: AnyPaintingOperator]\n)print flush @np exit}{currentflat 2 add setflat}ifelse}bd/@fill/fill ld/fill{currentflat{ {@fill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@eofill/eofill ld/eofill {currentflat{{@eofill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@clip /clip ld/clip{currentflat{{@clip}stopped{@ifl}{exit}ifelse}bind loop setflat} bd/@eoclip/eoclip ld/eoclip{currentflat{{@eoclip}stopped{@ifl}{exit}ifelse} bind loop setflat}bd/@stroke/stroke ld/stroke{currentflat{{@stroke}stopped {@ifl}{exit}ifelse}bind loop setflat}bd}if L2?{/@ssa{true setstrokeadjust}bd}{ /@ssa{}bd}ifelse/d/setdash ld/j/setlinejoin ld/J/setlinecap ld/M/setmiterlimit ld/w/setlinewidth ld/O{/$o xd}bd/R{/$O xd}bd/W/eoclip ld/c/curveto ld/C/c ld/l /lineto ld/L/l ld/rl/rlineto ld/m/moveto ld/n/newpath ld/N/newpath ld/P{11 rp} bd/u{}bd/U{}bd/A{pop}bd/q/@gs ld/Q/@gr ld/&{}bd/@j{@sv @np}bd/@J{@rs}bd/g{1 exch sub/$k xd/$c 0 def/$m 0 def/$y 0 def/$t 1 def/$n _ def/$fil 0 def}bd/G{1 sub neg/$K xd _ 1 0 0 0/$C xd/$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type /stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd /$fil 0 def}bd/K{1 index type/stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def} ifelse/$K xd/$Y xd/$M xd/$C xd}bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{ /s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{ dup 0 ne{setflat}{pop}ifelse}bd/v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2 copy c}bd/Y/y ld/@w{matrix rotate/$ptm xd matrix scale $ptm dup concatmatrix /$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd {/$scp xd/$sca xd/$scf xd}if}bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd} if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch load SetScr/$dsp xd/$dsa xd/$dsf xd}bd/$ngx{$SDF{$SCF SepMode_5 0 eq{$SCA} {$dsa}ifelse $SCP @ss}if}bd/p{/$pm xd 7 rp/$pyf xd/$pxf xd/$pn xd/$fil 1 def} bd/@MN{2 copy le{pop}{exch pop}ifelse}bd/@MX{2 copy ge{pop}{exch pop}ifelse}bd /InRange{3 -1 roll @MN @MX}bd/@sqr{dup 0 rl dup 0 exch rl neg 0 rl @cp}bd /currentscale{1 0 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt 0 1 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt}bd/@unscale{}bd/wDstChck{2 1 roll dup 3 -1 roll eq{1 add}if}bd /@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1 exch sub}bd /cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop}bd /rgb2cmyk{3{1 exch sub 3 1 roll}repeat 3 copy @MN @MN 3{dup 5 -1 roll sub neg exch}repeat}bd/rgb2g{2 index .299 mul 2 index .587 mul add 1 index .114 mul add 4 1 roll pop pop pop}bd/WaldoColor_5 where{pop}{/SetRgb/setrgbcolor ld/GetRgb /currentrgbcolor ld/SetGry/setgray ld/GetGry/currentgray ld/SetRgb2 systemdict /setrgbcolor get def/GetRgb2 systemdict/currentrgbcolor get def/SetHsb systemdict/sethsbcolor get def/GetHsb systemdict/currenthsbcolor get def /rgb2hsb{SetRgb2 GetHsb}bd/hsb2rgb{3 -1 roll dup floor sub 3 1 roll SetHsb GetRgb2}bd/setcmykcolor where{pop/SetCmyk_5/setcmykcolor ld}{/SetCmyk_5{ cmyk2rgb SetRgb}bd}ifelse/currentcmykcolor where{pop/GetCmyk/currentcmykcolor ld}{/GetCmyk{GetRgb rgb2cmyk}bd}ifelse/setoverprint where{pop}{/setoverprint{ /$op xd}bd}ifelse/currentoverprint where{pop}{/currentoverprint{$op}bd}ifelse /@tc_5{5 -1 roll dup 1 ge{pop}{4{dup 6 -1 roll mul exch}repeat pop}ifelse}bd /@trp{exch pop 5 1 roll @tc_5}bd/setprocesscolor_5{SepMode_5 0 eq{SetCmyk_5}{0 4 $ink_5 sub index exch pop 5 1 roll pop pop pop pop SepsColor true eq{$ink_5 3 gt{1 sub neg SetGry}{0 0 0 4 $ink_5 roll SetCmyk_5}ifelse}{1 sub neg SetGry} ifelse}ifelse}bd/findcmykcustomcolor where{pop}{/findcmykcustomcolor{5 array astore}bd}ifelse/setcustomcolor where{pop}{/setcustomcolor{exch aload pop SepMode_5 0 eq{pop @tc_5 setprocesscolor_5}{CurrentInkName_5 eq{4 index}{0} ifelse 6 1 roll 5 rp 1 sub neg SetGry}ifelse}bd}ifelse/@scc_5{dup type /booleantype eq{setoverprint}{1 eq setoverprint}ifelse dup _ eq{pop setprocesscolor_5 pop}{findcmykcustomcolor exch setcustomcolor}ifelse SepMode_5 0 eq{true}{GetGry 1 eq currentoverprint and not}ifelse}bd/colorimage where{pop /ColorImage{colorimage}def}{/ColorImage{/ncolors xd pop/dataaq xd{dataaq ncolors dup 3 eq{/$dat xd 0 1 $dat length 3 div 1 sub{dup 3 mul $dat 1 index get 255 div $dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div rgb2g 255 mul cvi exch pop $dat 3 1 roll put}for $dat 0 $dat length 3 idiv getinterval pop}{4 eq{/$dat xd 0 1 $dat length 4 div 1 sub{dup 4 mul $dat 1 index get 255 div $dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div $dat 4 index 3 add get 255 div cmyk2rgb rgb2g 255 mul cvi exch pop $dat 3 1 roll put}for $dat 0 $dat length ncolors idiv getinterval}if}ifelse}image}bd }ifelse/setcmykcolor{1 5 1 roll _ currentoverprint @scc_5/$ffpnt xd}bd /currentcmykcolor{0 0 0 0}bd/setrgbcolor{rgb2cmyk setcmykcolor}bd /currentrgbcolor{currentcmykcolor cmyk2rgb}bd/sethsbcolor{hsb2rgb setrgbcolor} bd/currenthsbcolor{currentrgbcolor rgb2hsb}bd/setgray{dup dup setrgbcolor}bd /currentgray{currentrgbcolor rgb2g}bd}ifelse/WaldoColor_5 true def/@sft{$tllx $pxf add dup $tllx gt{$pwid sub}if/$tx xd $tury $pyf sub dup $tury lt{$phei add}if/$ty xd}bd/@stb{pathbbox/$ury xd/$urx xd/$lly xd/$llx xd}bd/@ep{{cvx exec }forall}bd/@tp{@sv/$in true def 2 copy dup $lly le{/$in false def}if $phei sub $ury ge{/$in false def}if dup $urx ge{/$in false def}if $pwid add $llx le{/$in false def}if $in{@np 2 copy m $pwid 0 rl 0 $phei neg rl $pwid neg 0 rl 0 $phei rl clip @np $pn cvlit load aload pop 7 -1 roll 5 index sub 7 -1 roll 3 index sub Tl matrix currentmatrix/$ctm xd @ep pop pop pop pop}{pop pop}ifelse @rs}bd /@th{@sft 0 1 $tly 1 sub{dup $psx mul $tx add{dup $llx gt{$pwid sub}{exit} ifelse}loop exch $phei mul $ty exch sub 0 1 $tlx 1 sub{$pwid mul 3 copy 3 -1 roll add exch @tp pop}for pop pop}for}bd/@tv{@sft 0 1 $tlx 1 sub{dup $pwid mul $tx add exch $psy mul $ty exch sub{dup $ury lt{$phei add}{exit}ifelse}loop 0 1 $tly 1 sub{$phei mul 3 copy sub @tp pop}for pop pop}for}bd/$fm 0 def/wfill{1 $fm eq{fill}{eofill}ifelse}bd/wclip{1 $fm eq{clip}{eoclip}ifelse}bd/@pf{@gs $ctm setmatrix $pm concat @stb wclip Bburx Bbury $pm itransform/$tury xd/$turx xd Bbllx Bblly $pm itransform/$tlly xd/$tllx xd/$wid $turx $tllx sub def/$hei $tury $tlly sub def @gs $vectpat{1 0 0 0 0 _ $o 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b(arg)r(max)2043 1479 y Fi(i)p Fe(2N)2172 1487 y Fg(q)2211 1465 y Fk(k)p Fn(w)2321 1479 y Fi(q)2379 1465 y Fk(\000)20 b Fn(w)2535 1479 y Fi(i)2564 1465 y Fk(k)529 b Fr(\(2.35\))809 1659 y(then)30 b(split)f(this)g (connection)i(and)e(insert)h(a)g(new)g(neuron)g Fn(r)s Fr(.)1379 1853 y Fk(C)h Fr(=)25 b Fk(C)g([)20 b(f)p Fn(r)s Fk(g)p Fn(;)198 b(w)2130 1867 y Fi(r)2193 1853 y Fr(=)25 b(\()p Fn(w)2389 1867 y Fi(q)2448 1853 y Fr(+)19 b Fn(w)2603 1868 y Fi(f)2649 1853 y Fr(\))p Fn(=)p Fr(2)p Fn(:)339 b Fr(\(2.36\))711 2061 y Fp({)46 b Fr(Connect)23 b(this)f(new)h(neuron) f Fn(r)j Fr(to)f Fn(q)s Fr(,)g Fn(f)33 b Fr(and)22 b(all)g(common)h (simplices)e(\(that)809 2174 y(are)34 b(all)e(simplices,)f(whic)m(h)h (con)m(tain)i(the)f(edge)h(\()p Fn(q)s(;)15 b(f)10 b Fr(\).)49 b(Remo)m(v)m(e)35 b(the)e(old)809 2287 y(connection)e(\()p Fn(q)s(;)15 b(f)10 b Fr(\).)711 2430 y Fp({)46 b Fr(Mo)s(dify)29 b(the)i(error)f(v)-5 b(ariables)29 b(of)h Fn(r)j Fr(and)d(all)f(its)h (neigh)m(b)s(ors:)1197 2637 y 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figures/gcell.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 27 693 256 823 %%Creator: CorelDRAW 7 %%Title: %%CreationDate: Sat May 15 17:30:26 1999 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness true def /AutoSteps 0 def /CMYKMarks false def /UseLevel2 false def %Color profile: %%BeginResource: procset wCorel7Dict 7.0 0 /wCorel7Dict 300 dict def wCorel7Dict begin % Copyright (c)1992-96 Corel Corporation % All rights reserved. v7.0 r1.0 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/setscreen{pop pop pop}bd/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd /SepMode_5 where{pop}{/SepMode_5 0 def}ifelse/CurrentInkName_5 where{pop} {/CurrentInkName_5(Composite)def}ifelse/$ink_5 where{pop}{/$ink_5 -1 def} ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def/$t 1 def/$n _ def/$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def/$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def/s1t 0 def/s1n _ def/$bkg false def /SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd /$ptm matrix def/$ttm matrix def/$stm matrix def/$ffpnt true def /CorelDrawReencodeVect[16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /L2?/languagelevel where{pop languagelevel 2 ge}{false}ifelse def /@BeginSysCorelDict{systemdict/Corel30Dict known{systemdict/Corel30Dict get 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y Ff(X)1893 2420 y Fi(i)p Fe(2N)2022 2428 y Fg(c)2069 2317 y Fk(k)p Fn(w)2179 2331 y Fi(c)2235 2317 y Fk(\000)20 b Fn(w)2391 2331 y Fi(i)2419 2317 y Fk(k)p Fn(:)649 b Fr(\(2.41\))609 2604 y(#)p Fk(f\001g)29 b Fr(denotes)g(the)f(cardinalit)m(y)f(of)h(a)h(set.)41 b(As)28 b(great)h(V)-8 b(oronoi)29 b(regions)e(with)g(a)609 2717 y(small)h(but)g(v)m(ery)i(dense)e(cen)m(ter,)j(migh)m(t)e(b)s(e)g (underv)-5 b(alued,)27 b(the)i(appro)m(ximated)609 2830 y(densit)m(y)h(should)e(b)s(e)i(normalized)f(o)m(v)m(er)j(all)d(h)m(yp) s(ercub)s(es:)1764 3034 y(^)-53 b Fn(p)25 b Fr(=)33 b(~)-53 b Fn(p)1984 2954 y Ff(X)1985 3137 y Fi(c)p Fe(2C)2136 3010 y Fr(~)2119 3034 y Fn(f)2164 3048 y Fi(c)2198 3034 y Fn(:)915 b Fr(\(2.42\))609 3313 y(If)35 b(the)g(distribution)c(of)k (the)g(input)e(data)j(is)e(v)m(ery)h(lo)m(w)g(dimensional)d(in)i(com-) 609 3426 y(parison)f(to)h(the)h(input)d(space,)j(a)g(lo)m(w)m(er)f (dimensioned)d(h)m(yp)s(ercub)s(e)i(migh)m(t)g(b)s(e)609 3538 y(used.)382 3782 y Fj(2.4.2)112 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y(used)j(a)h(com)m(bination)f(of)h Fo(r)-5 b(adial)44 b(b)-5 b(asis)43 b(functions)48 b Fr(\(RBF\))42 b(and)e(the)h(gro)m (wing)f(cell)382 4970 y(structure)e(and)g(ac)m(hiev)m(ed)h(go)s(o)s(d)g (results)e(for)i(clustering)e(and)h(classi\014cation.)64 b(The)382 5083 y(b)s(ene\014t)31 b(of)i(his)d(approac)m(h)j(is,)f(that) g(the)h(clusters)e(lead)h(to)h(discrete)e(structures)h(that)382 5196 y(are)f(easy)g(to)g(pro)s(cess.)p eop %%Page: 26 32 26 31 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(26)382 548 y Fl(2.5)135 b(Gro)l(wing)46 b(Grid)382 751 y Fr(Dev)m(elop)s(ed)27 b(in)e(1995)k(b)m(y)e(F)-8 b(ritzk)m(e)28 b([26)q(],)g(this)d(mo)s(del) h(is)g(a)h(com)m(bination)f(of)h(ideas)f(from)382 864 y(the)36 b(gro)m(wing-cells)g(using)e(a)j(classic)f(grid,)g(as)h (underlying)c(structure.)58 b(Gro)m(wing)36 b(is)382 977 y(done,)27 b(b)m(y)f(inserting)e(a)j(whole)e(ro)m(w)h(or)g(column)f (at)i(the)g(p)s(oin)m(t)e(with)g(the)h(highest)f(lo)s(cal)382 1090 y(error)30 b(measure.)382 1327 y Fj(2.5.1)112 b(Algorithm)382 1499 y Fr(As)39 b(the)g(grid)e(is)h(gro)m(wing)g(v)m(ery)i(fast,)h(the) e(algorithm)f(is)g(divided)e(in)m(to)j(t)m(w)m(o)h(parts.)382 1611 y(The)30 b Fo(gr)-5 b(owing)33 b(phase)39 b Fr(and)30 b(the)g Fo(\014ne)j(tuning)f(phase)p Fr(.)523 1724 y(Starting)26 b(with)g(a)i(2)1195 1691 y Fi(n)1269 1724 y Fr(grid,)f(gro)m(wing)f(is) g(done)h(follo)m(wing)f(the)h(same)g(idea,)h(used)e(in)382 1837 y(the)34 b Fo(gr)-5 b(owing)37 b(neur)-5 b(al)37 b(gas)p Fr(.)51 b(After)34 b(determining)e(the)i Fo(b)-5 b(est)36 b(matching)h(weight)f(ve)-5 b(ctor)382 1950 y Fn(w)447 1964 y Fi(s)484 1950 y Fr(,)30 b(an)h(adaption)e(step)i(is)e (done.)382 2184 y Fp(Adaption)518 2356 y Fk(\017)46 b Fr(Up)s(date)30 b(W)-8 b(eigh)m(ts)32 b(I)609 2500 y(Using)e(the)h Fn(L)1084 2514 y Fh(1)1154 2500 y Fr(norm)f(\(witc)m(h)h(is)e(also)i (called)f Fo(city)j(blo)-5 b(ck)33 b(metric)5 b Fr(\),)32 b(the)f(w)m(eigh)m(t)609 2612 y(v)m(ectors)k(according)f(to)h(the)e (neigh)m(b)s(ors)g(of)h Fn(c)f Fr(are)i(shifted)d(to)m(w)m(ards)i Fn(x)g Fr(using)e(a)609 2725 y(Gaussian)g(neigh)m(b)s(orho)s(o)s(d)f (relation)h(\(see)i(equation)f(2.11)h(and)e(2.13\).)51 b Fn(\013)p Fr(\()p Fn(t)p Fr(\))34 b(is)609 2838 y(set)26 b(to)h(a)f(constan)m(t)h(v)-5 b(alue)25 b Fn(\013)1574 2852 y Fh(0)1639 2838 y Fr(while)f(gro)m(wing.)39 b(Then)25 b(the)h(lo)s(cal)f(error)g(v)-5 b(ariable)609 2951 y Fn(E)676 2965 y Fi(s)743 2951 y Fr(is)30 b(increased)f(b)m(y)i(1.)609 3095 y(This)k(learning)f(pro)s(cess)i(is)g(rep)s(eated,)i(un)m(til)c (all)i(neurons)f(w)m(ere)i(up)s(dated)e Fn(\025)3331 3109 y Fi(g)609 3208 y Fr(times)27 b(in)e(a)m(v)m(erage.)42 b(F)-8 b(or)28 b(a)f Fn(k)17 b Fk(\002)c Fn(m)26 b Fr(net)m(w)m(ork,)j (that)f(w)m(ould)d(b)s(e)i(after)g Fn(k)17 b Fk(\002)c Fn(m)g Fk(\002)g Fn(\025)3331 3222 y Fi(g)609 3321 y Fr(steps.)518 3495 y Fk(\017)46 b Fr(Up)s(date)30 b(net)h(structure)609 3639 y(If)25 b(enough)g(learning)f(steps)h(are)h(done,)g(the)g(neuron)e (with)g(the)i(maxim)m(um)e(error)609 3752 y(v)-5 b(alue)30 b(is)f(determined:)1634 3865 y Fn(q)f Fr(=)d(arg)r(max)2096 3879 y Fi(c)p Fe(2C)2219 3865 y Fn(E)2286 3879 y Fi(c)2321 3865 y Fn(:)792 b Fr(\(2.43\))609 4012 y(This)43 b(is)h(the)i(neuron,)h (that)f(represen)m(ts)f(the)g(most)g(input)e(samples.)84 b(No)m(w)609 4125 y(the)34 b(neigh)m(b)s(orho)s(o)s(d)c(neuron,)k(with) e(the)h(most)h(di\013eren)m(t)e(rev)m(erence)j(v)m(ector)g(is)609 4238 y(determined)1452 4351 y Fn(f)f Fr(=)25 b(arg)r(max)1924 4365 y Fi(c)p Fe(2N)2060 4373 y Fg(q)2098 4351 y Fk(k)p Fn(w)2208 4365 y Fi(c)2263 4351 y Fk(\000)20 b Fn(w)2419 4365 y Fi(q)2457 4351 y Fk(k)p Fn(:)611 b Fr(\(2.44\))609 4529 y(Without)39 b(loss)g(of)g(generalit)m(y)g(w)m(e)h(assume,)h(that) f Fn(q)i Fr(and)d Fn(f)48 b Fr(b)s(oth)39 b(b)s(elong)f(to)609 4642 y(the)31 b Fn(i)p Fr(-th)f(ro)m(w)h(\(otherwise)f(they)g(w)m(ould) f(b)s(elong)h(to)h(the)f(same)h(column,)e(witc)m(h)609 4755 y(can)d(b)s(e)f(treated)i(accordingly\).)39 b(F)-8 b(urthermore)25 b(w)m(e)h(assume,)h(that)f Fn(q)i Fr(=)d Fn(s)3138 4769 y Fi(ij)3224 4755 y Fr(and)609 4868 y Fn(f)35 b Fr(=)25 b Fn(s)828 4882 y Fi(ij)t Fh(+1)978 4868 y Fr(.)609 5012 y(A)33 b(new)f(column)f Fn(j)1261 4979 y Fe(0)1317 5012 y Fr(\(with)h Fn(k)j Fr(neurons\))d(is)f (inserted)g(b)s(et)m(w)m(een)i(the)g(columns)e Fn(j)609 5125 y Fr(and)39 b Fn(j)32 b Fr(+)25 b(1.)68 b(The)39 b(w)m(eigh)m(t)h(v)m(ectors)h(of)e(this)f(new)h(column)f(are)i(in)m (terp)s(olated)609 5238 y(from)30 b(its)g(neigh)m(b)s(ors:)1239 5403 y Fn(w)1304 5418 y Fi(r)r(j)1371 5399 y Fc(0)1422 5403 y Fr(=)25 b(\()p Fn(w)1618 5417 y Fi(r)r(j)1709 5403 y Fr(+)20 b Fn(w)1865 5417 y Fi(r)r(j)t Fh(+1)2026 5403 y Fr(\))p Fn(=)p Fr(2)183 b(1)25 b Fk(\024)g Fn(r)j Fk(\024)d Fn(k)s(:)398 b Fr(\(2.45\))p eop %%Page: 27 33 27 32 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(27)609 548 y(Then)30 b(all)f(lo)s(cal)g(error)h(v)-5 b(ariables)29 b(are)i(reset)g(to)g (zero.)609 693 y(While)26 b(a)h Fo(stopping)k(criteria)j Fr(is)26 b(not)i(ful\014lled)23 b(\(for)k(example)f(the)h(\014nal)f(n)m (um)m(b)s(er)609 806 y(of)31 b(neurons)e(is)g(not)i(reac)m(hed\),)h(a)e (new)g(learning)f(step)h(is)g(done.)980 1485 y currentpoint currentpoint translate 0.59612 0.59612 scale neg exch neg exch translate 980 1485 a 980 1485 a gsave currentpoint currentpoint translate 0 neg rotate neg exch neg exch translate 980 1485 a @beginspecial 20 @llx 698 @lly 381 @urx 812 @ury 3610 @rwi @setspecial %%BeginDocument: figures/ggrid.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 20 698 381 812 %%Creator: CorelDRAW 7 %%Title: %%CreationDate: Sat May 15 17:51:34 1999 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness true def /AutoSteps 0 def /CMYKMarks false def /UseLevel2 false def %Color profile: %%BeginResource: procset wCorel7Dict 7.0 0 /wCorel7Dict 300 dict def wCorel7Dict begin % Copyright (c)1992-96 Corel Corporation % All rights reserved. v7.0 r1.0 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/setscreen{pop pop pop}bd/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd /SepMode_5 where{pop}{/SepMode_5 0 def}ifelse/CurrentInkName_5 where{pop} {/CurrentInkName_5(Composite)def}ifelse/$ink_5 where{pop}{/$ink_5 -1 def} ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def/$t 1 def/$n _ def/$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def/$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def/s1t 0 def/s1n _ def/$bkg false def /SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd /$ptm matrix def/$ttm matrix def/$stm matrix def/$ffpnt true def /CorelDrawReencodeVect[16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /L2?/languagelevel where{pop languagelevel 2 ge}{false}ifelse def /@BeginSysCorelDict{systemdict/Corel30Dict known{systemdict/Corel30Dict get exec}if}bd/@EndSysCorelDict{systemdict/Corel30Dict known{end}if}bd AutoFlatness {/@ifl{dup currentflat exch sub 10 gt{ ([Error: PathTooComplex; OffendingCommand: AnyPaintingOperator]\n)print flush @np exit}{currentflat 2 add setflat}ifelse}bd/@fill/fill ld/fill{currentflat{ {@fill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@eofill/eofill ld/eofill {currentflat{{@eofill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@clip /clip ld/clip{currentflat{{@clip}stopped{@ifl}{exit}ifelse}bind loop setflat} bd/@eoclip/eoclip ld/eoclip{currentflat{{@eoclip}stopped{@ifl}{exit}ifelse} bind loop setflat}bd/@stroke/stroke ld/stroke{currentflat{{@stroke}stopped {@ifl}{exit}ifelse}bind loop setflat}bd}if L2?{/@ssa{true setstrokeadjust}bd}{ /@ssa{}bd}ifelse/d/setdash ld/j/setlinejoin ld/J/setlinecap ld/M/setmiterlimit ld/w/setlinewidth ld/O{/$o xd}bd/R{/$O xd}bd/W/eoclip ld/c/curveto ld/C/c ld/l /lineto ld/L/l ld/rl/rlineto ld/m/moveto ld/n/newpath ld/N/newpath ld/P{11 rp} bd/u{}bd/U{}bd/A{pop}bd/q/@gs ld/Q/@gr ld/&{}bd/@j{@sv @np}bd/@J{@rs}bd/g{1 exch sub/$k xd/$c 0 def/$m 0 def/$y 0 def/$t 1 def/$n _ def/$fil 0 def}bd/G{1 sub neg/$K xd _ 1 0 0 0/$C xd/$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type /stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd /$fil 0 def}bd/K{1 index type/stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def} ifelse/$K xd/$Y xd/$M xd/$C xd}bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{ /s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{ dup 0 ne{setflat}{pop}ifelse}bd/v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2 copy c}bd/Y/y ld/@w{matrix rotate/$ptm xd matrix scale $ptm dup concatmatrix /$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd {/$scp xd/$sca xd/$scf xd}if}bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd} if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch load SetScr/$dsp xd/$dsa xd/$dsf xd}bd/$ngx{$SDF{$SCF SepMode_5 0 eq{$SCA} {$dsa}ifelse $SCP @ss}if}bd/p{/$pm xd 7 rp/$pyf xd/$pxf xd/$pn xd/$fil 1 def} bd/@MN{2 copy le{pop}{exch pop}ifelse}bd/@MX{2 copy ge{pop}{exch pop}ifelse}bd /InRange{3 -1 roll @MN @MX}bd/@sqr{dup 0 rl dup 0 exch rl neg 0 rl @cp}bd /currentscale{1 0 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt 0 1 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt}bd/@unscale{}bd/wDstChck{2 1 roll dup 3 -1 roll eq{1 add}if}bd /@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1 exch sub}bd /cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop}bd /rgb2cmyk{3{1 exch sub 3 1 roll}repeat 3 copy @MN @MN 3{dup 5 -1 roll sub neg exch}repeat}bd/rgb2g{2 index .299 mul 2 index .587 mul add 1 index .114 mul add 4 1 roll pop pop pop}bd/WaldoColor_5 where{pop}{/SetRgb/setrgbcolor ld/GetRgb /currentrgbcolor ld/SetGry/setgray ld/GetGry/currentgray ld/SetRgb2 systemdict /setrgbcolor get def/GetRgb2 systemdict/currentrgbcolor get def/SetHsb systemdict/sethsbcolor get def/GetHsb systemdict/currenthsbcolor get def /rgb2hsb{SetRgb2 GetHsb}bd/hsb2rgb{3 -1 roll dup floor sub 3 1 roll SetHsb GetRgb2}bd/setcmykcolor where{pop/SetCmyk_5/setcmykcolor ld}{/SetCmyk_5{ cmyk2rgb SetRgb}bd}ifelse/currentcmykcolor where{pop/GetCmyk/currentcmykcolor ld}{/GetCmyk{GetRgb rgb2cmyk}bd}ifelse/setoverprint where{pop}{/setoverprint{ /$op xd}bd}ifelse/currentoverprint where{pop}{/currentoverprint{$op}bd}ifelse /@tc_5{5 -1 roll dup 1 ge{pop}{4{dup 6 -1 roll mul exch}repeat pop}ifelse}bd /@trp{exch pop 5 1 roll @tc_5}bd/setprocesscolor_5{SepMode_5 0 eq{SetCmyk_5}{0 4 $ink_5 sub index exch pop 5 1 roll pop pop pop pop SepsColor true eq{$ink_5 3 gt{1 sub neg SetGry}{0 0 0 4 $ink_5 roll SetCmyk_5}ifelse}{1 sub neg SetGry} ifelse}ifelse}bd/findcmykcustomcolor where{pop}{/findcmykcustomcolor{5 array astore}bd}ifelse/setcustomcolor where{pop}{/setcustomcolor{exch aload pop SepMode_5 0 eq{pop @tc_5 setprocesscolor_5}{CurrentInkName_5 eq{4 index}{0} ifelse 6 1 roll 5 rp 1 sub neg SetGry}ifelse}bd}ifelse/@scc_5{dup type /booleantype eq{setoverprint}{1 eq setoverprint}ifelse dup _ eq{pop setprocesscolor_5 pop}{findcmykcustomcolor exch setcustomcolor}ifelse SepMode_5 0 eq{true}{GetGry 1 eq currentoverprint and not}ifelse}bd/colorimage where{pop /ColorImage{colorimage}def}{/ColorImage{/ncolors xd pop/dataaq xd{dataaq ncolors dup 3 eq{/$dat xd 0 1 $dat length 3 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L 16642 21600 L 18811 28734 L 23442 28734 L @c F T %%PageTrailer @rs @rs %%Trailer @EndSysCorelDict end %%DocumentSuppliedResources: procset wCorel7Dict %%EOF %%EndDocument @endspecial 3988 1485 a currentpoint grestore moveto 3988 1485 a 980 1485 a currentpoint currentpoint translate 1 0.59612 div 1 0.59612 div scale neg exch neg exch translate 980 1485 a 976 1781 a Fr(Figure)f(2.7:)42 b(Gro)m(wing)30 b(Grid,)g(Adding)e(a)j (new)f(ro)m(w)518 2011 y Fk(\017)46 b Fr(Up)s(date)30 b(W)-8 b(eigh)m(ts)32 b(I)s(I)609 2156 y(If)k(the)g(net)g(con)m(tains)g (enough)f(neurons,)h(so)g(that)h(no)f(more)g(gro)m(wing)f(is)g(nec-)609 2269 y(essary)-8 b(,)42 b(a)e(few)f(\014ne-tuning)e(steps)j(can)f(b)s (e)g(p)s(erformed.)65 b(F)-8 b(ritzk)m(e)41 b(suggested)609 2382 y Fn(t)642 2349 y Fe(0)642 2404 y Fh(max)803 2382 y Fr(=)25 b Fn(k)e Fk(\002)d Fn(m)g Fk(\002)g Fn(\025)1304 2397 y Fi(f)1380 2382 y Fr(learning)28 b(steps,)j(with)e(a)i(learning)d (rate)1457 2568 y Fn(\013)p Fr(\()p Fn(t)1583 2531 y 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b(the)h(size)f(of)h(the)g(net)f(can)h(b)s(e)f(de\014ned)f(using)g(a)i (\015exible)382 3718 y(stopping)h(criteria,)j(it)d(is)h(p)s(ossible)d (to)k(adjust)f(the)g(qualit)m(y)g(of)g(the)g(net)h(v)m(ery)f(easy)-8 b(.)382 3831 y(On)34 b(the)h(other)g(hand,)h(if)d(the)j(densit)m(y)e (of)h(the)g(input)e(data)j(v)-5 b(aries)34 b(v)m(ery)h(m)m(uc)m(h)g(o)m (v)m(er)382 3943 y(the)h(input)d(space,)38 b(the)e(grid)e(ma)m(y)i(b)s (e)f(a)h(to)h(strong)e(structure,)i(to)g(represen)m(t)e(these)382 4056 y(inhomogeneities.)382 4338 y Fl(2.6)135 b(Gro)l(wing)46 b(Hyp)t(ercubial)f(Output)f(Space)h(\(GSOM\))382 4541 y Fr(The)25 b(GSOM)g(Mo)s(del,)h(dev)m(elop)s(ed)f(b)m(y)h(H.U.)g (Bauer)g(and)f(Th.)38 b(Villmann)23 b(in)h(1995)j([28)q(])382 4654 y(is)f(in)g(its)g(basic)g(idea)h(also)g(a)g(gro)m(wing)g(grid)f (algorithm.)38 b(As)27 b(opp)s(osed)g(to)g(F)-8 b(ritzk)m(e,)29 b(the)382 4767 y(GSOM)f(can)g(also)g(gro)m(w)g(in)m(to)g(a)g(new)f (dimension,)g(so)h(that)g(the)g(net)m(w)m(ork)h(dimension-)382 4880 y(alit)m(y)h(has)g(not)h(to)g(b)s(e)f(kno)m(wn)f(a)i(priori.)382 5119 y Fj(2.6.1)112 b(Mathematical)37 b(mo)s(del)382 5290 y Fr(Bauer)26 b(and)f(Villmann)e(suggested)j(to)g(use)g(the)g(a)m (v)m(erage)i(pro)5 b(jection)25 b(error,)i(to)f(decide)382 5403 y(where)32 b(to)i(add)f(a)g(new)f(ro)m(w)i(of)f(neurons)e(or)i(a)h (new)e(dimension.)46 b(F)-8 b(or)34 b(this)e(purp)s(ose,)p eop %%Page: 28 34 28 33 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(28)382 548 y(the)33 b(V)-8 b(oronoi-region)32 b(of)h(a)g(neurons)e Fn(r)k Fr(w)m(eigh)m(t-v)m(ector)h Fn(w)2384 562 y Fi(r)2454 548 y Fr(is)c(decomp)s(osed)g(in)m(to)g(the)382 661 y(di\013eren)m(t)41 b(directions,)j(induced)c(b)m(y)i(its)g(top)s(ological)g(neigh)m(b)s (ors)e(to)j(de\014ne)f(a)g(new)382 774 y(basis)29 b Fn(b)644 788 y Fh(1)699 774 y Fn(:)15 b(:)g(:)h(b)859 789 y Fi(d)899 774 y Fr(.)523 887 y(Supp)s(ose,)40 b Fn(r)h Fr(has)e(the)g(grid-co)s (ordinates)f(\()p Fn(r)2089 901 y Fi(i)2118 887 y Fn(;)15 b(r)2199 901 y Fi(j)2236 887 y Fn(;)g(r)2317 902 y Fi(k)2360 887 y Fn(;)g Fr(1)p Fn(;)g(:)g(:)g(:)r Fr(\),)42 b(with)c(net)m(w)m (ork)i(di-)382 1000 y(mensionalit)m(y)31 b Fn(d)d Fr(=)g(3)33 b(\(the)f(co)s(ordinates)g Fn(r)1886 1014 y Fi(i)1946 1000 y Fr(for)g Fn(l)e(>)e(d)33 b Fr(are)f(set)h(to)g(1)g(p)s(er)e (de\014nition.)382 1112 y(Then)i Fn(r)k Fr(has)e(t)m(w)m(o)h(neigh)m(b) s(ors)d(in)g(eac)m(h)i(dimension.)51 b(The)34 b(direction-v)m(ector)h (for)f(the)382 1225 y(\014rst-co)s(ordinate)f(can)h(b)s(e)f(computed)h (as)g(the)g(normalized)e(di\013erence)i(of)g(the)g(refer-)382 1338 y(ence)d(v)m(ectors)h(of)e(these)h(t)m(w)m(o)h(neigh)m(b)s(ors:) 1029 1571 y Fn(b)1068 1585 y Fh(1)1107 1571 y Fr(\()p Fn(r)s Fr(\))26 b(=)1398 1498 y Fn(w)1463 1516 y Fh(\()p Fi(r)1522 1526 y Fg(i)p Fd(+1)1626 1516 y Fi(;r)1678 1526 y Fg(j)1710 1516 y Fi(;r)1762 1528 y Fg(k)1799 1516 y Fi(;)p Fh(1)p Fi(;:::)o Fh(\))1985 1498 y Fk(\000)19 b Fn(w)2140 1516 y Fh(\()p Fi(r)2199 1526 y Fg(i)p Fc(\000)p Fd(1)2305 1516 y Fi(;r)2357 1526 y Fg(j)2389 1516 y Fi(;r)2441 1528 y Fg(k)2478 1516 y Fi(;)p Fh(1)p Fi(;:::)n Fh(\))p 1353 1550 1336 4 v 1353 1633 a Fk(k)p Fn(w)1463 1651 y Fh(\()p Fi(r)1522 1661 y Fg(i)p Fd(+1)1626 1651 y Fi(;r)1678 1661 y Fg(j)1710 1651 y Fi(;r)1762 1663 y Fg(k)1799 1651 y Fi(;)p Fh(1)p Fi(;:::)o Fh(\))1985 1633 y Fk(\000)g Fn(w)2140 1651 y Fh(\()p Fi(r)2199 1661 y Fg(i)p Fc(\000)p Fd(1)2305 1651 y Fi(;r)2357 1661 y Fg(j)2389 1651 y Fi(;r)2441 1663 y Fg(k)2478 1651 y Fi(;)p Fh(1)p Fi(;:::)n Fh(\))2643 1633 y Fk(k)2698 1571 y Fn(;)415 b Fr(\(2.47\))382 1823 y(for)30 b(the)h(other)f(t)m(w)m(o)i(dimensions)c(accordingly)-8 b(.)523 1936 y(In)22 b(general,)i(if)e(a)h(input)e(signal)g Fn(x)h Fr(is)g(mapp)s(ed)f(to)j(the)f(neuron)e Fn(r)s Fr(,)j(the)f(quan)m(tization)382 2049 y(error)30 b(\001)p Fn(w)746 2063 y Fi(r)784 2049 y Fr(\()p Fn(x)p Fr(\))c(=)f Fn(x)20 b Fk(\000)g Fn(w)1256 2063 y Fi(r)1324 2049 y Fr(can)31 b(no)m(w)f(b)s(e)g(written)f(as)947 2243 y(\001)p Fn(w)1088 2257 y Fi(r)1126 2243 y Fr(\()p Fn(x)p Fr(\))84 b(=)e Fn(a)1533 2257 y Fh(1)1573 2243 y Fr(\()p Fn(x)p Fr(\))p Fn(b)1734 2257 y Fh(1)1774 2243 y Fr(\()p Fn(r)s Fr(\))20 b(+)g Fn(a)2047 2257 y Fh(2)2087 2243 y Fr(\()p Fn(x)p Fr(\))p Fn(b)2248 2257 y Fh(2)2288 2243 y Fr(\()p Fn(r)s Fr(\))g(+)g Fn(:)15 b(:)g(:)21 b Fr(+)f Fn(x)2782 2205 y Fe(0)1332 2459 y Fr(=)1527 2353 y Fi(d)1485 2378 y Ff(X)1488 2560 y Fi(i)p Fh(=1)1620 2459 y Fn(a)1668 2473 y Fi(i)1697 2459 y Fr(\()p Fn(x)p Fr(\))p Fn(b)1858 2473 y Fi(i)1886 2459 y Fr(\()p Fn(r)s Fr(\))h(+)f Fn(x)2164 2421 y Fe(0)2187 2459 y Fn(:)926 b Fr(\(2.48\))382 2727 y(Where)23 b Fn(a)713 2741 y Fi(i)765 2727 y Fr(is)f(the)h(amplitude)e (of)j(the)f Fn(i)p Fr(-th)g(basis)g(v)m(ector)h(and)f Fn(x)2540 2694 y Fe(0)2586 2727 y Fr(the)g(remaining)f(error.)523 2840 y(The)42 b(idea)g(is,)j(to)f(insert)d(a)i(new)f(ro)m(w)h(in)f (that)h(dimension,)g(with)e(the)i(largest)382 2953 y(a)m(v)m(erage)31 b(error)d(amplitude)f Fn(a)1403 2967 y Fi(i)1431 2953 y Fr(.)40 b(T)-8 b(o)29 b(suggest,)h(whether)e(to)i(add)e(a)h(new)f (dimension)e(or)382 3066 y(not,)37 b(a)e(principle)d(comp)s(onen)m(t)j (analysis)f(is)g(done.)55 b(Let)35 b Fn(e)2422 3080 y Fi(P)10 b(GA)2624 3066 y Fr(b)s(e)34 b(the)i(direction)d(of)382 3179 y(largest)e(v)-5 b(ariabilit)m(y)d(,)28 b(then)i(the)h(remaining)d (error)i Fn(x)2188 3146 y Fe(0)2242 3179 y Fr(can)h(b)s(e)e(written)h (as)1408 3373 y Fn(x)1460 3335 y Fe(0)1508 3373 y Fr(=)25 b Fn(a)1652 3388 y Fi(d)p Fh(+1)1783 3373 y Fr(\()p Fn(x)p Fr(\))p Fn(e)1947 3387 y Fi(P)10 b(GA)2134 3373 y Fr(+)20 b Fn(x)2277 3335 y Fe(00)2320 3373 y Fn(:)793 b Fr(\(2.49\))523 3566 y(F)-8 b(or)37 b(eac)m(h)h(input)c(signal,)j(this)e(amplitude)f (is)i(normalized)f(and)g(then)h(a)m(v)m(eraged)382 3679 y(o)m(v)m(er)c(all)d(signals)981 3915 y(~)-46 b Fn(a)1028 3929 y Fi(i)1081 3915 y Fr(=)1177 3835 y Ff(X)1217 4009 y Fi(x)1461 3854 y Fk(j)p Fn(a)1534 3868 y Fi(i)1562 3854 y Fr(\()p Fn(x)p Fr(\))p Fk(j)p 1322 3894 527 4 v 1322 3911 a Ff(q)p 1405 3911 444 4 v 34 x(P)1493 3971 y Fi(d)p Fh(+1)1493 4035 y Fi(j)t Fh(=1)1639 4009 y Fn(a)1687 3978 y Fh(2)1687 4035 y Fi(j)1726 4009 y Fr(\()p Fn(x)p Fr(\))1859 3915 y Fn(;)197 b(i)26 b Fr(=)f(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(d;)g(d)21 b Fr(+)f(1)366 b(\(2.50\))382 4209 y(As)38 b(the)g(deviation)e(in)m(to)i(a)g(direction)f(will)e (decrease)j(b)m(y)g(adding)e(a)j(new)e(ro)m(w,)j(this)382 4322 y(factor)31 b(is)f(also)g(tak)m(en)h(in)m(to)g(accoun)m(t)h(as)e (its)g(ro)s(ot.)523 4435 y(The)g(gro)m(wing)g(direction)f Fn(i)i Fr(is)e(selected)i(according)f(to)872 4689 y Fn(i)c Fr(=)f(arg)r(max)1322 4703 y Fi(j)1373 4545 y Ff( )1439 4554 y(s)p 1522 4554 268 4 v 1610 4627 a Fn(n)1665 4641 y Fi(j)p 1532 4668 248 4 v 1532 4751 a Fn(n)1587 4765 y Fi(j)1643 4751 y Fr(+)20 b(1)1791 4689 y(~)-46 b Fn(a)1838 4703 y Fi(j)1874 4545 y Ff(!)1955 4689 y Fn(;)197 b(j)31 b Fr(=)25 b(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(d;)g(d)22 b Fr(+)e(1)258 b(\(2.51\))523 4951 y(As)45 b(no)g(lo)s(cal)f(error)h (measure)g(is)f(a)m(v)-5 b(ailable,)48 b(new)d(ro)m(ws)g(are)g (inserted)f(in)f(the)382 5064 y(middle)24 b(of)j(the)g(grid.)38 b(In)26 b(case)i(of)f(a)g(new)g(dimension,)e(the)i(net)g(should)d(gro)m (w)k(in)m(to)e(the)382 5177 y(direction)j(of)h(the)h(resp)s(ectiv)m(e)g (principal)c(comp)s(onen)m(ts.)523 5290 y(T)-8 b(o)38 b(enable)f(the)h(net)f(to)i(redistribute,)e(the)h(learning)d(rate)k Fn(\013)p Fr(\()p Fn(t)p Fr(\))f(should)e(b)s(e)h(in-)382 5403 y(creased)31 b(after)g(eac)m(h)g(gro)m(wing)f(step.)p eop %%Page: 29 35 29 34 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(29)382 548 y Fj(2.6.2)112 b(Algorithm)382 720 y Fp(Adaption)382 891 y Fr(Adaption)36 b(is)h(done)g(in)f(the)i(same)g(w)m(a)m(y)-8 b(,)40 b(as)e(suggested)g (b)m(y)f(Kohonen.)62 b(After)37 b(a)h(\014x)382 1004 y(n)m(um)m(b)s(er)h(of)i(learning)e(steps,)44 b(it)c(is)f(c)m(hec)m(k)m (ed,)46 b(whether)40 b(new)g(ro)m(ws)h(or)f(dimensions)382 1117 y(ha)m(v)m(e)d(to)f(b)s(e)g(added.)56 b(It)36 b(w)m(ould)e(also)i (b)s(e)f(p)s(ossible,)g(to)h(split)e(the)i(algorithm)f(in)f(the)382 1230 y(same)d(t)m(w)m(o)g(parts)f(as)h(in)e(the)i Fo(gr)-5 b(owing)33 b(grid)41 b Fr(algorithm.)523 1343 y(The)e(main)f (di\013erence)h(is,)i(that)f(no)f(lo)s(cal)f(error)h(measure)h(w)m(as)f (used)g(and)f(the)382 1456 y(learning)29 b(rate)i Fn(\013)p Fr(\()p Fn(t)p Fr(\))g(has)f(to)h(b)s(e)f(increased)g(after)h(eac)m(h)g (gro)m(wing)f(step.)382 1693 y Fj(2.6.3)112 b(Analysis)382 1865 y Fr(As)32 b(no)f(a)h(priori)e(kno)m(wledge)i(ab)s(out)f(the)h (dimension)d(or)j(size)g(of)g(the)g(grid)e(has)i(to)g(b)s(e)382 1978 y(kno)m(wn,)j(this)f(mo)s(del)f(allo)m(ws)h(v)m(ery)i(easy)f(ad)f (ho)s(c)h(pro)s(cessing)e(of)i(high)e(dimensional)382 2090 y(data.)79 b(T)-8 b(o)43 b(ac)m(hiev)m(e)h(this)e(goal,)k(a)e(lot) e(of)h(computation)g(is)f(necessary:)66 b(F)-8 b(or)43 b(eac)m(h)382 2203 y(neuron,)28 b(a)h(principle)c(comp)s(onen)m(t)k (analysis)e(o)m(v)m(er)j(all)d(sample)h(p)s(oin)m(ts)f(in)h(its)g(v)m (oronoi)382 2316 y(region)36 b(has)g(to)h(b)s(e)f(done)g(and)g (insertion)f(is)g(alw)m(a)m(ys)i(done)g(in)e(the)h(middle,)g(so)h(that) 382 2429 y(the)30 b(whole)g(net)h(has)f(to)h(redistribute.)523 2542 y(This)36 b(results)h(in)g(a)h(v)m(ery)h(complex)f(and)f(slo)m(w)h (algorithm,)h(so)f(that)h(the)f(added)382 2655 y(\015exibilit)m(y)21 b(is)i(quite)g(costly)-8 b(.)40 b(F)-8 b(ritzk)m(es)25 b(algorithms)d(deliv)m(er)h(comparable)h(results,)g(and)382 2768 y(allo)m(ws)33 b(a)h(real)f(incremen)m(tal)g(computation,)h(op)s (erating)f(only)g(with)f(lo)s(cal)h(measures)382 2881 y(and)d(mo)s(di\014cations.)382 3161 y Fl(2.7)135 b(Common)46 b(SOM)e(ob)7 b(jectiv)l(es)382 3364 y Fr(As)34 b(\015exible)f(as)i (comp)s(etitiv)m(e)f(learning)f(metho)s(ds)h(are,)i(as)f(di\013eren)m (t)e(are)i(the)g(goals,)382 3477 y(that)43 b(ma)m(y)g(b)s(e)e(aspired)g (b)m(y)i(this)e(metho)s(ds.)76 b(Dep)s(ending)41 b(on)h(the)g(p)s(oin)m (t)g(of)g(view,)382 3590 y(one)d(ma)m(y)g(try)g(to)g(minimize)d(the)j (a)m(v)m(erage)j(error)c(of)h(prediction,)g(or)g(on)f(the)h(other)382 3703 y(hand,)27 b(maximize)f(the)h(en)m(trop)m(y)-8 b(,)29 b(so)e(that)g(ev)m(ery)h(v)m(ector)h(has)d(the)h(same)h(c)m(hance)g(to) f(b)s(e)382 3816 y(c)m(hosen.)523 3928 y(It)33 b(is)f(clear,)j(that)e (these)h(goals)g(can)f(not)g(b)s(e)g(reac)m(hed)h(sim)m(ultaneously)d (in)h(man)m(y)382 4041 y(cases.)39 b(So)23 b(it)g(is)f(an)h(imp)s (ortan)m(t)g(part)g(of)g(the)h(design)e(pro)s(cess,)i(to)g(clear)f(the) h(ob)5 b(jectiv)m(es)382 4154 y(of)30 b(the)h(system.)427 4327 y(Error)f(Minimization)609 4471 y(An)j(imp)s(ortan)m(t)g(goal)g (in)f(man)m(y)i(cases)g(is)e(the)i(minimization)c(of)k(the)f Fo(exp)-5 b(e)g(cte)g(d)609 4584 y(quantization)37 b(err)-5 b(or)p Fr(.)51 b(This)32 b(means,)i(that)g(the)f(w)m(eigh)m(t)h(v)m (ectors)h(are)e(homoge-)609 4697 y(neous)26 b(distributed)e(o)m(v)m(er) k(the)f(input)e(data,)j(so)f(that)g(the)g(distance)f(of)h(an)g(input) 609 4810 y(signal,)j(to)h(the)f(nearest)h(w)m(eigh)m(t)g(v)m(ector)h (is)d(minimized)f(on)i(an)g(a)m(v)m(erage.)609 4953 y(Using)36 b(V)-8 b(oronoi)37 b(regions,)h(the)f Fo(exp)-5 b(e)g(cte)g(d)40 b(quantization)g(err)-5 b(or)48 b Fr(can)37 b(easily)f(b)s(e)609 5066 y(computed:)1239 5333 y(E\()p Fn(p)p Fr(\()p Fn(x)p Fr(\))p Fn(;)15 b Fk(A)p Fr(\))26 b(=)1781 5252 y Ff(X)1774 5436 y Fi(c)p Fe(2A)1923 5218 y Ff(Z)1969 5407 y Fi(V)2010 5415 y Fg(c)2062 5333 y Fk(k)p Fn(x)20 b Fk(\000)g Fn(w)2335 5347 y Fi(c)2370 5333 y Fk(k)2415 5296 y Fh(2)2455 5333 y Fn(p)p Fr(\()p Fn(x)p Fr(\))15 b(d)p Fn(x)397 b Fr(\(2.52\))p eop %%Page: 30 36 30 35 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(30)609 548 y(Where)26 b Fn(p)p Fr(\()p Fn(x)p Fr(\))h(is)e(the)h(input)e(signal)h (distribution)d(and)k Fn(w)2550 562 y Fi(c)2584 548 y Fn(;)15 b(c)26 b Fk(2)f(A)h Fr(is)f(the)h(w)m(eigh)m(t)609 661 y(v)m(ector)42 b(corresp)s(onding)d(to)i Fn(c)p Fr(.)71 b(A)40 b(V)-8 b(oronoi)41 b(region)f Fn(V)2536 675 y Fi(c)2571 661 y Fn(;)15 b(c)43 b Fk(2)e(A)f Fr(con)m(tains)h(all)609 774 y(p)s(oin)m(ts,)30 b(that)h(are)f(closer)h(to)g Fn(c)p Fr(,)g(than)f(to)h(an)m(y)g(other)f(p)s(oin)m(t)g(of)g Fk(A)p Fr(.)609 924 y(Minimizing)d(the)k(a)m(v)m(erage)h(error)e(is)f (v)m(ery)i(imp)s(ortan)m(t)e(in)g Fo(ve)-5 b(ctor)33 b(quantization)609 1037 y Fr(\(Linde)43 b(et)i(al.,)i(1980;)53 b(Gra)m(y)-8 b(,)49 b(1984\).)84 b(The)44 b(information)e(loss,)48 b(caused)c(b)m(y)609 1150 y(quan)m(tization,)k(is)c(equal)g(to)i(the)e (distance)h(to)g(the)g(nearest)g(w)m(eigh)m(t)g(v)m(ector)609 1263 y(\(called)30 b Fo(c)-5 b(o)g(deb)g(o)g(ok)35 b(ve)-5 b(ctor)10 b Fr(\).)427 1450 y(En)m(trop)m(y)31 b(Maximization)609 1601 y(If)e(ev)m(ery)g(w)m(eigh)m(t)h(v)m(ector)g(should)d(ha)m(v)m(e)k (the)e(same)g(c)m(hance)h(to)g(b)s(e)e(c)m(hosen)i(for)e(a)609 1714 y(randomly)i(selected)h(input)f(signal)f Fn(x)p Fr(,)j(ev)m(ery)g(v)m(oronoi)f(region)g(has)f(to)i(con)m(tain)609 1826 y(the)f(same)g(n)m(um)m(b)s(er)e(of)h(input)f(data)i(p)s(oin)m (ts.)1417 2105 y Fn(P)13 b Fr(\()p Fn(s)p Fr(\()p Fn(x)p Fr(\))25 b(=)g Fn(c)p Fr(\))h(=)2074 2044 y(1)p 2015 2084 164 4 v 2015 2168 a Fk(kAk)2219 2105 y Fr(\()p Fk(8)p Fn(c)f Fk(2)g(A)p Fr(\))575 b(\(2.53\))609 2353 y(The)30 b(adv)-5 b(an)m(tage)31 b(of)f(this)f(approac)m(h)h(is)f(the)h (robustness)f(of)h(the)g(resulting)e(sys-)609 2466 y(tem,)f(b)s(ecause) f(the)f(in\015uence)f(of)i(one)g(v)m(ector)h(is)d(limited)f(fraction)i (of)h(the)g(data.)427 2654 y(F)-8 b(eature)32 b(Mapping)609 2804 y(Some)i(net)m(w)m(orks)g(are)g(able,)h(to)f(reduce)g (high-dimensional)c(input)h(data,)36 b(to)e(a)609 2917 y(lo)m(w)m(er)g(dimensional)c(h)m(yp)s(er-plain)h(in)h(a)h Fo(top)-5 b(olo)g(gy)39 b(pr)-5 b(eserving)42 b Fr(manner.)48 b(This)609 3030 y(has)34 b(b)s(een)f(denoted)h Fo(fe)-5 b(atur)g(e)37 b(mapping)43 b Fr(and)34 b(is)f(also)g(v)m(ery)i(useful)d (in)h(data)h(vi-)609 3143 y(sualization.)609 3293 y(The)39 b(qualit)m(y)g(of)h(this)f(pro)5 b(jection)39 b(can)h(b)s(e)g (determined)e(using)g(a)i(v)-5 b(ariet)m(y)40 b(of)609 3406 y(measures.)h(See)30 b(section)h(2.8.1)h(and)e(2.8.2)i(for)e(more) h(information.)427 3593 y(Clustering)609 3744 y(In)39 b(a)h(few)f(cases,)k(it)c(is)g(p)s(ossible)e(to)j(cluster)f(the)h (input)e(data,)k(that)e(means,)609 3857 y(that)j(similar)d(data)j(is)e (group)s(ed)g(together.)78 b(F)-8 b(or)43 b(go)s(o)s(d)f(results,)i (the)f(in)m(tra-)609 3969 y(cluster)32 b(v)-5 b(ariance)33 b(should)d(b)s(e)i(small,)g(and)g(the)h(in)m(ter-cluster)f(v)-5 b(ariance)32 b(large.)609 4082 y(In)f(sup)s(ervised)e(en)m(vironmen)m (ts,)j(the)g(qualit)m(y)f(of)g(prediction)f(ma)m(y)i(b)s(e)g(used)e(as) 609 4195 y(a)h(qualit)m(y)e(measure.)382 4482 y Fl(2.8)135 b(V)-11 b(aluation)46 b(measures)382 4685 y Fr(T)-8 b(o)36 b(c)m(hec)m(k,)i(whether)d(the)h(top)s(ology)g(of)g(the)f(input)f (space)i(is)e(mapp)s(ed)g(in)h(a)g(correct)382 4798 y(manner)j(to)h (the)g(map,)h(t)m(w)m(o)g(functions)d(ha)m(v)m(e)j(b)s(een)e (suggested.)66 b(The)38 b Fo(top)-5 b(o)g(gr)g(aphic)382 4910 y(pr)g(o)g(duct)55 b Fr(in)m(tro)s(duced)42 b(b)m(y)i(Bauer)g(and) g(P)m(a)m(w)m(elzik)g(in)f(1992,)49 b(and)44 b(the)g Fo(top)-5 b(o)g(gr)g(aphic)382 5023 y(function)38 b Fr(from)31 b(Villmann)e(et)j(al.)43 b(\(1994\).)j(Both)32 b(try)f(to)h(compare)g (the)f(ordering)f(of)382 5136 y(neigh)m(b)s(ors,)f(instead)h(of)g (absolute)g(distances.)p eop %%Page: 31 37 31 36 bop 382 249 a Fs(CHAPTER)30 b(2.)71 b(ST)-8 b(A)g(TE)30 b(OF)g(THE)g(AR)-8 b(T)1362 b Fr(31)382 548 y Fj(2.8.1)112 b(T)-9 b(op)s(ographic)38 b(pro)s(duct)382 720 y Fr(This)31 b(metho)s(d)h(compares)h(the)g(neigh)m(b)s(orho)s(o)s(d)d(sequence)j (for)g(all)e(neurons)h(b)s(et)m(w)m(een)382 832 y(the)j(input)e(and)i (the)g(output)g(space.)55 b(F)-8 b(or)36 b(this)e(purp)s(ose,)h(t)m(w)m (o)i(sequence)e(functions)382 945 y Fn(n)437 912 y Fe(C)437 972 y Fi(k)482 945 y Fr(\()p Fn(s)p Fr(\))20 b(and)g Fn(n)837 912 y Fe(R)837 972 y Fi(k)901 945 y Fr(\()p Fn(s)p Fr(\))h(are)g(de\014ned,)g(where)f Fn(n)1815 912 y Fe(C)1815 972 y Fi(k)1860 945 y Fr(\()p Fn(s)p Fr(\))h(denotes)f(the) h Fn(k)s Fr(-th)g(neigh)m(b)s(or)e(of)i Fn(s)p Fr(,)h(with)382 1058 y(distances)k(measured)h(in)e(the)i(output)f(space.)41 b Fn(n)2052 1025 y Fe(R)2052 1085 y Fi(k)2116 1058 y Fr(\()p Fn(s)p Fr(\))27 b(denotes)g(the)g Fn(k)s Fr(-th)g(neigh)m(b)s (or)e(of)382 1171 y Fn(s)p Fr(,)30 b(with)f(distances)h(ev)-5 b(aluated)31 b(in)e(the)h(input)f(space)i(b)s(et)m(w)m(een)g Fn(w)2648 1185 y Fi(s)2715 1171 y Fr(and)f Fn(w)2957 1197 y Fi(n)3000 1174 y Fc(C)3000 1222 y Fg(k)3044 1171 y Fr(\()p Fn(s)p Fr(\).)523 1293 y(First)g(the)g(quan)m(tit)m(y)855 1615 y Fn(P)913 1629 y Fh(3)953 1615 y Fr(\()p Fn(s;)15 b(k)s Fr(\))26 b(=)1278 1446 y Ff(0)1278 1595 y(@)1387 1509 y Fi(k)1353 1534 y Ff(Y)1351 1718 y Fi(l)q Fh(=1)1496 1528 y Fn(d)1543 1495 y Fe(R)1623 1434 y Ff(\020)1672 1528 y Fn(w)1737 1542 y Fi(s)1774 1528 y Fn(;)15 b(w)1879 1554 y Fi(n)1922 1531 y Fc(C)1922 1579 y Fg(k)1967 1528 y Fr(\()p Fn(s)p Fr(\))2080 1434 y Ff(\021)p 1488 1594 651 4 v 1488 1701 a Fn(d)1535 1675 y Fe(R)1614 1607 y Ff(\020)1664 1701 y Fn(w)1729 1715 y Fi(s)1766 1701 y Fn(;)g(w)1871 1727 y Fi(n)1914 1704 y Fc(R)1914 1752 y Fg(k)1975 1701 y Fr(\()p Fn(s)p Fr(\))2088 1607 y Ff(\021)2168 1615 y Fk(\001)2224 1528 y Fn(d)2271 1495 y Fe(C)2331 1434 y Ff(\020)2381 1528 y Fn(s;)g(n)2519 1495 y Fe(C)2519 1555 y Fi(k)2564 1528 y Fr(\()p Fn(s)p Fr(\))2677 1434 y Ff(\021)p 2224 1594 503 4 v 2226 1680 a Fn(d)2273 1654 y Fe(C)2333 1611 y Ff(\000)2371 1680 y Fn(s;)g(n)2509 1649 y Fe(R)2509 1708 y Fi(k)2573 1680 y Fr(\()p Fn(s)p Fr(\))2686 1611 y Ff(\001)2737 1446 y(1)2737 1595 y(A)2836 1430 y Fd(1)p 2819 1442 65 4 v 2819 1483 a(2)p Fg(k)3138 1615 y Fr(\(2.54\))382 1903 y(is)30 b(de\014ned.)40 b(The)30 b(top)s(ographic)g(pro)s(duct)g(is)f(\014nally)g(computed)i(as)f(an)h (a)m(v)m(erage)i(o)m(v)m(er)382 2016 y(neigh)m(b)s(orho)s(o)s(d)28 b(orders)i Fn(k)j Fr(and)d(all)f(neurons)g Fn(s)1084 2284 y(P)38 b Fr(=)1509 2222 y(1)p 1286 2263 492 4 v 1286 2346 a Fn(N)1359 2360 y Fi(C)1418 2346 y Fr(\()p Fn(N)1526 2360 y Fi(C)1606 2346 y Fk(\000)20 b Fr(1\))1803 2203 y Ff(X)1803 2387 y Fi(s)p Fe(2C)1939 2174 y Fi(N)1995 2185 y Fg(C)2046 2174 y Fe(\000)p Fh(1)1977 2203 y Ff(X)1973 2388 y Fi(k)r Fh(=1)2151 2284 y Fn(l)r(og)s Fr(\()p Fn(P)2363 2298 y Fh(3)2404 2284 y Fr(\()p Fn(s;)15 b(k)s Fr(\)\))p Fn(:)471 b Fr(\(2.55\))382 2549 y(Where)31 b Fn(N)746 2563 y Fi(C)835 2549 y Fr(denotes)g(the)f(total)i(n)m(um)m(b)s(er)d(of) h(neurons)f(in)g(the)i(output)f(space.)523 2662 y(The)i(v)-5 b(alue)32 b(of)h Fn(P)45 b Fr(can)33 b(b)s(e)f(p)s(ositiv)m(e)g(or)g (negativ)m(e.)49 b(A)32 b(v)-5 b(alue)32 b(nearb)m(y)h(0)g(indicates) 382 2775 y(a)h(go)s(o)s(d)g(top)s(ological)g(appro)m(ximation)e(of)i (the)g(input)e(data,)k(a)e(to)s(o)h(high)d(or)i(to)s(o)h(lo)m(w)382 2888 y(v)-5 b(alue)30 b(indicates)f(some)i(violated)f(neigh)m(b)s(orho) s(o)s(d)e(relations.)382 3130 y Fj(2.8.2)112 b(T)-9 b(op)s(ographic)38 b(function)382 3301 y Fr(The)29 b(top)s(ographic)g(function)f(is)g (based)i(on)f(a)h(stringen)m(t)f(mathematical)h(foundation.)382 3414 y(It)h(uses)g(the)h(input)d(data)j(to)h(ev)-5 b(aluate)32 b(the)f(qualit)m(y)g(of)g(the)h(appro)m(ximation)e(instead)382 3527 y(of)g(a)h(neigh)m(b)s(orho)s(o)s(d)d(ordering.)40 b(F)-8 b(or)31 b(details)e(see)i(..)523 3640 y(The)43 b(basic)f(idea)h(is,)i(to)f(compare)f(t)m(w)m(o)i(neigh)m(b)s(orho)s(o) s(d)40 b(relations.)78 b(A)43 b(strong)382 3753 y(neigh)m(b)s(orho)s(o) s(d)c(relation)h(is)g(de\014ned)g(in)f(the)j(input)d(space,)44 b(using)c(the)h(Delauna)m(y-)382 3866 y(graph)33 b Fk(D)713 3880 y Fi(R)771 3866 y Fr(.)50 b(A)33 b(Delauna)m(y-graph)h(is)f (de\014ned,)g(b)m(y)g(connecting)h(all)e(neurons,)i(whose)382 3979 y(V)-8 b(oronoi-regions)31 b(share)h(an)f(edge.)44 b(Using)31 b(this)f(graph,)h(the)h(distance)f Fn(d)2921 3946 y Fe(D)2976 3957 y Fg(V)3033 3979 y Fr(\()p Fn(w)3133 3993 y Fi(s)3171 3979 y Fn(;)15 b(w)3276 3994 y Fi(s)3309 3975 y Fc(0)3335 3979 y Fr(\))382 4092 y(is)25 b(de\014ned)g(as)h(the)g (length)g(of)g(the)g(minimal)e(path)h(on)h(the)h(Delauna)m(y-graph)f(b) s(et)m(w)m(een)382 4205 y Fn(s)f Fr(and)g Fn(s)665 4172 y Fe(0)688 4205 y Fr(.)39 b(In)25 b(the)g(output)g(space,)j(a)e(w)m (eak)g(neigh)m(b)s(orho)s(o)s(d,)e(based)i(on)f(the)h(maxim)m(um)382 4317 y(norm)21 b Fk(k)p Fn(s)s Fk(\000)s Fn(s)819 4284 y Fe(0)842 4317 y Fk(k)887 4331 y Fi(max)1057 4317 y Fr(=)k Fn(max)1333 4284 y Fi(d)1333 4344 y(k)r Fh(=1)1465 4317 y Fk(j)p Fn(s)1533 4332 y Fi(k)1579 4317 y Fk(\000)s Fn(s)1696 4284 y Fe(0)1696 4344 y Fi(k)1739 4317 y Fk(j)d Fr(is)f(de\014ned.)36 b(A)22 b(map)g(is)f(called)g(neigh)m(b)s(orho)s (o)s(d)382 4430 y(preserving,)g(if)f(neurons)f Fn(w)1305 4444 y Fi(r)1343 4430 y Fn(;)c(w)1448 4446 y Fi(r)1482 4427 y Fc(0)1509 4430 y Fr(,)23 b(connected)e(b)m(y)g(the)g(Delauna)m (y-graph,)i(are)e(mapp)s(ed)382 4543 y(on)m(to)39 b(w)m(eakly)e(neigh)m (b)s(ored)f(neurons)h Fn(r)j Fr(and)d Fn(r)2029 4510 y Fe(0)2052 4543 y Fr(.)62 b(In)37 b(the)h(opp)s(osite)e(if)h(t)m(w)m (o)i(neurons)382 4656 y Fn(r)m(;)15 b(r)504 4623 y Fe(0)552 4656 y Fr(are)25 b(neigh)m(b)s(ored,)g(their)f(w)m(eigh)m(t)h(v)m (ectors)h Fn(w)2046 4670 y Fi(r)2084 4656 y Fn(;)15 b(w)2189 4671 y Fi(r)2223 4653 y Fc(0)2274 4656 y Fr(should)23 b(b)s(e)h(connected)i(b)m(y)e Fk(D)3284 4670 y Fi(V)3345 4656 y Fr(.)382 4769 y(W)-8 b(e)31 b(de\014ne)f(for)g(eac)m(h)i(neuron) d Fn(s)c Fk(2)g(C)844 4961 y Fn(f)889 4975 y Fi(s)925 4961 y Fr(\()p Fn(k)s Fr(\))84 b(=)f(#)p Fk(f)p Fn(s)1447 4923 y Fe(0)1500 4961 y Fk(j)31 b(k)p Fn(s)20 b Fk(\000)g Fn(s)1798 4923 y Fe(0)1821 4961 y Fk(k)1866 4975 y Fi(max)2036 4961 y Fn(>)25 b(k)18 b(;)30 b(d)2299 4923 y Fe(D)2354 4934 y Fg(V)2412 4961 y Fr(\()p Fn(w)2512 4975 y Fi(s)2549 4961 y Fn(;)15 b(w)2656 4923 y Fe(0)2654 4983 y Fi(s)2691 4961 y Fr(\))26 b(=)f(1)p 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y(putations.)40 b(Esp)s(ecially)28 b(the)i Fo(Neur)-5 b(al)34 b(Gas)k Fr(uses)30 b(a)h(completely)f(di\013eren)m (t)g(approac)m(h.)523 4844 y(Learning)36 b(Rate)j(\(see)f(section)g (3.2.4\))h(and)e(Neigh)m(b)s(orho)s(o)s(d)f(F)-8 b(unction)37 b(\(section)382 4957 y(3.2.5\))f(are)e(v)m(ery)g(imp)s(ortan)m(t)g (functions,)f(as)h(they)g(determine)f(the)h(b)s(eha)m(vior)f(of)h(the) 382 5070 y(learning)21 b(pro)s(cess.)37 b(They)22 b(ma)m(y)h(v)-5 b(ary)22 b(b)s(et)m(w)m(een)h(di\013eren)m(t)e(problems)g(and)g (algorithms.)382 5183 y(By)32 b(c)m(hanging)g(one)g(or)g(b)s(oth,)g (the)h(learning)d(pro)s(cess)i(can)g(b)s(e)f(con)m(trolled.)46 b(One)31 b(goal)382 5296 y(of)d(the)h(framew)m(ork)f(is)f(to)i(allo)m (w)f(easy)g(exc)m(hange)i(of)e(these)h(comp)s(onen)m(ts)f(to)h(compare) p eop %%Page: 39 45 39 44 bop 382 249 a Fs(CHAPTER)30 b(3.)71 b(COMPONENT)29 b(ANAL)-8 b(YSIS)1166 b Fr(39)382 548 y(the)24 b(results.)37 b(As)23 b(these)i(t)m(w)m(o)f(comp)s(onen)m(ts)g(are)g(similar)d(in)h 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Fr(\).)e(Blo)s(c)m(ks)39 b(b)s(elonging)e(together)j (are)382 4171 y(inden)m(ted.)382 4414 y Fj(4.2.1)112 b(class)38 b(Graph)382 4586 y Fr(This)33 b(general)h(graph)g(class)h (is)e(used)h(as)h(a)g(\015exible)d(basis)i(for)g(the)h(map.)53 b(As)34 b(\(undi-)382 4699 y(rected\))k(edges)h(can)f(b)s(e)f(created)h (b)s(et)m(w)m(een)g(an)m(y)g(no)s(des)f(without)g(restrictions,)h(an)m (y)382 4812 y(kind)28 b(of)j(structure)f(is)f(p)s(ossible.)523 4925 y(By)d(default)f(the)h Fb(GraphDistance)d Fr(metho)s(d)i(computes) h(the)g(length)f(of)i(the)f(mini-)382 5038 y(mal)i(path)g(b)s(et)m(w)m (een)h(t)m(w)m(o)g(no)s(des.)40 b(This)26 b(distance)i(measure)g(is)g (c)m(hosen,)h(b)s(ecause)g(this)382 5150 y(distance)h(measure)g(is)g (indep)s(enden)m(t)e(form)i(the)g(underlying)e(structure.)523 5263 y(F)-8 b(rom)33 b(the)g(graphs)f(p)s(oin)m(t)f(of)i(view,)g(this)e (distance)i(measure)f(is)g(de\014ned)f(for)h(the)382 5376 y(output)f(space)g(and)g(de\014nes)f(the)h(top)s(ology)g(of)g(the) h(graph.)42 b(Other)30 b(p)s(ossible)f(c)m(hoices)p eop %%Page: 50 56 50 55 bop 382 249 a Fs(CHAPTER)30 b(4.)71 b(IMPLEMENT)-8 b(A)g(TION)1402 b Fr(50)382 548 y(are,)31 b(e.g.)42 b(for)30 b(a)h(grid,)e(the)i(cit)m(y)f(blo)s(c)m(k)g(metric.)382 791 y Fj(4.2.2)112 b(class)38 b(Map)382 963 y Fr(The)d(map,)h(con)m (taining)f(the)g(data)h(and)f(the)g(metho)s(ds)g(to)h(train)e(the)h (net.)56 b(The)34 b(ad-)382 1076 y(ditional)j(features)i(are)h(a)f (generate)i(op)s(eration)e(to)g(set)h(up)e(the)h(net)h(structure,)h(an) 382 1189 y(op)s(eration)f(to)h(initialize)c(the)j(no)s(de)g(w)m(eigh)m (ts,)j(a)e(general)f(neigh)m(b)s(orho)s(o)s(d)e(function)382 1302 y(and)30 b(an)g(op)s(eration)g(to)h(train)e(the)i(net.)382 1415 y(See)g(\014gures)e(A.1)i(and)f(A.2)h(for)f(a)h(description)e(of)h (the)h(input)d(\014les.)382 1658 y Fj(4.2.3)112 b(class)38 b(SOMW)-9 b(eigh)m(t)382 1830 y Fr(Virtual)43 b(Class)h(that)h(allo)m (ws)f(basic)g(computations,)k(lik)m(e)c(addition,)j(subtraction,)382 1943 y(scalar)d(m)m(ultiplication)e(and)i(computation)g(of)h(a)g(norm.) 82 b(The)44 b(norm)g(is)f(used)h(to)382 2056 y(compute)31 b(a)f(distance)g(measure.)523 2168 y(Seen)f(from)g(the)h(maps)f(view,)g (the)h Fb(SOMWeight::Distance)24 b Fr(metho)s(d)29 b(is)g(de\014ned)382 2281 y(for)f(the)h(input)e(space)i(and)f(compares)h(t)m(w)m(o)h(ob)5 b(jects.)41 b(In)28 b(c)m(hapter)h(2.1)h(the)f(Euclidean)382 2394 y(distance)h Fk(k)21 b(\017)f(k)31 b Fr(w)m(as)g(used.)523 2507 y(As)c(an)m(y)g(kind)e(of)i(ob)5 b(ject)27 b(ma)m(y)h(b)s(e)e (used)g(as)h(an)f(input)f(for)i(the)g(map,)g(the)g(distance)382 2620 y(metho)s(d)43 b(is)f(an)h(imp)s(ortan)m(t)g(part)g(of)h(the)f(in) m(terface)h(b)s(et)m(w)m(een)g(these)g(ob)5 b(jects)44 b(and)382 2733 y(the)38 b(map.)62 b(It)37 b(de\014nes)g(the)h(top)s (ology)g(of)g(the)g(input)d(space)j(that)h(the)e(map)h(should)382 2846 y(learn.)h(It)30 b(should)d(b)s(e)i(men)m(tioned,)g(that)h(from)f (a)g(mathematical)h(p)s(oin)m(t)e(of)h(view,)g(the)382 2959 y(c)m(hoice)38 b(of)e(the)h(norm)f(metho)s(d)h(is)e(not)i (completely)g(free,)i(as)e(it)f(has)h(to)g(b)s(e)f(a)h(norm)382 3072 y(in)h(mathematical)i(sense)f(to)h(induce)e(a)i(consisten)m(t)f (top)s(ology)-8 b(.)69 b(F)-8 b(urthermore,)41 b(the)382 3185 y(distance)34 b(has)h(to)h(b)s(e)e(a)h(v)m(ector)i(space.)54 b(This)34 b(is)f(necessary)j(to)f(allo)m(w)g(computation)382 3298 y(with)29 b(its)h(elemen)m(ts.)382 3410 y(See)h(\014gures)e(A.5)i (and)f(A.6)h(for)f(a)h(description)e(of)h(the)h(input)d(\014les.)382 3654 y Fj(4.2.4)112 b(class)38 b(SOMT)-9 b(rainNet)382 3826 y Fr(This)25 b(metho)s(d)h(is)g(used,)h(to)h(run)d(a)i(learning)f (pro)s(cess.)39 b(In)26 b(eac)m(h)i(learning)d(step,)j(a)f(new)382 3938 y(sample)f(is)g(tak)m(en)h(from)g Fb(SOMGetNewSample)22 b Fr(and)k(giv)m(en)h(to)h Fb(SOMAdaption)23 b Fr(un)m(til)i(the)382 4051 y Fb(SOMBreakCondition)h Fr(is)j(ful\014lled.)523 4164 y(In)40 b(the)h(prepro)s(cessing)e(step,)44 b(some)d(initial)d (calculations)i(ma)m(y)i(b)s(e)e(done,)j(lik)m(e)382 4277 y(a)36 b(computation)h(of)f(the)g Fb(GraphDistance)d Fr(b)s(et)m(w)m(een)k(all)e(no)s(des)g(to)i(sp)s(eed)e(up)h(later)382 4390 y(computation.)p 382 4471 2989 4 v 382 4634 a Fb(if)47 b(\(DoPreProcess\))525 4747 y(PreProcess)382 4860 y(while)f(\(NOT)h (SOMBreakCondition\(\)\))525 4973 y(sample)f(=)i(SOMGetNewSample\(\)) 525 5085 y(SOMAdaption\(\))382 5198 y(end)p 382 5301 V eop %%Page: 51 57 51 56 bop 382 249 a Fs(CHAPTER)30 b(4.)71 b(IMPLEMENT)-8 b(A)g(TION)1402 b Fr(51)1376 548 y(Algorithm)29 b(SOMT)-8 b(rainNet)382 817 y(See)31 b(\014gures)e(A.7)i(and)f(A.8)h(for)f(a)h (description)e(of)h(the)h(input)d(\014les.)382 1060 y Fj(4.2.5)112 b(class)38 b(SOMBreakCondition)382 1231 y Fr(Returns)29 b(1)i(if)e(the)i Fo(Br)-5 b(e)g(akCondition)40 b Fr(is)29 b(not)i(ful\014lled,)c(0)k(otherwise.)523 1344 y(The)36 b(basic)h(implemen)m(tation)e(is)h(a)h(test,)j(whether)c Fn(t)g(>)g(t)2561 1358 y Fi(max)2705 1344 y Fr(.)60 b(Other)36 b(p)s(ossible)382 1457 y(test)28 b(w)m(ould)e(b)s(e)h(if)f(a)i(n)m(um)m (b)s(er)e(of)h(no)s(des)f(has)h(b)s(een)g(generated)h(already)f(or)g (if)g(a)g(global)382 1570 y(error)j(measure)g(is)g(minimized.)p 382 1633 2989 4 v 382 1792 a Fb(if)47 b(\(t)g(<)h(t_max\))525 1905 y(result)e(=)i(true)382 2018 y(else)525 2131 y(result)e(=)i(false) p 382 2230 V 1242 2389 a Fr(Algorithm)29 b(SOMBreakCondition)382 2658 y(See)i(\014gures)e(A.9)i(and)f(A.10)i(for)e(a)h(description)d(of) j(the)f(input)f(\014les.)382 2901 y Fj(4.2.6)112 b(class)38 b(SOMAdaption)382 3072 y Fr(This)j(metho)s(d)h(is)g(the)i(k)m(ernel)e (of)h(the)g(learning)f(pro)s(cess.)78 b(Giv)m(en)43 b(a)g(sample,)i (the)382 3185 y Fo(BestMatchingNo)-5 b(de)36 b Fr(is)28 b(computed)h(using)e(the)i Fb(SOMBestMatching)24 b Fr(metho)s(d.)40 b(Then)382 3298 y(all)29 b(no)s(des)h(are)g(mo)s(di\014ed,)f(using)g (the)h Fb(SOMUpdateNode)d Fr(metho)s(d.)p 382 3382 V 382 3541 a Fb(if)47 b(\(DoPreProcess\))525 3654 y (UpdateNode->PreProcess)382 3767 y(bmNode)f(=)h(SOMBestMatching\(\))382 3879 y(for)g(all)g(nodes)525 3992 y(SOMUpdateNode\(curNode\))382 4105 y(end)p 382 4204 V 1370 4363 a Fr(Algorithm)29 b(SOMAdaption)382 4632 y(See)i(\014gures)e(A.11)j(and)e(A.12)h(for)f(a)h(description)e (of)h(the)h(input)d(\014les.)382 4875 y Fj(4.2.7)112 b(class)38 b(SOMBestMatc)m(hing)382 5047 y Fr(Giv)m(en)24 b(a)h(sample)f(w)m(eigh)m(t,)i(this)d(metho)s(d)h(computes)h(the)g(in)e (some)i(w)m(a)m(y)g(\\b)s(est)f(matc)m(h-)382 5160 y(ing")33 b(no)s(de)g(of)h(the)g(map.)51 b(Usually)32 b(the)i(w)m(eigh)m(ts)g Fb(Distance\(\))d Fr(metho)s(d)i(is)g(used)g(to)382 5272 y(determine)27 b(the)g(minim)m(um)e(distance)i(b)s(et)m(w)m(een)i(the)e (giv)m(en)h(sample)f(and)g(the)g(w)m(eigh)m(ts)382 5385 y(of)j(the)h(map-no)s(des.)p eop %%Page: 52 58 52 57 bop 382 249 a Fs(CHAPTER)30 b(4.)71 b(IMPLEMENT)-8 b(A)g(TION)1402 b Fr(52)p 382 548 2989 4 v 382 703 a Fb(for)47 b(all)g(nodes)525 816 y(curDist)f(=)h (node.Weight->Distance\(samp)o(le.)o(Weig)o(ht\))525 929 y(if)g(\(curDist)f(<)h(bestDist\))668 1042 y(bestDist)f(=)h (curDist)382 1155 y(end)p 382 1258 V 1281 1420 a Fr(Algorithm)29 b(SOMBestMatc)m(hing)382 1824 y Fj(4.2.8)112 b(class)38 b(SOMUp)s(dateNo)s(de)382 1995 y Fr(The)30 b(giv)m(en)g(no)s(de)g(is)f (shifted)g(to)m(w)m(ards)i(the)g(sample,)f(according)g(to)h(the)382 2108 y Fb(SOMNeighborhoodFunction)24 b Fr(and)30 b(the)g Fb(SOMLearningRate)p Fr(.)p 382 2192 V 382 2354 a Fb(hsi)47 b(=)g(SOMNeighborhoodFunction\(\))382 2467 y(alpha)f(=)i (SOMLearningRate\(\))382 2580 y(factor)e(=)h(alpha)g(*)g(hsi;)382 2693 y(if)g(\(factor)f(!=)h(0\))525 2806 y(node.Weight)e(=)i (node.Weight)e(+)1193 2919 y(factor)h(*)i(\(sample)e(-)h(node.Weight\)) 382 3032 y(end)p 382 3135 V 1305 3298 a Fr(Algorithm)30 b(SOMUp)s(dateNo)s(de)382 3701 y Fj(4.2.9)112 b(class)38 b(SOMLearningRate)382 3872 y Fr(This)27 b(op)s(erator)i(returns)f(a)i (real)e(v)-5 b(alue)29 b Fn(\013)g Fr(b)s(et)m(w)m(een)h(0)f(and)g(1,)h (that)f(is)f(used)h(to)g(de\014ne)382 3985 y(the)42 b(strength)g(of)h (the)f(learning)f(pro)s(cess.)76 b(If)41 b Fn(\013)i Fr(is)e(1,)46 b(the)c(up)s(date)g(is)f(done)h(with)382 4098 y(maxim)m(um)29 b(strength,)i(if)e(it)h(is)f(0,)i(no)g(up)s(date)e (is)h(done)g(at)h(all.)523 4211 y(The)23 b Fb(SOMLearningRate)c Fn(\013)p Fr(\()p Fn(t)p Fr(\))25 b(is)d(a)i(\(usually\))e (monotonically)h(decreasing)g(func-)382 4324 y(tion)30 b(in)f(time.)40 b(See)31 b(equation)f(2.14)i(for)e(a)h(simple)d (example.)p 382 4405 V 382 4568 a Fb(result)46 b(=)h(alpha_start)e(*)j (pow)f(\(\(alpha_end)d(/)k(alpha_start\),)c(t/t_max\))p 382 4691 V 1286 4854 a Fr(Algorithm)30 b(SOMLearningRate)382 5126 y(See)h(\014gures)e(A.15)j(and)e(A.16)h(for)f(a)h(description)e (of)h(the)h(input)d(\014les.)p eop %%Page: 53 59 53 58 bop 382 249 a Fs(CHAPTER)30 b(4.)71 b(IMPLEMENT)-8 b(A)g(TION)1402 b Fr(53)382 548 y Fj(4.2.10)112 b(class)38 b(SOMNeigh)m(b)s(orho)s(o)s(dF)-9 b(unction)382 720 y Fr(Computes)20 b(a)h(real)f(v)-5 b(alued)20 b(w)m(eigh)m(t)h(that)g (describ)s(es)e(the)i(in\015uence)e(of)h(the)h(b)s(est)f(matc)m(h-)382 832 y(ing)38 b(no)s(de)f(to)j(a)f(giv)m(en)f(no)s(de.)65 b(F)-8 b(or)39 b(that)g(purp)s(ose,)g(the)g Fb(Graph::GraphDistance)382 945 y Fr(metho)s(d)d(is)f(used)h(to)h(determine)f(the)g(distance)g (\(in)g(the)g(graph\))h(b)s(et)m(w)m(een)g(the)f(t)m(w)m(o)382 1058 y(no)s(des.)523 1171 y(The)j Fb(Graph::GraphDistance)33 b Fr(metho)s(d)39 b(de\014nes)f(\(together)j(with)c(the)j(struc-)382 1284 y(ture)34 b(of)g(the)g(map\))g(the)g(top)s(ology)h(of)f(the)g (net.)52 b(It)34 b(is)f(a)h(distance)g(function)f(de\014ned)382 1397 y(for)d(the)h(output)f(space)h(\(the)f(map\).)523 1510 y(A)21 b(lot)f(of)h(di\013eren)m(t)f(c)m(hoices)h(for)f(the)h (neigh)m(b)s(orho)s(o)s(d)d(exist.)37 b(With)20 b(this)g(framew)m(ork) 382 1623 y(it)32 b(is)g(extremely)g(easy)i(to)f(c)m(hange)h(the)f (neigh)m(b)s(orho)s(o)s(d)d(function)h(and)h(compare)h(the)382 1736 y(results.)39 b(In)30 b(equation)g(2.12)i(and)e(2.13)i(t)m(w)m(o)g (sample)d(de\014nitions)f(are)j(giv)m(en.)523 1849 y(The)23 b(radius)e(of)i(in\015uence)e(should)g(b)s(e)i(time)f(decreasing,)j(to) f(ensure)e(con)m(v)m(ergence.)523 1962 y(The)43 b(basic)g(set)i(up)d (is)h(a)h(linear)f(decreasing)g(k)m(ernel)g(com)m(bined)g(with)g(a)h ([1)p Fn(;)15 b Fr(0])382 2074 y(step)34 b(function,)f(starting)h(at)h Fn(t)30 b Fr(=)h(0)j(width)e(radius)g Fb(kwidth)h Fr(and)g(ending)f (with)h(0)h(at)382 2187 y Fn(t)25 b Fr(=)g Fn(t)569 2201 y Fi(max)713 2187 y Fr(.)p 382 2264 2989 4 v 382 2427 a Fb(dist)47 b(=)g(map->graphDistance\(curre)o(ntNo)o(de,)41 b(bestMatchingNode\))382 2540 y(radius)46 b(=)h(\(1.1*t_max-t\)*kwidth) c(/)k(t_max)382 2653 y(if)g(\(dist)f(<)i(radius\))525 2766 y(result)e(=)i(1)382 2879 y(else)525 2992 y(result)e(=)i(0)p 382 3095 V 1101 3257 a Fr(Algorithm)30 b(SOMNeigh)m(b)s(orho)s(o)s(dF) -8 b(unction)382 3530 y(See)31 b(\014gures)e(A.17)j(and)e(A.18)h(for)f (a)h(description)e(of)h(the)h(input)d(\014les.)p eop %%Page: 54 60 54 59 bop 382 1171 a Fm(Chapter)65 b(5)382 1586 y Fq(Extensions)382 2031 y Fr(Based)35 b(up)s(on)e(the)i(implemen)m(tation)f(of)h(Kohonens) f(algorithm)f(in)h(c)m(hapter)h(4)g(some)382 2144 y(extensions)i(to)g (his)f(mo)s(del)g(are)i(discussed.)59 b(They)37 b(are)g(set)h(in)m(to)f (relation)f(and)h(use)382 2257 y(the)46 b(adv)-5 b(an)m(tages)47 b(of)e(the)h(ob)5 b(ject-orien)m(ted)47 b(design)d(to)i(minimize)d(and) i(ease)i(later)382 2369 y(mo)s(di\014cations.)37 b(In)25 b(opp)s(osite)g(to)h(the)f(previously)f(describ)s(ed)f(metho)s(ds,)j (all)e(metho)s(ds)382 2482 y(presen)m(ted)h(in)f(this)g(c)m(hapter)h (use)g(a)g(\015exible)f(net)h(top)s(ology)-8 b(.)40 b(This)23 b(is)h(done,)i(b)m(y)f(adding)382 2595 y(no)s(des)44 b(and)h(edges)g(on)g(demand.)84 b(Though,)48 b(the)d(metho)s(ds)g (di\013er)e(a)j(lot)f(in)f(ho)m(w)382 2708 y(this)33 b(is)g(done,)j(they)e(w)m(ere)h(implemen)m(ted)d(inside)g(the)j(framew) m(ork)f(with)f(only)g(a)i(few)382 2821 y(mo)s(di\014cations.)52 b(Esp)s(ecially)32 b(a)j(new)g(gro)m(wing-step)g(had)f(to)h(b)s(e)f (inserted,)h(that)g(can)382 2934 y(b)s(e)29 b(mo)s(di\014ed)f(on)i (demand.)40 b(As)30 b(an)g(criteria)f(where)g(to)i(insert)e(new)h(no)s (des)f(or)h(edges,)382 3047 y(usually)j(a)j(lo)s(cal)e(error)h(measure) g(is)f(used.)55 b(This)33 b(error)i(measure)g(is)g(stored)g(lo)s(cally) 382 3160 y(inside)c(the)i(no)s(des)g(and)g(remo)m(v)m(ed)h(afterw)m (ards.)50 b(As)33 b(of)g(the)h(great)g(\015exibilit)m(y)d(of)i(the)382 3273 y(framew)m(ork)23 b(and)g(great)h(brough)m(t)f(of)h(di\013eren)m (t)e(metho)s(ds,)j(these)e(extensions)g(can)h(only)382 3386 y(act)31 b(as)g(samples)e(for)i(further)e(dev)m(elopmen)m(t.)382 3672 y Fl(5.1)135 b(Class)46 b(structure)382 3875 y Fr(Extending)27 b(the)h(basic)f(class)g(structure,)i(eac)m(h)g(algorithm)e(is)f (describ)s(ed)g(in)h(his)f(mo)s(d-)382 3988 y(i\014cations)j(and)g (extension.)41 b(UML)30 b(diagrams)f(will)f(sho)m(w,)i(ho)m(w)g(the)g (ob)5 b(ject-orien)m(ted)382 4101 y(design)29 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I51 It 34 58 5104 6550 I53 It 59 60 5135 6548 I35 It 17 77 5200 6550 I55 It 78 80 5220 6548 I57 It 59 60 5301 6548 I35 It 58 60 5358 6548 I59 It 5760 6490 m 3440 6490 l S 5760 6390 m 3440 6390 l S 1860 6910 m 1930 7040 l S 2000 6910 m 1930 7040 l S 2000 6910 m 1860 6910 l S 1930 6570 m 1930 6890 l S 2860 5440 m 2930 5570 l S 3000 5440 m 2930 5570 l S 3000 5440 m 2860 5440 l S 2930 5230 m 2930 5420 l S 4470 6930 m 4530 7060 l S 4600 6930 m 4530 7060 l S 4600 6930 m 4470 6930 l S 4530 6690 m 4530 6910 l S 2210 6950 m 2160 6830 l 2110 6950 l 2160 7040 l h S 2210 6950 m 2160 6830 l 2110 6950 l 2160 7040 l f* 2160 6730 m 2160 6830 l S 2920 6730 m 2160 6730 l S 2920 5960 m 2920 6730 l S 1 g 2920 5960 0 0 re f 0 g 2920 5960 0 0 re S 1 g 2160 7040 0 0 re f 0 g 2160 7040 0 0 re S 1130 5480 m 1090 5360 l 1040 5480 l 1090 5570 l h S 1090 5250 m 1090 5360 l S 1 g 1090 5250 0 0 re f 0 g 1090 5250 0 0 re S 1 g 1090 5570 0 0 re f 0 g 1090 5570 0 0 re S 1800 6950 m 1750 6830 l 1700 6950 l 1750 7040 l 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y(It)j(is)g(used)f(to)i(sho)m(w)f(the)h(b)s(ene\014ts)e(of)h(the)h (ob)5 b(ject-orien)m(ted)31 b(approac)m(h.)41 b(The)30 b(k)m(ernel,)382 4532 y(referring)39 b(to)h(Kohonens)g(algorithm)f(has) h(b)s(een)f(reused,)j(while)c(the)i(new)g(gro)m(wing)382 4645 y(steps)30 b(w)m(ere)h(added)f(seamless)g(in)m(to)g(the)h(system.) 382 4885 y Fp(Ov)m(erview)382 5057 y Fr(The)f Fo(Gr)-5 b(owing)34 b(Neur)-5 b(al)33 b(Gas)g(algorithm)40 b Fr(is)29 b(a)i(classical)e(gro)m(wing)i(algorithm)e(in)g(that)382 5170 y(sense,)48 b(as)d(it)f(splits)e(the)j(learning)d(pro)s(cess)i(in) g(t)m(w)m(o)h(parts:)69 b(A)44 b(w)m(eigh)m(t)h(adaption)382 5283 y(step,)35 b(as)g(kno)m(wn)f(from)f(Kohonen,)i(and)f(a)g(net)h (adaption)f(step,)h(that)g(is)e(p)s(erformed)p eop %%Page: 57 63 57 62 bop 382 249 a Fs(CHAPTER)30 b(5.)71 b(EXTENSIONS)1668 b Fr(57)382 548 y(frequen)m(tly)39 b(to)h(up)s(date)f(the)h(net)g (structure.)69 b(While)39 b(the)h Fo(Neur)-5 b(al)41 b(Gas)49 b Fr(algorithm)382 661 y(only)29 b(added)h(new)g(edges,)h (this)e(one)i(is)e(also)i(able)f(to)h(create)h(new)e(no)s(des.)523 774 y(The)25 b(new)g(learning)f(step)h(is)f(co)m(v)m(ered)j(b)m(y)f (the)f(new)h Fb(SOMTrainNetGrowing)21 b Fr(op)s(er-)382 887 y(ator,)33 b(that)f(extends)g(the)g(standard)f(op)s(erator.)45 b(The)31 b(learning)f(step)i(itself)e(is)h(imple-)382 1000 y(men)m(ted)h(in)f(the)i Fb(SOMAdaptionNetGNG)27 b Fr(class,)33 b(whic)m(h)e(is)g(deriv)m(ed)g(from)h(the)g(virtual)382 1112 y Fb(SOMAdaptionNet)i Fr(class.)62 b(This)36 b(allo)m(ws)h(easy)h (exc)m(hange)h(of)f(this)f(step,)j(b)m(y)d(simply)382 1225 y(creating)j(a)f(new)g(deriv)m(ed)f(class.)67 b(F)-8 b(or)40 b(example,)i(the)d(criteria)g(for)g(adding)e(a)j(new)382 1338 y(no)s(de)28 b(ma)m(y)h(b)s(e)f(c)m(hanged,)i(so)e(that)h(an)g (external)f(error)g(measure)h(can)g(b)s(e)f(considered.)523 1451 y(The)33 b(adaption)h(step)g(for)f(the)h(w)m(eigh)m(ts)g(extends)g (the)g Fb(SOMAdaptionNeuralGas)382 1564 y Fr(op)s(eration,)44 b(but)c(reuses)h(most)h(of)f(it.)74 b(Only)39 b(a)j(new)f(temp)s(oral)f (error)h(measure)h(is)382 1677 y(computed,)30 b(that)h(is)f(later)g (used)g(to)h(decide)f(where)f(to)j(add)d(new)h(neurons.)523 1790 y(The)f Fo(br)-5 b(e)g(ak)33 b(c)-5 b(ondition)39 b Fr(w)m(as)30 b(also)g(mo)s(di\014ed,)e(to)j(stop)f(the)g(gro)m(wing)g (pro)s(cess,)f(if)g(a)382 1903 y(prede\014ned)23 b(n)m(um)m(b)s(er)g (of)i(no)s(des)f(has)g(b)s(een)g(created,)j(or)e(a)g(upp)s(er)e(b)s (ound)f(of)j(iterations)382 2016 y(w)m(as)31 b(reac)m(hed.)523 2129 y(The)20 b Fb(SOMLearningRateTwoStep)14 b Fr(op)s(erator)21 b(w)m(as)f(used)g(from)f(the)i(utilities)d(pac)m(k-)382 2242 y(age)24 b(and)e(the)h Fb(SOMBestMatchingNeuralGas)16 b Fr(op)s(erator)24 b(w)m(as)f(reused)f(from)g(the)h(Neu-)382 2355 y(ralGas)30 b(algorithm.)382 2595 y Fp(Classes)382 4715 y currentpoint currentpoint translate 0.68973 0.68973 scale neg exch neg exch translate 382 4715 a 382 4715 a gsave currentpoint currentpoint translate 0 neg rotate neg 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