(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips 5.516 Copyright 1986, 1993 Radical Eye Software %%Title: top.dvi %%CreationDate: Sat May 13 22:55:31 1995 %%Pages: 36 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o top.ps top %DVIPSSource: TeX output 1995.05.13:2252 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N 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192 a Fu(allo)o(ws)18 b(the)g(algorithm)g(to)f(b)q(e)h (insensitiv)o(e)i(to)d(the)h(particular)g(c)o(hoice)h(of)e(co)q (ordinates)h(used)150 248 y(to)e(enco)q(de)h(the)g(data)e(\(an)h(impro) o(v)o(emen)o(t)g(o)o(v)o(er)g(metho)q(ds)g(suc)o(h)h(as)f(MARS)h(and)f (ID3,)g(whic)o(h)150 304 y(are)f(co)q(ordinate-dep)q(enden)o(t\).)241 361 y(The)j(w)o(ork)f(that)g(w)o(e)g(describ)q(e)j(here)e(mak)o(es)f (con)o(tact)g(with)h(a)g(n)o(um)o(b)q(er)g(of)f(branc)o(hes)h(of)150 417 y(statistical)i(theory)l(.)33 b(First,)20 b(as)f(in)h(our)f (earlier)i(w)o(ork)d(\(Jacobs,)j(Jordan,)f(No)o(wlan,)g(&)g(Hin-)150 474 y(ton,)14 b(1991\),)f(w)o(e)h(form)o(ulate)g(the)g(learning)i (problem)f(as)f(a)g(mixture)h(estimation)f(problem)i(\(cf.)150 530 y(Cheeseman,)e(et)f(al,)g(1988;)g(Duda)g(&)h(Hart,)e(1973;)h(No)o (wlan,)g(1991;)f(Redner)j(&)e(W)l(alk)o(er,)g(1984;)150 587 y(Titterington,)g(Smith,)i(&)e(Mak)o(o)o(v,)f(1985\).)18 b(W)l(e)c(sho)o(w)e(that)h(the)h(algorithm)f(that)g(is)h(generally)150 643 y(emplo)o(y)o(ed)20 b(for)f(the)h Ft(unsup)n(ervise)n(d)k Fu(learning)d(of)e(mixture)h(parameters|the)f(Exp)q(ectation-)150 700 y(Maximization)25 b(\(EM\))e(algorithm)h(of)f(Dempster,)j(Laird)f (and)f(Rubin)i(\(1977\)|can)d(also)150 756 y(b)q(e)e(exploited)g(for)f Ft(sup)n(ervise)n(d)k Fu(learning.)36 b(Second,)22 b(w)o(e)e(utilize)i (generalized)g(linear)f(mo)q(del)150 813 y(\(GLIM\))c(theory)f (\(McCullagh)i(&)g(Nelder,)g(1983\))e(to)g(pro)o(vide)i(the)g(basic)f (statistical)h(struc-)150 869 y(ture)d(for)f(the)g(comp)q(onen)o(ts)h (of)f(the)h(arc)o(hitecture.)20 b(In)15 b(particular,)g(the)g(\\soft)e (splits")j(referred)150 925 y(to)11 b(ab)q(o)o(v)o(e)g(are)h(mo)q (deled)h(as)e Ft(multinomial)i(lo)n(git)j Fu(mo)q(dels|a)d(sp)q (eci\014c)g(form)e(of)h(GLIM.)f(W)l(e)h(also)150 982 y(sho)o(w)18 b(that)g(the)h(algorithm)g(dev)o(elop)q(ed)h(for)e (\014tting)h(GLIM's|the)g(iterativ)o(ely)h(rew)o(eigh)o(ted)150 1038 y(least)c(squares)g(\(IRLS\))i(algorithm|can)e(b)q(e)h(usefully)h 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y(Exp)q(ectation-Maximization)j(\(EM\))d(framew)o(ork)g(of)h (Dempster,)g(Laird)h(and)g(Rubin)h(\(1977\).)150 1546 y(W)l(e)j(also)g(describ)q(e)h(a)f(least-squares)g(v)o(ersion)g(of)f (this)h(algorithm)g(that)f(leads)i(to)e(a)h(partic-)150 1603 y(ularly)h(e\016cien)o(t)g(implemen)o(tation.)35 b(Both)20 b(of)g(the)g(latter)f(algorithms)h(are)g(batc)o(h)g(learning) 150 1659 y(algorithms.)27 b(In)18 b(the)f(\014nal)i(section,)f(w)o(e)f (presen)o(t)h(an)f(on-line)i(v)o(ersion)f(of)f(the)h(least-squares)150 1716 y(algorithm)g(that)f(in)i(practice)g(app)q(ears)f(to)g(b)q(e)h (the)f(most)f(e\016cien)o(t)i(of)e(the)i(algorithms)f(that)150 1772 y(w)o(e)d(ha)o(v)o(e)g(studied.)150 1915 y Fs(Hierarc)n(hical)20 b(mixtures)i(of)h(exp)r(erts)150 2017 y Fu(The)d(algorithms)f(that)g(w) o(e)h(discuss)g(in)h(this)f(pap)q(er)g(are)f(sup)q(ervised)j(learning)e (algorithms.)150 2073 y(W)l(e)14 b(explicitly)i(address)e(the)f(case)h (of)f(regression,)h(in)g(whic)o(h)h(the)f(input)g(v)o(ectors)f(are)g (elemen)o(ts)150 2130 y(of)18 b Fn(<)238 2113 y Fm(m)289 2130 y Fu(and)h(the)f(output)g(v)o(ectors)g(are)g(elemen)o(ts)h(of)f Fn(<)1128 2113 y Fm(n)1151 2130 y Fu(.)29 b(W)l(e)19 b(also)f(consider)h(classi\014cation)150 2186 y(mo)q(dels)d(and)f(coun) o(ting)g(mo)q(dels)h(in)g(whic)o(h)g(the)f(outputs)f(are)h(in)o (teger-v)m(alued.)22 b(The)15 b(data)f(are)150 2243 y(assumed)k(to)f (form)h(a)f(coun)o(table)i(set)f(of)f(paired)i(observ)m(ations)g Fn(X)24 b Fu(=)18 b Fn(f)p Fu(\()p Fl(x)1464 2226 y Fk(\()p Fm(t)p Fk(\))1505 2243 y Fj(;)8 b Fl(y)1555 2226 y Fk(\()p Fm(t)p Fk(\))1596 2243 y Fu(\))p Fn(g)p Fu(.)27 b(In)19 b(the)150 2299 y(case)c(of)f(the)g Ft(b)n(atch)k Fu(algorithms)d (discussed)h(b)q(elo)o(w,)e(this)h(set)g(is)g(assumed)f(to)g(b)q(e)h (\014nite;)h(in)f(the)150 2356 y(case)g(of)g(the)g Ft(on-line)j Fu(algorithms,)d(the)g(set)g(ma)o(y)f(b)q(e)i(in\014nite.)241 2412 y(W)l(e)e(prop)q(ose)g(to)f(solv)o(e)h(nonlinear)h(sup)q(ervised)g 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def /ar { %% sa ea sx sy rot tx ty matrix currentmatrix 8 1 roll translate rotate scale n 0 0 1 5 3 roll arc setmatrix } bind def /arn { %% sa ea sx sy rot tx ty matrix currentmatrix 8 1 roll translate rotate scale n 0 0 1 5 3 roll arcn setmatrix } bind def /el { %% sx sy rot tx ty matrix currentmatrix 6 1 roll translate rotate scale n 0 0 1 0 360 arc setmatrix cl } bind def /bp {setlinejoin setlinewidth setrgbcolor} bind def /bpbw {setlinejoin setlinewidth setgray} bind def /lw {setlinewidth} bind def /lj {setlinejoin} bind def /gr {setgray} bind def /BPSIDE 32 def %% pixels per pattern side /PATFREQ 3.0 def %% pattern pixels per mm /dp_mat [PATFREQ 0 0 PATFREQ 0 0] def /dp_pw BPSIDE def %% pattern pixel width /dp_ph BPSIDE def %% pattern pixel height /dp_w dp_pw PATFREQ div def %% pattern mm width /dp_h dp_ph PATFREQ div def %% pattern mm height /dp_bs 1 def %% pattern bits per pixel /savemat matrix def /topmat matrix def /patmat matrix def /patpath { /inv exch def topmat setmatrix pathbbox %% get lo - hi indecies /hy exch dp_h div floor cvi def /hx exch dp_w div floor cvi def /ly exch dp_h div floor cvi def /lx exch dp_w div floor cvi def lx 1 hx { dp_w mul ly 1 hy { dp_h mul exch dup 3 1 roll exch patmat currentmatrix pop translate dp_pw dp_ph inv dp_mat dp_proc imagemask patmat setmatrix } for pop } for } bind def /setpattern { /blue exch def /green exch def /red exch def /freq exch def /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def freq 0 {/y exch def /x exch def /xindex x 1 add 2 div bpside mul cvi def /yindex y 1 add 2 div bpside mul cvi def bstring yindex bwidth mul xindex 8 idiv add get not 1 7 xindex 8 mod sub bitshift and 0 ne {/onbits onbits 1 add def 1} {/offbits offbits 1 add def 0} ifelse } setscreen {} settransfer systemdict /setcmykcolor known { /fact 1 onbits offbits onbits add div sub def 1 red sub fact mul 1 green sub fact mul 1 blue sub fact mul 0 setcmykcolor } { offbits offbits onbits add div setgray } ifelse } bind def /dmatrix matrix def /dpi 72 0 dmatrix defaultmatrix dtransform dup mul exch dup mul add sqrt def /B {gsave bp stroke grestore} bind def %% brush: gr lw lj /Bbw {gsave bpbw stroke grestore} bind def %% brush: gr lw lj /F {gsave setrgbcolor eofill grestore} bind def %% fill: gr /Fbw {gsave setgray eofill grestore} bind def %% fill: gr /PB {gsave setlinejoin setlinewidth setpattern stroke grestore} bind def /PF {gsave eoclip patpath grestore} bind def /BB { gsave setrgbcolor setlinejoin setlinewidth strokepath clip patpath grestore } bind def /BLACK { 0.0 } bind def /CP {closepath} bind def /FI {eofill} bind def /E {exch} bind def /FF {findfont} bind def /GR {grestore} bind def /GS {gsave} bind def /MF {makefont} bind def /NP {newpath} bind def /RO {rotate} bind def /ST {stroke} bind def /SC {scale} bind def /SF {setfont} bind def /SG {setgray} bind def /SLC {setlinecap} bind def /SLJ {setlinejoin} bind def /SLW {setlinewidth} bind def /TR {translate} bind def /WHITE { 1.0 } bind def /m {moveto} bind def /r {rmoveto} bind def /l {lineto} bind def /sp {x 0 rmoveto} bind def /rl {rlineto} bind def /s {show} bind def /box { NP m l l l CP } bind def /pageboundary { NP m l l l CP } bind def /BS { % black stroke GS SLJ SLW BLACK SG ST GR } bind def /WS { % white stroke GS SLJ SLW WHITE SG ST GR } bind def /reencode_small_dict 12 dict def /ReencodeSmall { reencode_small_dict begin /new_codes_and_names E def /new_font_name E def /base_font_name E def /base_font_dict base_font_name FF def /newfont base_font_dict maxlength dict def base_font_dict { E dup /FID ne { dup /Encoding eq { E dup length array copy newfont 3 1 roll put } { E newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName new_font_name put new_codes_and_names aload pop new_codes_and_names length 2 idiv { newfont /Encoding get 3 1 roll put } repeat new_font_name newfont definefont pop end %reencode_small_dict } def /extended_Zapf [ 8#223 /a89 8#224 /a90 8#225 /a93 8#226 /a94 8#227 /a91 8#230 /a92 8#231 /a205 8#232 /a85 8#233 /a206 8#234 /a86 8#235 /a87 8#236 /a88 8#237 /a95 8#240 /a96 ] def /extended_Standard [ 29 /thorn 30 /yacute 31 /divide 128 /Acircumflex 129 /Adieresis 130 /Agrave 131 /Aring 132 /Atilde 133 /Ccedilla 134 /Eacute 135 /Ecircumflex 136 /Edieresis 137 /Egrave 138 /Iacute 139 /Icircumflex 140 /Idieresis 141 /Igrave 142 /Ntilde 143 /Oacute 144 /Ocircumflex 145 /Odieresis 146 /Ograve 147 /Otilde 148 /Scaron 149 /Uacute 150 /Ucircumflex 151 /Udieresis 152 /Ugrave 153 /Ydieresis 154 /Zcaron 155 /aacute 156 /acircumflex 157 /adieresis 158 /agrave 159 /aring 160 /atilde 161 /exclamdown 162 /cent 163 /sterling 164 /fraction 165 /yen 166 /florin 167 /section 168 /currency 169 /quotesingle 170 /quotedblleft 171 /guillemotleft 172 /guilsinglleft 173 /guilsinglright 174 /fi 175 /fl 176 /plusminus 177 /endash 178 /dagger 179 /daggerdbl 180 /periodcentered 181 /twosuperior 182 /paragraph 183 /bullet 184 /quotesinglebase 185 /quotedblbase 186 /quotedblright 187 /guillemotright 188 /ellipsis 189 /perthousand 190 /threesuperior 191 /questiondown 192 /mu 193 /grave 194 /acute 195 /circumflex 196 /tilde 197 /macron 198 /breve 199 /dotaccent 200 /dieresis 201 /onesuperior 202 /ring 203 /cedilla 204 /onequarter 205 /hungarumlaut 206 /ogonek 207 /caron 208 /emdash 209 /ccedilla 210 /copyright 211 /eacute 212 /ecircumflex 213 /edieresis 214 /egrave 215 /iacute 216 /icircumflex 217 /idieresis 218 /igrave 219 /logicalnot 220 /minus 221 /ntilde 222 /oacute 223 /ocircumflex 224 /odieresis 225 /AE 226 /onehalf 227 /ordfeminine 228 /ograve 229 /otilde 230 /registered 231 /scaron 232 /Lslash 233 /Oslash 234 /OE 235 /ordmasculine 236 /trademark 237 /uacute 238 /ucircumflex 239 /udieresis 240 /ugrave 241 /ae 242 /ydieresis 243 /zcaron 244 /Aacute 245 /dotlessi 246 /threequarters 247 /Eth 248 /lslash 249 /oslash 250 /oe 251 /germandbls 252 /multiply 253 /Yacute 254 /Thorn 255 /eth ] def /extended_Symbol [ ] def /extend_font { % stack: fontname newfontname E dup (ZapfDingbats) eq { cvn E cvn extended_Zapf ReencodeSmall } { dup (Symbol) eq { cvn E cvn extended_Symbol ReencodeSmall } { cvn E cvn extended_Standard ReencodeSmall } ifelse } ifelse } bind def /getfont { /f E def f cvn where { begin f cvn load exec SF end } { f 0 f length 8 sub getinterval dup dup length 1 add string /localfont exch def localfont exch 0 exch putinterval localfont dup length 1 sub (X) putinterval localfont extend_font localfont FF /xsz f f length 4 sub 4 getinterval cvi def /ysz f f length 8 sub 4 getinterval cvi def [ xsz 0 0 ysz neg 0 0 ] MF dup f cvn E def SF } ifelse } bind def /ul { % space drop thickness GS currentpoint currentlinewidth currentpoint NP m 6 -3 roll SLW 0 E r 0 rl ST SLW m GR } bind def /ss { currentpoint pop E m } bind def /image_raster { % sw sh dw dh xs ys TR SC /sh E def /sw E def /imagebuf sw 7 add 8 idiv string def sw sh 1 [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop } image } bind def /imagemask_raster { TR SC /sh E def /sw E def /imagebuf sw 7 add 8 idiv string def sw sh false [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop } imagemask } bind def /image_color_raster { % sw sh sd dw dh xs ys systemdict /colorimage known not { /colorimage /colimg load def } if TR SC /sd E def /sh E def /sw E def /imagebuf sw 3 mul sd mul 7 add 8 idiv string def sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop } false 3 colorimage } bind def /nx { /x E def } bind def 0. nx .75 dup scale gsave 2.83465 -2.83465 scale 0 -279.4 translate topmat currentmatrix pop n 109.31 93.722 m 201.89 93.722 l 201.89 156.97 l 109.31 156.97 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n 7.028 93.806 m 99.611 93.806 l 99.611 157.06 l 7.028 157.06 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n 104.81 39.417 m 104.81 13.444 l 1 Fbw n 104.81 13.444 m 105.99 17 l 103.62 17 l cl 0 Fbw n 104.81 39.417 m 104.81 17 l gsave 0 0.352 0 Bbw grestore n 62.333 102.06 m 102.36 48.278 l 1 Fbw n 102.36 48.278 m 101.19 51.838 l 99.287 50.423 l cl 0 Fbw n 62.333 102.06 m 100.24 51.131 l gsave 0 0.352 0 Bbw grestore n 146.97 101.75 m 107.86 48.278 l 1 Fbw n 107.86 48.278 m 110.92 50.448 l 109 51.848 l cl 0 Fbw n 146.97 101.75 m 109.96 51.148 l gsave 0 0.352 0 Bbw grestore 5.466 5.466 90 105.03 43.306 el 1 Fbw gsave 0 0.352 0 Bbw grestore n 30.861 115.81 m 52.861 115.81 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 52.778 112.67 m 52.778 118.78 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 28.722 122.53 m 70.583 122.53 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 70.583 119.47 m 70.583 125.58 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 10.444 112.19 m 38.333 112.19 l 38.333 126.03 l 10.444 126.03 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n savemat currentmatrix pop [0.955 0 0 1 16.983 118.167] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Gating) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 15.194 123.972] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Network) s savemat setmatrix n 84.097 141.08 m 66.306 106.64 l 1 Fbw n 66.306 106.64 m 68.991 109.25 l 66.885 110.34 l cl 0 Fbw n 84.097 141.08 m 67.938 109.8 l gsave 0 0.352 0 Bbw grestore n 68.361 140.17 m 96.25 140.17 l 96.25 154 l 68.361 154 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n savemat currentmatrix pop [1 0 0 1 75.083 145.972] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Expert) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 73.556 151.778] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Network) s savemat setmatrix 5.466 5.466 0 63.861 101.75 el 1 Fbw gsave 0 0.352 0 Bbw grestore n 44 141.78 m 61.417 106.64 l 1 Fbw n 61.417 106.64 m 60.9 110.35 l 58.776 109.3 l cl 0 Fbw n 44 141.78 m 59.838 109.83 l gsave 0 0.352 0 Bbw grestore n 32.528 140.08 m 60.417 140.08 l 60.417 153.92 l 32.528 153.92 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n savemat currentmatrix pop [1 0 0 1 39.25 145.889] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Expert) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 37.722 151.694] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Network) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 83.417 134.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Symbol03600360) getfont () s 0.00 SG (m) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 86.778 135.417] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (12) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 54.389 135.667] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (11) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 51.083 134.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Symbol03600360) getfont () s 0.00 SG (m) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 123.139 135.139] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (21) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 119.139 134.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Symbol03600360) getfont () s 0.00 SG (m) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 155.222 135.222] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (22) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 151.444 134.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Symbol03600360) getfont () s 0.00 SG (m) s savemat setmatrix n 164.99 141 m 147.19 106.56 l 1 Fbw n 147.19 106.56 m 149.88 109.17 l 147.77 110.26 l cl 0 Fbw n 164.99 141 m 148.83 109.72 l gsave 0 0.352 0 Bbw grestore n 149.25 140.08 m 177.14 140.08 l 177.14 153.92 l 149.25 153.92 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n savemat currentmatrix pop [1 0 0 1 155.972 145.889] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Expert) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 154.444 151.694] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Network) s savemat setmatrix 5.466 5.466 0 144.75 101.67 el 1 Fbw gsave 0 0.352 0 Bbw grestore n 124.89 141.69 m 142.31 106.56 l 1 Fbw n 142.31 106.56 m 141.79 110.27 l 139.66 109.22 l cl 0 Fbw n 124.89 141.69 m 140.73 109.74 l gsave 0 0.352 0 Bbw grestore n 113.42 140 m 141.31 140 l 141.31 153.83 l 113.42 153.83 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n savemat currentmatrix pop [1 0 0 1 120.139 145.806] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Expert) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 118.611 151.611] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Network) s savemat setmatrix n 178.06 115.81 m 156.06 115.81 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 156.14 112.67 m 156.14 118.78 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 180.19 122.53 m 138.33 122.53 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 138.33 119.47 m 138.33 125.58 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 170.47 112.19 m 198.36 112.19 l 198.36 126.03 l 170.47 126.03 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n savemat currentmatrix pop [0.955 0 0 1 177.011 118.167] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Gating) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 175.222 123.972] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Network) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 108.222 30.917] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Symbol03600360) getfont () s 0.00 SG (m) s savemat setmatrix n 69.583 55.222 m 91.583 55.222 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 91.5 52.083 m 91.5 58.194 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 62.639 61.944 m 112.97 61.944 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 113.28 58.889 m 113.28 65 l 1 Fbw gsave 0 0.352 0 Bbw grestore n 45.5 51.611 m 73.389 51.611 l 73.389 65.444 l 45.5 65.444 l cl 1 Fbw gsave 0 0.352 0 Bbw grestore n savemat currentmatrix pop [0.955 0 0 1 52.039 57.583] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Gating) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 50.25 63.389] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Helvetica02800280) getfont () s 0.00 SG (Network) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 78.139 88.528] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Symbol03600360) getfont () s 0.00 SG (m) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 81.278 89.528] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (1) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 127.25 88.528] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Symbol03600360) getfont () s 0.00 SG (m) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 131.083 89.611] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (2) s savemat setmatrix n 57.75 83.111 m 57.75 65.694 l 1 Fbw n 57.75 65.694 m 58.935 69.25 l 56.565 69.25 l cl 0 Fbw n 57.75 83.111 m 57.75 69.25 l gsave 0 0.352 0 Bbw grestore n 21.917 143.53 m 21.917 126.11 l 1 Fbw n 21.917 126.11 m 23.102 129.67 l 20.732 129.67 l cl 0 Fbw n 21.917 143.53 m 21.917 129.67 l gsave 0 0.352 0 Bbw grestore n 81.778 171.61 m 81.778 154.19 l 1 Fbw n 81.778 154.19 m 82.963 157.75 l 80.593 157.75 l cl 0 Fbw n 81.778 171.61 m 81.778 157.75 l gsave 0 0.352 0 Bbw grestore n 126.31 171.61 m 126.31 154.19 l 1 Fbw n 126.31 154.19 m 127.49 157.75 l 125.12 157.75 l cl 0 Fbw n 126.31 171.61 m 126.31 157.75 l gsave 0 0.352 0 Bbw grestore n 162.28 171.61 m 162.28 154.19 l 1 Fbw n 162.28 154.19 m 163.46 157.75 l 161.09 157.75 l cl 0 Fbw n 162.28 171.61 m 162.28 157.75 l gsave 0 0.352 0 Bbw grestore n 185.42 143.56 m 185.42 126.14 l 1 Fbw n 185.42 126.14 m 186.6 129.69 l 184.23 129.69 l cl 0 Fbw n 185.42 143.56 m 185.42 129.69 l gsave 0 0.352 0 Bbw grestore n 45.639 171.61 m 45.639 154.19 l 1 Fbw n 45.639 154.19 m 46.824 157.75 l 44.454 157.75 l cl 0 Fbw n 45.639 171.61 m 45.639 157.75 l gsave 0 0.352 0 Bbw grestore n savemat currentmatrix pop [1 0 0 1 84.333 167.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-BoldItalic03600360) getfont () s 0.00 SG (x) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 47.889 167.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-BoldItalic03600360) getfont () s 0.00 SG (x) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 16.639 138.556] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-BoldItalic03600360) getfont () s 0.00 SG (x) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 120.75 167.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-BoldItalic03600360) getfont () s 0.00 SG (x) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 156.722 167.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-BoldItalic03600360) getfont () s 0.00 SG (x) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 187.806 138.556] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-BoldItalic03600360) getfont () s 0.00 SG (x) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 52.389 77.75] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-BoldItalic03600360) getfont () s 0.00 SG (x) s savemat setmatrix n savemat currentmatrix pop [1 0 -0 1 78.833 51.639] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic03600360) getfont () s 0.00 SG (g) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 81.583 53.167] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (1) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 39.944 127.028] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic03600360) getfont () s 0.00 SG (g) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 46.5 128.389] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (1) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 43.083 128.333] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (2) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 45.444 128.861] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Roman02800280) getfont () s 0.00 SG (|) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 43 114.194] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (1) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 46.889 114.111] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (1) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 45.361 114.417] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Roman02800280) getfont () s 0.00 SG (|) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 39.861 112.583] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic03600360) getfont () s 0.00 SG (g) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 163.778 128.639] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (1) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 165.917 128.945] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Roman02800280) getfont () s 0.00 SG (|) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 167.667 128.861] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (2) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 160.472 127.167] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic03600360) getfont () s 0.00 SG (g) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 163.611 114.111] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (2) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 167.5 114.028] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (2) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 165.889 114.556] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Roman02800280) getfont () s 0.00 SG (|) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 160.556 112.583] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic03600360) getfont () s 0.00 SG (g) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 81.556 68.417] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic02400240) getfont () s 0.00 SG (2) s savemat setmatrix n savemat currentmatrix pop [1 0 0 1 78.278 66.667] concat 25.4 1440 div 1.000000 mul dup scale 0 0 m 0 0 m 0 ss (Times-Italic03600360) getfont () s 0.00 SG (g) s savemat setmatrix userdict /#copies 1 put showpage grestore end restore %%EndDocument endTexFig 150 1690 a Fu(Figure)17 b(1:)22 b(A)16 b(t)o(w)o(o-lev)o(el)h(hierarc) o(hical)h(mixture)f(of)f(exp)q(erts.)24 b(T)l(o)16 b(form)g(a)g(deep)q (er)i(tree,)e(eac)o(h)150 1746 y(exp)q(ert)g(is)f(expanded)i(recursiv)o 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b(will)h(refer)f(to)f(suc)o(h)h (a)g(net)o(w)o(ork)e(as)i(\\generalized)h(linear,")f(b)q(orro)o(wing)g (the)g(terminology)150 304 y(from)21 b(statistics)h(\(McCullagh)g(&)g (Nelder,)j(1983\).)38 b(Exp)q(ert)22 b(net)o(w)o(ork)f(\()p Fj(i;)8 b(j)s Fu(\))19 b(pro)q(duces)k(its)150 361 y(output)15 b Fi(\026)331 368 y Fm(ij)377 361 y Fu(as)f(a)h(generalized)i(linear)g (function)f(of)e(the)i(input)g Fl(x)p Fu(:)828 461 y Fi(\026)860 468 y Fm(ij)903 461 y Fu(=)d Fj(f)5 b Fu(\()p Fj(U)1027 468 y Fm(ij)1057 461 y Fl(x)p Fu(\))p Fj(;)626 b Fu(\(1\))150 561 y(where)15 b Fj(U)312 568 y Fm(ij)357 561 y Fu(is)g(a)f(w)o(eigh)o(t)g(matrix)h(and)f Fj(f)20 b Fu(is)15 b(a)f(\014xed)i(con)o(tin)o(uous)e(nonlinearit)o(y)l(.)22 b(The)15 b(v)o(ector)e Fl(x)150 617 y Fu(is)j(assumed)f(to)g(include)i (a)e(\014xed)h(comp)q(onen)o(t)f(of)g(one)g(to)g(allo)o(w)g(for)g(an)g (in)o(tercept)h(term.)241 674 y(F)l(or)11 b(regression)g(problems,)i Fj(f)5 b Fu(\()p Fn(\001)p Fu(\))10 b(is)i(generally)g(c)o(hosen)g(to)f 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b(De\014ne)17 b(in)o(termediate)g(v)m(ariables)150 1125 y Fj(\030)170 1132 y Fm(i)199 1125 y Fu(as)e(follo)o(ws:)876 1182 y Fj(\030)896 1189 y Fm(i)922 1182 y Fu(=)e Fl(v)999 1163 y Fm(T)998 1193 y(i)1026 1182 y Fl(x)p Fj(;)675 b Fu(\(2\))150 1264 y(where)16 b Fl(v)310 1271 y Fm(i)340 1264 y Fu(is)g(a)f(w)o(eigh)o(t)h(v)o(ector.)21 b(Then)c(the)f Fj(i)940 1248 y Fm(th)990 1264 y Fu(output)g(of)f(the)h(top-lev)o(el)h (gating)f(net)o(w)o(ork)e(is)150 1321 y(the)h(\\softmax")f(function)i (of)f(the)g Fj(\030)774 1328 y Fm(i)803 1321 y Fu(\(Bridle,)h(1989;)e (McCullagh)i(&)f(Nelder,)h(1983\):)850 1450 y Fj(g)872 1457 y Fm(i)899 1450 y Fu(=)991 1419 y Fj(e)1012 1403 y Fm(\030)1028 1408 y Ff(i)p 952 1439 131 2 v 952 1449 a Fe(P)995 1493 y Fm(k)1024 1481 y Fj(e)1045 1468 y Fm(\030)1061 1474 y Ff(k)1087 1450 y Fj(:)642 b Fu(\(3\))150 1572 y(Note)17 b(that)f(the)h Fj(g)464 1579 y Fm(i)495 1572 y Fu(are)f(p)q(ositiv)o(e)i(and)f(sum)g(to)f(one)h(for)g(eac)o(h)g Fl(x)p Fu(.)24 b(They)18 b(can)f(b)q(e)g(in)o(terpreted)150 1628 y(as)e(pro)o(viding)h(a)f(\\soft")f(partitioning)i(of)f(the)g (input)h(space.)241 1685 y(Similarly)l(,)e(the)d(gating)g(net)o(w)o (orks)g(at)f(lo)o(w)o(er)h(lev)o(els)i(are)e(also)g(generalized)i (linear)g(systems.)150 1741 y(De\014ne)j Fj(\030)311 1748 y Fm(ij)356 1741 y Fu(as)f(follo)o(ws:)871 1798 y Fj(\030)891 1805 y Fm(ij)933 1798 y Fu(=)e Fl(v)1010 1779 y Fm(T)1009 1809 y(ij)1039 1798 y Fl(x)p Fj(:)662 b Fu(\(4\))150 1880 y(Then)838 1954 y Fj(g)860 1964 y Fm(j)r Fd(j)p Fm(i)912 1954 y Fu(=)1003 1924 y Fj(e)1024 1907 y Fm(\030)1040 1912 y Ff(ij)p 965 1944 142 2 v 965 1953 a Fe(P)1009 1997 y Fm(k)1038 1986 y Fj(e)1059 1972 y Fm(\030)1075 1978 y Ff(ik)1742 1954 y Fu(\(5\))150 2067 y(is)20 b(the)f(output)h(of)e(the)i Fj(j)596 2050 y Fm(th)650 2067 y Fu(unit)g(in)g(the)g Fj(i)906 2050 y Fm(th)960 2067 y Fu(gating)f(net)o(w)o(ork)f(at)h(the)g(second)h(lev) o(el)h(of)e(the)150 2123 y(arc)o(hitecture.)h(Once)15 b(again,)g(the)f Fj(g)767 2132 y Fm(j)r Fd(j)p Fm(i)821 2123 y Fu(are)h(p)q(ositiv)o(e)g(and)g(sum)f(to)g(one)h(for)e(eac)o(h)i Fl(x)p Fu(.)k(They)c(can)150 2180 y(b)q(e)k(in)o(terpreted)g(as)e(pro)o (viding)i(a)f(nested)h(\\soft")d(partitioning)j(of)f(the)g(input)h (space)g(within)150 2236 y(the)c(partitioning)h(pro)o(viding)h(b)o(y)e (the)g(higher-lev)o(el)j(gating)c(net)o(w)o(ork.)241 2292 y(The)f(output)g(v)o(ector)f(at)h(eac)o(h)g(non)o(terminal)h(of)e (the)i(tree)e(is)i(the)f(w)o(eigh)o(ted)g(output)g(of)g(the)150 2349 y(exp)q(erts)k(b)q(elo)o(w)g(that)f(non)o(terminal.)25 b(That)16 b(is,)i(the)e(output)h(at)f(the)h Fj(i)1376 2332 y Fm(th)1427 2349 y Fu(non)o(terminal)g(in)h(the)150 2405 y(second)e(la)o(y)o(er)f(of)f(the)i(t)o(w)o(o-lev)o(el)f(tree)g (is:)826 2505 y Fi(\026)858 2512 y Fm(i)885 2505 y Fu(=)933 2465 y Fe(X)954 2556 y Fm(j)1000 2505 y Fj(g)1022 2515 y Fm(j)r Fd(j)p Fm(i)1062 2505 y Fi(\026)1094 2512 y Fm(ij)964 2775 y Fu(5)p eop %%Page: 6 6 6 5 bop 150 192 a Fu(and)15 b(the)h(output)f(at)f(the)h(top)g(lev)o(el) i(of)e(the)g(tree)g(is:)848 294 y Fi(\026)d Fu(=)940 253 y Fe(X)964 344 y Fm(i)1008 294 y Fj(g)1030 301 y Fm(i)1044 294 y Fi(\026)1076 301 y Fm(i)1090 294 y Fj(:)150 430 y Fu(Note)j(that)f(b)q(oth)i(the)f Fj(g)r Fu('s)f(and)h(the)h Fi(\026)p Fu('s)f(dep)q(end)h(on)g(the)f(input)h Fl(x)p Fu(,)f(th)o(us)g(the)g(total)f(output)h(is)150 486 y(a)g(nonlinear)i (function)f(of)e(the)i(input.)150 608 y Fp(Regression)h(surface)150 694 y Fu(Giv)o(en)c(the)h(de\014nitions)g(of)f(the)g(exp)q(ert)h(net)o (w)o(orks)e(and)h(the)g(gating)g(net)o(w)o(orks,)f(the)h(regression)150 750 y(surface)e(de\014ned)h(b)o(y)f(the)g(hierarc)o(h)o(y)g(is)g(a)f (piecewise)j(blend)g(of)d(the)h(regression)g(surfaces)g(de\014ned)150 807 y(b)o(y)19 b(the)g(exp)q(erts.)32 b(The)20 b(gating)f(net)o(w)o (orks)f(pro)o(vide)h(a)g(nested,)h(\\soft")e(partitioning)i(of)f(the) 150 863 y(input)i(space)f(and)h(the)f(exp)q(ert)g(net)o(w)o(orks)f(pro) 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Fc(v)1123 1299 y Fh(2)1142 1294 y Fk(\))1156 1285 y Ff(T)1180 1294 y Fc(x)1210 1273 y Fj(;)519 b Fu(\(7\))150 1392 y(whic)o(h)18 b(is)g(a)f(logistic)i(ridge)f(function)g(whose)f (orien)o(tation)g(is)h(determined)h(b)o(y)e(the)h(direction)150 1449 y(of)e(the)h(v)o(ector)f Fl(v)449 1456 y Fk(1)479 1449 y Fn(\000)c Fl(v)554 1456 y Fk(2)573 1449 y Fu(.)24 b(The)17 b(gating)f(output)g Fj(g)1017 1456 y Fk(2)1053 1449 y Fu(is)h(equal)h(to)e(1)11 b Fn(\000)g Fj(g)1381 1456 y Fk(1)1400 1449 y Fu(.)24 b(F)l(or)16 b(a)h(giv)o(en)g Fl(x)p Fu(,)f(the)150 1505 y(total)c(output)h Fi(\026)g Fu(is)g(the)g(con)o(v)o(ex)g(com)o(bination)g Fj(g)992 1512 y Fk(1)1012 1505 y Fi(\026)1044 1516 y Fk(1)1069 1505 y Fu(+)5 b Fj(g)1131 1512 y Fk(2)1151 1505 y Fi(\026)1183 1516 y Fk(2)1203 1505 y Fu(.)19 b(This)14 b(is)f(a)g(w)o(eigh)o(ted)g (a)o(v)o(erage)e(of)150 1562 y(the)17 b(exp)q(erts,)g(where)f(the)h(w)o (eigh)o(ts)f(are)h(determined)h(b)o(y)e(the)h(v)m(alues)g(of)g(the)f (ridge)h(function.)150 1618 y(Along)f(the)g(ridge,)g Fj(g)511 1625 y Fk(1)544 1618 y Fu(=)e Fj(g)615 1625 y Fk(2)648 1618 y Fu(=)702 1600 y Fk(1)p 702 1607 18 2 v 702 1634 a(2)725 1618 y Fu(,)h(and)h(b)q(oth)g(exp)q(erts)g(con)o (tribute)h(equally)l(.)23 b(Aw)o(a)o(y)15 b(from)g(the)150 1675 y(ridge,)e(one)g(exp)q(ert)g(or)f(the)g(other)h(dominates.)19 b(The)13 b(amoun)o(t)e(of)h(smo)q(othing)h(across)f(the)g(ridge)150 1731 y(is)k(determined)h(b)o(y)e(the)g(magnitude)h(of)f(the)h(v)o (ector)f Fl(v)1096 1738 y Fk(2)1125 1731 y Fn(\000)c Fl(v)1199 1738 y Fk(1)1218 1731 y Fu(.)21 b(If)15 b Fl(v)1325 1738 y Fk(2)1355 1731 y Fn(\000)10 b Fl(v)1428 1738 y Fk(1)1463 1731 y Fu(is)16 b(large,)f(then)h(the)150 1788 y(ridge)c(function)h(b)q(ecomes)f(a)f(sharp)h(split)h(and)e(the)h(w)o (eigh)o(ted)g(output)f(of)h(the)f(exp)q(erts)h(b)q(ecomes)150 1844 y(piecewise)k(\(generalized\))g(linear.)21 b(If)14 b Fl(v)832 1851 y Fk(2)860 1844 y Fn(\000)9 b Fl(v)932 1851 y Fk(1)966 1844 y Fu(is)15 b(small,)g(then)g(eac)o(h)f(exp)q(ert)h (con)o(tributes)f(to)g(a)150 1901 y(signi\014can)o(t)k(degree)f(on)g (eac)o(h)g(side)g(of)g(the)g(ridge,)g(thereb)o(y)g(smo)q(othing)g(the)g (piecewise)i(map.)150 1957 y(In)e(the)f(limit)i(of)e(a)g(zero)g (di\013erence)h(v)o(ector,)f Fj(g)963 1964 y Fk(1)997 1957 y Fu(=)f Fj(g)1069 1964 y Fk(2)1102 1957 y Fu(=)1157 1939 y Fk(1)p 1157 1946 V 1157 1973 a(2)1196 1957 y Fu(for)h(all)h Fl(x)p Fu(,)f(and)g(the)g(total)g(output)150 2013 y(is)g(the)f(same)g (\014xed)h(a)o(v)o(erage)e(of)h(the)g(exp)q(erts)g(on)g(b)q(oth)h (sides)g(of)f(the)g(\014ctitious)h(\\split.")241 2070 y(In)i(general,)h(a)f(giv)o(en)g(gating)g(net)o(w)o(ork)e(induces)k(a)e (smo)q(othed)f(planar)i(partitioning)f(of)150 2126 y(the)13 b(input)h(space.)20 b(Lo)o(w)o(er-lev)o(el)14 b(gating)e(net)o(w)o (orks)g(induce)j(a)e(partition)g(within)h(the)g(partition)150 2183 y(induced)22 b(b)o(y)e(higher-lev)o(el)j(gating)d(net)o(w)o(orks.) 34 b(The)20 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y(the)e(parameters.)25 b(The)17 b(decisions)i(form)d(a)h(decision)i(tree.)25 b(W)l(e)17 b(use)g(a)g(statistical)h(mo)q(del)g(to)150 729 y(mo)q(del)i(this)g(decision)g(tree;)h(in)f(particular,)g(our)f(c)o (hoice)h(of)f(parameterization)g(\(cf.)31 b(Equa-)150 785 y(tions)19 b(2,)f(4,)h(3)f(and)h(5\))f(corresp)q(onds)g(to)g(a)g Ft(multinomial)i(lo)n(git)i Fu(probabilit)o(y)e(mo)q(del)f(at)f(eac)o (h)150 842 y(non)o(terminal)h(of)f(the)g(tree)g(\(see)g(App)q(endix)i (2\).)28 b(A)18 b(m)o(ultinomial)i(logit)f(mo)q(del)g(is)f(a)g(sp)q (ecial)150 898 y(case)12 b(of)g(a)f(GLIM)h(that)g(is)g(commonly)h(used) f(for)g(\\soft")e(m)o(ultiw)o(a)o(y)i(classi\014cation)h(\(McCullagh) 150 955 y(&)k(Nelder,)i(1983\).)24 b(Under)18 b(the)f(m)o(ultinomial)i (logit)e(mo)q(del,)h(w)o(e)f(in)o(terpret)h(the)f(gating)g(net-)150 1011 y(w)o(orks)f(as)h(mo)q(deling)i(the)f(input-dep)q(enden)o(t,)i(m)o (ultinomial)f(probabilities)h(asso)q(ciated)e(with)150 1068 y(decisions)f(at)d(particular)i(lev)o(els)h(of)e(resolution)h(in)g (a)e(tree-structured)i(mo)q(del)g(of)f(the)g(data.)241 1124 y(Once)20 b(a)e(particular)i(sequence)g(of)e(decisions)j(has)e(b)q (een)h(made,)f(resulting)h(in)g(a)e(c)o(hoice)150 1181 y(of)g(regressiv)o(e)h(pro)q(cess)g(\()p Fj(i;)8 b(j)s Fu(\),)16 b(output)j Fl(y)g Fu(is)g(assumed)g(to)f(b)q(e)h(generated)g (according)g(to)f(the)150 1237 y(follo)o(wing)e(statistical)f(mo)q (del.)21 b(First,)15 b(a)g(linear)h(predictor)g Fi(\021)1212 1244 y Fm(ij)1257 1237 y Fu(is)g(formed:)864 1332 y Fi(\021)893 1313 y Fk(0)893 1343 y Fm(ij)936 1332 y Fu(=)d Fj(U)1020 1313 y Fk(0)1015 1343 y Fm(ij)1045 1332 y Fl(x)p Fj(:)150 1427 y Fu(The)h(exp)q(ected)h(v)m(alue)f(of)g Fl(y)g Fu(is)g(obtained)g(b)o(y)f(passing)h(the)g(linear)h(predictor)f (through)f(the)h Ft(link)150 1484 y(function)k Fj(f)5 b Fu(:)371 1467 y Fk(2)846 1540 y Fi(\026)878 1521 y Fk(0)878 1551 y Fm(ij)921 1540 y Fu(=)13 b Fj(f)5 b Fu(\()p Fi(\021)1043 1521 y Fk(0)1043 1551 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2081 y Fm(ij)1076 1969 y Fe(#)1108 2041 y Fj(:)150 2154 y Fu(W)l(e)g(assume)g(the)h(densit)o(y)g Fj(P)18 b Fu(to)12 b(b)q(e)h(a)f(mem)o(b)q(er)g(of)g(the)g(exp)q(onen)o (tial)i(family)f(of)e(densities)j(\(Mc-)150 2210 y(Cullagh)k(&)g (Nelder,)h(1983\).)24 b(The)18 b(in)o(terpretation)f(of)g(the)h(disp)q (ersion)h(parameter)d(dep)q(ends)150 2267 y(on)h(the)h(particular)g(c)o (hoice)g(of)f(densit)o(y)l(.)28 b(F)l(or)16 b(example,)j(in)f(the)g (case)f(of)g(the)g Fj(n)p Fu(-dimensional)150 2323 y(Gaussian,)e(the)g (disp)q(ersion)i(parameter)e(is)g(the)h(co)o(v)m(ariance)f(matrix)g (\006)1392 2307 y Fk(0)1392 2335 y Fm(ij)1422 2323 y Fu(.)1435 2307 y Fk(3)p 150 2370 660 2 v 202 2397 a Fh(2)219 2413 y Fg(W)m(e)f(utilize)h(the)f(neural)h(net)o(w)o(ork)e(con)o(v)o (en)o(tion)i(in)g(de\014ning)g(links.)20 b(In)13 b(GLIM)h(theory)m(,)f (the)h(con)o(v)o(en)o(tion)h(is)150 2459 y(that)e(the)g(link)i (function)f(relates)g Fb(\021)h Fg(to)e Fb(\026)p Fg(;)f(th)o(us,)h Fb(\021)g Fg(=)d Fb(h)p Fg(\()p 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1560 y Fj(:)150 1755 y Ft(Example)k(\(binary)g(classi\014c)n(ation\)) 241 1837 y Fu(In)21 b(binary)g(classi\014cation)h(problems)g(the)e (output)h Fj(y)h Fu(is)g(a)e(discrete)h(random)g(v)m(ariable)150 1893 y(ha)o(ving)c(p)q(ossible)i(outcomes)d(of)g(\\failure")i(and)f (\\success.")25 b(The)17 b(probabilistic)i(comp)q(onen)o(t)150 1950 y(of)d(the)g(mo)q(del)h(is)g(generally)h(assumed)e(to)g(b)q(e)h (the)f(Bernoulli)j(distribution)f(\(Co)o(x,)d(1970\).)21 b(In)150 2006 y(this)h(case,)g(the)f(mean)g Fj(\026)599 2013 y Fm(ij)651 2006 y Fu(is)h(the)f(conditional)i(probabilit)o(y)f (of)f(classifying)i(the)e(input)h(as)150 2063 y(\\success.")33 b(The)20 b(resulting)h(hierarc)o(hical)g(probabilit)o(y)g(mo)q(del)f (is)g(a)g(mixture)g(of)f(Bernoulli)150 2119 y(densities:)580 2175 y Fj(P)6 b Fu(\()p Fj(y)r Fn(j)p Fl(x)p Fj(;)i Fi(\022)o Fu(\))13 b(=)823 2135 y Fe(X)847 2226 y Fm(i)890 2175 y Fj(g)912 2182 y Fm(i)933 2135 y Fe(X)955 2226 y Fm(j)1001 2175 y Fj(g)1023 2185 y Fm(j)r Fd(j)p Fm(i)1063 2175 y Fj(\026)1090 2153 y Fm(y)1090 2188 y(ij)1120 2175 y Fu(\(1)d Fn(\000)g Fj(\026)1243 2182 y Fm(ij)1274 2175 y Fu(\))1292 2157 y Fk(1)p Fd(\000)p Fm(y)1358 2175 y Fj(:)150 2339 y Fp(P)n(osterior)18 b(probabilities)150 2425 y Fu(In)13 b(dev)o(eloping)i(the)e(learning)g(algorithms)g(to)f(b) q(e)i(presen)o(ted)f(in)g(the)g(remainder)h(of)e(the)h(pap)q(er,)150 2481 y(it)h(will)i(pro)o(v)o(e)e(useful)h(to)f(de\014ne)h(p)q(osterior) f(probabilities)i(asso)q(ciated)f(with)f(the)g(no)q(des)h(of)f(the)150 2538 y(tree.)29 b(The)18 b(terms)g(\\p)q(osterior")g(and)g(\\prior")g (ha)o(v)o(e)g(meaning)g(in)h(this)g(con)o(text)f(during)h(the)964 2775 y(8)p eop %%Page: 9 9 9 8 bop 150 192 a Fu(training)12 b(of)g(the)f(system.)19 b(W)l(e)11 b(refer)h(to)f(the)h(probabilities)i Fj(g)1191 199 y Fm(i)1216 192 y Fu(and)e Fj(g)1323 201 y Fm(j)r Fd(j)p Fm(i)1374 192 y Fu(as)g Ft(prior)17 b Fu(probabilities,)150 248 y(b)q(ecause)j(they)e(are)h(computed)g(based)g(only)g(on)g(the)f (input)i Fl(x)p Fu(,)f(without)f(kno)o(wledge)i(of)e(the)150 304 y(corresp)q(onding)j(target)e(output)g Fl(y)q Fu(.)34 b(A)20 b Ft(p)n(osterior)25 b Fu(probabilit)o(y)c(is)f(de\014ned)i (once)e(b)q(oth)g(the)150 361 y(input)d(and)g(the)f(target)f(output)h (are)g(kno)o(wn.)22 b(Using)17 b(Ba)o(y)o(es')f(rule,)g(w)o(e)g (de\014ne)i(the)e(p)q(osterior)150 417 y(probabilities)h(at)e(the)g(no) q(des)h(of)f(the)g(tree)g(as)g(follo)o(ws:)738 545 y Fj(h)764 552 y Fm(i)791 545 y Fu(=)877 510 y Fj(g)899 517 y Fm(i)920 478 y Fe(P)964 522 y Fm(j)990 510 y Fj(g)1012 519 y Fm(j)r Fd(j)p Fm(i)1051 510 y Fj(P)1080 517 y Fm(ij)1111 510 y Fu(\()p Fl(y)q Fu(\))p 844 535 364 2 v 844 544 a Fe(P)888 588 y Fm(i)909 576 y Fj(g)931 583 y Fm(i)953 544 y Fe(P)996 588 y Fm(j)1022 576 y Fj(g)1044 586 y Fm(j)r Fd(j)p Fm(i)1084 576 y Fj(P)1113 583 y Fm(ij)1143 576 y Fu(\()p Fl(y)q Fu(\))1719 545 y(\(10\))150 676 y(and)773 749 y Fj(h)799 758 y Fm(j)r Fd(j)p Fm(i)852 749 y Fu(=)940 714 y Fj(g)962 723 y Fm(j)r Fd(j)p Fm(i)1001 714 y 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y(hierarc)o(hical)g(arc)o(hitecture.)22 b(The)16 b(algorithm)g(w)o(as)f(based)h(on)f(earlier)i(w)o(ork)e(b)o(y) g(Jacobs,)h(Jor-)150 1724 y(dan,)h(No)o(wlan,)g(and)g(Hin)o(ton)g (\(1991\),)e(who)i(treated)f(the)h(problem)g(of)g(learning)h(in)f (mixture-)150 1781 y(of-exp)q(erts)d(arc)o(hitectures)g(as)g(a)g(maxim) o(um)g(lik)o(eliho)q(o)q(d)j(estimation)d(problem.)20 b(The)15 b(log)f(lik)o(e-)150 1837 y(liho)q(o)q(d)i(of)d(a)h(data)g (set)f Fn(X)20 b Fu(=)13 b Fn(f)p Fu(\()p Fl(x)716 1821 y Fk(\()p Fm(t)p Fk(\))757 1837 y Fj(;)8 b Fl(y)807 1821 y Fk(\()p Fm(t)p Fk(\))848 1837 y Fu(\))p Fn(g)889 1821 y Fm(N)889 1849 y Fk(1)936 1837 y Fu(is)15 b(obtained)f(b)o(y)g(taking) h(the)f(log)g(of)f(the)i(pro)q(duct)150 1894 y(of)g Fj(N)k Fu(densities)e(of)e(the)g(form)g(of)g(Equation)g(9,)f(whic)o(h)i (yields)h(the)e(follo)o(wing)h(log)g(lik)o(eliho)q(o)q(d:)594 2001 y Fj(l)q Fu(\()p Fi(\022)p Fu(;)8 b Fn(X)f Fu(\))12 b(=)791 1961 y Fe(X)815 2051 y Fm(t)858 2001 y Fu(ln)904 1961 y Fe(X)928 2052 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Fu(ln)i Fj(P)1178 689 y Fm(ij)1208 682 y Fu(\()p Fl(y)1255 663 y Fk(\()p Fm(t)p Fk(\))1296 682 y Fu(\))p Fj(;)392 b Fu(\(27\))631 834 y Fl(v)660 810 y Fk(\()p Fm(p)p Fk(+1\))659 847 y Fm(i)765 834 y Fu(=)13 b(arg)7 b(max)908 859 y Fc(v)931 864 y Ff(i)976 793 y Fe(X)1000 883 y Fm(t)1044 793 y Fe(X)1064 886 y Fm(k)1111 834 y Fj(h)1137 810 y Fk(\()p Fm(t)p Fk(\))1137 848 y Fm(k)1187 834 y Fu(ln)i Fj(g)1257 810 y Fk(\()p Fm(t)p Fk(\))1255 848 y Fm(k)1298 834 y Fj(;)408 b Fu(\(28\))150 952 y(and)563 1009 y Fl(v)592 985 y Fk(\()p Fm(p)p Fk(+1\))591 1022 y Fm(ij)697 1009 y Fu(=)13 b(arg)7 b(max)833 1034 y Fc(v)856 1039 y Ff(ij)908 968 y Fe(X)932 1058 y Fm(t)976 968 y Fe(X)996 1061 y Fm(k)1043 1009 y Fj(h)1069 985 y Fk(\()p Fm(t)p Fk(\))1069 1023 y Fm(k)1119 968 y Fe(X)1144 1061 y Fm(l)1187 1009 y Fj(h)1213 985 y Fk(\()p Fm(t)p Fk(\))1213 1025 y Fm(l)p Fd(j)p Fm(k)1263 1009 y Fu(ln)h Fj(g)1332 985 y Fk(\()p Fm(t)p Fk(\))1330 1025 y Fm(l)p Fd(j)p Fm(k)1374 1009 y Fj(:)332 b Fu(\(29\))150 1128 y(Eac)o(h)12 b(of)g(these)h(maximization)g(problems)g(are)f(themselv)o(es)h(maxim)o (um)f(lik)o(eliho)q(o)q(d)k(problems.)150 1184 y(Equation)11 b(27)g(is)g(simply)i(the)e(general)g(form)g(of)f(a)h(w)o(eigh)o(ted)g (maxim)o(um)g(lik)o(eliho)q(o)q(d)j(problem)e(in)150 1240 y(the)i(probabilit)o(y)i(densit)o(y)f Fj(P)644 1247 y Fm(ij)675 1240 y Fu(.)k(Giv)o(en)c(our)f(parameterization)g(of)g Fj(P)1348 1247 y Fm(ij)1379 1240 y Fu(,)g(the)g(log)h(lik)o(eliho)q(o)q (d)i(in)150 1297 y(Equation)f(27)f(is)h(a)f(w)o(eigh)o(ted)g(log)h(lik) o(eliho)q(o)q(d)i(for)d(a)g(GLIM.)h(An)f(e\016cien)o(t)i(algorithm)e (kno)o(wn)150 1353 y(as)20 b(iterativ)o(ely)i(rew)o(eigh)o(ted)f (least-squares)f(\(IRLS\))i(is)f(a)o(v)m(ailable)h(to)e(solv)o(e)h(the) g(maxim)o(um)150 1410 y(lik)o(eliho)q(o)q(d)e(problem)d(for)f(suc)o(h)i (mo)q(dels)f(\(McCullagh)h(&)f(Nelder,)g(1983\).)21 b(W)l(e)16 b(discuss)g(IRLS)150 1466 y(in)g(App)q(endix)h(A.)241 1523 y(Equation)c(28)g(in)o(v)o(olv)o(es)g(maximizing)i(the)e(cross-en) o(trop)o(y)f(b)q(et)o(w)o(een)i(the)f(p)q(osterior)g(prob-)150 1585 y(abilities)k Fj(h)345 1561 y Fk(\()p Fm(t)p Fk(\))345 1599 y Fm(k)401 1585 y Fu(and)e(the)f(prior)h(probabilities)h Fj(g)960 1561 y Fk(\()p Fm(t)p Fk(\))958 1599 y Fm(k)1002 1585 y Fu(.)j(This)c(cross-en)o(trop)o(y)f(is)g(the)h(log)f(lik)o (eliho)q(o)q(d)150 1652 y(asso)q(ciated)k(with)f(a)g(m)o(ultinomial)i (logit)f(probabilit)o(y)g(mo)q(del)h(in)f(whic)o(h)g(the)f Fj(h)1541 1628 y Fk(\()p Fm(t)p Fk(\))1541 1666 y Fm(k)1601 1652 y Fu(act)g(as)g(the)150 1708 y(output)i(observ)m(ations)h(\(see)f (App)q(endix)j(B\).)d(Th)o(us)g(the)h(maximization)g(in)g(Equation)g (28)e(is)150 1765 y(also)c(a)f(maxim)o(um)h(lik)o(eliho)q(o)q(d)j (problem)d(for)g(a)f(GLIM)h(and)g(can)g(b)q(e)h(solv)o(ed)f(using)h (IRLS.)g(The)150 1821 y(same)f(is)g(true)g(of)g(Equation)g(29,)f(whic)o (h)i(is)g(a)f(w)o(eigh)o(ted)g(maxim)o(um)g(lik)o(eliho)q(o)q(d)j (problem)e(with)150 1883 y(output)g(observ)m(ations)g Fj(h)584 1859 y Fk(\()p Fm(t)p Fk(\))584 1899 y Fm(l)p Fd(j)p Fm(k)642 1883 y Fu(and)g(observ)m(ation)h(w)o(eigh)o(ts)f Fj(h)1160 1859 y Fk(\()p Fm(t)p Fk(\))1160 1897 y Fm(k)1202 1883 y Fu(.)241 1939 y(In)k(summary)l(,)g(the)g(EM)g(algorithm)g(that)f (w)o(e)g(ha)o(v)o(e)h(obtained)g(in)o(v)o(olv)o(es)h(a)e(calculation) 150 1996 y(of)g(p)q(osterior)g(probabilities)i(in)f(the)f(outer)g(lo)q (op)h(\(the)e(E)h(step\),)g(and)h(the)f(solution)h(of)e(a)h(set)150 2052 y(of)e(IRLS)i(problems)f(in)h(the)e(inner)i(lo)q(op)f(\(the)f(M)g (step\).)24 b(W)l(e)16 b(summarize)h(the)g(algorithm)f(as)150 2109 y(follo)o(ws:)952 2775 y(13)p eop %%Page: 14 14 14 13 bop 150 192 1650 5 v 241 273 a Ft(A)o(lgorithm)16 b(1)206 370 y Fu(1.)22 b(F)l(or)12 b(eac)o(h)h(data)g(pair)g(\()p Fl(x)682 354 y Fk(\()p Fm(t)p Fk(\))724 370 y Fj(;)8 b Fl(y)774 354 y Fk(\()p Fm(t)p Fk(\))814 370 y Fu(\),)13 b(compute)g(the)g(p)q(osterior)g(probabilities)j Fj(h)1590 346 y Fk(\()p Fm(t)p Fk(\))1590 383 y Fm(i)1645 370 y Fu(and)e Fj(h)1758 346 y Fk(\()p Fm(t)p Fk(\))1758 386 y Fm(j)r Fd(j)p Fm(i)264 427 y Fu(using)i(the)f(curren)o(t)g(v)m(alues) h(of)f(the)g(parameters.)206 520 y(2.)22 b(F)l(or)12 b(eac)o(h)i(exp)q(ert)g(\()p Fj(i;)8 b(j)s Fu(\),)j(solv)o(e)i(an)g (IRLS)i(problem)g(with)e(observ)m(ations)h Fn(f)p Fu(\()p Fl(x)1594 503 y Fk(\()p Fm(t)p Fk(\))1635 520 y Fj(;)8 b Fl(y)1685 503 y Fk(\()p Fm(t)p Fk(\))1726 520 y Fu(\))p Fn(g)1767 503 y Fm(N)1767 532 y Fk(1)264 585 y Fu(and)15 b(observ)m(ation)h(w)o(eigh)o(ts)e Fn(f)p Fj(h)804 561 y Fk(\()p Fm(t)p Fk(\))804 597 y Fm(ij)847 585 y Fn(g)870 568 y Fm(N)870 596 y Fk(1)903 585 y Fu(.)206 678 y(3.)22 b(F)l(or)17 b(eac)o(h)g(top-lev)o(el)i(gating)f(net)o(w)o(ork,)f(solv)o (e)g(an)h(IRLS)h(problem)g(with)f(observ)m(ations)264 740 y Fn(f)p Fu(\()p Fl(x)333 723 y Fk(\()p Fm(t)p Fk(\))374 740 y Fj(;)8 b(h)421 716 y Fk(\()p Fm(t)p Fk(\))421 754 y Fm(k)463 740 y Fu(\))p Fn(g)504 723 y Fm(N)504 751 y Fk(1)537 740 y Fu(.)206 833 y(4.)22 b(F)l(or)15 b(eac)o(h)g(lo)o(w)o (er-lev)o(el)i(gating)e(net)o(w)o(ork,)g(solv)o(e)g(a)h(w)o(eigh)o(ted) g(IRLS)h(problem)f(with)g(ob-)264 895 y(serv)m(ations)f Fn(f)p Fu(\()p Fl(x)544 878 y Fk(\()p Fm(t)p Fk(\))585 895 y Fj(;)8 b(h)632 871 y Fk(\()p Fm(t)p Fk(\))632 910 y Fm(l)p Fd(j)p Fm(k)674 895 y Fu(\))p Fn(g)715 878 y Fm(N)715 906 y Fk(1)763 895 y Fu(and)16 b(observ)m(ation)f(w)o(eigh)o (ts)g Fn(f)p Fj(h)1304 871 y Fk(\()p Fm(t)p Fk(\))1304 909 y Fm(k)1346 895 y Fn(g)1369 878 y Fm(N)1369 906 y Fk(1)1403 895 y Fu(.)206 988 y(5.)22 b(Iterate)14 b(using)i(the)g(up)q (dated)g(parameter)e(v)m(alues.)p 150 1086 V 150 1184 a Fp(A)19 b(least-squares)e(algorithm)150 1270 y Fu(In)j(the)g(case)f (of)g(regression,)h(in)g(whic)o(h)h(a)e(Gaussian)g(probabilit)o(y)i(mo) q(del)f(and)g(an)f(iden)o(tit)o(y)150 1326 y(link)f(function)g(are)e (used,)i(the)f(IRLS)h(lo)q(op)f(for)f(the)h(exp)q(ert)g(net)o(w)o(orks) f(reduces)i(to)e(w)o(eigh)o(ted)150 1383 y(least)d(squares,)g(whic)o(h) g(can)g(b)q(e)h(solv)o(ed)f(\(in)g(one)g(pass\))f(b)o(y)h(an)o(y)f(of)h (the)g(standard)f(least-squares)150 1439 y(algorithms)18 b(\(Golub)g(&)h(v)m(an)f(Loan,)g(1989\).)27 b(The)19 b(gating)e(net)o(w)o(orks)g(still)j(require)f(iterativ)o(e)150 1496 y(pro)q(cessing.)33 b(Supp)q(ose,)20 b(ho)o(w)o(ev)o(er,)f(that)g (w)o(e)g(\014t)g(the)g(parameters)f(of)h(the)g(gating)g(net)o(w)o(orks) 150 1552 y(using)c(least)e(squares)h(rather)f(than)h(maxim)o(um)f(lik)o (eliho)q(o)q(d.)23 b(In)14 b(this)g(case,)g(w)o(e)g(migh)o(t)f(hop)q(e) i(to)150 1609 y(obtain)j(an)f(algorithm)g(in)h(whic)o(h)g(the)g(gating) f(net)o(w)o(ork)f(parameters)g(are)h(\014t)h(b)o(y)f(a)g(one-pass)150 1665 y(algorithm.)k(T)l(o)15 b(motiv)m(ate)h(this)g(approac)o(h,)f (note)h(that)e(w)o(e)i(can)g(express)f(the)h(IRLS)h(problem)150 1721 y(for)f(the)g(gating)g(net)o(w)o(orks)g(as)g(follo)o(ws.)23 b(Di\013eren)o(tiating)17 b(the)f(cross-en)o(trop)o(y)g(\(Equation)g (28\))150 1778 y(with)f(resp)q(ect)g(to)f(the)g(parameters)g Fl(v)803 1785 y Fm(i)831 1778 y Fu(\(using)h(the)g(fact)f(that)g Fj(@)s(g)1281 1785 y Fm(i)1294 1778 y Fj(=@)s(\030)1364 1785 y Fm(j)1394 1778 y Fu(=)f Fj(g)1464 1785 y Fm(i)1478 1778 y Fu(\()p Fj(\016)1516 1785 y Fm(ij)1555 1778 y Fn(\000)c Fj(g)1621 1785 y Fm(j)1639 1778 y Fu(\),)14 b(where)150 1834 y Fj(\016)170 1841 y Fm(ij)218 1834 y Fu(is)19 b(the)f(Kronec)o(k)o(er)g(delta\))g(and)g(setting)g(the)g (deriv)m(ativ)o(es)h(to)f(zero)g(yields)h(the)f(follo)o(wing)150 1891 y(equations:)681 1912 y Fe(X)705 2002 y Fm(t)741 1953 y Fu(\()p Fj(h)785 1929 y Fk(\()p Fm(t)p Fk(\))785 1966 y Fm(i)837 1953 y Fn(\000)11 b Fj(g)905 1960 y Fm(i)918 1953 y Fu(\()p Fl(x)964 1934 y Fk(\()p Fm(t)p Fk(\))1006 1953 y Fj(;)d Fl(v)1055 1960 y Fm(i)1068 1953 y Fu(\)\))p Fl(x)1132 1934 y Fk(\()p Fm(t)p Fk(\))1186 1953 y Fu(=)13 b(0)p Fj(;)449 b Fu(\(30\))150 2068 y(whic)o(h)18 b(are)e(a)g(coupled)j (set)d(of)h(equations)g(that)f(m)o(ust)g(b)q(e)h(solv)o(ed)g(for)g(eac) o(h)f Fj(i)p Fu(.)24 b(Similarly)l(,)c(for)150 2124 y(eac)o(h)11 b(gating)f(net)o(w)o(ork)g(at)g(the)h(second)g(lev)o(el)h(of)e(the)h (tree,)g(w)o(e)g(obtain)g(the)g(follo)o(wing)g(equations:)622 2189 y Fe(X)646 2279 y Fm(t)689 2230 y Fj(h)715 2206 y Fk(\()p Fm(t)p Fk(\))715 2243 y Fm(i)758 2230 y Fu(\()p Fj(h)802 2206 y Fk(\()p Fm(t)p Fk(\))802 2245 y Fm(j)r Fd(j)p Fm(i)854 2230 y Fn(\000)g Fj(g)922 2239 y Fm(j)r Fd(j)p Fm(i)961 2230 y Fu(\()p Fl(x)1007 2211 y Fk(\()p Fm(t)p Fk(\))1049 2230 y Fj(;)d Fl(v)1098 2237 y Fm(ij)1127 2230 y Fu(\)\))p Fl(x)1191 2211 y Fk(\()p Fm(t)p Fk(\))1245 2230 y Fu(=)13 b(0)p Fj(;)390 b Fu(\(31\))150 2362 y(whic)o(h)22 b(m)o(ust)f(b)q(e)h(solv)o(ed)g(for)f(eac)o(h)g Fj(i)h Fu(and)f Fj(j)s Fu(.)38 b(There)22 b(is)g(one)f(asp)q(ect)h(of)f(these) h(equations)150 2424 y(that)15 b(renders)h(them)g(un)o(usual.)22 b(Recall)17 b(that)e(if)h(the)g(lab)q(els)h Fj(z)1221 2400 y Fk(\()p Fm(t)p Fk(\))1219 2437 y Fm(i)1279 2424 y Fu(and)f Fj(z)1391 2400 y Fk(\()p Fm(t)p Fk(\))1389 2440 y Fm(j)r Fd(j)p Fm(i)1449 2424 y Fu(w)o(ere)f(kno)o(wn,)h(then)150 2481 y(the)22 b(gating)f(net)o(w)o(orks)f(w)o(ould)i(b)q(e)g(essen)o (tially)h(solving)f(a)f(set)g(of)g(m)o(ultiw)o(a)o(y)g (classi\014cation)150 2542 y(problems.)27 b(The)17 b(sup)q(ervised)i (errors)e(\()p Fj(z)862 2518 y Fk(\()p Fm(t)p Fk(\))860 2555 y Fm(i)915 2542 y Fn(\000)12 b Fj(g)986 2518 y Fk(\()p Fm(t)p Fk(\))984 2555 y Fm(i)1028 2542 y Fu(\))k(and)i(\()p Fj(z)1194 2518 y Fk(\()p Fm(t)p Fk(\))1192 2558 y Fm(j)r Fd(j)p Fm(i)1247 2542 y Fn(\000)12 b Fj(g)1318 2518 y Fk(\()p Fm(t)p Fk(\))1316 2558 y Fm(j)r Fd(j)p Fm(i)1360 2542 y Fu(\))17 b(w)o(ould)g(app)q(ear)h(in)g(the)952 2775 y(14)p eop %%Page: 15 15 15 14 bop 150 192 a Fu(algorithm)13 b(for)g(solving)h(these)f (problems.)20 b(Note)13 b(that)g(these)g(errors)f(are)h(di\013erences)i (b)q(et)o(w)o(een)150 248 y(indicator)k(v)m(ariables)g(and)f (probabilities.)30 b(In)19 b(Equations)f(30)f(and)h(31,)g(on)f(the)h (other)g(hand,)150 310 y(the)e(errors)e(that)h(driv)o(e)h(the)f (algorithm)h(are)f(the)g(di\013erences)i(\()p Fj(h)1277 286 y Fk(\()p Fm(t)p Fk(\))1277 323 y Fm(i)1329 310 y Fn(\000)11 b Fj(g)1399 286 y Fk(\()p Fm(t)p Fk(\))1397 323 y Fm(i)1440 310 y Fu(\))16 b(and)f(\()p Fj(h)1606 286 y Fk(\()p Fm(t)p Fk(\))1606 326 y Fm(j)r Fd(j)p Fm(i)1659 310 y Fn(\000)10 b Fj(g)1728 286 y Fk(\()p Fm(t)p Fk(\))1726 326 y Fm(j)r Fd(j)p Fm(i)1770 310 y Fu(\),)150 366 y(whic)o(h)23 b(are)f(di\013erences)h(b)q(et)o(w)o(een)g(probabilities.)43 b(The)23 b(EM)e(algorithm)i(e\013ectiv)o(ely)g(\\\014lls)150 423 y(in")e(the)g(missing)g(lab)q(els)h(with)f(estimated)f (probabilities)j Fj(h)1228 430 y Fm(i)1263 423 y Fu(and)d Fj(h)1382 432 y Fm(j)r Fd(j)p Fm(i)1422 423 y Fu(.)36 b(These)21 b(estimated)150 479 y(probabilities)c(can)d(b)q(e)i(though)o (t)d(of)h(as)g(targets)g(for)f(the)i Fj(g)1149 486 y Fm(i)1177 479 y Fu(and)g(the)f Fj(g)1364 488 y Fm(j)r Fd(j)p Fm(i)1404 479 y Fu(.)20 b(This)15 b(suggests)f(that)150 536 y(w)o(e)g(can)g(compute)g(\\virtual)g(targets")f(for)g(the)h (underlying)i(linear)f(predictors)f Fj(\030)1550 543 y Fm(i)1578 536 y Fu(and)g Fj(\030)1685 545 y Fm(j)r Fd(j)p Fm(i)1725 536 y Fu(,)g(b)o(y)150 592 y(in)o(v)o(erting)19 b(the)f(softmax)f(function.)29 b(\(Note)18 b(that)f(this)i(option)f(w)o (ould)g(not)g(b)q(e)h(a)o(v)m(ailable)h(for)150 649 y(the)15 b Fj(z)249 656 y Fm(i)279 649 y Fu(and)g Fj(z)388 658 y Fm(j)r Fd(j)p Fm(i)429 649 y Fu(,)g(ev)o(en)g(if)h(they)f(w)o(ere)h (kno)o(wn,)e(b)q(ecause)j(zero)e(and)g(one)h(are)f(not)g(in)h(the)f (range)150 705 y(of)g(the)g(softmax)f(function.\))20 b(Th)o(us)c(the)f(targets)f(for)g(the)i Fj(\030)1179 712 y Fm(i)1208 705 y Fu(are)f(the)g(v)m(alues:)845 815 y(ln)8 b Fj(h)916 791 y Fk(\()p Fm(t)p Fk(\))916 828 y Fm(i)969 815 y Fn(\000)i Fu(ln)f Fj(C)q(;)150 917 y Fu(where)20 b Fj(C)k Fu(=)399 885 y Fe(P)443 929 y Fm(k)472 917 y Fj(e)493 901 y Fm(\030)509 907 y Ff(k)550 917 y Fu(is)c(the)g(normalization)h(constan)o(t)e(in)i(the)f(softmax)f (function.)36 b(Note,)150 974 y(ho)o(w)o(ev)o(er,)17 b(that)f(constan)o(ts)g(that)h(are)g(common)f(to)h(all)h(of)f(the)g Fj(\030)1282 981 y Fm(i)1313 974 y Fu(can)h(b)q(e)g(omitted,)f(b)q (ecause)150 1035 y(suc)o(h)d(constan)o(ts)f(disapp)q(ear)h(when)g Fj(\030)792 1042 y Fm(i)820 1035 y Fu(are)f(con)o(v)o(erted)g(to)g Fj(g)1174 1042 y Fm(i)1188 1035 y Fu(.)19 b(Th)o(us)14 b(the)f(v)m(alues)i(ln)9 b Fj(h)1616 1012 y Fk(\()p Fm(t)p Fk(\))1616 1048 y Fm(i)1672 1035 y Fu(can)k(b)q(e)150 1101 y(used)k(as)f(targets)f(for)h(the)g Fj(\030)635 1108 y Fm(i)649 1101 y Fu(.)24 b(A)16 b(similar)i(argumen)o(t)d(sho)o (ws)h(that)g(the)g(v)m(alues)i(ln)8 b Fj(h)1610 1077 y Fk(\()p Fm(t)p Fk(\))1610 1117 y Fm(l)p Fd(j)p Fm(k)1669 1101 y Fu(can)16 b(b)q(e)150 1179 y(used)g(as)f(targets)e(for)i(the)g Fj(\030)629 1186 y Fm(ij)660 1179 y Fu(,)f(with)i(observ)m(ation)f(w)o (eigh)o(ts)g Fj(h)1220 1155 y Fk(\()p Fm(t)p Fk(\))1220 1193 y 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j 1 1 j 2 1 j 1 1 j 1 2 j 1 1 j 2 1 j 1 1 j 1 2 j 1 1 j 2 1 j 1 1 j 1 1 j 1 2 j 1 1 j 2 1 j 1 1 j 1 1 j 1 2 j 2 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 2 1 j 1 1 j 1 1 j 1 2 j 1 1 j 2 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 2 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 2 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 2 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 0 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 0 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 2 j 1 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 0 1 j 1 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 1 2 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 1 1 j 0 2 j 1 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 2 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 1 2 j 0 1 j 1 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 2 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 2 j 1 1 j 1 1 j 0 1 j stroke % to avoid limitcheck 4053 3222 M 1 1 j 1 1 j 1 1 j 0 1 j 1 2 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 2 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 2 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 0 2 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 2 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 1 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 1 j 1 0 h 0 1 j 1 1 j 1 1 j 1 1 j 1 1 j 0 1 j 1 0 h 1 1 j 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 1 1 j 1 1 j 1 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 1 1 j 1 1 j 1 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 1 1 j 1 1 j 1 0 h 0 1 j 1 0 h 1 1 j 1 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 1 1 j 1 0 h 0 1 j 1 0 h 1 1 j 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 1 0 h 0 1 j 1 0 h stroke % to avoid limitcheck 4359 2811 M 0 1 j 1 0 h 1 0 h 0 1 j 1 0 h 0 1 j 1 0 h 1 0 h 0 1 j 1 0 h 1 1 j 1 0 h 0 1 j 1 0 h 1 0 h 0 1 j 1 0 h 1 1 j 1 0 h 0 1 j 1 0 h 1 0 h 0 1 j 1 0 h 1 1 j 1 0 h 1 1 j 1 0 h 1 1 j 1 0 h 1 1 j 1 0 h 0 1 j 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 1 1 j 1 0 h 1 1 j 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 1 1 j 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 1 0 h 1 1 j 1 0 h 1 0 h 1 0 h 0 1 j 1 0 h 1 0 h 1 0 h 1 1 j 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 1 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h 1 0 h 1 0 h 1 0 h 0 1 h 1 0 h 1 0 h (Epochs) 2840 2067 1 0 0.5 Stext (Relative error) 549 3670 1 90 0.5 Stext 1173 2557 M 0 44 j 2043 2557 M 0 44 j 2912 2557 M 0 44 j 3782 2557 M 0 44 j 1173 2557 M 2609 0 h (1) 1173 2347 1 0 0.5 Stext (10) 2043 2347 1 0 0.5 Stext (100) 2912 2347 1 0 0.5 Stext (1000) 3782 2347 1 0 0.5 Stext 1040 2639 M 45 0 k 1040 2934 M 45 0 k 1040 3228 M 45 0 k 1040 3522 M 45 0 k 1040 3817 M 45 0 k 1040 4111 M 45 0 k 1040 4405 M 45 0 k 1040 4700 M 45 0 k 1040 2639 M 0 2061 h (0.0) 830 2639 1 90 0.5 Stext (0.4) 830 3228 1 90 0.5 Stext (0.8) 830 3817 1 90 0.5 Stext (1.2) 830 4405 1 90 0.5 Stext 1040 4782 M 0 2225 j 3600 0 h 0 2225 h 3600 0 k stroke [5 40] 0 setdash 1173 4111 M 262 1063 j 153 2 j 108 8 j 85 17 j 69 26 j 58 26 j 50 22 j 45 19 j 40 16 j 36 14 j 32 12 j 31 10 j 28 7 j 26 6 j 24 6 j 23 4 j 22 4 j 20 4 j 19 3 j 19 3 j 17 3 j 17 3 j 16 2 j 16 3 j 14 2 j 15 2 j 13 2 j 14 2 j 13 2 j 12 2 j 12 1 j 12 2 j 11 1 j 11 1 j 10 1 j 11 1 j 10 1 j 10 0 h 9 1 j 10 0 h 9 1 j 8 1 j 9 0 h 9 0 h 8 1 j 8 0 h 8 0 h 8 0 h 7 1 j 8 0 h 7 0 h 7 0 h 8 0 h 6 1 j 7 0 h 7 0 h 7 0 h 6 1 j 6 0 h 7 0 h 6 0 h 6 0 h 6 0 h 6 0 h 5 0 h 6 0 h 6 0 h 5 0 h 6 1 h 5 0 h 5 0 h 5 0 h 6 1 h 5 0 h 5 0 h 5 1 h 4 0 h 5 0 h 5 1 h 5 0 h 4 0 h 5 0 h 4 1 h 5 0 h 4 0 h 5 0 h 4 0 h 4 0 h 4 0 h 5 1 h 4 0 h 4 0 h 4 0 h 4 0 h 4 1 h 4 0 h 4 0 h 3 0 h 4 0 h 4 1 h 4 0 h 3 0 h 4 1 h 4 0 h 3 0 h 4 0 h 3 1 h 4 0 h 3 0 h 4 0 h 3 0 h 3 1 h 4 0 h 3 0 h 3 0 h 4 1 h 3 0 h 3 0 h 3 0 h stroke 1.000000 S [] 0 setdash N 2304 4238 M 2304 4700 L 4464 4700 L 4464 4238 L C 1.000000 S F 1.000000 S 0.000000 S 2304 4238 M 0 462 h 4464 4238 M 0 462 h 2304 4238 M 2160 0 h 2304 4700 M 2160 0 h 2464 4560 M 320 0 h (Backpropagation) 2944 4560 1 0 0 Stext stroke [5 40] 0 setdash 2464 4406 M 320 0 h stroke [] 0 setdash (HME \(Algorithm 2\)) 2944 4406 1 0 0 Stext showpage grestore %%EndDocument endTexFig 150 1448 a Fu(Figure)16 b(2:)22 b(Relativ)o(e)c(error)d(on)h(the)h (test)e(set)h(for)g(a)g(bac)o(kpropagation)g(net)o(w)o(ork)f(and)h(a)g (four-)150 1504 y(lev)o(el)i(HME)d(arc)o(hitecture)i(trained)g(with)f (batc)o(h)g(algorithms.)23 b(The)17 b(standard)f(errors)f(at)h(the)150 1561 y(minima)g(of)f(the)g(curv)o(es)g(are)g(0.013)f(for)h(bac)o(kprop) g(and)g(0.002)f(for)g(HME.)150 1694 y(w)o(ere)19 b(c)o(hosen)h(for)f (these)h(parameters.\))31 b(There)20 b(w)o(ere)f(di\016culties)j(with)e (lo)q(cal)g(minima)h(\(or)150 1751 y(plateaus\))11 b(using)h(the)f(bac) o(kpropagation)f(algorithm:)18 b(Fiv)o(e)11 b(of)f(ten)h(runs)h(failed) g(to)e(con)o(v)o(erge)h(to)150 1807 y(\\reasonable")h(error)e(v)m (alues.)20 b(\(As)12 b(w)o(e)f(rep)q(ort)g(in)i(the)f(next)f(section,)i (no)e(suc)o(h)h(di\016culties)i(w)o(ere)150 1864 y(encoun)o(tered)e(in) h(the)f(case)f(of)g Ft(on-line)k Fu(bac)o(kpropagation\).)i(W)l(e)12 b(rep)q(ort)f(a)o(v)o(erage)g(con)o(v)o(ergence)150 1920 y(times)j(and)f(a)o(v)o(erage)f(relativ)o(e)i(errors)f(only)h(for)f (those)g(runs)g(that)g(con)o(v)o(erged)g(to)g(\\reasonable")150 1977 y(error)e(v)m(alues.)20 b(All)13 b(ten)f(runs)g(for)f(b)q(oth)h (of)f(the)h(HME)g(algorithms)f(con)o(v)o(erged)h(to)f(\\reasonable")150 2033 y(error)j(v)m(alues.)241 2089 y(Figure)h(2)f(sho)o(ws)f(the)i(p)q (erformance)g(of)f(the)g(hierarc)o(h)o(y)h(and)f(the)h(bac)o (kpropagation)f(net-)150 2146 y(w)o(ork.)37 b(The)21 b(horizon)o(tal)g(axis)h(of)f(the)g(graph)g(giv)o(es)g(the)g(training)h (time)f(in)h(ep)q(o)q(c)o(hs.)39 b(The)150 2202 y(v)o(ertical)18 b(axis)g(giv)o(es)g(generalization)h(p)q(erformance)f(as)f(measured)h (b)o(y)f(the)h(a)o(v)o(erage)e(relativ)o(e)150 2259 y(error)e(on)i(the) f(test)g(set.)241 2315 y(T)l(able)j(1)e(rep)q(orts)h(the)g(a)o(v)o (erage)e(relativ)o(e)j(errors)e(for)g(b)q(oth)h(arc)o(hitectures)g (measured)h(at)150 2372 y(the)12 b(minima)h(of)f(the)g(relativ)o(e)g (error)g(curv)o(es.)19 b(\(Minima)12 b(w)o(ere)g(de\014ned)h(b)o(y)f(a) g(sequence)h(of)f(three)150 2428 y(successiv)o(e)j(increases)g(in)g (the)g(relativ)o(e)f(error.\))19 b(W)l(e)14 b(also)g(rep)q(ort)g(v)m (alues)i(of)d(relativ)o(e)i(error)f(for)150 2485 y(the)22 b(b)q(est)g(linear)h(appro)o(ximation,)g(the)e(CAR)l(T)i(algorithm,)f (and)g(the)g(MARS)g(algorithm.)150 2541 y(Both)15 b(CAR)l(T)g(and)g (MARS)h(w)o(ere)e(run)i(four)e(times,)h(once)g(for)g(eac)o(h)g(of)f (the)h(output)g(v)m(ariables.)952 2775 y(17)p eop %%Page: 18 18 18 17 bop 461 201 1029 2 v 460 258 2 57 v 485 241 a Fu(Arc)o(hitecture) p 908 258 V 202 w(Relativ)o(e)16 b(Error)p 1242 258 V 49 w(#)f(Ep)q(o)q(c)o(hs)p 1489 258 V 461 260 1029 2 v 461 270 V 460 326 2 57 v 485 309 a(linear)p 908 326 V 452 w(.31)p 1242 326 V 249 w(1)p 1489 326 V 460 382 V 485 366 a(bac)o(kprop)p 908 382 V 380 w(.09)p 1242 382 V 209 w(5,500)p 1489 382 V 460 439 V 485 422 a(HME)g(\(Algorithm)g (1\))p 908 439 V 163 w(.10)p 1242 439 V 238 w(35)p 1489 439 V 460 495 V 485 478 a(HME)g(\(Algorithm)g(2\))p 908 495 V 163 w(.12)p 1242 495 V 238 w(39)p 1489 495 V 460 552 V 485 535 a(CAR)l(T)p 908 552 V 433 w(.17)p 1242 552 V 226 w(NA)p 1489 552 V 460 608 V 485 591 a(CAR)l(T)h(\(linear\))p 908 608 V 271 w(.13)p 1242 608 V 226 w(NA)p 1489 608 V 460 665 V 485 648 a(MARS)p 908 665 V 428 w(.16)p 1242 665 V 226 w(NA)p 1489 665 V 461 666 1029 2 v 150 793 a(T)l(able)f(1:)k(Av)o(erage)14 b(v)m(alues)h(of)f(relativ)o(e)h(error) f(and)g(n)o(um)o(b)q(er)h(of)f(ep)q(o)q(c)o(hs)h(required)g(for)f(con)o (v)o(er-)150 850 y(gence)i(for)e(the)i(batc)o(h)f(algorithms.)150 983 y(W)l(e)e(com)o(bined)h(the)e(results)i(from)e(these)h(four)f (computations)g(to)h(compute)f(the)h(total)f(relativ)o(e)150 1040 y(error.)26 b(Tw)o(o)16 b(v)o(ersions)i(of)f(CAR)l(T)h(w)o(ere)f (run;)i(one)e(in)h(whic)o(h)h(the)e(splits)i(w)o(ere)e(restricted)h(to) 150 1096 y(b)q(e)f(parallel)g(to)e(the)h(axes)g(and)g(one)g(in)h(whic)o (h)g(linear)g(com)o(binations)f(of)g(the)g(input)h(v)m(ariables)150 1153 y(w)o(ere)e(allo)o(w)o(ed.)241 1209 y(The)h(MARS)h(algorithm)f (requires)h(c)o(hoices)g(to)e(b)q(e)i(made)f(for)f(the)h(v)m(alues)i (of)d(t)o(w)o(o)g(struc-)150 1266 y(tural)f(parameters:)19 b(the)14 b(maxim)o(um)g(n)o(um)o(b)q(er)h(of)e(basis)i(functions)g(and) f(the)h(maxim)o(um)f(n)o(um-)150 1322 y(b)q(er)e(of)f(in)o(teraction)h (terms.)18 b(Eac)o(h)11 b(basis)h(function)g(in)h(MARS)e(yields)i(a)f (linear)g(surface)g(de\014ned)150 1378 y(o)o(v)o(er)i(a)h(rectangular)g (region)g(of)g(the)g(input)h(space,)f(corresp)q(onding)h(roughly)g(to)e (the)h(function)150 1435 y(implemen)o(ted)22 b(b)o(y)f(a)f(single)i (exp)q(ert)e(in)i(the)e(HME)g(arc)o(hitecture.)36 b(Therefore)21 b(w)o(e)f(c)o(hose)g(a)150 1491 y(maxim)o(um)g(of)f(16)h(basis)g (functions)h(to)e(corresp)q(ond)i(to)e(the)h(16)g(exp)q(erts)g(in)h (the)f(four-lev)o(el)150 1548 y(hierarc)o(h)o(y)l(.)k(T)l(o)16 b(c)o(ho)q(ose)h(the)f(maxim)o(um)g(n)o(um)o(b)q(er)h(of)f(in)o (teractions)h(\()p Fj(mi)p Fu(\),)e(w)o(e)h(compared)h(the)150 1604 y(p)q(erformance)e(of)g(MARS)g(for)f Fj(mi)e Fu(=)h(1,)i(2,)f(3,)g (6,)g(and)i(12,)e(and)h(c)o(hose)g(the)g(v)m(alue)h(that)e(yielded)150 1661 y(the)h(b)q(est)h(p)q(erformance)f(\()p Fj(mi)d Fu(=)h(3\).)241 1717 y(F)l(or)22 b(the)g(iterativ)o(e)g(algorithms,)i (w)o(e)e(also)g(rep)q(ort)g(the)g(n)o(um)o(b)q(er)h(of)f(ep)q(o)q(c)o (hs)h(required)150 1774 y(for)c(con)o(v)o(ergence.)33 b(Because)20 b(the)g(learning)g(curv)o(es)g(for)f(these)g(algorithms)h (generally)g(ha)o(v)o(e)150 1830 y(length)o(y)d(tails,)h(w)o(e)e (de\014ned)j(con)o(v)o(ergence)e(as)f(the)h(\014rst)g(ep)q(o)q(c)o(h)g (at)g(whic)o(h)g(the)g(relativ)o(e)h(error)150 1887 y(drops)d(within)i (\014v)o(e)e(p)q(ercen)o(t)h(of)e(the)i(minim)o(um.)241 1943 y(All)23 b(of)e(the)h(arc)o(hitectures)g(that)e(w)o(e)i(studied)g (p)q(erformed)g(signi\014can)o(tly)h(b)q(etter)f(than)150 1999 y(the)f(b)q(est)g(linear)h(appro)o(ximation.)37 b(As)20 b(exp)q(ected,)j(the)e(CAR)l(T)g(arc)o(hitecture)h(with)f (linear)150 2056 y(com)o(binations)14 b(p)q(erformed)g(b)q(etter)g (than)f(CAR)l(T)h(with)g(axis-parallel)h(splits.)1492 2039 y Fk(5)1532 2056 y Fu(The)f(HME)f(ar-)150 2112 y(c)o(hitecture)i (yielded)g(a)f(mo)q(dest)f(impro)o(v)o(emen)o(t)h(o)o(v)o(er)f(MARS)h (and)g(CAR)l(T.)g(Bac)o(kpropagation)150 2169 y(pro)q(duced)i(the)f(lo) o(w)o(est)g(relativ)o(e)h(error)e(of)h(the)g(algorithms)g(tested)g (\(ignoring)h(the)f(di\016culties)150 2225 y(with)h(con)o(v)o (ergence\).)241 2282 y(These)j(di\013erences)g(in)h(relativ)o(e)f (error)e(should)j(b)q(e)f(treated)f(with)h(some)f(caution.)30 b(The)150 2338 y(need)13 b(to)e(set)h(free)h(parameters)e(for)g(some)h (of)g(the)g(arc)o(hitectures)g(\(e.g.,)g(bac)o(kpropagation\))f(and)p 150 2380 660 2 v 202 2407 a 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75.379 l 79.149 75.264 l 78.665 75.264 l 78.665 75.379 l gsave 0 0.038 0 Bbw grestore n 77.401 75.379 m 77.885 75.379 l 77.401 75.379 l cl 0 Fbw n 77.401 75.379 m 77.885 75.379 l 77.401 75.379 l gsave 0 0.038 0 Bbw grestore n 93.5 75.472 m 93.5 75.385 l gsave 0 0.05 0 Bbw grestore n 92.235 75.472 m 92.235 75.385 l gsave 0 0.05 0 Bbw grestore n 90.971 75.472 m 90.971 75.385 l gsave 0 0.05 0 Bbw grestore n 89.707 75.472 m 89.707 75.385 l gsave 0 0.05 0 Bbw grestore n 88.443 75.472 m 88.443 75.385 l gsave 0 0.05 0 Bbw grestore n 87.174 75.472 m 87.174 75.385 l gsave 0 0.05 0 Bbw grestore n 93.5 75.385 m 87.174 75.385 l gsave 0 0.05 0 Bbw grestore n 86.927 68.351 m 87.006 68.351 l gsave 0 0.05 0 Bbw grestore n 86.927 70.696 m 87.006 70.696 l gsave 0 0.05 0 Bbw grestore n 86.927 73.04 m 87.006 73.04 l gsave 0 0.05 0 Bbw grestore n 86.927 75.379 m 87.006 75.379 l gsave 0 0.05 0 Bbw grestore n 87.006 68.351 m 87.006 75.379 l gsave 0 0.05 0 Bbw grestore n 92.625 75.379 m 93.109 75.379 l 92.625 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58.425 m 22.2 58.425 l 21.716 58.425 l cl 0 Fbw n 21.716 58.425 m 22.2 58.425 l 21.716 58.425 l gsave 0 0.038 0 Bbw grestore n 20.452 58.425 m 20.936 58.425 l 20.452 58.425 l cl 0 Fbw n 20.452 58.425 m 20.936 58.425 l 20.452 58.425 l gsave 0 0.038 0 Bbw grestore n 46.724 58.518 m 46.724 58.431 l gsave 0 0.05 0 Bbw grestore n 45.46 58.518 m 45.46 58.431 l gsave 0 0.05 0 Bbw grestore n 44.195 58.518 m 44.195 58.431 l gsave 0 0.05 0 Bbw grestore n 42.931 58.518 m 42.931 58.431 l gsave 0 0.05 0 Bbw grestore n 41.667 58.518 m 41.667 58.431 l gsave 0 0.05 0 Bbw grestore n 40.398 58.518 m 40.398 58.431 l gsave 0 0.05 0 Bbw grestore n 46.724 58.431 m 40.398 58.431 l gsave 0 0.05 0 Bbw grestore n 40.151 51.397 m 40.23 51.397 l gsave 0 0.05 0 Bbw grestore n 40.151 53.742 m 40.23 53.742 l gsave 0 0.05 0 Bbw grestore n 40.151 56.086 m 40.23 56.086 l gsave 0 0.05 0 Bbw grestore n 40.151 58.425 m 40.23 58.425 l gsave 0 0.05 0 Bbw grestore n 40.23 51.397 m 40.23 58.425 l gsave 0 0.05 0 Bbw grestore n 45.85 58.425 m 46.334 58.425 l 45.85 58.425 l cl 0 Fbw n 45.85 58.425 m 46.334 58.425 l 45.85 58.425 l gsave 0 0.038 0 Bbw grestore n 44.586 58.425 m 45.07 58.425 l 44.586 58.425 l cl 0 Fbw n 44.586 58.425 m 45.07 58.425 l 44.586 58.425 l gsave 0 0.038 0 Bbw grestore n 43.321 58.425 m 43.805 58.425 l 43.805 51.518 l 43.321 51.518 l 43.321 58.425 l cl 0 Fbw n 43.321 58.425 m 43.805 58.425 l 43.805 51.518 l 43.321 51.518 l 43.321 58.425 l gsave 0 0.038 0 Bbw grestore n 42.057 58.425 m 42.541 58.425 l 42.057 58.425 l cl 0 Fbw n 42.057 58.425 m 42.541 58.425 l 42.057 58.425 l gsave 0 0.038 0 Bbw grestore n 40.793 58.425 m 41.277 58.425 l 40.793 58.425 l cl 0 Fbw n 40.793 58.425 m 41.277 58.425 l 40.793 58.425 l gsave 0 0.038 0 Bbw grestore n 67.06 58.518 m 67.06 58.431 l gsave 0 0.05 0 Bbw grestore n 65.796 58.518 m 65.796 58.431 l gsave 0 0.05 0 Bbw grestore n 64.531 58.518 m 64.531 58.431 l gsave 0 0.05 0 Bbw grestore n 63.267 58.518 m 63.267 58.431 l gsave 0 0.05 0 Bbw grestore n 62.003 58.518 m 62.003 58.431 l gsave 0 0.05 0 Bbw grestore n 60.734 58.518 m 60.734 58.431 l gsave 0 0.05 0 Bbw grestore n 67.06 58.431 m 60.734 58.431 l gsave 0 0.05 0 Bbw grestore n 60.487 51.397 m 60.566 51.397 l gsave 0 0.05 0 Bbw grestore n 60.487 53.742 m 60.566 53.742 l gsave 0 0.05 0 Bbw grestore n 60.487 56.086 m 60.566 56.086 l gsave 0 0.05 0 Bbw grestore n 60.487 58.425 m 60.566 58.425 l gsave 0 0.05 0 Bbw grestore n 60.566 51.397 m 60.566 58.425 l gsave 0 0.05 0 Bbw grestore n 66.186 58.425 m 66.67 58.425 l 66.186 58.425 l cl 0 Fbw n 66.186 58.425 m 66.67 58.425 l 66.186 58.425 l gsave 0 0.038 0 Bbw grestore n 64.922 58.425 m 65.406 58.425 l 64.922 58.425 l cl 0 Fbw n 64.922 58.425 m 65.406 58.425 l 64.922 58.425 l gsave 0 0.038 0 Bbw grestore n 63.657 58.425 m 64.141 58.425 l 64.141 51.518 l 63.657 51.518 l 63.657 58.425 l cl 0 Fbw n 63.657 58.425 m 64.141 58.425 l 64.141 51.518 l 63.657 51.518 l 63.657 58.425 l gsave 0 0.038 0 Bbw grestore n 62.393 58.425 m 62.877 58.425 l 62.393 58.425 l cl 0 Fbw n 62.393 58.425 m 62.877 58.425 l 62.393 58.425 l gsave 0 0.038 0 Bbw grestore n 61.129 58.425 m 61.613 58.425 l 61.129 58.425 l cl 0 Fbw n 61.129 58.425 m 61.613 58.425 l 61.129 58.425 l gsave 0 0.038 0 Bbw grestore n 87.396 58.518 m 87.396 58.431 l gsave 0 0.05 0 Bbw grestore n 86.132 58.518 m 86.132 58.431 l gsave 0 0.05 0 Bbw grestore n 84.867 58.518 m 84.867 58.431 l gsave 0 0.05 0 Bbw grestore n 83.603 58.518 m 83.603 58.431 l gsave 0 0.05 0 Bbw grestore n 82.339 58.518 m 82.339 58.431 l gsave 0 0.05 0 Bbw grestore n 81.07 58.518 m 81.07 58.431 l gsave 0 0.05 0 Bbw grestore n 87.396 58.431 m 81.07 58.431 l gsave 0 0.05 0 Bbw grestore n 80.823 51.397 m 80.902 51.397 l gsave 0 0.05 0 Bbw grestore n 80.823 53.742 m 80.902 53.742 l gsave 0 0.05 0 Bbw grestore n 80.823 56.086 m 80.902 56.086 l gsave 0 0.05 0 Bbw grestore n 80.823 58.425 m 80.902 58.425 l gsave 0 0.05 0 Bbw grestore n 80.902 51.397 m 80.902 58.425 l gsave 0 0.05 0 Bbw grestore n 86.522 58.425 m 87.006 58.425 l 86.522 58.425 l cl 0 Fbw n 86.522 58.425 m 87.006 58.425 l 86.522 58.425 l gsave 0 0.038 0 Bbw grestore n 85.258 58.425 m 85.742 58.425 l 85.258 58.425 l cl 0 Fbw n 85.258 58.425 m 85.742 58.425 l 85.258 58.425 l gsave 0 0.038 0 Bbw grestore n 83.993 58.425 m 84.477 58.425 l 84.477 51.518 l 83.993 51.518 l 83.993 58.425 l cl 0 Fbw n 83.993 58.425 m 84.477 58.425 l 84.477 51.518 l 83.993 51.518 l 83.993 58.425 l gsave 0 0.038 0 Bbw grestore n 82.729 58.425 m 83.213 58.425 l 82.729 58.425 l cl 0 Fbw n 82.729 58.425 m 83.213 58.425 l 82.729 58.425 l gsave 0 0.038 0 Bbw grestore n 81.465 58.425 m 81.949 58.425 l 81.465 58.425 l cl 0 Fbw n 81.465 58.425 m 81.949 58.425 l 81.465 58.425 l gsave 0 0.038 0 Bbw grestore n 36.551 41.559 m 36.551 41.471 l gsave 0 0.05 0 Bbw grestore n 35.287 41.559 m 35.287 41.471 l gsave 0 0.05 0 Bbw grestore n 34.023 41.559 m 34.023 41.471 l gsave 0 0.05 0 Bbw grestore n 32.758 41.559 m 32.758 41.471 l gsave 0 0.05 0 Bbw grestore n 31.494 41.559 m 31.494 41.471 l gsave 0 0.05 0 Bbw grestore n 30.225 41.559 m 30.225 41.471 l gsave 0 0.05 0 Bbw grestore n 36.551 41.471 m 30.225 41.471 l gsave 0 0.05 0 Bbw grestore n 29.978 34.438 m 30.057 34.438 l gsave 0 0.05 0 Bbw grestore n 29.978 36.782 m 30.057 36.782 l gsave 0 0.05 0 Bbw grestore n 29.978 39.127 m 30.057 39.127 l gsave 0 0.05 0 Bbw grestore n 29.978 41.465 m 30.057 41.465 l gsave 0 0.05 0 Bbw grestore n 30.057 34.438 m 30.057 41.465 l gsave 0 0.05 0 Bbw grestore n 35.677 41.465 m 36.161 41.465 l 35.677 41.465 l cl 0 Fbw n 35.677 41.465 m 36.161 41.465 l 35.677 41.465 l gsave 0 0.038 0 Bbw grestore n 34.413 41.465 m 34.897 41.465 l 34.413 41.465 l cl 0 Fbw n 34.413 41.465 m 34.897 41.465 l 34.413 41.465 l gsave 0 0.038 0 Bbw grestore n 33.148 41.465 m 33.632 41.465 l 33.632 34.559 l 33.148 34.559 l 33.148 41.465 l cl 0 Fbw n 33.148 41.465 m 33.632 41.465 l 33.632 34.559 l 33.148 34.559 l 33.148 41.465 l gsave 0 0.038 0 Bbw grestore n 31.884 41.465 m 32.368 41.465 l 31.884 41.465 l cl 0 Fbw n 31.884 41.465 m 32.368 41.465 l 31.884 41.465 l gsave 0 0.038 0 Bbw grestore n 30.62 41.465 m 31.104 41.465 l 30.62 41.465 l cl 0 Fbw n 30.62 41.465 m 31.104 41.465 l 30.62 41.465 l gsave 0 0.038 0 Bbw grestore n 77.228 41.559 m 77.228 41.471 l gsave 0 0.05 0 Bbw grestore n 75.964 41.559 m 75.964 41.471 l gsave 0 0.05 0 Bbw grestore n 74.699 41.559 m 74.699 41.471 l gsave 0 0.05 0 Bbw grestore n 73.435 41.559 m 73.435 41.471 l gsave 0 0.05 0 Bbw grestore n 72.171 41.559 m 72.171 41.471 l gsave 0 0.05 0 Bbw grestore n 70.902 41.559 m 70.902 41.471 l gsave 0 0.05 0 Bbw grestore n 77.228 41.471 m 70.902 41.471 l gsave 0 0.05 0 Bbw grestore n 70.655 34.438 m 70.734 34.438 l gsave 0 0.05 0 Bbw grestore n 70.655 36.782 m 70.734 36.782 l gsave 0 0.05 0 Bbw grestore n 70.655 39.127 m 70.734 39.127 l gsave 0 0.05 0 Bbw grestore n 70.655 41.465 m 70.734 41.465 l gsave 0 0.05 0 Bbw grestore n 70.734 34.438 m 70.734 41.465 l gsave 0 0.05 0 Bbw grestore n 76.354 41.465 m 76.838 41.465 l 76.354 41.465 l cl 0 Fbw n 76.354 41.465 m 76.838 41.465 l 76.354 41.465 l gsave 0 0.038 0 Bbw grestore n 75.089 41.465 m 75.573 41.465 l 75.089 41.465 l cl 0 Fbw n 75.089 41.465 m 75.573 41.465 l 75.089 41.465 l gsave 0 0.038 0 Bbw grestore n 73.826 41.465 m 74.309 41.465 l 74.309 34.559 l 73.826 34.559 l 73.826 41.465 l cl 0 Fbw n 73.826 41.465 m 74.309 41.465 l 74.309 34.559 l 73.826 34.559 l 73.826 41.465 l gsave 0 0.038 0 Bbw grestore n 72.561 41.465 m 73.045 41.465 l 72.561 41.465 l cl 0 Fbw n 72.561 41.465 m 73.045 41.465 l 72.561 41.465 l gsave 0 0.038 0 Bbw grestore n 71.297 41.465 m 71.781 41.465 l 71.297 41.465 l cl 0 Fbw n 71.297 41.465 m 71.781 41.465 l 71.297 41.465 l gsave 0 0.038 0 Bbw grestore n 56.892 24.599 m 56.892 24.511 l gsave 0 0.05 0 Bbw grestore n 55.628 24.599 m 55.628 24.511 l gsave 0 0.05 0 Bbw grestore n 54.363 24.599 m 54.363 24.511 l gsave 0 0.05 0 Bbw grestore n 53.099 24.599 m 53.099 24.511 l gsave 0 0.05 0 Bbw grestore n 51.835 24.599 m 51.835 24.511 l gsave 0 0.05 0 Bbw grestore n 50.566 24.599 m 50.566 24.511 l gsave 0 0.05 0 Bbw grestore n 56.892 24.511 m 50.566 24.511 l gsave 0 0.05 0 Bbw grestore n 50.319 17.478 m 50.398 17.478 l gsave 0 0.05 0 Bbw grestore n 50.319 19.823 m 50.398 19.823 l gsave 0 0.05 0 Bbw grestore n 50.319 22.167 m 50.398 22.167 l gsave 0 0.05 0 Bbw grestore n 50.319 24.506 m 50.398 24.506 l gsave 0 0.05 0 Bbw grestore n 50.398 17.478 m 50.398 24.506 l gsave 0 0.05 0 Bbw grestore n 56.018 24.506 m 56.502 24.506 l 56.018 24.506 l cl 0 Fbw n 56.018 24.506 m 56.502 24.506 l 56.018 24.506 l gsave 0 0.038 0 Bbw grestore n 54.753 24.506 m 55.237 24.506 l 54.753 24.506 l cl 0 Fbw n 54.753 24.506 m 55.237 24.506 l 54.753 24.506 l gsave 0 0.038 0 Bbw grestore n 53.489 24.506 m 53.973 24.506 l 53.973 17.478 l 53.489 17.478 l 53.489 24.506 l cl 0 Fbw n 53.489 24.506 m 53.973 24.506 l 53.973 17.478 l 53.489 17.478 l 53.489 24.506 l gsave 0 0.038 0 Bbw grestore n 52.225 24.506 m 52.709 24.506 l 52.225 24.506 l cl 0 Fbw n 52.225 24.506 m 52.709 24.506 l 52.225 24.506 l gsave 0 0.038 0 Bbw grestore n 50.961 24.506 m 51.445 24.506 l 50.961 24.506 l cl 0 Fbw n 50.961 24.506 m 51.445 24.506 l 50.961 24.506 l gsave 0 0.038 0 Bbw grestore n 116.61 75.472 m 116.61 75.384 l gsave 0 0.05 0 Bbw grestore n 115.34 75.472 m 115.34 75.384 l gsave 0 0.05 0 Bbw grestore n 114.07 75.472 m 114.07 75.384 l gsave 0 0.05 0 Bbw grestore n 112.8 75.472 m 112.8 75.384 l gsave 0 0.05 0 Bbw grestore n 111.52 75.472 m 111.52 75.384 l gsave 0 0.05 0 Bbw grestore n 110.25 75.472 m 110.25 75.384 l gsave 0 0.05 0 Bbw grestore n 116.61 75.384 m 110.25 75.384 l gsave 0 0.05 0 Bbw grestore n 110 68.306 m 110.08 68.306 l gsave 0 0.05 0 Bbw grestore n 110 70.666 m 110.08 70.666 l gsave 0 0.05 0 Bbw grestore n 110 73.025 m 110.08 73.025 l gsave 0 0.05 0 Bbw grestore n 110 75.379 m 110.08 75.379 l gsave 0 0.05 0 Bbw grestore n 110.08 68.306 m 110.08 75.379 l gsave 0 0.05 0 Bbw grestore n 115.73 75.379 m 116.22 75.379 l 116.22 75.263 l 115.73 75.263 l 115.73 75.379 l cl 0 Fbw n 115.73 75.379 m 116.22 75.379 l 116.22 75.263 l 115.73 75.263 l 115.73 75.379 l gsave 0 0.038 0 Bbw grestore n 114.46 75.379 m 114.95 75.379 l 114.95 74.085 l 114.46 74.085 l 114.46 75.379 l cl 0 Fbw n 114.46 75.379 m 114.95 75.379 l 114.95 74.085 l 114.46 74.085 l 114.46 75.379 l gsave 0 0.038 0 Bbw grestore n 113.19 75.379 m 113.68 75.379 l 113.68 71.257 l 113.19 71.257 l 113.19 75.379 l cl 0 Fbw n 113.19 75.379 m 113.68 75.379 l 113.68 71.257 l 113.19 71.257 l 113.19 75.379 l gsave 0 0.038 0 Bbw grestore n 111.92 75.379 m 112.4 75.379 l 112.4 74.202 l 111.92 74.202 l 111.92 75.379 l cl 0 Fbw n 111.92 75.379 m 112.4 75.379 l 112.4 74.202 l 111.92 74.202 l 111.92 75.379 l gsave 0 0.038 0 Bbw grestore n 110.65 75.379 m 111.13 75.379 l 111.13 75.263 l 110.65 75.263 l 110.65 75.379 l cl 0 Fbw n 110.65 75.379 m 111.13 75.379 l 111.13 75.263 l 110.65 75.263 l 110.65 75.379 l gsave 0 0.038 0 Bbw grestore n 127.76 33.145 m 148.22 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75.379 m 186.56 75.379 l 186.56 73.61 l 186.07 73.61 l 186.07 75.379 l gsave 0 0.038 0 Bbw grestore n 184.8 75.379 m 185.29 75.379 l 185.29 73.378 l 184.8 73.378 l 184.8 75.379 l cl 0 Fbw n 184.8 75.379 m 185.29 75.379 l 185.29 73.378 l 184.8 73.378 l 184.8 75.379 l gsave 0 0.038 0 Bbw grestore n 183.53 75.379 m 184.01 75.379 l 184.01 73.964 l 183.53 73.964 l 183.53 75.379 l cl 0 Fbw n 183.53 75.379 m 184.01 75.379 l 184.01 73.964 l 183.53 73.964 l 183.53 75.379 l gsave 0 0.038 0 Bbw grestore n 182.25 75.379 m 182.74 75.379 l 182.74 74.671 l 182.25 74.671 l 182.25 75.379 l cl 0 Fbw n 182.25 75.379 m 182.74 75.379 l 182.74 74.671 l 182.25 74.671 l 182.25 75.379 l gsave 0 0.038 0 Bbw grestore n 120.7 58.411 m 120.7 58.323 l gsave 0 0.05 0 Bbw grestore n 119.43 58.411 m 119.43 58.323 l gsave 0 0.05 0 Bbw grestore n 118.16 58.411 m 118.16 58.323 l gsave 0 0.05 0 Bbw grestore n 116.89 58.411 m 116.89 58.323 l gsave 0 0.05 0 Bbw grestore n 115.61 58.411 m 115.61 58.323 l gsave 0 0.05 0 Bbw 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57.024 l 178.66 57.024 l 178.66 58.317 l cl 0 Fbw n 178.66 58.317 m 179.15 58.317 l 179.15 57.024 l 178.66 57.024 l 178.66 58.317 l gsave 0 0.038 0 Bbw grestore n 177.39 58.317 m 177.87 58.317 l 177.87 56.671 l 177.39 56.671 l 177.39 58.317 l cl 0 Fbw n 177.39 58.317 m 177.87 58.317 l 177.87 56.671 l 177.39 56.671 l 177.39 58.317 l gsave 0 0.038 0 Bbw grestore n 176.11 58.317 m 176.6 58.317 l 176.6 56.079 l 176.11 56.079 l 176.11 58.317 l cl 0 Fbw n 176.11 58.317 m 176.6 58.317 l 176.6 56.079 l 176.11 56.079 l 176.11 58.317 l gsave 0 0.038 0 Bbw grestore n 130.93 41.344 m 130.93 41.256 l gsave 0 0.05 0 Bbw grestore n 129.66 41.344 m 129.66 41.256 l gsave 0 0.05 0 Bbw grestore n 128.39 41.344 m 128.39 41.256 l gsave 0 0.05 0 Bbw grestore n 127.12 41.344 m 127.12 41.256 l gsave 0 0.05 0 Bbw grestore n 125.84 41.344 m 125.84 41.256 l gsave 0 0.05 0 Bbw grestore n 124.57 41.344 m 124.57 41.256 l gsave 0 0.05 0 Bbw grestore n 130.93 41.256 m 124.57 41.256 l gsave 0 0.05 0 Bbw grestore n 124.32 34.179 m 124.4 34.179 l gsave 0 0.05 0 Bbw grestore n 124.32 36.538 m 124.4 36.538 l gsave 0 0.05 0 Bbw grestore n 124.32 38.897 m 124.4 38.897 l gsave 0 0.05 0 Bbw grestore n 124.32 41.25 m 124.4 41.25 l gsave 0 0.05 0 Bbw grestore n 124.4 34.179 m 124.4 41.25 l gsave 0 0.05 0 Bbw grestore n 130.05 41.25 m 130.54 41.25 l 130.54 40.19 l 130.05 40.19 l 130.05 41.25 l cl 0 Fbw n 130.05 41.25 m 130.54 41.25 l 130.54 40.19 l 130.05 40.19 l 130.05 41.25 l gsave 0 0.038 0 Bbw grestore n 128.78 41.25 m 129.27 41.25 l 129.27 39.482 l 128.78 39.482 l 128.78 41.25 l cl 0 Fbw n 128.78 41.25 m 129.27 41.25 l 129.27 39.482 l 128.78 39.482 l 128.78 41.25 l gsave 0 0.038 0 Bbw grestore n 127.51 41.25 m 128 41.25 l 128 39.72 l 127.51 39.72 l 127.51 41.25 l cl 0 Fbw n 127.51 41.25 m 128 41.25 l 128 39.72 l 127.51 39.72 l 127.51 41.25 l gsave 0 0.038 0 Bbw grestore n 126.24 41.25 m 126.72 41.25 l 126.72 39.72 l 126.24 39.72 l 126.24 41.25 l cl 0 Fbw n 126.24 41.25 m 126.72 41.25 l 126.72 39.72 l 126.24 39.72 l 126.24 41.25 l gsave 0 0.038 0 Bbw grestore n 124.96 41.25 m 125.45 41.25 l 125.45 40.543 l 124.96 40.543 l 124.96 41.25 l cl 0 Fbw n 124.96 41.25 m 125.45 41.25 l 125.45 40.543 l 124.96 40.543 l 124.96 41.25 l gsave 0 0.038 0 Bbw grestore n 171.85 41.344 m 171.85 41.256 l gsave 0 0.05 0 Bbw grestore n 170.58 41.344 m 170.58 41.256 l gsave 0 0.05 0 Bbw grestore n 169.31 41.344 m 169.31 41.256 l gsave 0 0.05 0 Bbw grestore n 168.04 41.344 m 168.04 41.256 l gsave 0 0.05 0 Bbw grestore n 166.76 41.344 m 166.76 41.256 l gsave 0 0.05 0 Bbw grestore n 165.49 41.344 m 165.49 41.256 l gsave 0 0.05 0 Bbw grestore n 171.85 41.256 m 165.49 41.256 l gsave 0 0.05 0 Bbw grestore n 165.24 34.179 m 165.32 34.179 l gsave 0 0.05 0 Bbw grestore n 165.24 36.538 m 165.32 36.538 l gsave 0 0.05 0 Bbw grestore n 165.24 38.897 m 165.32 38.897 l gsave 0 0.05 0 Bbw grestore n 165.24 41.25 m 165.32 41.25 l gsave 0 0.05 0 Bbw grestore n 165.32 34.179 m 165.32 41.25 l gsave 0 0.05 0 Bbw grestore n 170.97 41.25 m 171.46 41.25 l 171.46 39.72 l 170.97 39.72 l 170.97 41.25 l cl 0 Fbw n 170.97 41.25 m 171.46 41.25 l 171.46 39.72 l 170.97 39.72 l 170.97 41.25 l gsave 0 0.038 0 Bbw grestore n 169.7 41.25 m 170.19 41.25 l 170.19 39.72 l 169.7 39.72 l 169.7 41.25 l cl 0 Fbw n 169.7 41.25 m 170.19 41.25 l 170.19 39.72 l 169.7 39.72 l 169.7 41.25 l gsave 0 0.038 0 Bbw grestore n 168.43 41.25 m 168.92 41.25 l 168.92 39.957 l 168.43 39.957 l 168.43 41.25 l cl 0 Fbw n 168.43 41.25 m 168.92 41.25 l 168.92 39.957 l 168.43 39.957 l 168.43 41.25 l gsave 0 0.038 0 Bbw grestore n 167.16 41.25 m 167.64 41.25 l 167.64 40.073 l 167.16 40.073 l 167.16 41.25 l cl 0 Fbw n 167.16 41.25 m 167.64 41.25 l 167.64 40.073 l 167.16 40.073 l 167.16 41.25 l gsave 0 0.038 0 Bbw grestore n 165.89 41.25 m 166.37 41.25 l 166.37 39.957 l 165.89 39.957 l 165.89 41.25 l cl 0 Fbw n 165.89 41.25 m 166.37 41.25 l 166.37 39.957 l 165.89 39.957 l 165.89 41.25 l gsave 0 0.038 0 Bbw grestore n 151.39 24.278 m 151.39 24.189 l gsave 0 0.05 0 Bbw grestore n 150.12 24.278 m 150.12 24.189 l gsave 0 0.05 0 Bbw grestore n 148.85 24.278 m 148.85 24.189 l gsave 0 0.05 0 Bbw grestore n 147.58 24.278 m 147.58 24.189 l gsave 0 0.05 0 Bbw grestore n 146.31 24.278 m 146.31 24.189 l gsave 0 0.05 0 Bbw grestore n 145.03 24.278 m 145.03 24.189 l gsave 0 0.05 0 Bbw grestore n 151.39 24.189 m 145.03 24.189 l gsave 0 0.05 0 Bbw grestore n 144.78 17.111 m 144.86 17.111 l gsave 0 0.05 0 Bbw grestore n 144.78 19.47 m 144.86 19.47 l gsave 0 0.05 0 Bbw grestore n 144.78 21.83 m 144.86 21.83 l gsave 0 0.05 0 Bbw grestore n 144.78 24.183 m 144.86 24.183 l gsave 0 0.05 0 Bbw grestore n 144.86 17.111 m 144.86 24.183 l gsave 0 0.05 0 Bbw grestore n 150.51 24.183 m 151 24.183 l 151 23.244 l 150.51 23.244 l 150.51 24.183 l cl 0 Fbw n 150.51 24.183 m 151 24.183 l 151 23.244 l 150.51 23.244 l 150.51 24.183 l gsave 0 0.038 0 Bbw grestore n 149.24 24.183 m 149.73 24.183 l 149.73 22.416 l 149.24 22.416 l 149.24 24.183 l cl 0 Fbw n 149.24 24.183 m 149.73 24.183 l 149.73 22.416 l 149.24 22.416 l 149.24 24.183 l gsave 0 0.038 0 Bbw grestore n 147.97 24.183 m 148.46 24.183 l 148.46 22.537 l 147.97 22.537 l 147.97 24.183 l cl 0 Fbw n 147.97 24.183 m 148.46 24.183 l 148.46 22.537 l 147.97 22.537 l 147.97 24.183 l gsave 0 0.038 0 Bbw grestore n 146.7 24.183 m 147.19 24.183 l 147.19 22.652 l 146.7 22.652 l 146.7 24.183 l cl 0 Fbw n 146.7 24.183 m 147.19 24.183 l 147.19 22.652 l 146.7 22.652 l 146.7 24.183 l gsave 0 0.038 0 Bbw grestore n 145.43 24.183 m 145.91 24.183 l 145.91 23.36 l 145.43 23.36 l 145.43 24.183 l cl 0 Fbw n 145.43 24.183 m 145.91 24.183 l 145.91 23.36 l 145.43 23.36 l 145.43 24.183 l gsave 0 0.038 0 Bbw grestore n 19.311 151.86 m 19.311 151.77 l gsave 0 0.05 0 Bbw grestore n 17.934 151.86 m 17.934 151.77 l gsave 0 0.05 0 Bbw grestore n 16.557 151.86 m 16.557 151.77 l gsave 0 0.05 0 Bbw grestore n 15.18 151.86 m 15.18 151.77 l gsave 0 0.05 0 Bbw grestore n 13.802 151.86 m 13.802 151.77 l gsave 0 0.05 0 Bbw grestore n 12.42 151.86 m 12.42 151.77 l gsave 0 0.05 0 Bbw grestore n 19.311 151.77 m 12.42 151.77 l gsave 0 0.05 0 Bbw grestore n 12.151 144.57 m 12.237 144.57 l gsave 0 0.05 0 Bbw grestore n 12.151 146.97 m 12.237 146.97 l gsave 0 0.05 0 Bbw grestore n 12.151 149.37 m 12.237 149.37 l gsave 0 0.05 0 Bbw grestore n 12.151 151.77 m 12.237 151.77 l gsave 0 0.05 0 Bbw grestore n 12.237 144.57 m 12.237 151.77 l gsave 0 0.05 0 Bbw grestore n 18.359 151.77 m 18.887 151.77 l 18.887 149.61 l 18.359 149.61 l 18.359 151.77 l cl 0 Fbw n 18.359 151.77 m 18.887 151.77 l 18.887 149.61 l 18.359 149.61 l 18.359 151.77 l gsave 0 0.038 0 Bbw grestore n 16.983 151.77 m 17.509 151.77 l 17.509 150.93 l 16.983 150.93 l 16.983 151.77 l cl 0 Fbw n 16.983 151.77 m 17.509 151.77 l 17.509 150.93 l 16.983 150.93 l 16.983 151.77 l gsave 0 0.038 0 Bbw grestore n 15.605 151.77 m 16.132 151.77 l 16.132 151.05 l 15.605 151.05 l 15.605 151.77 l cl 0 Fbw n 15.605 151.77 m 16.132 151.77 l 16.132 151.05 l 15.605 151.05 l 15.605 151.77 l gsave 0 0.038 0 Bbw grestore n 14.227 151.77 m 14.754 151.77 l 14.754 150.93 l 14.227 150.93 l 14.227 151.77 l cl 0 Fbw n 14.227 151.77 m 14.754 151.77 l 14.754 150.93 l 14.227 150.93 l 14.227 151.77 l gsave 0 0.038 0 Bbw grestore n 12.85 151.77 m 13.378 151.77 l 13.378 149.61 l 12.85 149.61 l 12.85 151.77 l cl 0 Fbw n 12.85 151.77 m 13.378 151.77 l 13.378 149.61 l 12.85 149.61 l 12.85 151.77 l gsave 0 0.038 0 Bbw grestore n 31.379 108.78 m 53.545 100.66 l gsave 0 0.074 0 Bbw grestore n 75.125 108.58 m 53.545 100.66 l gsave 0 0.074 0 Bbw grestore n 20.205 126.2 m 31.379 118.08 l gsave 0 0.074 0 Bbw grestore n 42.172 126.2 m 31.568 118.08 l gsave 0 0.074 0 Bbw grestore n 64.332 126.6 m 75.313 118.03 l gsave 0 0.074 0 Bbw grestore n 86.299 126.2 m 75.313 118.03 l gsave 0 0.074 0 Bbw grestore n 16.041 143.42 m 20.205 134.91 l gsave 0 0.074 0 Bbw grestore n 25.322 143.62 m 20.205 134.94 l gsave 0 0.074 0 Bbw grestore n 38.379 143.23 m 42.543 134.94 l gsave 0 0.074 0 Bbw grestore n 60.351 143.03 m 64.515 134.94 l gsave 0 0.074 0 Bbw grestore n 82.695 143.03 m 86.859 134.94 l gsave 0 0.074 0 Bbw grestore n 47.848 143.42 m 42.58 134.94 l gsave 0 0.074 0 Bbw grestore n 69.82 143.63 m 64.515 134.94 l gsave 0 0.074 0 Bbw grestore n 91.976 143.43 m 86.859 134.94 l gsave 0 0.074 0 Bbw grestore n 30.389 151.86 m 30.389 151.77 l gsave 0 0.05 0 Bbw grestore n 29.012 151.86 m 29.012 151.77 l gsave 0 0.05 0 Bbw grestore n 27.635 151.86 m 27.635 151.77 l gsave 0 0.05 0 Bbw grestore n 26.257 151.86 m 26.257 151.77 l gsave 0 0.05 0 Bbw grestore n 24.881 151.86 m 24.881 151.77 l gsave 0 0.05 0 Bbw grestore n 23.498 151.86 m 23.498 151.77 l gsave 0 0.05 0 Bbw grestore n 30.389 151.77 m 23.498 151.77 l gsave 0 0.05 0 Bbw grestore n 23.228 144.57 m 23.315 144.57 l gsave 0 0.05 0 Bbw grestore n 23.228 146.97 m 23.315 146.97 l gsave 0 0.05 0 Bbw grestore n 23.228 149.37 m 23.315 149.37 l gsave 0 0.05 0 Bbw grestore n 23.228 151.77 m 23.315 151.77 l gsave 0 0.05 0 Bbw grestore n 23.315 144.57 m 23.315 151.77 l gsave 0 0.05 0 Bbw grestore n 29.437 151.77 m 29.965 151.77 l 29.965 147.21 l 29.437 147.21 l 29.437 151.77 l cl 0 Fbw n 29.437 151.77 m 29.965 151.77 l 29.965 147.21 l 29.437 147.21 l 29.437 151.77 l gsave 0 0.038 0 Bbw grestore n 28.059 151.77 m 28.587 151.77 l 28.587 151.29 l 28.059 151.29 l 28.059 151.77 l cl 0 Fbw n 28.059 151.77 m 28.587 151.77 l 28.587 151.29 l 28.059 151.29 l 28.059 151.77 l gsave 0 0.038 0 Bbw grestore n 26.683 151.77 m 27.209 151.77 l 27.209 151.41 l 26.683 151.41 l 26.683 151.77 l cl 0 Fbw n 26.683 151.77 m 27.209 151.77 l 27.209 151.41 l 26.683 151.41 l 26.683 151.77 l gsave 0 0.038 0 Bbw grestore n 25.305 151.77 m 25.833 151.77 l 25.833 151.41 l 25.305 151.41 l 25.305 151.77 l cl 0 Fbw n 25.305 151.77 m 25.833 151.77 l 25.833 151.41 l 25.305 151.41 l 25.305 151.77 l gsave 0 0.038 0 Bbw grestore n 23.928 151.77 m 24.455 151.77 l 24.455 150.57 l 23.928 150.57 l 23.928 151.77 l cl 0 Fbw n 23.928 151.77 m 24.455 151.77 l 24.455 150.57 l 23.928 150.57 l 23.928 151.77 l gsave 0 0.038 0 Bbw grestore n 41.468 151.86 m 41.468 151.77 l gsave 0 0.05 0 Bbw grestore n 40.09 151.86 m 40.09 151.77 l gsave 0 0.05 0 Bbw grestore n 38.712 151.86 m 38.712 151.77 l gsave 0 0.05 0 Bbw grestore n 37.335 151.86 m 37.335 151.77 l gsave 0 0.05 0 Bbw grestore n 35.958 151.86 m 35.958 151.77 l gsave 0 0.05 0 Bbw grestore n 34.575 151.86 m 34.575 151.77 l gsave 0 0.05 0 Bbw grestore n 41.468 151.77 m 34.575 151.77 l gsave 0 0.05 0 Bbw grestore n 34.306 144.57 m 34.392 144.57 l gsave 0 0.05 0 Bbw grestore n 34.306 146.97 m 34.392 146.97 l gsave 0 0.05 0 Bbw grestore n 34.306 149.37 m 34.392 149.37 l gsave 0 0.05 0 Bbw grestore n 34.306 151.77 m 34.392 151.77 l gsave 0 0.05 0 Bbw grestore n 34.392 144.57 m 34.392 151.77 l gsave 0 0.05 0 Bbw grestore n 40.515 151.77 m 41.042 151.77 l 41.042 150.33 l 40.515 150.33 l 40.515 151.77 l cl 0 Fbw n 40.515 151.77 m 41.042 151.77 l 41.042 150.33 l 40.515 150.33 l 40.515 151.77 l gsave 0 0.038 0 Bbw grestore n 39.138 151.77 m 39.665 151.77 l 39.665 151.41 l 39.138 151.41 l 39.138 151.77 l cl 0 Fbw n 39.138 151.77 m 39.665 151.77 l 39.665 151.41 l 39.138 151.41 l 39.138 151.77 l gsave 0 0.038 0 Bbw grestore n 37.76 151.77 m 38.287 151.77 l 38.287 151.41 l 37.76 151.41 l 37.76 151.77 l cl 0 Fbw n 37.76 151.77 m 38.287 151.77 l 38.287 151.41 l 37.76 151.41 l 37.76 151.77 l gsave 0 0.038 0 Bbw grestore n 36.383 151.77 m 36.91 151.77 l 36.91 151.29 l 36.383 151.29 l 36.383 151.77 l cl 0 Fbw n 36.383 151.77 m 36.91 151.77 l 36.91 151.29 l 36.383 151.29 l 36.383 151.77 l gsave 0 0.038 0 Bbw grestore n 35.006 151.77 m 35.533 151.77 l 35.533 147.45 l 35.006 147.45 l 35.006 151.77 l cl 0 Fbw n 35.006 151.77 m 35.533 151.77 l 35.533 147.45 l 35.006 147.45 l 35.006 151.77 l gsave 0 0.038 0 Bbw grestore n 52.544 151.86 m 52.544 151.77 l gsave 0 0.05 0 Bbw grestore n 51.168 151.86 m 51.168 151.77 l gsave 0 0.05 0 Bbw grestore n 49.79 151.86 m 49.79 151.77 l gsave 0 0.05 0 Bbw grestore n 48.413 151.86 m 48.413 151.77 l gsave 0 0.05 0 Bbw grestore n 47.035 151.86 m 47.035 151.77 l gsave 0 0.05 0 Bbw grestore n 45.653 151.86 m 45.653 151.77 l gsave 0 0.05 0 Bbw grestore n 52.544 151.77 m 45.653 151.77 l gsave 0 0.05 0 Bbw grestore n 45.384 144.57 m 45.47 144.57 l gsave 0 0.05 0 Bbw grestore n 45.384 146.97 m 45.47 146.97 l gsave 0 0.05 0 Bbw grestore n 45.384 149.37 m 45.47 149.37 l gsave 0 0.05 0 Bbw grestore n 45.384 151.77 m 45.47 151.77 l gsave 0 0.05 0 Bbw grestore n 45.47 144.57 m 45.47 151.77 l gsave 0 0.05 0 Bbw grestore n 51.592 151.77 m 52.12 151.77 l 52.12 150.57 l 51.592 150.57 l 51.592 151.77 l cl 0 Fbw n 51.592 151.77 m 52.12 151.77 l 52.12 150.57 l 51.592 150.57 l 51.592 151.77 l gsave 0 0.038 0 Bbw grestore n 50.215 151.77 m 50.742 151.77 l 50.742 149.97 l 50.215 149.97 l 50.215 151.77 l cl 0 Fbw n 50.215 151.77 m 50.742 151.77 l 50.742 149.97 l 50.215 149.97 l 50.215 151.77 l gsave 0 0.038 0 Bbw grestore n 48.838 151.77 m 49.365 151.77 l 49.365 150.21 l 48.838 150.21 l 48.838 151.77 l cl 0 Fbw n 48.838 151.77 m 49.365 151.77 l 49.365 150.21 l 48.838 150.21 l 48.838 151.77 l gsave 0 0.038 0 Bbw grestore n 47.46 151.77 m 47.988 151.77 l 47.988 150.45 l 47.46 150.45 l 47.46 151.77 l cl 0 Fbw n 47.46 151.77 m 47.988 151.77 l 47.988 150.45 l 47.46 150.45 l 47.46 151.77 l gsave 0 0.038 0 Bbw grestore n 46.083 151.77 m 46.611 151.77 l 46.611 150.69 l 46.083 150.69 l 46.083 151.77 l cl 0 Fbw n 46.083 151.77 m 46.611 151.77 l 46.611 150.69 l 46.083 150.69 l 46.083 151.77 l gsave 0 0.038 0 Bbw grestore n 63.623 151.86 m 63.623 151.77 l gsave 0 0.05 0 Bbw grestore n 62.245 151.86 m 62.245 151.77 l gsave 0 0.05 0 Bbw grestore n 60.868 151.86 m 60.868 151.77 l gsave 0 0.05 0 Bbw grestore n 59.491 151.86 m 59.491 151.77 l gsave 0 0.05 0 Bbw grestore n 58.113 151.86 m 58.113 151.77 l gsave 0 0.05 0 Bbw grestore n 56.73 151.86 m 56.73 151.77 l gsave 0 0.05 0 Bbw grestore n 63.623 151.77 m 56.73 151.77 l gsave 0 0.05 0 Bbw grestore n 56.461 144.57 m 56.548 144.57 l gsave 0 0.05 0 Bbw grestore n 56.461 146.97 m 56.548 146.97 l gsave 0 0.05 0 Bbw grestore n 56.461 149.37 m 56.548 149.37 l gsave 0 0.05 0 Bbw grestore n 56.461 151.77 m 56.548 151.77 l gsave 0 0.05 0 Bbw grestore n 56.548 144.57 m 56.548 151.77 l gsave 0 0.05 0 Bbw grestore n 62.671 151.77 m 63.197 151.77 l 63.197 149.61 l 62.671 149.61 l 62.671 151.77 l cl 0 Fbw n 62.671 151.77 m 63.197 151.77 l 63.197 149.61 l 62.671 149.61 l 62.671 151.77 l gsave 0 0.038 0 Bbw grestore n 61.293 151.77 m 61.82 151.77 l 61.82 150.33 l 61.293 150.33 l 61.293 151.77 l cl 0 Fbw n 61.293 151.77 m 61.82 151.77 l 61.82 150.33 l 61.293 150.33 l 61.293 151.77 l gsave 0 0.038 0 Bbw grestore n 59.915 151.77 m 60.443 151.77 l 60.443 150.69 l 59.915 150.69 l 59.915 151.77 l cl 0 Fbw n 59.915 151.77 m 60.443 151.77 l 60.443 150.69 l 59.915 150.69 l 59.915 151.77 l gsave 0 0.038 0 Bbw grestore n 58.538 151.77 m 59.066 151.77 l 59.066 150.81 l 58.538 150.81 l 58.538 151.77 l cl 0 Fbw n 58.538 151.77 m 59.066 151.77 l 59.066 150.81 l 58.538 150.81 l 58.538 151.77 l gsave 0 0.038 0 Bbw grestore n 57.161 151.77 m 57.688 151.77 l 57.688 150.57 l 57.161 150.57 l 57.161 151.77 l cl 0 Fbw n 57.161 151.77 m 57.688 151.77 l 57.688 150.57 l 57.161 150.57 l 57.161 151.77 l gsave 0 0.038 0 Bbw grestore n 74.706 151.86 m 74.706 151.77 l gsave 0 0.05 0 Bbw grestore n 73.328 151.86 m 73.328 151.77 l gsave 0 0.05 0 Bbw grestore n 71.951 151.86 m 71.951 151.77 l gsave 0 0.05 0 Bbw grestore n 70.573 151.86 m 70.573 151.77 l gsave 0 0.05 0 Bbw grestore n 69.197 151.86 m 69.197 151.77 l gsave 0 0.05 0 Bbw grestore n 67.814 151.86 m 67.814 151.77 l gsave 0 0.05 0 Bbw grestore n 74.706 151.77 m 67.814 151.77 l gsave 0 0.05 0 Bbw grestore n 67.544 144.57 m 67.631 144.57 l gsave 0 0.05 0 Bbw grestore n 67.544 146.97 m 67.631 146.97 l gsave 0 0.05 0 Bbw grestore n 67.544 149.37 m 67.631 149.37 l gsave 0 0.05 0 Bbw grestore n 67.544 151.77 m 67.631 151.77 l gsave 0 0.05 0 Bbw grestore n 67.631 144.57 m 67.631 151.77 l gsave 0 0.05 0 Bbw grestore n 73.753 151.77 m 74.281 151.77 l 74.281 151.05 l 73.753 151.05 l 73.753 151.77 l cl 0 Fbw n 73.753 151.77 m 74.281 151.77 l 74.281 151.05 l 73.753 151.05 l 73.753 151.77 l gsave 0 0.038 0 Bbw grestore n 72.376 151.77 m 72.903 151.77 l 72.903 150.69 l 72.376 150.69 l 72.376 151.77 l cl 0 Fbw n 72.376 151.77 m 72.903 151.77 l 72.903 150.69 l 72.376 150.69 l 72.376 151.77 l gsave 0 0.038 0 Bbw grestore n 70.999 151.77 m 71.526 151.77 l 71.526 150.57 l 70.999 150.57 l 70.999 151.77 l cl 0 Fbw n 70.999 151.77 m 71.526 151.77 l 71.526 150.57 l 70.999 150.57 l 70.999 151.77 l gsave 0 0.038 0 Bbw grestore n 69.621 151.77 m 70.149 151.77 l 70.149 150.21 l 69.621 150.21 l 69.621 151.77 l cl 0 Fbw n 69.621 151.77 m 70.149 151.77 l 70.149 150.21 l 69.621 150.21 l 69.621 151.77 l gsave 0 0.038 0 Bbw grestore n 68.244 151.77 m 68.771 151.77 l 68.771 149.61 l 68.244 149.61 l 68.244 151.77 l cl 0 Fbw n 68.244 151.77 m 68.771 151.77 l 68.771 149.61 l 68.244 149.61 l 68.244 151.77 l gsave 0 0.038 0 Bbw grestore n 85.784 151.86 m 85.784 151.77 l gsave 0 0.05 0 Bbw grestore n 84.406 151.86 m 84.406 151.77 l gsave 0 0.05 0 Bbw grestore n 83.028 151.86 m 83.028 151.77 l gsave 0 0.05 0 Bbw grestore n 81.651 151.86 m 81.651 151.77 l gsave 0 0.05 0 Bbw grestore n 80.274 151.86 m 80.274 151.77 l gsave 0 0.05 0 Bbw grestore n 78.891 151.86 m 78.891 151.77 l gsave 0 0.05 0 Bbw grestore n 85.784 151.77 m 78.891 151.77 l gsave 0 0.05 0 Bbw grestore n 78.622 144.57 m 78.708 144.57 l gsave 0 0.05 0 Bbw grestore n 78.622 146.97 m 78.708 146.97 l gsave 0 0.05 0 Bbw grestore n 78.622 149.37 m 78.708 149.37 l gsave 0 0.05 0 Bbw grestore n 78.622 151.77 m 78.708 151.77 l gsave 0 0.05 0 Bbw grestore n 78.708 144.57 m 78.708 151.77 l gsave 0 0.05 0 Bbw grestore n 84.831 151.77 m 85.358 151.77 l 85.358 149.61 l 84.831 149.61 l 84.831 151.77 l cl 0 Fbw n 84.831 151.77 m 85.358 151.77 l 85.358 149.61 l 84.831 149.61 l 84.831 151.77 l gsave 0 0.038 0 Bbw grestore n 83.454 151.77 m 83.981 151.77 l 83.981 151.29 l 83.454 151.29 l 83.454 151.77 l cl 0 Fbw n 83.454 151.77 m 83.981 151.77 l 83.981 151.29 l 83.454 151.29 l 83.454 151.77 l gsave 0 0.038 0 Bbw grestore n 82.076 151.77 m 82.603 151.77 l 82.603 151.29 l 82.076 151.29 l 82.076 151.77 l cl 0 Fbw n 82.076 151.77 m 82.603 151.77 l 82.603 151.29 l 82.076 151.29 l 82.076 151.77 l gsave 0 0.038 0 Bbw grestore n 80.699 151.77 m 81.226 151.77 l 81.226 151.17 l 80.699 151.17 l 80.699 151.77 l cl 0 Fbw n 80.699 151.77 m 81.226 151.77 l 81.226 151.17 l 80.699 151.17 l 80.699 151.77 l gsave 0 0.038 0 Bbw grestore n 79.322 151.77 m 79.849 151.77 l 79.849 148.53 l 79.322 148.53 l 79.322 151.77 l cl 0 Fbw n 79.322 151.77 m 79.849 151.77 l 79.849 148.53 l 79.322 148.53 l 79.322 151.77 l gsave 0 0.038 0 Bbw grestore n 96.861 151.86 m 96.861 151.77 l gsave 0 0.05 0 Bbw grestore n 95.484 151.86 m 95.484 151.77 l gsave 0 0.05 0 Bbw grestore n 94.106 151.86 m 94.106 151.77 l gsave 0 0.05 0 Bbw grestore n 92.728 151.86 m 92.728 151.77 l gsave 0 0.05 0 Bbw grestore n 91.352 151.86 m 91.352 151.77 l gsave 0 0.05 0 Bbw grestore n 89.969 151.86 m 89.969 151.77 l gsave 0 0.05 0 Bbw grestore n 96.861 151.77 m 89.969 151.77 l gsave 0 0.05 0 Bbw grestore n 89.7 144.57 m 89.786 144.57 l gsave 0 0.05 0 Bbw grestore n 89.7 146.97 m 89.786 146.97 l gsave 0 0.05 0 Bbw grestore n 89.7 149.37 m 89.786 149.37 l gsave 0 0.05 0 Bbw grestore n 89.7 151.77 m 89.786 151.77 l gsave 0 0.05 0 Bbw grestore n 89.786 144.57 m 89.786 151.77 l gsave 0 0.05 0 Bbw grestore n 95.908 151.77 m 96.436 151.77 l 96.436 148.05 l 95.908 148.05 l 95.908 151.77 l cl 0 Fbw n 95.908 151.77 m 96.436 151.77 l 96.436 148.05 l 95.908 148.05 l 95.908 151.77 l gsave 0 0.038 0 Bbw grestore n 94.531 151.77 m 95.058 151.77 l 95.058 151.29 l 94.531 151.29 l 94.531 151.77 l cl 0 Fbw n 94.531 151.77 m 95.058 151.77 l 95.058 151.29 l 94.531 151.29 l 94.531 151.77 l gsave 0 0.038 0 Bbw grestore n 93.154 151.77 m 93.682 151.77 l 93.682 151.41 l 93.154 151.41 l 93.154 151.77 l cl 0 Fbw n 93.154 151.77 m 93.682 151.77 l 93.682 151.41 l 93.154 151.41 l 93.154 151.77 l gsave 0 0.038 0 Bbw grestore n 91.776 151.77 m 92.304 151.77 l 92.304 151.29 l 91.776 151.29 l 91.776 151.77 l cl 0 Fbw n 91.776 151.77 m 92.304 151.77 l 92.304 151.29 l 91.776 151.29 l 91.776 151.77 l gsave 0 0.038 0 Bbw grestore n 90.399 151.77 m 90.926 151.77 l 90.926 149.85 l 90.399 149.85 l 90.399 151.77 l cl 0 Fbw n 90.399 151.77 m 90.926 151.77 l 90.926 149.85 l 90.399 149.85 l 90.399 151.77 l gsave 0 0.038 0 Bbw grestore n 23.739 134.5 m 23.739 134.41 l gsave 0 0.05 0 Bbw grestore n 22.363 134.5 m 22.363 134.41 l gsave 0 0.05 0 Bbw grestore n 20.985 134.5 m 20.985 134.41 l gsave 0 0.05 0 Bbw grestore n 19.608 134.5 m 19.608 134.41 l gsave 0 0.05 0 Bbw grestore n 18.23 134.5 m 18.23 134.41 l gsave 0 0.05 0 Bbw grestore n 16.848 134.5 m 16.848 134.41 l gsave 0 0.05 0 Bbw grestore n 23.739 134.41 m 16.848 134.41 l gsave 0 0.05 0 Bbw grestore n 16.578 127.2 m 16.664 127.2 l gsave 0 0.05 0 Bbw grestore n 16.578 129.6 m 16.664 129.6 l gsave 0 0.05 0 Bbw grestore n 16.578 132.01 m 16.664 132.01 l gsave 0 0.05 0 Bbw grestore n 16.578 134.4 m 16.664 134.4 l gsave 0 0.05 0 Bbw grestore n 16.664 127.2 m 16.664 134.4 l gsave 0 0.05 0 Bbw grestore n 22.787 134.4 m 23.315 134.4 l 23.315 132.01 l 22.787 132.01 l 22.787 134.4 l cl 0 Fbw n 22.787 134.4 m 23.315 134.4 l 23.315 132.01 l 22.787 132.01 l 22.787 134.4 l gsave 0 0.038 0 Bbw grestore n 21.41 134.4 m 21.937 134.4 l 21.937 133.8 l 21.41 133.8 l 21.41 134.4 l cl 0 Fbw n 21.41 134.4 m 21.937 134.4 l 21.937 133.8 l 21.41 133.8 l 21.41 134.4 l gsave 0 0.038 0 Bbw grestore n 20.033 134.4 m 20.56 134.4 l 20.56 133.92 l 20.033 133.92 l 20.033 134.4 l cl 0 Fbw n 20.033 134.4 m 20.56 134.4 l 20.56 133.92 l 20.033 133.92 l 20.033 134.4 l gsave 0 0.038 0 Bbw grestore n 18.655 134.4 m 19.182 134.4 l 19.182 133.8 l 18.655 133.8 l 18.655 134.4 l cl 0 Fbw n 18.655 134.4 m 19.182 134.4 l 19.182 133.8 l 18.655 133.8 l 18.655 134.4 l gsave 0 0.038 0 Bbw grestore n 17.278 134.4 m 17.806 134.4 l 17.806 131.65 l 17.278 131.65 l 17.278 134.4 l cl 0 Fbw n 17.278 134.4 m 17.806 134.4 l 17.806 131.65 l 17.278 131.65 l 17.278 134.4 l gsave 0 0.038 0 Bbw grestore n 45.9 134.5 m 45.9 134.41 l gsave 0 0.05 0 Bbw grestore n 44.523 134.5 m 44.523 134.41 l gsave 0 0.05 0 Bbw grestore n 43.146 134.5 m 43.146 134.41 l gsave 0 0.05 0 Bbw grestore n 41.768 134.5 m 41.768 134.41 l gsave 0 0.05 0 Bbw grestore n 40.391 134.5 m 40.391 134.41 l gsave 0 0.05 0 Bbw grestore n 39.009 134.5 m 39.009 134.41 l gsave 0 0.05 0 Bbw grestore n 45.9 134.41 m 39.009 134.41 l gsave 0 0.05 0 Bbw grestore n 38.739 127.2 m 38.825 127.2 l gsave 0 0.05 0 Bbw grestore n 38.739 129.6 m 38.825 129.6 l gsave 0 0.05 0 Bbw grestore n 38.739 132.01 m 38.825 132.01 l gsave 0 0.05 0 Bbw grestore n 38.739 134.4 m 38.825 134.4 l gsave 0 0.05 0 Bbw grestore n 38.825 127.2 m 38.825 134.4 l gsave 0 0.05 0 Bbw grestore n 44.948 134.4 m 45.475 134.4 l 45.475 132.72 l 44.948 132.72 l 44.948 134.4 l cl 0 Fbw n 44.948 134.4 m 45.475 134.4 l 45.475 132.72 l 44.948 132.72 l 44.948 134.4 l gsave 0 0.038 0 Bbw grestore n 43.57 134.4 m 44.098 134.4 l 44.098 133.44 l 43.57 133.44 l 43.57 134.4 l cl 0 Fbw n 43.57 134.4 m 44.098 134.4 l 44.098 133.44 l 43.57 133.44 l 43.57 134.4 l gsave 0 0.038 0 Bbw grestore n 42.194 134.4 m 42.721 134.4 l 42.721 133.44 l 42.194 133.44 l 42.194 134.4 l cl 0 Fbw n 42.194 134.4 m 42.721 134.4 l 42.721 133.44 l 42.194 133.44 l 42.194 134.4 l gsave 0 0.038 0 Bbw grestore n 40.816 134.4 m 41.343 134.4 l 41.343 133.32 l 40.816 133.32 l 40.816 134.4 l cl 0 Fbw n 40.816 134.4 m 41.343 134.4 l 41.343 133.32 l 40.816 133.32 l 40.816 134.4 l gsave 0 0.038 0 Bbw grestore n 39.439 134.4 m 39.965 134.4 l 39.965 132.12 l 39.439 132.12 l 39.439 134.4 l cl 0 Fbw n 39.439 134.4 m 39.965 134.4 l 39.965 132.12 l 39.439 132.12 l 39.439 134.4 l gsave 0 0.038 0 Bbw grestore n 68.056 134.5 m 68.056 134.41 l gsave 0 0.05 0 Bbw grestore n 66.679 134.5 m 66.679 134.41 l gsave 0 0.05 0 Bbw grestore n 65.301 134.5 m 65.301 134.41 l gsave 0 0.05 0 Bbw grestore n 63.924 134.5 m 63.924 134.41 l gsave 0 0.05 0 Bbw grestore n 62.546 134.5 m 62.546 134.41 l gsave 0 0.05 0 Bbw grestore n 61.164 134.5 m 61.164 134.41 l gsave 0 0.05 0 Bbw grestore n 68.056 134.41 m 61.164 134.41 l gsave 0 0.05 0 Bbw grestore n 60.894 127.2 m 60.98 127.2 l gsave 0 0.05 0 Bbw grestore n 60.894 129.6 m 60.98 129.6 l gsave 0 0.05 0 Bbw grestore n 60.894 132.01 m 60.98 132.01 l gsave 0 0.05 0 Bbw grestore n 60.894 134.4 m 60.98 134.4 l gsave 0 0.05 0 Bbw grestore n 60.98 127.2 m 60.98 134.4 l gsave 0 0.05 0 Bbw grestore n 67.104 134.4 m 67.631 134.4 l 67.631 132.96 l 67.104 132.96 l 67.104 134.4 l cl 0 Fbw n 67.104 134.4 m 67.631 134.4 l 67.631 132.96 l 67.104 132.96 l 67.104 134.4 l gsave 0 0.038 0 Bbw grestore n 65.726 134.4 m 66.253 134.4 l 66.253 133.08 l 65.726 133.08 l 65.726 134.4 l cl 0 Fbw n 65.726 134.4 m 66.253 134.4 l 66.253 133.08 l 65.726 133.08 l 65.726 134.4 l gsave 0 0.038 0 Bbw grestore n 64.349 134.4 m 64.876 134.4 l 64.876 132.84 l 64.349 132.84 l 64.349 134.4 l cl 0 Fbw n 64.349 134.4 m 64.876 134.4 l 64.876 132.84 l 64.349 132.84 l 64.349 134.4 l gsave 0 0.038 0 Bbw grestore n 62.971 134.4 m 63.498 134.4 l 63.498 132.84 l 62.971 132.84 l 62.971 134.4 l cl 0 Fbw n 62.971 134.4 m 63.498 134.4 l 63.498 132.84 l 62.971 132.84 l 62.971 134.4 l gsave 0 0.038 0 Bbw grestore n 61.595 134.4 m 62.122 134.4 l 62.122 133.2 l 61.595 133.2 l 61.595 134.4 l cl 0 Fbw n 61.595 134.4 m 62.122 134.4 l 62.122 133.2 l 61.595 133.2 l 61.595 134.4 l gsave 0 0.038 0 Bbw grestore n 90.211 134.5 m 90.211 134.41 l gsave 0 0.05 0 Bbw grestore n 88.834 134.5 m 88.834 134.41 l gsave 0 0.05 0 Bbw grestore n 87.456 134.5 m 87.456 134.41 l gsave 0 0.05 0 Bbw grestore n 86.08 134.5 m 86.08 134.41 l gsave 0 0.05 0 Bbw grestore n 84.702 134.5 m 84.702 134.41 l gsave 0 0.05 0 Bbw grestore n 83.319 134.5 m 83.319 134.41 l gsave 0 0.05 0 Bbw grestore n 90.211 134.41 m 83.319 134.41 l gsave 0 0.05 0 Bbw grestore n 83.05 127.2 m 83.136 127.2 l gsave 0 0.05 0 Bbw grestore n 83.05 129.6 m 83.136 129.6 l gsave 0 0.05 0 Bbw grestore n 83.05 132.01 m 83.136 132.01 l gsave 0 0.05 0 Bbw grestore n 83.05 134.4 m 83.136 134.4 l gsave 0 0.05 0 Bbw grestore n 83.136 127.2 m 83.136 134.4 l gsave 0 0.05 0 Bbw grestore n 89.258 134.4 m 89.786 134.4 l 89.786 133.44 l 89.258 133.44 l 89.258 134.4 l cl 0 Fbw n 89.258 134.4 m 89.786 134.4 l 89.786 133.44 l 89.258 133.44 l 89.258 134.4 l gsave 0 0.038 0 Bbw grestore n 87.882 134.4 m 88.409 134.4 l 88.409 134.04 l 87.882 134.04 l 87.882 134.4 l cl 0 Fbw n 87.882 134.4 m 88.409 134.4 l 88.409 134.04 l 87.882 134.04 l 87.882 134.4 l gsave 0 0.038 0 Bbw grestore n 86.504 134.4 m 87.032 134.4 l 87.032 134.04 l 86.504 134.04 l 86.504 134.4 l cl 0 Fbw n 86.504 134.4 m 87.032 134.4 l 87.032 134.04 l 86.504 134.04 l 86.504 134.4 l gsave 0 0.038 0 Bbw grestore n 85.127 134.4 m 85.654 134.4 l 85.654 133.8 l 85.127 133.8 l 85.127 134.4 l cl 0 Fbw n 85.127 134.4 m 85.654 134.4 l 85.654 133.8 l 85.127 133.8 l 85.127 134.4 l gsave 0 0.038 0 Bbw grestore n 83.75 134.4 m 84.277 134.4 l 84.277 129.85 l 83.75 129.85 l 83.75 134.4 l cl 0 Fbw n 83.75 134.4 m 84.277 134.4 l 84.277 129.85 l 83.75 129.85 l 83.75 134.4 l gsave 0 0.038 0 Bbw grestore n 34.817 117.13 m 34.817 117.04 l gsave 0 0.05 0 Bbw grestore n 33.439 117.13 m 33.439 117.04 l gsave 0 0.05 0 Bbw grestore n 32.063 117.13 m 32.063 117.04 l gsave 0 0.05 0 Bbw grestore n 30.685 117.13 m 30.685 117.04 l gsave 0 0.05 0 Bbw grestore n 29.308 117.13 m 29.308 117.04 l gsave 0 0.05 0 Bbw grestore n 27.925 117.13 m 27.925 117.04 l gsave 0 0.05 0 Bbw grestore n 34.817 117.04 m 27.925 117.04 l gsave 0 0.05 0 Bbw grestore n 27.656 109.83 m 27.743 109.83 l gsave 0 0.05 0 Bbw grestore n 27.656 112.23 m 27.743 112.23 l gsave 0 0.05 0 Bbw grestore n 27.656 114.63 m 27.743 114.63 l gsave 0 0.05 0 Bbw grestore n 27.656 117.03 m 27.743 117.03 l gsave 0 0.05 0 Bbw grestore n 27.743 109.83 m 27.743 117.03 l gsave 0 0.05 0 Bbw grestore n 33.865 117.03 m 34.392 117.03 l 34.392 114.51 l 33.865 114.51 l 33.865 117.03 l cl 0 Fbw n 33.865 117.03 m 34.392 117.03 l 34.392 114.51 l 33.865 114.51 l 33.865 117.03 l gsave 0 0.038 0 Bbw grestore n 32.487 117.03 m 33.015 117.03 l 33.015 115.95 l 32.487 115.95 l 32.487 117.03 l cl 0 Fbw n 32.487 117.03 m 33.015 117.03 l 33.015 115.95 l 32.487 115.95 l 32.487 117.03 l gsave 0 0.038 0 Bbw grestore n 31.111 117.03 m 31.637 117.03 l 31.637 116.19 l 31.111 116.19 l 31.111 117.03 l cl 0 Fbw n 31.111 117.03 m 31.637 117.03 l 31.637 116.19 l 31.111 116.19 l 31.111 117.03 l gsave 0 0.038 0 Bbw grestore n 29.733 117.03 m 30.26 117.03 l 30.26 116.19 l 29.733 116.19 l 29.733 117.03 l cl 0 Fbw n 29.733 117.03 m 30.26 117.03 l 30.26 116.19 l 29.733 116.19 l 29.733 117.03 l gsave 0 0.038 0 Bbw grestore n 28.355 117.03 m 28.883 117.03 l 28.883 115.35 l 28.355 115.35 l 28.355 117.03 l 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115.83 l 78.181 115.83 l 78.181 117.03 l gsave 0 0.038 0 Bbw grestore n 76.804 117.03 m 77.331 117.03 l 77.331 116.31 l 76.804 116.31 l 76.804 117.03 l cl 0 Fbw n 76.804 117.03 m 77.331 117.03 l 77.331 116.31 l 76.804 116.31 l 76.804 117.03 l gsave 0 0.038 0 Bbw grestore n 75.427 117.03 m 75.953 117.03 l 75.953 116.43 l 75.427 116.43 l 75.427 117.03 l cl 0 Fbw n 75.427 117.03 m 75.953 117.03 l 75.953 116.43 l 75.427 116.43 l 75.427 117.03 l gsave 0 0.038 0 Bbw grestore n 74.049 117.03 m 74.577 117.03 l 74.577 116.07 l 74.049 116.07 l 74.049 117.03 l cl 0 Fbw n 74.049 117.03 m 74.577 117.03 l 74.577 116.07 l 74.049 116.07 l 74.049 117.03 l gsave 0 0.038 0 Bbw grestore n 72.671 117.03 m 73.199 117.03 l 73.199 113.67 l 72.671 113.67 l 72.671 117.03 l cl 0 Fbw n 72.671 117.03 m 73.199 117.03 l 73.199 113.67 l 72.671 113.67 l 72.671 117.03 l gsave 0 0.038 0 Bbw grestore n 56.978 99.754 m 56.978 99.664 l gsave 0 0.05 0 Bbw grestore n 55.601 99.754 m 55.601 99.664 l gsave 0 0.05 0 Bbw grestore n 54.223 99.754 m 54.223 99.664 l gsave 0 0.05 0 Bbw grestore n 52.846 99.754 m 52.846 99.664 l gsave 0 0.05 0 Bbw grestore n 51.468 99.754 m 51.468 99.664 l gsave 0 0.05 0 Bbw grestore n 50.086 99.754 m 50.086 99.664 l gsave 0 0.05 0 Bbw grestore n 56.978 99.664 m 50.086 99.664 l gsave 0 0.05 0 Bbw grestore n 49.817 92.461 m 49.904 92.461 l gsave 0 0.05 0 Bbw grestore n 49.817 94.862 m 49.904 94.862 l gsave 0 0.05 0 Bbw grestore n 49.817 97.263 m 49.904 97.263 l gsave 0 0.05 0 Bbw grestore n 49.817 99.659 m 49.904 99.659 l gsave 0 0.05 0 Bbw grestore n 49.904 92.461 m 49.904 99.659 l gsave 0 0.05 0 Bbw grestore n 56.025 99.659 m 56.553 99.659 l 56.553 97.623 l 56.025 97.623 l 56.025 99.659 l cl 0 Fbw n 56.025 99.659 m 56.553 99.659 l 56.553 97.623 l 56.025 97.623 l 56.025 99.659 l gsave 0 0.038 0 Bbw grestore n 54.648 99.659 m 55.176 99.659 l 55.176 98.939 l 54.648 98.939 l 54.648 99.659 l cl 0 Fbw n 54.648 99.659 m 55.176 99.659 l 55.176 98.939 l 54.648 98.939 l 54.648 99.659 l gsave 0 0.038 0 Bbw grestore n 53.271 99.659 m 53.798 99.659 l 53.798 99.063 l 53.271 99.063 l 53.271 99.659 l cl 0 Fbw n 53.271 99.659 m 53.798 99.659 l 53.798 99.063 l 53.271 99.063 l 53.271 99.659 l gsave 0 0.038 0 Bbw grestore n 51.894 99.659 m 52.422 99.659 l 52.422 98.939 l 51.894 98.939 l 51.894 99.659 l cl 0 Fbw n 51.894 99.659 m 52.422 99.659 l 52.422 98.939 l 51.894 98.939 l 51.894 99.659 l gsave 0 0.038 0 Bbw grestore n 50.516 99.659 m 51.044 99.659 l 51.044 96.903 l 50.516 96.903 l 50.516 99.659 l cl 0 Fbw n 50.516 99.659 m 51.044 99.659 l 51.044 96.903 l 50.516 96.903 l 50.516 99.659 l gsave 0 0.038 0 Bbw grestore n 113.9 151.86 m 113.9 151.77 l gsave 0 0.05 0 Bbw grestore n 112.51 151.86 m 112.51 151.77 l gsave 0 0.05 0 Bbw grestore n 111.14 151.86 m 111.14 151.77 l gsave 0 0.05 0 Bbw grestore n 109.75 151.86 m 109.75 151.77 l gsave 0 0.05 0 Bbw grestore n 108.38 151.86 m 108.38 151.77 l gsave 0 0.05 0 Bbw grestore n 106.99 151.86 m 106.99 151.77 l gsave 0 0.05 0 Bbw grestore n 113.9 151.77 m 106.99 151.77 l gsave 0 0.05 0 Bbw grestore n 106.72 144.58 m 106.81 144.58 l gsave 0 0.05 0 Bbw grestore n 106.72 146.98 m 106.81 146.98 l gsave 0 0.05 0 Bbw grestore n 106.72 149.38 m 106.81 149.38 l gsave 0 0.05 0 Bbw grestore n 106.72 151.77 m 106.81 151.77 l gsave 0 0.05 0 Bbw grestore n 106.81 144.58 m 106.81 151.77 l gsave 0 0.05 0 Bbw grestore n 112.94 151.77 m 113.47 151.77 l 113.47 149.38 l 112.94 149.38 l 112.94 151.77 l cl 0 Fbw n 112.94 151.77 m 113.47 151.77 l 113.47 149.38 l 112.94 149.38 l 112.94 151.77 l gsave 0 0.038 0 Bbw grestore n 111.56 151.77 m 112.09 151.77 l 112.09 151.41 l 111.56 151.41 l 111.56 151.77 l cl 0 Fbw n 111.56 151.77 m 112.09 151.77 l 112.09 151.41 l 111.56 151.41 l 111.56 151.77 l gsave 0 0.038 0 Bbw grestore n 110.18 151.77 m 110.71 151.77 l 110.71 151.41 l 110.18 151.41 l 110.18 151.77 l cl 0 Fbw n 110.18 151.77 m 110.71 151.77 l 110.71 151.41 l 110.18 151.41 l 110.18 151.77 l gsave 0 0.038 0 Bbw grestore n 108.8 151.77 m 109.33 151.77 l 109.33 151.29 l 108.8 151.29 l 108.8 151.77 l cl 0 Fbw n 108.8 151.77 m 109.33 151.77 l 109.33 151.29 l 108.8 151.29 l 108.8 151.77 l gsave 0 0.038 0 Bbw grestore n 107.42 151.77 m 107.95 151.77 l 107.95 148.42 l 107.42 148.42 l 107.42 151.77 l cl 0 Fbw n 107.42 151.77 m 107.95 151.77 l 107.95 148.42 l 107.42 148.42 l 107.42 151.77 l gsave 0 0.038 0 Bbw grestore n 125.98 108.87 m 148.19 100.77 l gsave 0 0.074 0 Bbw grestore n 169.81 108.67 m 148.19 100.77 l gsave 0 0.074 0 Bbw grestore n 114.79 126.25 m 125.98 118.15 l gsave 0 0.074 0 Bbw grestore n 136.8 126.25 m 126.17 118.15 l gsave 0 0.074 0 Bbw grestore n 159 126.65 m 170 118.1 l gsave 0 0.074 0 Bbw grestore n 181 126.25 m 170 118.1 l gsave 0 0.074 0 Bbw grestore n 110.62 143.44 m 114.79 134.94 l gsave 0 0.074 0 Bbw grestore n 119.92 143.63 m 114.79 134.98 l gsave 0 0.074 0 Bbw grestore n 133 143.25 m 137.17 134.98 l gsave 0 0.074 0 Bbw grestore n 155.01 143.05 m 159.18 134.98 l gsave 0 0.074 0 Bbw grestore n 177.39 143.05 m 181.56 134.98 l gsave 0 0.074 0 Bbw grestore n 142.48 143.44 m 137.21 134.98 l gsave 0 0.074 0 Bbw grestore n 164.49 143.65 m 159.18 134.98 l gsave 0 0.074 0 Bbw grestore n 186.69 143.44 m 181.56 134.98 l gsave 0 0.074 0 Bbw grestore n 124.99 151.86 m 124.99 151.77 l gsave 0 0.05 0 Bbw grestore n 123.61 151.86 m 123.61 151.77 l gsave 0 0.05 0 Bbw grestore n 122.23 151.86 m 122.23 151.77 l gsave 0 0.05 0 Bbw grestore n 120.85 151.86 m 120.85 151.77 l gsave 0 0.05 0 Bbw grestore n 119.47 151.86 m 119.47 151.77 l gsave 0 0.05 0 Bbw grestore n 118.09 151.86 m 118.09 151.77 l gsave 0 0.05 0 Bbw grestore n 124.99 151.77 m 118.09 151.77 l gsave 0 0.05 0 Bbw grestore n 117.82 144.58 m 117.9 144.58 l gsave 0 0.05 0 Bbw grestore n 117.82 146.98 m 117.9 146.98 l gsave 0 0.05 0 Bbw grestore n 117.82 149.38 m 117.9 149.38 l gsave 0 0.05 0 Bbw grestore n 117.82 151.77 m 117.9 151.77 l gsave 0 0.05 0 Bbw grestore n 117.9 144.58 m 117.9 151.77 l gsave 0 0.05 0 Bbw grestore n 124.04 151.77 m 124.57 151.77 l 124.57 146.62 l 124.04 146.62 l 124.04 151.77 l cl 0 Fbw n 124.04 151.77 m 124.57 151.77 l 124.57 146.62 l 124.04 146.62 l 124.04 151.77 l gsave 0 0.038 0 Bbw grestore n 122.66 151.77 m 123.19 151.77 l 123.19 151.53 l 122.66 151.53 l 122.66 151.77 l cl 0 Fbw n 122.66 151.77 m 123.19 151.77 l 123.19 151.53 l 122.66 151.53 l 122.66 151.77 l gsave 0 0.038 0 Bbw grestore n 121.28 151.77 m 121.81 151.77 l 121.81 151.53 l 121.28 151.53 l 121.28 151.77 l cl 0 Fbw n 121.28 151.77 m 121.81 151.77 l 121.81 151.53 l 121.28 151.53 l 121.28 151.77 l gsave 0 0.038 0 Bbw grestore n 119.9 151.77 m 120.43 151.77 l 120.43 151.53 l 119.9 151.53 l 119.9 151.77 l cl 0 Fbw n 119.9 151.77 m 120.43 151.77 l 120.43 151.53 l 119.9 151.53 l 119.9 151.77 l gsave 0 0.038 0 Bbw grestore n 118.52 151.77 m 119.05 151.77 l 119.05 150.81 l 118.52 150.81 l 118.52 151.77 l cl 0 Fbw n 118.52 151.77 m 119.05 151.77 l 119.05 150.81 l 118.52 150.81 l 118.52 151.77 l gsave 0 0.038 0 Bbw grestore n 136.09 151.86 m 136.09 151.77 l gsave 0 0.05 0 Bbw grestore n 134.71 151.86 m 134.71 151.77 l gsave 0 0.05 0 Bbw grestore n 133.33 151.86 m 133.33 151.77 l gsave 0 0.05 0 Bbw grestore n 131.95 151.86 m 131.95 151.77 l gsave 0 0.05 0 Bbw grestore n 130.57 151.86 m 130.57 151.77 l gsave 0 0.05 0 Bbw grestore n 129.18 151.86 m 129.18 151.77 l gsave 0 0.05 0 Bbw grestore n 136.09 151.77 m 129.18 151.77 l gsave 0 0.05 0 Bbw grestore n 128.92 144.58 m 129 144.58 l gsave 0 0.05 0 Bbw grestore n 128.92 146.98 m 129 146.98 l gsave 0 0.05 0 Bbw grestore n 128.92 149.38 m 129 149.38 l gsave 0 0.05 0 Bbw grestore n 128.92 151.77 m 129 151.77 l gsave 0 0.05 0 Bbw grestore n 129 144.58 m 129 151.77 l gsave 0 0.05 0 Bbw grestore n 135.14 151.77 m 135.66 151.77 l 135.66 150.81 l 135.14 150.81 l 135.14 151.77 l cl 0 Fbw n 135.14 151.77 m 135.66 151.77 l 135.66 150.81 l 135.14 150.81 l 135.14 151.77 l gsave 0 0.038 0 Bbw grestore n 133.76 151.77 m 134.28 151.77 l 134.28 151.53 l 133.76 151.53 l 133.76 151.77 l cl 0 Fbw n 133.76 151.77 m 134.28 151.77 l 134.28 151.53 l 133.76 151.53 l 133.76 151.77 l gsave 0 0.038 0 Bbw grestore n 132.38 151.77 m 132.9 151.77 l 132.9 151.53 l 132.38 151.53 l 132.38 151.77 l cl 0 Fbw n 132.38 151.77 m 132.9 151.77 l 132.9 151.53 l 132.38 151.53 l 132.38 151.77 l gsave 0 0.038 0 Bbw grestore n 131 151.77 m 131.52 151.77 l 131.52 151.53 l 131 151.53 l 131 151.77 l cl 0 Fbw n 131 151.77 m 131.52 151.77 l 131.52 151.53 l 131 151.53 l 131 151.77 l gsave 0 0.038 0 Bbw grestore n 129.62 151.77 m 130.15 151.77 l 130.15 146.5 l 129.62 146.5 l 129.62 151.77 l cl 0 Fbw n 129.62 151.77 m 130.15 151.77 l 130.15 146.5 l 129.62 146.5 l 129.62 151.77 l gsave 0 0.038 0 Bbw grestore n 147.19 151.86 m 147.19 151.77 l gsave 0 0.05 0 Bbw grestore n 145.81 151.86 m 145.81 151.77 l gsave 0 0.05 0 Bbw grestore n 144.43 151.86 m 144.43 151.77 l gsave 0 0.05 0 Bbw grestore n 143.05 151.86 m 143.05 151.77 l gsave 0 0.05 0 Bbw grestore n 141.67 151.86 m 141.67 151.77 l gsave 0 0.05 0 Bbw grestore n 140.28 151.86 m 140.28 151.77 l gsave 0 0.05 0 Bbw grestore n 147.19 151.77 m 140.28 151.77 l gsave 0 0.05 0 Bbw grestore n 140.01 144.58 m 140.1 144.58 l gsave 0 0.05 0 Bbw grestore n 140.01 146.98 m 140.1 146.98 l gsave 0 0.05 0 Bbw grestore n 140.01 149.38 m 140.1 149.38 l gsave 0 0.05 0 Bbw grestore n 140.01 151.77 m 140.1 151.77 l gsave 0 0.05 0 Bbw grestore n 140.1 144.58 m 140.1 151.77 l gsave 0 0.05 0 Bbw grestore n 146.23 151.77 m 146.76 151.77 l 146.76 148.17 l 146.23 148.17 l 146.23 151.77 l cl 0 Fbw n 146.23 151.77 m 146.76 151.77 l 146.76 148.17 l 146.23 148.17 l 146.23 151.77 l gsave 0 0.038 0 Bbw grestore n 144.85 151.77 m 145.38 151.77 l 145.38 151.29 l 144.85 151.29 l 144.85 151.77 l cl 0 Fbw n 144.85 151.77 m 145.38 151.77 l 145.38 151.29 l 144.85 151.29 l 144.85 151.77 l gsave 0 0.038 0 Bbw grestore n 143.47 151.77 m 144 151.77 l 144 151.41 l 143.47 151.41 l 143.47 151.77 l cl 0 Fbw n 143.47 151.77 m 144 151.77 l 144 151.41 l 143.47 151.41 l 143.47 151.77 l gsave 0 0.038 0 Bbw grestore n 142.09 151.77 m 142.62 151.77 l 142.62 151.29 l 142.09 151.29 l 142.09 151.77 l cl 0 Fbw n 142.09 151.77 m 142.62 151.77 l 142.62 151.29 l 142.09 151.29 l 142.09 151.77 l gsave 0 0.038 0 Bbw grestore n 140.71 151.77 m 141.24 151.77 l 141.24 149.85 l 140.71 149.85 l 140.71 151.77 l cl 0 Fbw n 140.71 151.77 m 141.24 151.77 l 141.24 149.85 l 140.71 149.85 l 140.71 151.77 l gsave 0 0.038 0 Bbw grestore n 158.28 151.86 m 158.28 151.77 l gsave 0 0.05 0 Bbw grestore n 156.91 151.86 m 156.91 151.77 l gsave 0 0.05 0 Bbw grestore n 155.52 151.86 m 155.52 151.77 l gsave 0 0.05 0 Bbw grestore n 154.15 151.86 m 154.15 151.77 l gsave 0 0.05 0 Bbw grestore n 152.77 151.86 m 152.77 151.77 l gsave 0 0.05 0 Bbw grestore n 151.38 151.86 m 151.38 151.77 l gsave 0 0.05 0 Bbw grestore n 158.28 151.77 m 151.38 151.77 l gsave 0 0.05 0 Bbw grestore n 151.11 144.58 m 151.2 144.58 l gsave 0 0.05 0 Bbw grestore n 151.11 146.98 m 151.2 146.98 l gsave 0 0.05 0 Bbw grestore n 151.11 149.38 m 151.2 149.38 l gsave 0 0.05 0 Bbw grestore n 151.11 151.77 m 151.2 151.77 l gsave 0 0.05 0 Bbw grestore n 151.2 144.58 m 151.2 151.77 l gsave 0 0.05 0 Bbw grestore n 157.33 151.77 m 157.86 151.77 l 157.86 148.17 l 157.33 148.17 l 157.33 151.77 l cl 0 Fbw n 157.33 151.77 m 157.86 151.77 l 157.86 148.17 l 157.33 148.17 l 157.33 151.77 l gsave 0 0.038 0 Bbw grestore n 155.95 151.77 m 156.48 151.77 l 156.48 151.29 l 155.95 151.29 l 155.95 151.77 l cl 0 Fbw n 155.95 151.77 m 156.48 151.77 l 156.48 151.29 l 155.95 151.29 l 155.95 151.77 l gsave 0 0.038 0 Bbw grestore n 154.57 151.77 m 155.1 151.77 l 155.1 151.41 l 154.57 151.41 l 154.57 151.77 l cl 0 Fbw n 154.57 151.77 m 155.1 151.77 l 155.1 151.41 l 154.57 151.41 l 154.57 151.77 l gsave 0 0.038 0 Bbw grestore n 153.19 151.77 m 153.72 151.77 l 153.72 151.41 l 153.19 151.41 l 153.19 151.77 l cl 0 Fbw n 153.19 151.77 m 153.72 151.77 l 153.72 151.41 l 153.19 151.41 l 153.19 151.77 l gsave 0 0.038 0 Bbw grestore n 151.81 151.77 m 152.34 151.77 l 152.34 149.61 l 151.81 149.61 l 151.81 151.77 l cl 0 Fbw n 151.81 151.77 m 152.34 151.77 l 152.34 149.61 l 151.81 149.61 l 151.81 151.77 l gsave 0 0.038 0 Bbw grestore n 169.39 151.86 m 169.39 151.77 l gsave 0 0.05 0 Bbw grestore n 168.01 151.86 m 168.01 151.77 l gsave 0 0.05 0 Bbw grestore n 166.63 151.86 m 166.63 151.77 l gsave 0 0.05 0 Bbw grestore n 165.25 151.86 m 165.25 151.77 l gsave 0 0.05 0 Bbw grestore n 163.87 151.86 m 163.87 151.77 l gsave 0 0.05 0 Bbw grestore n 162.48 151.86 m 162.48 151.77 l gsave 0 0.05 0 Bbw grestore n 169.39 151.77 m 162.48 151.77 l gsave 0 0.05 0 Bbw grestore n 162.22 144.58 m 162.3 144.58 l gsave 0 0.05 0 Bbw grestore n 162.22 146.98 m 162.3 146.98 l gsave 0 0.05 0 Bbw grestore n 162.22 149.38 m 162.3 149.38 l gsave 0 0.05 0 Bbw grestore n 162.22 151.77 m 162.3 151.77 l gsave 0 0.05 0 Bbw grestore n 162.3 144.58 m 162.3 151.77 l gsave 0 0.05 0 Bbw grestore n 168.43 151.77 m 168.96 151.77 l 168.96 150.21 l 168.43 150.21 l 168.43 151.77 l cl 0 Fbw n 168.43 151.77 m 168.96 151.77 l 168.96 150.21 l 168.43 150.21 l 168.43 151.77 l gsave 0 0.038 0 Bbw grestore n 167.06 151.77 m 167.58 151.77 l 167.58 151.53 l 167.06 151.53 l 167.06 151.77 l cl 0 Fbw n 167.06 151.77 m 167.58 151.77 l 167.58 151.53 l 167.06 151.53 l 167.06 151.77 l gsave 0 0.038 0 Bbw grestore n 165.67 151.77 m 166.2 151.77 l 166.2 151.53 l 165.67 151.53 l 165.67 151.77 l cl 0 Fbw n 165.67 151.77 m 166.2 151.77 l 166.2 151.53 l 165.67 151.53 l 165.67 151.77 l gsave 0 0.038 0 Bbw grestore n 164.3 151.77 m 164.82 151.77 l 164.82 151.53 l 164.3 151.53 l 164.3 151.77 l cl 0 Fbw n 164.3 151.77 m 164.82 151.77 l 164.82 151.53 l 164.3 151.53 l 164.3 151.77 l gsave 0 0.038 0 Bbw grestore n 162.92 151.77 m 163.44 151.77 l 163.44 147.1 l 162.92 147.1 l 162.92 151.77 l cl 0 Fbw n 162.92 151.77 m 163.44 151.77 l 163.44 147.1 l 162.92 147.1 l 162.92 151.77 l gsave 0 0.038 0 Bbw grestore n 180.49 151.86 m 180.49 151.77 l gsave 0 0.05 0 Bbw grestore n 179.11 151.86 m 179.11 151.77 l gsave 0 0.05 0 Bbw grestore n 177.73 151.86 m 177.73 151.77 l gsave 0 0.05 0 Bbw grestore n 176.35 151.86 m 176.35 151.77 l gsave 0 0.05 0 Bbw grestore n 174.97 151.86 m 174.97 151.77 l gsave 0 0.05 0 Bbw grestore n 173.58 151.86 m 173.58 151.77 l gsave 0 0.05 0 Bbw grestore n 180.49 151.77 m 173.58 151.77 l gsave 0 0.05 0 Bbw grestore n 173.31 144.58 m 173.4 144.58 l gsave 0 0.05 0 Bbw grestore n 173.31 146.98 m 173.4 146.98 l gsave 0 0.05 0 Bbw grestore n 173.31 149.38 m 173.4 149.38 l gsave 0 0.05 0 Bbw grestore n 173.31 151.77 m 173.4 151.77 l gsave 0 0.05 0 Bbw grestore n 173.4 144.58 m 173.4 151.77 l gsave 0 0.05 0 Bbw grestore n 179.53 151.77 m 180.06 151.77 l 180.06 149.97 l 179.53 149.97 l 179.53 151.77 l cl 0 Fbw n 179.53 151.77 m 180.06 151.77 l 180.06 149.97 l 179.53 149.97 l 179.53 151.77 l gsave 0 0.038 0 Bbw grestore n 178.15 151.77 m 178.68 151.77 l 178.68 151.41 l 178.15 151.41 l 178.15 151.77 l cl 0 Fbw n 178.15 151.77 m 178.68 151.77 l 178.68 151.41 l 178.15 151.41 l 178.15 151.77 l gsave 0 0.038 0 Bbw grestore n 176.77 151.77 m 177.3 151.77 l 177.3 151.41 l 176.77 151.41 l 176.77 151.77 l cl 0 Fbw n 176.77 151.77 m 177.3 151.77 l 177.3 151.41 l 176.77 151.41 l 176.77 151.77 l gsave 0 0.038 0 Bbw grestore n 175.39 151.77 m 175.92 151.77 l 175.92 151.29 l 175.39 151.29 l 175.39 151.77 l cl 0 Fbw n 175.39 151.77 m 175.92 151.77 l 175.92 151.29 l 175.39 151.29 l 175.39 151.77 l gsave 0 0.038 0 Bbw grestore n 174.01 151.77 m 174.54 151.77 l 174.54 147.82 l 174.01 147.82 l 174.01 151.77 l cl 0 Fbw n 174.01 151.77 m 174.54 151.77 l 174.54 147.82 l 174.01 147.82 l 174.01 151.77 l gsave 0 0.038 0 Bbw grestore n 191.58 151.86 m 191.58 151.77 l gsave 0 0.05 0 Bbw grestore n 190.2 151.86 m 190.2 151.77 l gsave 0 0.05 0 Bbw grestore n 188.82 151.86 m 188.82 151.77 l gsave 0 0.05 0 Bbw grestore n 187.44 151.86 m 187.44 151.77 l gsave 0 0.05 0 Bbw grestore n 186.06 151.86 m 186.06 151.77 l gsave 0 0.05 0 Bbw grestore n 184.68 151.86 m 184.68 151.77 l gsave 0 0.05 0 Bbw grestore n 191.58 151.77 m 184.68 151.77 l gsave 0 0.05 0 Bbw grestore n 184.41 144.58 m 184.5 144.58 l gsave 0 0.05 0 Bbw grestore n 184.41 146.98 m 184.5 146.98 l gsave 0 0.05 0 Bbw grestore n 184.41 149.38 m 184.5 149.38 l gsave 0 0.05 0 Bbw grestore n 184.41 151.77 m 184.5 151.77 l gsave 0 0.05 0 Bbw grestore n 184.5 144.58 m 184.5 151.77 l gsave 0 0.05 0 Bbw grestore n 190.63 151.77 m 191.16 151.77 l 191.16 147.82 l 190.63 147.82 l 190.63 151.77 l cl 0 Fbw n 190.63 151.77 m 191.16 151.77 l 191.16 147.82 l 190.63 147.82 l 190.63 151.77 l gsave 0 0.038 0 Bbw grestore n 189.25 151.77 m 189.78 151.77 l 189.78 151.41 l 189.25 151.41 l 189.25 151.77 l cl 0 Fbw n 189.25 151.77 m 189.78 151.77 l 189.78 151.41 l 189.25 151.41 l 189.25 151.77 l gsave 0 0.038 0 Bbw grestore n 187.87 151.77 m 188.4 151.77 l 188.4 151.53 l 187.87 151.53 l 187.87 151.77 l cl 0 Fbw n 187.87 151.77 m 188.4 151.77 l 188.4 151.53 l 187.87 151.53 l 187.87 151.77 l gsave 0 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111.24 132.05 l gsave 0 0.05 0 Bbw grestore n 111.16 134.44 m 111.24 134.44 l gsave 0 0.05 0 Bbw grestore n 111.24 127.25 m 111.24 134.44 l gsave 0 0.05 0 Bbw grestore n 117.38 134.44 m 117.9 134.44 l 117.9 132.88 l 117.38 132.88 l 117.38 134.44 l cl 0 Fbw n 117.38 134.44 m 117.9 134.44 l 117.9 132.88 l 117.38 132.88 l 117.38 134.44 l gsave 0 0.038 0 Bbw grestore n 116 134.44 m 116.52 134.44 l 116.52 133.96 l 116 133.96 l 116 134.44 l cl 0 Fbw n 116 134.44 m 116.52 134.44 l 116.52 133.96 l 116 133.96 l 116 134.44 l gsave 0 0.038 0 Bbw grestore n 114.62 134.44 m 115.15 134.44 l 115.15 133.96 l 114.62 133.96 l 114.62 134.44 l cl 0 Fbw n 114.62 134.44 m 115.15 134.44 l 115.15 133.96 l 114.62 133.96 l 114.62 134.44 l gsave 0 0.038 0 Bbw grestore n 113.24 134.44 m 113.76 134.44 l 113.76 133.72 l 113.24 133.72 l 113.24 134.44 l cl 0 Fbw n 113.24 134.44 m 113.76 134.44 l 113.76 133.72 l 113.24 133.72 l 113.24 134.44 l gsave 0 0.038 0 Bbw grestore n 111.86 134.44 m 112.39 134.44 l 112.39 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2389 y Fm(ij)1114 2376 y Fl(x)1142 2360 y Fk(\()p Fm(t)p Fk(\))1184 2376 y Fl(x)1212 2360 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1279 2376 y Fj(R)1314 2352 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))1314 2389 y Fm(ij)p 941 2402 511 2 v 941 2456 a Fj(\025)p Fu([)p Fj(h)1007 2432 y Fk(\()p Fm(t)p Fk(\))1007 2469 y Fm(ij)1049 2456 y Fu(])1062 2443 y Fd(\000)p Fk(1)1119 2456 y Fu(+)f Fl(x)1192 2443 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1259 2456 y Fj(R)1294 2432 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))1294 2469 y Fm(ij)1382 2456 y Fl(x)1410 2443 y Fk(\()p Fm(t)p Fk(\))1456 2413 y Fj(;)250 b Fu(\(33\))150 2550 y(where)15 b Fj(\025)g Fu(is)h(the)f(deca)o(y)g(parameter.)952 2775 y(21)p eop %%Page: 22 22 22 21 bop 241 192 a Fu(It)19 b(is)h(in)o(teresting)g(to)f(note)g(the)h (similarit)o(y)g(b)q(et)o(w)o(een)g(the)f(parameter)g(up)q(date)h(rule) g(in)150 248 y(Equation)11 b(32)f(and)g(the)h(gradien)o(t)g(rule)g (presen)o(ted)g(earlier)g(\(cf.)18 b(Equation)11 b(14\).)18 b(These)10 b(up)q(dates)150 312 y(are)19 b(essen)o(tially)i(the)f (same,)f(except)h(that)f(the)g(scalar)h Fj(\032)f Fu(is)h(replaced)g(b) o(y)g(the)f(matrix)g Fj(R)1745 288 y Fk(\()p Fm(t)p Fk(\))1745 325 y Fm(ij)1787 312 y Fu(.)150 385 y(It)h(can)g(b)q(e)h(sho)o(wn,)f (ho)o(w)o(ev)o(er,)h(that)e Fj(R)847 361 y Fk(\()p Fm(t)p Fk(\))847 398 y Fm(ij)909 385 y Fu(is)h(an)g(estimate)g(of)g(the)g(in)o (v)o(erse)h(Hessian)f(of)g(the)150 441 y(least-squares)15 b(cost)f(function)h(\(Ljung)g(&)g(S\177)-23 b(oderstr\177)g(om,)13 b(1986\),)g(th)o(us)h(Equation)h(32)f(is)h(in)h(fact)150 498 y(a)j(sto)q(c)o(hastic)g(appro)o(ximation)g(to)g(a)g (Newton-Raphson)h(metho)q(d)f(rather)g(than)g(a)g(gradien)o(t)150 554 y(metho)q(d.)313 538 y Fk(7)241 611 y Fu(Similar)h(equations)f (apply)h(for)e(the)h(up)q(dates)g(of)g(the)f(gating)h(net)o(w)o(orks.) 29 b(The)19 b(up)q(date)150 667 y(rule)i(for)f(the)g(parameters)g(of)g (the)g(top-lev)o(el)i(gating)e(net)o(w)o(ork)f(is)i(giv)o(en)g(b)o(y)f (the)g(follo)o(wing)150 723 y(equation)c(\(for)e(the)h Fj(i)517 707 y Fm(th)567 723 y Fu(output)g(of)g(the)g(gating)g(net)o(w) o(ork\):)612 834 y Fl(v)641 810 y Fk(\()p Fm(t)p Fk(+1\))640 846 y Fm(i)740 834 y Fu(=)e Fl(v)817 810 y Fk(\()p Fm(t)p Fk(\))816 846 y Fm(i)869 834 y Fu(+)d Fj(S)945 810 y Fk(\()p Fm(t)p Fk(\))942 846 y Fm(i)987 834 y Fu(\(ln)e Fj(h)1076 810 y Fk(\()p Fm(t)p Fk(\))1076 846 y Fm(i)1128 834 y Fn(\000)j Fj(\030)1196 810 y Fk(\()p Fm(t)p Fk(\))1194 846 y Fm(i)1238 834 y Fu(\))p Fl(x)1284 815 y Fk(\()p Fm(t)p Fk(\))1326 834 y Fj(;)380 b Fu(\(34\))150 936 y(where)15 b(the)h(in)o(v)o(erse)f(co)o(v)m(ariance)h(matrix)f Fj(S)906 943 y Fm(i)935 936 y Fu(is)h(up)q(dated)g(b)o(y:)536 1074 y Fj(S)567 1050 y Fk(\()p Fm(t)p Fk(\))564 1087 y Fm(i)622 1074 y Fu(=)d Fj(\025)697 1055 y Fd(\000)p Fk(1)743 1074 y Fj(S)774 1050 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))771 1087 y Fm(i)871 1074 y Fn(\000)e Fj(\025)944 1055 y Fd(\000)p Fk(1)995 1043 y Fj(S)1026 1019 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))1023 1056 y Fm(i)1113 1043 y Fl(x)1141 1027 y Fk(\()p Fm(t)p Fk(\))1183 1043 y Fl(x)1211 1027 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1278 1043 y Fj(S)1309 1019 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))1306 1056 y Fm(i)p 995 1063 401 2 v 1013 1117 a Fj(\025)f Fu(+)g Fl(x)1123 1104 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1191 1117 y Fj(S)1222 1093 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))1219 1130 y Fm(i)1308 1117 y Fl(x)1336 1104 y Fk(\()p Fm(t)p Fk(\))1401 1074 y Fj(:)305 b Fu(\(35\))150 1212 y(Finally)l(,)17 b(the)e(up)q(date)h (rule)g(for)e(the)i(parameters)e(of)h(the)g(lo)o(w)o(er-lev)o(el)h (gating)f(net)o(w)o(ork)f(are)h(as)150 1268 y(follo)o(ws:)577 1324 y Fl(v)606 1300 y Fk(\()p Fm(t)p Fk(+1\))605 1337 y Fm(ij)706 1324 y Fu(=)e Fl(v)783 1300 y Fk(\()p Fm(t)p Fk(\))782 1337 y Fm(ij)834 1324 y Fu(+)e Fj(S)911 1300 y Fk(\()p Fm(t)p Fk(\))908 1337 y Fm(ij)953 1324 y Fj(h)979 1300 y Fk(\()p Fm(t)p Fk(\))979 1337 y Fm(i)1021 1324 y Fu(\(ln)d Fj(h)1110 1300 y Fk(\()p Fm(t)p Fk(\))1110 1340 y Fm(j)r Fd(j)p Fm(i)1163 1324 y Fn(\000)i Fj(\030)1230 1300 y Fk(\()p Fm(t)p Fk(\))1228 1337 y Fm(ij)1272 1324 y Fu(\))p Fl(x)1318 1306 y Fk(\()p Fm(t)p Fk(\))1360 1324 y Fj(;)346 b Fu(\(36\))150 1408 y(where)15 b(the)h(in)o(v)o(erse)f (co)o(v)m(ariance)h(matrix)f Fj(S)906 1415 y Fm(i)935 1408 y Fu(is)h(up)q(dated)g(b)o(y:)484 1552 y Fj(S)515 1528 y Fk(\()p Fm(t)p Fk(\))512 1565 y Fm(ij)569 1552 y Fu(=)d Fj(\025)644 1533 y Fd(\000)p Fk(1)691 1552 y Fj(S)722 1528 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))719 1565 y Fm(ij)819 1552 y Fn(\000)d Fj(\025)891 1533 y Fd(\000)p Fk(1)995 1515 y Fj(S)1026 1491 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))1023 1528 y Fm(ij)1113 1515 y Fl(x)1141 1499 y Fk(\()p Fm(t)p Fk(\))1183 1515 y Fl(x)1211 1499 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1278 1515 y Fj(S)1309 1491 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))1306 1528 y Fm(ij)p 943 1541 506 2 v 943 1595 a Fj(\025)p Fu([)p Fj(h)1009 1571 y Fk(\()p Fm(t)p Fk(\))1009 1608 y Fm(i)1050 1595 y Fu(])1063 1582 y Fd(\000)p Fk(1)1120 1595 y Fu(+)h Fl(x)1194 1582 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1261 1595 y Fj(S)1292 1571 y Fk(\()p Fm(t)p Fd(\000)p Fk(1\))1289 1608 y Fm(ij)1379 1595 y Fl(x)1407 1582 y Fk(\()p Fm(t)p Fk(\))1454 1552 y Fj(:)252 b Fu(\(37\))150 1714 y Fl(Sim)o(ulation)19 b(results)150 1800 y Fu(The)e(on-line)i(algorithm)e(w)o(as)f(tested)g (on)h(the)g(rob)q(ot)f(dynamics)i(problem)f(describ)q(ed)i(in)f(the)150 1856 y(previous)g(section.)28 b(Preliminary)19 b(sim)o(ulations)g(con)o (vinced)g(us)f(of)f(the)h(necessit)o(y)g(of)g(the)f(de-)150 1913 y(ca)o(y)g(parameter)g(\()p Fj(\025)p Fu(\).)25 b(W)l(e)17 b(also)h(found)f(that)g(this)h(parameter)f(should)h(b)q(e)g (slo)o(wly)g(increased)150 1969 y(as)i(training)i(pro)q(ceeds|on)f(the) g(early)g(trials)g(the)g(p)q(osterior)f(probabilities)j(are)d(c)o (hanging)150 2026 y(rapidly)g(so)e(that)g(the)g(co)o(v)m(ariances)h (should)h(b)q(e)f(deca)o(y)o(ed)g(rapidly)l(,)h(whereas)f(on)f(later)h (trials)150 2082 y(the)12 b(p)q(osterior)g(probabilities)i(ha)o(v)o(e)d (stabilized)j(and)e(the)g(co)o(v)m(ariances)h(should)g(b)q(e)f(deca)o (y)o(ed)g(less)150 2139 y(rapidly)l(.)23 b(W)l(e)16 b(used)h(a)f (simple)h(\014xed)f(sc)o(hedule:)23 b Fj(\025)16 b Fu(w)o(as)f (initialized)k(to)c(0.99)g(and)h(increased)h(a)150 2195 y(\014xed)f(fraction)f(\(0.6\))e(of)i(the)h(remaining)g(distance)g(to)e (1.0)h(ev)o(ery)g(1000)f(time)h(steps.)241 2251 y(The)20 b(p)q(erformance)g(of)f(the)h(on-line)i(algorithm)e(w)o(as)f(compared)g (to)h(an)f(on-line)j(bac)o(k-)150 2308 y(propagation)f(net)o(w)o(ork.) 38 b(P)o(arameter)20 b(settings)h(for)g(the)h(bac)o(kpropagation)f(net) o(w)o(ork)f(w)o(ere)p 150 2347 660 2 v 202 2374 a Fh(7)219 2390 y Fg(This)12 b(is)g(true)g(for)f(\014xed)i(v)n(alues)g(of)e(the)h (p)q(osterior)g(probabili)q(tie)q(s.)19 b(These)12 b(p)q(osterior)h (probabiliti)q(es)h(are)e(also)150 2436 y(c)o(hanging)18 b(o)o(v)o(er)e(time,)g(ho)o(w)o(ev)o(er,)h(as)f(required)h(b)o(y)f(the) g(EM)g(algorithm.)27 b(The)16 b(o)o(v)o(erall)h(con)o(v)o(ergence)g (rate)f(of)150 2482 y(the)e(algorithm)j(is)e(determined)g(b)o(y)g(the)f 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3 9 k 6 7 k 10 2 k 9 2 l 6 7 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 1484 3483 M 3 10 k 7 6 k 9 2 k 9 2 l 6 7 l 2 9 l 3 9 j 7 7 j 9 1 j 9 3 h 6 7 h 2 7 h 1530 3404 M 2 9 k 7 6 k 9 3 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 1574 3364 M 2 9 k 7 7 k 9 2 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 1615 3324 M 2 9 k 7 7 k 9 2 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 7 7 h 1 8 h 1654 3286 M 2 9 k 7 7 k 9 2 k 9 3 l 6 6 l 3 10 l 3 9 j 7 6 j 9 2 j 9 3 h 7 7 h 1 8 h 1691 3262 M 2 9 k 7 6 k 9 3 k 9 3 l 6 7 l 3 9 l 3 9 j 7 6 j 10 2 j 8 3 h 7 7 h 1 8 h 1984 3210 M 2 9 k 7 7 k 9 2 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 7 7 h 1 8 h 2192 3115 M 2 9 k 7 7 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 8 h 2 7 h 2353 3050 M 2 10 k 7 6 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 1 j 9 3 h 6 7 h 2 7 h 2485 3003 M 3 9 k 6 7 k 9 2 k 9 3 l 7 6 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 2596 2972 M 2 9 k 7 7 k 9 2 k 9 2 l 7 7 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 2693 2948 M 3 9 k 6 7 k 10 2 k 9 2 l 6 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 8 h 2 7 h 2778 2929 M 3 9 k 6 7 k 10 2 k 9 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 2854 2918 M 3 9 k 6 6 k 9 3 k 10 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 2923 2912 M 3 10 k 6 6 k 10 2 k 9 2 l 6 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 8 h 2 7 h 2986 2909 M 3 9 k 7 7 k 9 2 k 9 3 l 6 6 l 2 10 l 2 9 j 7 6 j 10 2 j 9 3 h 6 7 h 2 8 h 3043 2905 M 2 10 k 7 6 k 9 2 k 9 2 l 6 7 l 3 9 l 3 9 j 7 7 j 9 1 j 9 3 h 7 7 h 1 7 h 3097 2901 M 3 10 k 6 6 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 8 h 2 7 h 3147 2898 M 3 9 k 6 6 k 10 3 k 9 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3193 2895 M 2 9 k 7 6 k 9 3 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 7 7 h 1 8 h 3237 2893 M 2 9 k 7 6 k 9 3 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3278 2891 M 2 9 k 7 7 k 9 2 k 9 3 l 6 6 l 3 10 l 3 9 j 7 6 j 10 2 j 8 3 h 7 7 h 1 8 h 3317 2890 M 2 9 k 7 6 k 9 3 k 9 3 l 6 7 l 2 9 l 2 9 j stroke % to avoid limitcheck 3284 2880 M 7 6 j 10 2 j 9 3 h 6 7 h 1 8 h 3355 2888 M 3 9 k 7 7 k 9 2 k 9 2 l 6 7 l 2 10 l 2 9 j 7 6 j 10 2 j 9 3 h 6 7 h 2 8 h 3390 2886 M 3 9 k 6 7 k 10 2 k 9 3 l 6 6 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3423 2883 M 2 10 k 7 6 k 9 2 k 9 2 l 6 7 l 3 9 l 3 9 j 7 7 j 9 1 j 9 3 h 7 7 h 1 7 h 3455 2880 M 2 9 k 7 7 k 9 2 k 9 2 l 6 7 l 2 9 l 2 9 j 7 7 j 10 2 j 8 3 h 7 7 h 1 8 h 3486 2876 M 2 10 k 7 6 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 1 j 9 3 h 6 7 h 2 7 h 3516 2872 M 3 10 k 6 6 k 10 2 k 9 2 l 6 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 8 h 2 7 h 3544 2868 M 2 9 k 7 7 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 7 h 2 8 h 3571 2864 M 2 9 k 7 7 k 9 2 k 9 3 l 7 6 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3598 2860 M 3 9 k 7 7 k 9 2 k 9 3 l 6 7 l 2 9 l 2 9 j 7 6 j 10 2 j 9 3 h 6 7 h 2 8 h 3623 2856 M 3 9 k 6 7 k 9 2 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3647 2852 M 2 9 k 7 6 k 9 3 k 9 3 l 6 7 l 3 9 l 3 9 j 7 6 j 9 2 j 9 3 h 7 7 h 1 8 h 3671 2848 M 2 9 k 7 6 k 9 3 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3694 2844 M 2 9 k 7 7 k 9 2 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3716 2840 M 2 9 k 7 7 k 9 2 k 9 2 l 7 7 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3738 2837 M 3 9 k 6 6 k 10 3 k 9 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3759 2833 M 3 10 k 6 6 k 10 2 k 9 2 l 6 7 l 2 9 l 3 9 j 7 7 j 9 1 j 9 3 h 6 7 h 2 7 h 3779 2830 M 2 10 k 7 6 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 8 h 2 7 h 3799 2828 M 3 9 k 6 6 k 9 3 k 10 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3818 2825 M 2 9 k 7 7 k 9 2 k 9 3 l 7 6 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3837 2823 M 3 9 k 6 7 k 9 2 k 10 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3855 2821 M 2 9 k 7 7 k 9 2 k 9 2 l 7 7 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3873 2820 M 2 9 k 7 6 k 9 3 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3890 2819 M 2 9 k 7 6 k 9 3 k 9 3 l 6 7 l 3 9 l 3 9 j 7 6 j 9 2 j 9 3 h 7 7 h 1 8 h 3907 2818 M 2 9 k 7 7 k 9 2 k 9 3 l 6 7 l 3 9 l 3 9 j 7 6 j 9 2 j 9 3 h 7 7 h 1 8 h 3924 2817 M 2 9 k 7 7 k 9 2 k 9 3 l 7 6 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3940 2816 M 2 9 k 7 7 k 9 2 k 9 3 l 7 6 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3956 2815 M 2 9 k 7 7 k 9 2 k 9 3 l 7 6 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 3972 2814 M 3 9 k 7 7 k 9 2 k 9 2 l 6 7 l 2 10 l 2 9 j 7 6 j 10 2 j 9 3 h 6 7 h 2 8 h 3987 2813 M 3 10 k 6 6 k 9 2 k 10 2 l 6 7 l 2 9 l 3 9 j 7 7 j 9 1 j 9 3 h 6 7 h 2 7 h 4002 2813 M 3 9 k 6 7 k 10 2 k 9 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 4016 2812 M 2 9 k 7 7 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 7 7 h 1 8 h 4031 2812 M 3 9 k 7 6 k 9 3 k 9 3 l 6 7 l 2 9 l 2 9 j 7 6 j 10 2 j 9 3 h 6 7 h 2 8 h 4045 2811 M 3 9 k 7 7 k 9 2 k 9 2 l 6 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 7 h 2 8 h 4058 2811 M 2 9 k 7 7 k 9 2 k 9 3 l 6 7 l 3 9 l 3 9 j 7 6 j 9 2 j 9 3 h 7 7 h 1 8 h 4072 2811 M 3 9 k 6 6 k 9 3 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 4085 2810 M 2 10 k 7 6 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 1 j 9 3 h 6 7 h 2 7 h 4098 2810 M 2 9 k 7 7 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 8 h 2 7 h 4111 2810 M 2 9 k 7 7 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 7 h 2 8 h 4124 2810 M stroke % to avoid limitcheck 4124 2810 M 3 9 k 7 7 k 9 2 k 9 3 l 6 6 l 2 10 l 2 9 j 7 6 j 10 2 j 9 3 h 6 7 h 2 8 h 4136 2810 M 3 9 k 6 7 k 9 2 k 10 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 4148 2810 M 2 9 k 7 6 k 9 3 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 4160 2809 M 3 9 k 6 7 k 9 2 k 9 2 l 7 7 l 2 9 l 3 9 j 7 7 j 9 2 j 9 3 h 6 8 h 2 7 h 4172 2809 M 3 9 k 6 7 k 10 2 k 9 3 l 6 6 l 2 10 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 4183 2809 M 2 9 k 7 6 k 9 3 k 9 3 l 7 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 8 h 4195 2808 M 3 9 k 7 7 k 9 2 k 9 2 l 6 7 l 2 9 l 2 9 j 7 7 j 10 2 j 9 3 h 6 7 h 2 8 h 4206 2808 M 3 9 k 6 7 k 10 2 k 9 3 l 6 7 l 2 9 l 3 9 j 7 6 j 9 2 j 9 3 h 6 7 h 2 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(ervised)h(learning)g(w)o(orth)d(exploring)i(is)g(the)f(non-asso)q (ciativ)o(e)150 1434 y(v)o(ersion)i(of)f(the)h(hierarc)o(hical)i(arc)o (hitecture.)k(Suc)o(h)16 b(a)g(mo)q(del)g(w)o(ould)g(b)q(e)h(a)e (recursiv)o(e)i(v)o(ersion)150 1490 y(of)d(classical)j(mixture-lik)o (eliho)q(o)q(d)h(clustering)e(and)f(ma)o(y)f(ha)o(v)o(e)g(in)o (teresting)h(ties)h(to)e(hierarc)o(hi-)150 1546 y(cal)j(clustering)h (mo)q(dels.)26 b(Finally)l(,)18 b(it)f(is)h(also)e(of)h(in)o(terest)g (to)f(note)g(that)g(the)h(recursiv)o(e)h(least)150 1603 y(squares)d(algorithm)f(that)g(w)o(e)h(utilized)i(in)f(obtaining)f(an)g (on-line)i(v)m(arian)o(t)d(of)h(Algorithm)g(2)f(is)150 1659 y(not)j(the)h(only)h(p)q(ossible)h(on-line)f(approac)o(h.)28 b(An)o(y)18 b(of)f(the)h(fast)f(\014lter)h(algorithms)g(\(Ha)o(ykin,) 150 1716 y(1991\))e(could)j(also)e(b)q(e)h(utilized,)i(giving)f(rise)f (to)e(a)i(family)g(of)f(on-line)i(algorithms.)27 b(Also,)18 b(it)150 1772 y(is)f(w)o(orth)f(studying)i(the)f(application)h(of)f (the)g(recursiv)o(e)g(algorithms)g(to)f(PRESS-lik)o(e)j(cross-)150 1829 y(v)m(alidation)j(calculations)g(to)d(e\016cien)o(tly)j(compute)e (the)h(c)o(hanges)f(in)h(lik)o(eliho)q(o)q(d)i(that)d(arise)150 1885 y(from)14 b(adding)i(or)f(deleting)i(parameters)d(or)h(data)f(p)q (oin)o(ts.)150 2003 y Fl(References)150 2089 y Fu(Bourlard,)i(H.,)f(&)h (Kamp,)g(Y.)f(\(1988\).)20 b(Auto-asso)q(ciation)c(b)o(y)f(m)o(ultila)o (y)o(er)i(p)q(erceptrons)f(and)241 2145 y(singular)g(v)m(alue)g(decomp) q(osition.)22 b Ft(Biolo)n(gic)n(al)15 b(Cyb)n(ernetics,)f(59)p Fu(,)i(291-294.)150 2235 y(Breiman,)h(L.,)f(F)l(riedman,)h(J.)f(H.,)g (Olshen,)i(R.)f(A.,)f(&)g(Stone,)h(C.)e(J.)i(\(1984\).)k Ft(Classi\014c)n(ation)241 2291 y(and)16 b(R)n(e)n(gr)n(ession)e(T)m(r) n(e)n(es)p Fu(.)k(Belmon)o(t,)d(CA:)g(W)l(adsw)o(orth)f(In)o (ternational)i(Group.)150 2381 y(Bridle,)i(J.)f(\(1989\).)22 b(Probabilistic)c(in)o(terpretation)f(of)f(feedforw)o(ard)g (classi\014cation)i(net)o(w)o(ork)241 2437 y(outputs,)d(with)h (relationships)h(to)e(statistical)i(pattern)e(recognition.)22 b(In)16 b(F.)f(F)l(ogelman{)241 2494 y(Soulie)20 b(&)f(J.)f(H)o(\023) -21 b(erault)18 b(\(Eds.\),)f Ft(Neur)n(o-c)n(omputing:)28 b(A)o(lgorithms,)19 b(A)o(r)n(chite)n(ctur)n(es,)g(and)241 2550 y(Applic)n(ations)p Fu(.)g(New)c(Y)l(ork:)20 b(Springer-V)l (erlag.)952 2775 y(27)p eop %%Page: 28 28 28 27 bop 150 192 a Fu(Bun)o(tine,)20 b(W.)e(\(1991\).)27 b Ft(L)n(e)n(arning)17 b(classi\014c)n(ation)h(tr)n(e)n(es.)28 b Fu(NASA)19 b(Ames)f(T)l(ec)o(hnical)j(Rep)q(ort)241 248 y(FIA-90-12-19-01,)13 b(Mo\013ett)h(Field,)i(CA.)150 342 y(Cheeseman,)21 b(P)l(.,)f(Kelly)l(,)i(J.,)e(Self,)h(M.,)f(Stutz,)g (J.,)g(T)l(a)o(ylor,)g(W.,)f(&)h(F)l(reeman,)g(D.)f(\(1988\).)241 398 y(Auto)q(class:)i(A)16 b(Ba)o(y)o(esian)f(classi\014cation)i (system.)k(In)16 b Ft(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)j(the)g(Fifth)f (Inter-)241 455 y(national)g(Confer)n(enc)n(e)e(on)i(Machine)g(L)n(e)n (arning)p Fu(,)d(Ann)i(Arb)q(or,)g(MI.)150 549 y(Co)o(x,)f(D.)h(R.)g (\(1970\).)j Ft(The)e(A)o(nalysis)e(of)j(Binary)f(Data)p Fu(.)21 b(London:)f(Chapman-Hall.)150 642 y(Co)o(x,)11 b(D.)f(R.,)i(&)f(Hinkley)l(,)j(D.)c(V.)h(\(1974\).)17 b Ft(The)n(or)n(etic)n(al)11 b(Statistics)p Fu(.)18 b(London:)g (Chapman-Hall.)150 736 y(Dempster,)e(A.)h(P)l(.,)f(Laird,)i(N.)e(M.,)g (&)i(Rubin,)g(D.)e(B.)h(\(1977\).)23 b(Maxim)o(um)16 b(lik)o(eliho)q(o)q(d)k(from)241 793 y(incomplete)14 b(data)e(via)h(the)f(EM)g(algorithm.)19 b Ft(Journal)14 b(of)g(the)g(R)n(oyal)g(Statistic)n(al)f(So)n(ciety,)241 849 y(B)p Fu(,)i Ft(39)p Fu(,)g(1-38.)150 943 y(Duda,)f(R.)g(O.,)f(&)h (Hart,)f(P)l(.)h(E.)f(\(1973\).)18 b Ft(Pattern)d(Classi\014c)n(ation)e (and)i(Sc)n(ene)e(A)o(nalysis)p Fu(.)18 b(New)241 999 y(Y)l(ork:)i(John)15 b(Wiley)l(.)150 1093 y(Finney)l(,)h(D.)e(J.)i (\(1973\).)i Ft(Statistic)n(al)d(Metho)n(ds)h(in)g(Biolo)n(gic)n(al)f (Assay)p Fu(.)k(New)c(Y)l(ork:)20 b(Hafner.)150 1187 y(F)l(riedman,)h(J.)e(H.)g(\(1991\).)31 b(Multiv)m(ariate)20 b(adaptiv)o(e)g(regression)f(splines.)35 b Ft(The)20 b(A)o(nnals)e(of)241 1243 y(Statistics,)d(19)p Fu(,)g(1-141.)150 1337 y(F)l(un,)h(W.)f(&)h(Jordan,)f(M.)g(I.)h(\(1993\).)j Ft(The)e(moving)f(b)n(asin:)21 b(E\013e)n(ctive)15 b(action)i(se)n(ar)n (ch)f(in)g(for-)241 1394 y(war)n(d)i(mo)n(dels)p Fu(.)k(MIT)16 b(Computational)g(Cognitiv)o(e)h(Science)h(T)l(ec)o(h)f(Rep)q(ort)f (9205,)f(Cam-)241 1450 y(bridge,)h(MA.)150 1544 y(Geman,)26 b(S.,)g(Bienensto)q(c)o(k,)h(E.,)f(&)f(Doursat,)g(R.)f(\(1992\).)45 b(Neural)25 b(net)o(w)o(orks)e(and)i(the)241 1600 y(bias/v)m(ariance)16 b(dilemma.)21 b Ft(Neur)n(al)16 b(Computation,)h(4)p Fu(,)e(1-52.)150 1694 y(Golub,)j(G.)f(H.,)h(&)g(V)l(an)g(Loan,)f(G.)g (F.)g(\(1989\).)26 b Ft(Matrix)18 b(Computations)p Fu(.)28 b(Baltimore,)18 b(MD:)241 1751 y(The)d(Johns)h(Hopkins)g(Univ)o(ersit)o (y)g(Press.)150 1844 y(Hastie,)k(T.)f(J.,)h(&)f(Tibshirani,)j(R.)d(J.)g (\(1990\).)31 b Ft(Gener)n(alize)n(d)18 b(A)n(dditive)i(Mo)n(dels)p Fu(.)31 b(London:)241 1901 y(Chapman)15 b(and)g(Hall.)150 1995 y(Ha)o(ykin,)g(S.)g(\(1991\).)j Ft(A)n(daptive)f(Filter)e(The)n (ory)p Fu(.)20 b(Englew)o(o)q(o)q(d)c(Cli\013s,)f(NJ:)g(Pren)o (tice-Hall.)150 2089 y(Hin)o(ton,)20 b(G.)f(E.)g(&)h(Sejno)o(wski,)g (T.)f(J.)g(\(1986\).)31 b(Learning)20 b(and)g(relearning)g(in)g (Boltzmann)241 2145 y(mac)o(hines.)k(In)17 b(D.)f(E.)g(Rumelhart)h(&)g (J.)f(L.)g(McClelland)j(\(Eds.\),)c Ft(Par)n(al)r(lel)i(distribute)n(d) 241 2201 y(pr)n(o)n(c)n(essing)p Fu(:)h Ft(V)m(olume)e(1)p Fu(,)f(282-317.)j(Cam)o(bridge,)d(MA:)f(MIT)h(Press.)150 2295 y(Hornik,)f(K.,)f(Stinc)o(hcom)o(b)q(e,)i(M.,)d(&)i(White,)g(H.)f (\(1989\).)k(Multila)o(y)o(er)d(feedforw)o(ard)f(net)o(w)o(orks)241 2352 y(are)i(univ)o(ersal)h(appro)o(ximators.)j Ft(Neur)n(al)d (Networks,)g(2)p Fu(,)f(359-366.)150 2446 y(Jacobs,)23 b(R.)e(A,)g(Jordan,)h(M.)f(I.,)h(No)o(wlan,)g(S.)f(J.,)i(&)e(Hin)o (ton,)i(G.)d(E.)h(\(1991\).)36 b(Adaptiv)o(e)241 2502 y(mixtures)15 b(of)g(lo)q(cal)i(exp)q(erts.)j Ft(Neur)n(al)c (Computation,)h(3)p Fu(,)e(79-87.)952 2775 y(28)p eop %%Page: 29 29 29 28 bop 150 192 a Fu(Jordan,)12 b(M.)g(I.,)g(&)g(Jacobs,)h(R.)f(A.)g (\(1992\).)17 b(Hierarc)o(hies)c(of)e(adaptiv)o(e)i(exp)q(erts.)19 b(In)12 b(J.)g(Mo)q(o)q(dy)l(,)241 248 y(S.)i(Hanson,)g(&)g(R.)g (Lippmann)i(\(Eds.\),)d Ft(A)n(dvanc)n(es)h(in)g(Neur)n(al)h (Information)g(Pr)n(o)n(c)n(essing)241 304 y(Systems)g(4)p Fu(.)20 b(San)c(Mateo,)d(CA:)i(Morgan)f(Kaufmann.)20 b(pp.)h(985-993.)150 398 y(Jordan,)h(M.)e(I.,)i(&)f(Xu,)i(L.)e (\(1993\).)35 b Ft(Conver)n(genc)n(e)19 b(pr)n(op)n(erties)j(of)f(the)h (EM)f(appr)n(o)n(ach)i(to)241 455 y(le)n(arning)16 b(in)i(mixtur)n (e-of-exp)n(erts)g(ar)n(chite)n(ctur)n(es)p Fu(.)25 b(Computational)17 b(Cognitiv)o(e)g(Science)241 511 y(T)l(ec)o(h.)j(Rep.)h(9301,)14 b(MIT,)g(Cam)o(bridge,)h(MA.)150 605 y(Little,)f(R.)e(J.)h(A.,)f(&)h (Rubin,)i(D.)c(B.)i(\(1987\).)k Ft(Statistic)n(al)c(A)o(nalysis)f(with) i(Missing)e(Data)p Fu(.)20 b(New)241 661 y(Y)l(ork:)g(John)15 b(Wiley)l(.)150 755 y(Ljung,)g(L.)h(&)f(S\177)-23 b(oderstr\177)g(om,) 14 b(T.)h(\(1986\).)j Ft(The)n(ory)e(and)g(pr)n(actic)n(e)g(of)h(r)n(e) n(cursive)e(identi\014c)n(ation)p Fu(.)241 812 y(Cam)o(bridge:)20 b(MIT)15 b(Press.)150 906 y(McCullagh,)j(P)l(.)e(&)h(Nelder,)h(J.A.)e (\(1983\).)23 b Ft(Gener)n(alize)n(d)16 b(Line)n(ar)h(Mo)n(dels)p Fu(.)23 b(London:)h(Chap-)241 962 y(man)15 b(and)g(Hall.)150 1056 y(Mo)q(o)q(dy)l(,)h(J.)h(\(1989\).)22 b(F)l(ast)15 b(learning)j(in)f(m)o(ulti-resolution)h(hierarc)o(hies.)25 b(In)17 b(D.S.)f(T)l(ouretzky)241 1112 y(\(Ed.\),)i Ft(A)n(dvanc)n(es)g (in)h(Neur)n(al)g(Information)g(Pr)n(o)n(c)n(essing)e(Systems)p Fu(.)30 b(San)18 b(Mateo,)g(CA:)241 1169 y(Morgan)c(Kaufmann)h (Publishers.)150 1262 y(Murth)o(y)l(,)21 b(S.)g(K.,)h(Kasif,)g(S.,)g(&) f(Salzb)q(erg,)i(S.)d(\(1993\).)35 b Ft(OC1:)c(A)21 b(r)n(andomize)n(d) g(algorithm)241 1319 y(for)16 b(building)f(oblique)g(de)n(cision)g(tr)n (e)n(es)p Fu(.)j(T)l(ec)o(hnical)e(Rep)q(ort,)f(Departmen)o(t)e(of)h (Computer)241 1375 y(Science,)j(Johns)e(Hopkins)h(Univ)o(ersit)o(y)l(.) 150 1469 y(No)o(wlan,)f(S.J.)h(\(1990\).)j(Maxim)o(um)d(lik)o(eliho)q (o)q(d)i(comp)q(etitiv)o(e)f(learning.)22 b(In)16 b(D.S.)f(T)l (ouretzky)241 1526 y(\(Ed.\),)f Ft(A)n(dvanc)n(es)g(in)i(Neur)n(al)g (Information)g(Pr)n(o)n(c)n(essing)d(Systems)j(2)p Fu(.)k(San)15 b(Mateo,)e(CA:)241 1582 y(Morgan)h(Kaufmann)h(Publishers.)150 1676 y(No)o(wlan,)21 b(S.J.)g(\(1991\).)34 b Ft(Soft)21 b(c)n(omp)n(etitive)f(adaptation:)32 b(Neur)n(al)21 b(network)g(le)n (arning)f(algo-)241 1732 y(rithms)14 b(b)n(ase)n(d)f(on)g(\014tting)g (statistic)n(al)g(mixtur)n(es)p Fu(.)19 b(T)l(ec)o(h.)h(Rep.)f (CMU-CS-91-126,)12 b(CMU,)241 1789 y(Pittsburgh,)j(P)l(A.)150 1883 y(Quandt,)21 b(R.E.,)g(&)f(Ramsey)l(,)h(J.B.)f(\(1972\).)32 b(A)20 b(new)g(approac)o(h)g(to)f(estimating)h(switc)o(hing)241 1939 y(regressions.)g Ft(Journal)c(of)h(the)f(A)o(meric)n(an)g (Statistic)n(al)f(So)n(ciety,)g(67)p Fu(,)h(306-310.)150 2033 y(Quinlan,)h(J.)e(R.)g(\(1986\).)j(Induction)f(of)e(decision)i (trees.)j Ft(Machine)15 b(L)n(e)n(arning,)g(1)p Fu(,)g(81-106.)150 2127 y(Quinlan,)i(J.)e(R.,)g(&)h(Riv)o(est,)f(R.)h(L.)f(\(1989\).)j (Inferring)f(decision)g(trees)e(using)h(the)f(Minim)o(um)241 2183 y(Description)h(Length)g(Principle.)22 b Ft(Information)16 b(and)g(Computation,)h(80)p Fu(,)e(227-248.)150 2277 y(Redner,)g(R.)f(A.,)f(&)h(W)l(alk)o(er,)f(H.)h(F.)f(\(1984\).)k (Mixture)d(densities,)i(maxim)o(um)d(lik)o(eliho)q(o)q(d)k(and)241 2333 y(the)e(EM)g(algorithm.)20 b Ft(SIAM)15 b(R)n(eview,)h(26)p Fu(,)f(195-239.)150 2427 y(Sanger,)h(T.)g(D.)g(\(1991\).)22 b(A)16 b(tree-structured)h(adaptiv)o(e)g(net)o(w)o(ork)e(for)h (function)h(appro)o(xima-)241 2484 y(tion)i(in)h(high)h(dimensional)g (spaces.)32 b Ft(IEEE)19 b(T)m(r)n(ansactions)f(on)i(Neur)n(al)f (Networks,)i(2)p Fu(,)241 2540 y(285-293.)952 2775 y(29)p eop %%Page: 30 30 30 29 bop 150 192 a Fu(Scott,)14 b(D.)h(W.)f(\(1992\).)k Ft(Multivariate)f(Density)f(Estimation)p Fu(.)j(New)c(Y)l(ork:)20 b(John)c(Wiley)l(.)150 285 y(Sp)q(ec)o(h)o(t,)h(D.)f(F.)f(\(1991\).)22 b(A)17 b(general)g(regression)g(neural)g(net)o(w)o(ork.)22 b Ft(IEEE)17 b(T)m(r)n(ansactions)e(on)241 342 y(Neur)n(al)h(Networks)p Fu(,)f Ft(2)p Fu(,)g(568-576.)150 435 y(Stone,)f(C.)g(J.)g(\(1977\).)j (Consisten)o(t)d(nonparametric)g(regression.)20 b Ft(The)15 b(A)o(nnals)f(of)h(Statistics,)241 492 y(5)p Fu(,)g(595-645.)150 585 y(Str\177)-23 b(om)o(b)q(erg,)11 b(J.)g(E.,)f(Zrida,)i(J.,)f(&)h (Isaksson,)f(A.)f(\(1991\).)17 b(Neural)12 b(trees|using)f(neural)h (nets)f(in)241 642 y(a)i(tree)h(classi\014er)h(structure.)20 b Ft(IEEE)14 b(International)f(Confer)n(enc)n(e)g(on)i(A)n(c)n (oustics,)f(Sp)n(e)n(e)n(ch)241 698 y(and)i(Signal)f(Pr)n(o)n(c)n (essing)p Fu(,)e(pp.)20 b(137-140.)150 792 y(Titterington,)c(D.)g(M.,)f (Smith,)i(A.)f(F.)g(M.,)f(&)i(Mak)o(o)o(v,)d(U.)i(E.)g(\(1985\).)22 b Ft(Statistic)n(al)16 b(A)o(nalysis)241 848 y(of)g(Finite)g(Mixtur)n (e)g(Distributions)p Fu(.)k(New)15 b(Y)l(ork:)20 b(John)15 b(Wiley)l(.)150 942 y(Utgo\013,)k(P)l(.)g(E.,)h(&)f(Bro)q(dley)l(,)j (C.)c(E.)h(\(1990\).)31 b(An)20 b(incremen)o(tal)g(metho)q(d)g(for)f (\014nding)h(m)o(ul-)241 998 y(tiv)m(ariate)f(splits)g(for)e(decision)j (trees.)29 b(In)18 b Ft(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)i(the)g (Seventh)f(International)241 1055 y(Confer)n(enc)n(e)c(on)i(Machine)g (L)n(e)n(arning)p Fu(,)d(Los)i(Altos,)g(CA.)150 1148 y(W)l(ah)o(ba,)20 b(G.,)f(Gu,)h(C.,)g(W)l(ang,)g(Y.,)g(&)g(Chapp)q (ell,)i(R.)d(\(1993\).)31 b Ft(Soft)21 b(classi\014c)n(ation,)e(a.k.a.) 241 1205 y(risk)i(estimation,)i(via)f(p)n(enalize)n(d)e(lo)n(g)h (likeliho)n(o)n(d)g(and)h(smo)n(othing)f(spline)f(analysis)h(of)241 1261 y(varianc)n(e)p Fu(.)k(T)l(ec)o(h.)g(Rep.)h(899,)16 b(Departmen)o(t)g(of)h(Statistics,)g(Univ)o(ersit)o(y)h(of)e (Wisconsin,)241 1318 y(Madison.)150 1411 y(W)l(u,)21 b(C.)f(F.)f(J.)i(\(1983\).)33 b(On)21 b(the)f(con)o(v)o(ergence)g(prop) q(erties)h(of)f(the)h(EM)e(algorithm.)35 b Ft(The)241 1468 y(A)o(nnals)14 b(of)j(Statistics,)e(11)p Fu(,)g(95-103.)150 1590 y Fp(App)r(endix)j(A)h({)f(Iterativ)n(ely)f(rew)n(eigh)n(ted)h (least)g(squares)150 1675 y Fu(The)g(iterativ)o(ely)g(rew)o(eigh)o(ted) g(least)g(squares)g(\(IRLS\))g(algorithm)g(is)g(the)g(inner)g(lo)q(op)h (of)e(the)150 1732 y(algorithm)k(that)f(w)o(e)g(ha)o(v)o(e)g(prop)q (osed)i(for)e(the)g(HME)h(arc)o(hitecture.)36 b(In)22 b(this)f(section,)h(w)o(e)150 1788 y(describ)q(e)i(the)e(IRLS)h (algorithm,)h(deriving)f(it)f(as)g(a)g(sp)q(ecial)i(case)e(of)f(the)h (Fisher)h(scoring)150 1845 y(metho)q(d)d(for)g(generalized)i(linear)f (mo)q(dels.)36 b(Our)21 b(presen)o(tation)f(deriv)o(es)h(from)e (McCullagh)150 1901 y(and)c(Nelder)i(\(1983\).)241 1958 y(IRLS)c(is)g(an)f(iterativ)o(e)g(algorithm)g(for)g(computing)g(the)g (maxim)o(um)g(lik)o(eliho)q(o)q(d)j(estimates)150 2014 y(of)k(the)h(parameters)e(of)h(a)g(generalized)j(linear)e(mo)q(del.)34 b(It)19 b(is)h(a)g(sp)q(ecial)h(case)e(of)g(a)g(general)150 2071 y(algorithm)d(for)g(maxim)o(um)g(lik)o(eliho)q(o)q(d)j(estimation) e(kno)o(wn)f(as)f(the)i(Fisher)f(scoring)h(metho)q(d)150 2127 y(\(Finney)l(,)c(1973\).)k(Let)11 b Fj(l)q Fu(\()p Fi(\014)q Fu(;)d Fn(X)f Fu(\))j(b)q(e)i(a)f(log)g(lik)o(eliho)q(o)q(d)j (function|a)e(function)h(of)d(the)i(parameter)150 2183 y(v)o(ector)h Fi(\014)r Fu(|and)i(let)g(\()p Fj(@)s(l)o(=@)s Fi(\014)q Fj(@)s Fi(\014)712 2165 y Fm(T)740 2183 y Fu(\))f(denote)g (the)g(Hessian)h(of)f(the)g(log)h(lik)o(eliho)q(o)q(d.)22 b(The)15 b(Fisher)150 2240 y(scoring)g(metho)q(d)h(up)q(dates)g(the)f (parameter)f(estimates)h Fi(\014)i Fu(as)e(follo)o(ws:)646 2360 y Fi(\014)677 2371 y Fm(r)q Fk(+1)754 2360 y Fu(=)e Fi(\014)833 2371 y Fm(r)863 2360 y Fn(\000)d(f)p Fj(E)s Fu([)1036 2330 y Fj(@)s(l)p 986 2350 144 2 v 986 2396 a(@)s Fi(\014)p Fj(@)s Fi(\014)1101 2377 y Fm(T)1133 2360 y Fu(])p Fn(g)1169 2342 y Fd(\000)p Fk(1)1229 2330 y Fj(@)s(l)p 1221 2350 59 2 v 1221 2392 a(@)s Fi(\014)1284 2360 y Fj(;)422 b Fu(\(38\))150 2494 y(where)22 b Fi(\014)320 2504 y Fm(r)360 2494 y Fu(denotes)g(the)g(parameter)f(estimate)h(at)f (the)h Fj(r)1198 2477 y Fm(th)1255 2494 y Fu(iteration)g(and)g Fj(@)s(l=@)s Fi(\014)h Fu(is)f(the)150 2550 y(gradien)o(t)15 b(v)o(ector.)k(Note)c(that)f(the)h(Fisher)h(scoring)f(metho)q(d)g(is)h (essen)o(tially)g(the)f(same)g(as)g(the)952 2775 y(30)p eop %%Page: 31 31 31 30 bop 150 192 a Fu(Newton-Raphson)15 b(algorithm,)g(except)g(that)f (the)h(exp)q(ected)h(v)m(alue)g(of)e(the)h(Hessian)g(replaces)150 248 y(the)j(Hessian.)28 b(There)18 b(are)f(statistical)h(reasons)g(for) f(preferring)h(the)g(exp)q(ected)h(v)m(alue)g(of)e(the)150 304 y(Hessian|and)22 b(the)f(exp)q(ected)g(v)m(alue)h(of)e(the)h (Hessian)g(is)g(often)f(easier)h(to)f(compute|but)150 361 y(Newton-Raphson)c(can)f(also)g(b)q(e)h(used)g(in)g(man)o(y)f (cases.)241 417 y(The)f(lik)o(eliho)q(o)q(d)j(in)d(generalized)i (linear)f(mo)q(del)g(theory)e(is)h(a)g(pro)q(duct)g(of)f(densities)j (from)150 474 y(the)11 b(exp)q(onen)o(tial)i(family)e(of)g (distributions.)20 b(This)12 b(family)g(is)f(an)g(imp)q(ortan)o(t)g (class)g(in)h(statistics)150 530 y(and)18 b(includes)h(man)o(y)e (useful)h(densities,)i(suc)o(h)d(as)g(the)h(normal,)f(the)h(P)o (oisson,)f(the)g(binomial)150 587 y(and)j(the)h(gamma.)33 b(The)21 b(general)f(form)g(of)g(a)g(densit)o(y)g(in)h(the)g(exp)q (onen)o(tial)g(family)g(is)g(the)150 643 y(follo)o(wing:)559 700 y Fj(P)6 b Fu(\()p Fj(y)r(;)i(\021)r(;)g(\036)p Fu(\))i(=)j(exp)q Fn(f)p Fu(\()p Fj(\021)r(y)d Fn(\000)h Fj(b)p Fu(\()p Fj(\021)r Fu(\)\))p Fj(=\036)c Fu(+)k Fj(c)p Fu(\()p Fj(y)r(;)d(\036)p Fu(\))p Fn(g)p Fj(;)333 b Fu(\(39\))150 783 y(where)15 b Fj(\021)i Fu(is)e(kno)o(wn)g(as)g(the)h(\\natural)e (parameter")h(and)g Fj(\036)g Fu(is)h(the)f(disp)q(ersion)i(parameter.) 1751 767 y Fk(8)150 889 y Ft(Example)f(\(Bernoul)r(li)g(density\))241 946 y Fu(The)f(Bernoulli)j(densit)o(y)e(with)f(mean)g Fj(\031)i Fu(has)e(the)g(follo)o(wing)h(form:)563 1048 y Fj(P)6 b Fu(\()p Fj(y)r(;)i(\031)r Fu(\))40 b(=)i Fj(\031)852 1029 y Fm(y)872 1048 y Fu(\(1)10 b Fn(\000)g Fj(\031)r Fu(\))1014 1029 y Fk(1)p Fd(\000)p Fm(y)747 1126 y Fu(=)42 b(exp)q Fn(f)p Fu(ln\()1016 1095 y Fj(\031)p 977 1116 106 2 v 977 1157 a Fu(1)10 b Fn(\000)g Fj(\031)1088 1126 y Fu(\))p Fj(y)h Fu(+)g(ln)q(\(1)e Fn(\000)i Fj(\031)r Fu(\))p Fn(g)747 1212 y Fu(=)42 b(exp)q Fn(f)p Fj(\021)r(y)11 b Fn(\000)f Fu(ln)q(\(1)f(+)i Fj(e)1175 1193 y Fm(\021)1196 1212 y Fu(\))p Fn(g)p Fj(;)469 b Fu(\(40\))150 1314 y(where)22 b Fj(\021)i Fu(=)f(ln)q(\()p Fj(\031)r(=)p Fu(1)13 b Fn(\000)h Fj(\031)r Fu(\))21 b(is)h(the)f(natural)h(parameter)e(of)h (the)h(Bernoulli)h(densit)o(y)l(.)39 b(This)150 1370 y(parameter)13 b(has)g(the)h(in)o(terpretation)f(as)g(the)h(log)g(o)q (dds)f(of)h(\\success")f(in)h(a)g(random)f(Bernoulli)150 1427 y(exp)q(erimen)o(t.)241 1533 y(In)k(a)e(generalized)j(linear)f(mo) q(del,)g(the)f(parameter)f Fj(\021)j Fu(is)e(mo)q(deled)i(as)d(a)h (linear)h(function)150 1590 y(of)e(the)g(input)h Fl(x)p Fu(:)883 1646 y Fj(\021)e Fu(=)f Fi(\014)999 1627 y Fm(T)1027 1646 y Fl(x)p Fj(;)150 1729 y Fu(where)k Fi(\014)i Fu(is)e(a)g (parameter)g(v)o(ector.)24 b(Substituting)19 b(this)e(expression)h(in)o (to)f(Equation)g(39)g(and)150 1786 y(taking)e(the)f(pro)q(duct)h(of)f Fj(N)19 b Fu(suc)o(h)c(densities)h(yields)g(the)f(follo)o(wing)g(log)g (lik)o(eliho)q(o)q(d)i(for)d(a)h(data)150 1842 y(set)g Fn(X)20 b Fu(=)13 b Fn(f)p Fu(\()p Fl(x)390 1826 y Fk(\()p Fm(t)p Fk(\))431 1842 y Fj(;)8 b(y)476 1826 y Fk(\()p Fm(t)p Fk(\))517 1842 y Fu(\))p Fn(g)558 1826 y Fm(N)558 1854 y Fk(1)591 1842 y Fu(:)444 1945 y Fj(l)q Fu(\()p Fi(\014)p Fj(;)g Fn(X)f Fu(\))12 b(=)645 1904 y Fe(X)668 1994 y Fm(t)705 1945 y Fn(f)p Fu(\()p Fi(\014)777 1926 y Fm(T)804 1945 y Fl(x)832 1926 y Fk(\()p Fm(t)p Fk(\))874 1945 y Fj(y)898 1926 y Fk(\()p Fm(t)p Fk(\))950 1945 y Fn(\000)f Fj(b)p Fu(\()p Fi(\014)1065 1926 y Fm(T)1092 1945 y Fl(x)1120 1926 y Fk(\()p Fm(t)p Fk(\))1162 1945 y Fu(\)\))p Fj(=\036)e Fu(+)i Fj(c)p Fu(\()p Fj(y)1365 1926 y Fk(\()p Fm(t)p Fk(\))1406 1945 y Fj(;)d(\036)p Fu(\))p Fn(g)p Fj(:)150 2089 y Fu(The)25 b(observ)m(ations)f Fj(y)545 2073 y Fk(\()p Fm(t)p Fk(\))612 2089 y Fu(are)g(assumed)g(to)g (b)q(e)h(sampled)g(indep)q(enden)o(tly)j(from)23 b(densities)150 2146 y Fj(P)6 b Fu(\()p Fj(y)r(;)i(\021)273 2129 y Fk(\()p Fm(t)p Fk(\))314 2146 y Fj(;)g(\036)p Fu(\),)13 b(where)j Fj(\021)563 2129 y Fk(\()p Fm(t)p Fk(\))617 2146 y Fu(=)d Fi(\014)697 2127 y Fm(T)724 2146 y Fl(x)752 2129 y Fk(\()p Fm(t)p Fk(\))794 2146 y Fu(.)241 2202 y(W)l(e)i(no)o(w)g(compute)g(the) h(gradien)o(t)f(of)f(the)i(log)f(lik)o(eliho)q(o)q(d:)658 2292 y Fj(@)s(l)p 650 2312 59 2 v 650 2354 a(@)s Fi(\014)725 2323 y Fu(=)773 2283 y Fe(X)797 2372 y Fm(t)833 2323 y Fu(\()p Fj(y)875 2304 y Fk(\()p Fm(t)p Fk(\))927 2323 y Fn(\000)c Fj(b)993 2304 y Fd(0)1004 2323 y Fu(\()p Fi(\014)1053 2304 y Fm(T)1081 2323 y Fl(x)1109 2304 y Fk(\()p Fm(t)p Fk(\))1150 2323 y Fu(\)\))p Fl(x)1214 2304 y Fk(\()p Fm(t)p Fk(\))1256 2323 y Fj(=\036)413 b Fu(\(41\))p 150 2407 660 2 v 202 2434 a Fh(8)219 2450 y Fg(W)m(e)16 b(restrict)g(ourselv)o(es)h(to)e(scalar-v)n(alued)j (random)e(v)n(ariables)i(to)d(simplify)j(the)d(presen)o(tation,)j(and)e (de-)150 2495 y(scrib)q(e)e(the)f(\(straigh)o(tforw)o(ard\))h (extension)h(to)e(v)o(ector-v)n(alued)i(random)f(v)n(ariables)h(at)e (the)g(end)g(of)g(the)g(section.)952 2775 y Fu(31)p eop %%Page: 32 32 32 31 bop 150 192 a Fu(and)15 b(the)h(Hessian)g(of)e(the)i(log)f(lik)o (eliho)q(o)q(d:)653 282 y Fj(@)s(l)p 601 302 144 2 v 601 348 a(@)s Fi(\014)q Fj(@)s Fi(\014)718 330 y Fm(T)763 312 y Fu(=)e Fn(\000)854 272 y Fe(X)877 362 y Fm(t)921 312 y Fj(b)941 294 y Fd(00)962 312 y Fu(\()p Fi(\014)1011 294 y Fm(T)1039 312 y Fl(x)1067 294 y Fk(\()p Fm(t)p Fk(\))1108 312 y Fu(\))p Fl(x)1154 294 y Fk(\()p Fm(t)p Fk(\))1196 312 y Fl(x)1224 294 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1291 312 y Fj(=\036:)365 b Fu(\(42\))150 447 y(These)19 b(quan)o(tities)g(could)h(b)q(e)f(substituted)h(directly)g(in)o(to)e (Equation)h(38,)g(ho)o(w)o(ev)o(er)f(there)g(is)150 504 y(additional)h(mathematical)f(structure)g(that)f(can)h(b)q(e)g (exploited.)29 b(First)18 b(note)f(the)h(follo)o(wing)150 560 y(iden)o(tit)o(y)l(,)e(whic)o(h)g(is)g(true)f(of)f(an)o(y)h(log)g (lik)o(eliho)q(o)q(d:)862 681 y Fj(E)s Fu([)925 650 y Fj(@)s(l)p 917 671 59 2 v 917 712 a(@)s Fi(\014)979 681 y Fu(])d(=)h(0)p Fj(:)150 805 y Fu(\(This)e(fact)g(can)g(b)q(e)h(pro)o (v)o(ed)f(b)o(y)g(di\013eren)o(tiating)h(b)q(oth)f(sides)h(of)e(the)i (iden)o(tit)o(y)1475 771 y Fe(R)1511 805 y Fj(P)6 b Fu(\()p Fj(y)r(;)i Fi(\014)p Fj(;)g(\036)p Fu(\))p Fj(dy)14 b Fu(=)150 862 y(1)19 b(with)h(resp)q(ect)h(to)e Fi(\014)q Fu(\).)33 b(Because)21 b(this)f(iden)o(tit)o(y)g(is)g(true)g(for)f(an)o (y)g(set)h(of)f(observ)o(ed)h(data,)150 918 y(including)e(all)e (subsets)f(of)g Fn(X)7 b Fu(,)15 b(w)o(e)g(ha)o(v)o(e)g(the)g(follo)o (wing:)777 1020 y Fj(E)s Fu([)p Fj(y)851 1002 y Fk(\()p Fm(t)p Fk(\))892 1020 y Fu(])d(=)h Fj(b)985 1002 y Fd(0)996 1020 y Fu(\()p Fi(\014)1045 1002 y Fm(T)1073 1020 y Fl(x)1101 1002 y Fk(\()p Fm(t)p Fk(\))1143 1020 y Fu(\))p Fj(;)150 1122 y Fu(for)k(all)i Fj(t)p Fu(.)29 b(This)19 b(equation)g(implies)h (that)d(the)h(mean)g(of)g Fj(y)1184 1106 y Fk(\()p Fm(t)p Fk(\))1226 1122 y Fu(,)h(whic)o(h)g(w)o(e)e(denote)i(as)e Fj(\026)1695 1106 y Fk(\()p Fm(t)p Fk(\))1738 1122 y Fu(,)h(is)150 1179 y(a)i(function)h(of)g Fj(\021)459 1162 y Fk(\()p Fm(t)p Fk(\))500 1179 y Fu(.)36 b(W)l(e)20 b(therefore)g(include)j(in)f(the)e(generalized)i(linear)g(mo)q(del)f (the)g Ft(link)150 1235 y(function)p Fu(,)14 b(whic)o(h)i(mo)q(dels)g Fj(\026)g Fu(as)f(a)g(function)h(of)f Fj(\021)r Fu(:)839 1338 y Fj(\026)866 1319 y Fk(\()p Fm(t)p Fk(\))921 1338 y Fu(=)e Fj(f)5 b Fu(\()p Fj(\021)1039 1319 y Fk(\()p Fm(t)p Fk(\))1081 1338 y Fu(\))p Fj(:)150 1489 y Ft(Example)16 b(\(Bernoul)r(li)g(density\))241 1546 y Fu(Equation)f(40)g(sho)o(ws)f (that)h Fj(b)p Fu(\()p Fj(\021)r Fu(\))c(=)i(ln\(1)d(+)g Fj(e)1021 1529 y Fm(\021)1043 1546 y Fu(\))k(for)h(the)g(Bernoulli)j (densit)o(y)l(.)i(Th)o(us)784 1669 y Fj(\026)13 b Fu(=)g Fj(b)892 1650 y Fd(0)903 1669 y Fu(\()p Fj(\021)r Fu(\))e(=)1067 1638 y Fj(e)1088 1621 y Fm(\021)p 1028 1658 121 2 v 1028 1700 a Fu(1)f(+)g Fj(e)1127 1687 y Fm(\021)1154 1669 y Fj(;)150 1787 y Fu(whic)o(h)15 b(is)g(the)f(logistic)h(function.)21 b(In)o(v)o(erting)14 b(the)h(logistic)g(function)g(yields)h Fj(\021)d Fu(=)g(ln)q(\()p Fj(\026=)p Fu(1)8 b Fn(\000)g Fj(\026)p Fu(\);)150 1844 y(th)o(us,)15 b Fj(\026)g Fu(equals)h Fj(\031)r Fu(,)e(as)h(it)h(m)o(ust.)241 1950 y(The)h(link)h(function)g Fj(f)5 b Fu(\()p Fj(\021)r Fu(\))14 b(=)h Fj(b)779 1934 y Fd(0)790 1950 y Fu(\()p Fj(\021)r Fu(\))h(is)h(kno)o(wn)f(in)i (generalized)g(linear)g(mo)q(del)g(theory)e(as)150 2007 y(the)j Ft(c)n(anonic)n(al)g(link)p Fu(.)30 b(By)20 b(parameterizing)f 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(link)g(is)f(used,)g(the)g(results)g(presen)o(ted)g(in)g(the)g (remainder)g(of)f(this)i(section)150 2473 y(are)i(correct)h(for)f(the)g (Fisher)h(scoring)h(metho)q(d.)24 b(If)15 b(noncanonical)j(links)f(are) f(used,)g(then)f(Newton-Raphson)150 2519 y(will)g(include)g(additional) h(terms)d(\(terms)g(that)g(v)n(anish)i(under)f(the)f(exp)q(ectation)i (op)q(erator\).)952 2775 y Fu(32)p eop %%Page: 33 33 33 32 bop 241 192 a Fu(T)l(o)14 b(con)o(tin)o(ue)h(the)f(dev)o(elopmen) o(t,)h(w)o(e)f(need)h(an)f(additional)i(fact)d(ab)q(out)h(log)h(lik)o (eliho)q(o)q(ds.)150 248 y(By)d(di\013eren)o(tiating)h(the)f(iden)o (tit)o(y)748 214 y Fe(R)783 248 y Fj(P)6 b Fu(\()p Fj(y)r(;)i Fi(\014)q Fu(\))p Fj(dy)14 b Fu(=)f(1)e(t)o(wice)h(with)h(resp)q(ect)f (to)f Fi(\014)r Fu(,)h(the)g(follo)o(wing)150 304 y(iden)o(tit)o(y)k (can)f(b)q(e)h(established:)688 421 y Fj(E)s Fu([)793 390 y Fj(@)s(l)p 743 411 144 2 v 743 457 a(@)s Fi(\014)p Fj(@)s Fi(\014)858 438 y Fm(T)890 421 y Fu(])d(=)g Fn(\000)p Fj(E)s Fu([)1061 390 y Fj(@)s(l)p 1054 411 59 2 v 1054 452 a(@)s Fi(\014)1116 421 y Fu(][)1155 390 y Fj(@)s(l)p 1147 411 V 1147 452 a(@)s Fi(\014)1209 421 y Fu(])1222 402 y Fm(T)1250 421 y Fj(:)150 543 y Fu(This)19 b(iden)o(tit)o(y)h(can) f(b)q(e)g(used)h(to)e(obtain)h(a)f(relationship)j(b)q(et)o(w)o(een)e (the)g(v)m(ariance)g(of)g Fj(\021)h Fu(and)150 599 y(the)13 b(function)h Fj(b)p Fu(\()p Fj(\021)r Fu(\))d(in)j(the)f(exp)q(onen)o (tial)i(family)e(densit)o(y)l(.)20 b(Beginning)15 b(with)f(Equation)f (42,)f(w)o(e)150 656 y(ha)o(v)o(e:)150 763 y Fn(\000)p Fj(E)s Fu([)235 723 y Fe(X)258 813 y Fm(t)302 763 y Fj(b)322 745 y Fd(00)342 763 y Fu(\()p Fi(\014)392 745 y Fm(T)419 763 y Fl(x)447 745 y Fk(\()p Fm(t)p Fk(\))489 763 y Fu(\))p Fl(x)535 745 y Fm(t)549 763 y Fl(x)577 745 y Fk(\()p Fm(t)p Fk(\))p Fm(T)644 763 y Fj(=\036)p Fu(])41 b(=)h Fj(E)s Fu([)930 733 y Fj(@)s(l)p 880 753 144 2 v 880 799 a(@)s Fi(\014)p Fj(@)s Fi(\014)995 781 y Fm(T)1028 763 y Fu(])748 896 y(=)g Fn(\000)p Fj(E)s Fu([)923 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Fe(X)990 1192 y Fm(t)1026 1142 y Fu(\()p Fj(y)1068 1124 y Fk(\()p Fm(t)p Fk(\))1120 1142 y Fn(\000)10 b Fj(b)1185 1124 y Fd(0)1197 1142 y Fu(\()p Fi(\014)1246 1124 y Fm(T)1273 1142 y Fl(x)1301 1124 y Fk(\()p Fm(t)p Fk(\))1343 1142 y Fu(\)\))1379 1124 y Fk(2)1398 1142 y Fl(x)1426 1124 y Fk(\()p Fm(t)p Fk(\))1468 1142 y Fl(x)1496 1124 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1563 1142 y Fu(])748 1272 y(=)42 b Fn(\000)878 1242 y Fu(1)p 865 1262 V 865 1304 a Fj(\036)892 1290 y Fk(2)925 1232 y Fe(X)948 1322 y Fm(t)992 1272 y Fu(V)l(ar[)p Fj(y)1100 1254 y Fk(\()p Fm(t)p Fk(\))1142 1272 y Fu(])p Fl(x)1183 1254 y Fk(\()p Fm(t)p Fk(\))1225 1272 y Fl(x)1253 1254 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1320 1272 y Fj(;)150 1403 y Fu(where)19 b(w)o(e)f(ha)o(v)o(e)f(used)i(the)g(indep)q(endence)i (assumption)e(in)g(the)f(fourth)g(step.)29 b(Comparing)150 1459 y(Equation)15 b(42)g(with)g(the)h(last)f(equation,)g(w)o(e)g (obtain)g(the)h(follo)o(wing)g(relationship:)741 1557 y(V)l(ar[)p Fj(y)849 1538 y Fk(\()p Fm(t)p Fk(\))891 1557 y Fu(])c(=)h Fj(\036b)1011 1538 y Fd(00)1032 1557 y Fu(\()p Fi(\014)1081 1538 y Fm(T)1109 1557 y Fl(x)1137 1538 y Fk(\()p Fm(t)p Fk(\))1178 1557 y Fu(\))p Fj(:)150 1655 y Fu(Moreo)o(v)o(er,)g(b)q(ecause)k Fj(f)5 b Fu(\()p Fj(\021)r Fu(\))11 b(=)i Fj(b)698 1639 y Fd(0)709 1655 y Fu(\()p Fj(\021)r Fu(\),)g(w)o(e)i(ha)o(v)o(e)742 1753 y(V)l(ar[)p Fj(y)850 1734 y Fk(\()p Fm(t)p Fk(\))892 1753 y Fu(])d(=)h Fj(\036f)1019 1734 y Fd(0)1031 1753 y Fu(\()p Fi(\014)1080 1734 y Fm(T)1108 1753 y Fl(x)1136 1734 y Fk(\()p Fm(t)p Fk(\))1178 1753 y Fu(\))p Fj(:)510 b Fu(\(43\))241 1851 y(W)l(e)12 b(no)o(w)f(assem)o(ble)h(the)f(v)m (arious)h(pieces.)20 b(First)12 b(note)f(that)g(Equation)h(43)f(can)g (b)q(e)i(utilized)150 1907 y(to)i(express)g(the)g(Hessian)h(\(Equation) f(42\))g(in)h(the)f(follo)o(wing)h(form:)742 1996 y Fj(@)s(l)p 691 2016 144 2 v 691 2062 a(@)s Fi(\014)q Fj(@)s Fi(\014)807 2044 y Fm(T)852 2026 y Fu(=)d Fn(\000)943 1986 y Fe(X)967 2076 y Fm(t)1011 2026 y Fl(x)1039 2008 y Fk(\()p Fm(t)p Fk(\))1080 2026 y Fl(x)1108 2008 y Fk(\()p Fm(t)p Fk(\))p Fm(T)1176 2026 y Fj(w)1210 2008 y Fk(\()p Fm(t)p Fk(\))1252 2026 y Fj(;)150 2167 y Fu(where)i(the)h(w)o(eigh)o(t)f Fj(w)538 2150 y Fk(\()p Fm(t)p Fk(\))595 2167 y Fu(is)h(de\014ned)g(as) f(follo)o(ws:)814 2293 y Fj(w)848 2275 y Fk(\()p Fm(t)p Fk(\))903 2293 y Fu(=)957 2263 y Fj(f)984 2246 y Fd(0)996 2263 y Fu(\()p Fj(\021)1039 2246 y Fk(\()p Fm(t)p Fk(\))1080 2263 y Fu(\))1098 2246 y Fk(2)p 956 2283 163 2 v 956 2326 a Fu(V)l(ar[)p Fj(y)1064 2313 y Fk(\()p Fm(t)p Fk(\))1106 2326 y Fu(])1123 2293 y Fj(:)150 2414 y Fu(In)h(matrix)f(notation)f(w)o (e)h(ha)o(v)o(e:)818 2491 y Fj(@)s(l)p 767 2512 144 2 v 767 2558 a(@)s Fi(\014)q Fj(@)s Fi(\014)883 2539 y Fm(T)928 2522 y Fu(=)e Fn(\000)p Fj(X)1053 2503 y Fm(T)1080 2522 y Fj(W)6 b(X)q(;)538 b Fu(\(44\))952 2775 y(33)p eop %%Page: 34 34 34 33 bop 150 192 a Fu(where)18 b Fj(X)i Fu(is)e(the)f(matrix)g(whose)g (ro)o(ws)f(are)h(the)h(input)g(v)o(ectors)f Fl(x)1330 175 y 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Fj(W)g Fl(e)392 b Fu(\(46\))760 1293 y(=)42 b(\()p Fj(X)897 1274 y Fm(T)924 1293 y Fj(W)6 b(X)t Fu(\))1033 1274 y Fd(\000)p Fk(1)1079 1293 y Fj(X)1121 1274 y Fm(T)1148 1293 y Fj(W)g Fl(z)p Fj(;)486 b Fu(\(47\))150 1395 y(where)19 b Fl(z)h Fu(=)f Fj(X)t Fi(\014)455 1406 y Fm(r)487 1395 y Fu(+)13 b Fl(e)p Fu(.)572 1379 y Fk(10)640 1395 y Fu(These)20 b(equations)f(are)g(the)g(normal)g(equations)h(for)e(a)h(w)o(eigh)o (ted)150 1452 y(least)13 b(squares)f(problem)i(with)f(observ)m(ations)g Fn(f)p Fu(\()p Fl(x)1014 1435 y Fk(\()p Fm(t)p Fk(\))1055 1452 y Fj(;)8 b(z)1099 1435 y Fk(\()p Fm(t)p Fk(\))1140 1452 y Fu(\))p Fn(g)1181 1435 y Fm(N)1181 1463 y Fk(1)1227 1452 y Fu(and)13 b(observ)m(ation)g(w)o(eigh)o(ts)f Fj(w)1745 1435 y Fk(\()p Fm(t)p Fk(\))1787 1452 y Fu(.)150 1508 y(The)17 b(w)o(eigh)o(ts)g(c)o(hange)g(from)f(iteration)h(to)f (iteration,)i(b)q(ecause)f(they)g(are)g(a)g(function)g(of)g(the)150 1565 y(parameters)c Fi(\014)416 1576 y Fm(r)435 1565 y Fu(.)19 b(The)c(resulting)g(iterativ)o(e)f(algorithm)g(is)h(kno)o(wn) f(as)g(iterativ)o(ely)h(rew)o(eigh)o(ted)150 1621 y(least)g(squares)g (\(IRLS\).)241 1678 y(It)23 b(is)g(easy)g(to)f(generalize)i(the)f (deriv)m(ation)h(to)e(allo)o(w)h(additional)h(\014xed)g(observ)m(ation) 150 1734 y(w)o(eigh)o(ts)c(to)h(b)q(e)g(asso)q(ciated)g(with)g(the)g (data)f(pairs.)37 b(Suc)o(h)21 b(w)o(eigh)o(ts)g(simply)h(m)o(ultiply)g (the)150 1790 y(iterativ)o(ely)17 b(v)m(arying)g(w)o(eigh)o(ts)f Fj(w)727 1774 y Fk(\()p Fm(t)p Fk(\))769 1790 y Fu(,)g(leading)i(to)d (an)h(iterativ)o(ely)h(rew)o(eigh)o(ted)g Ft(weighte)n(d)k Fu(least)150 1847 y(squares)c(algorithm.)27 b(Suc)o(h)18 b(a)f(generalization)h(is)g(in)h(fact)d(necessary)i(in)g(our)f (application)i(of)150 1903 y(IRLS:)f(The)g(EM)f(algorithm)g(de\014nes)i (observ)m(ation)e(w)o(eigh)o(ts)g(in)i(the)e(outer)g(lo)q(op)h(that)e (IRLS)150 1960 y(m)o(ust)f(treat)f(as)h(\014xed)h(during)g(the)f(inner) h(lo)q(op.)241 2016 y(Finally)l(,)h(it)e(is)h(also)g(straigh)o(tforw)o (ard)d(to)i(generalize)h(the)g(deriv)m(ation)h(in)f(this)g(section)g (to)150 2073 y(the)i(case)h(of)f(v)o(ector)g(outputs.)29 b(In)19 b(the)f(case)h(of)f(v)o(ector)f(outputs,)i(eac)o(h)f(ro)o(w)g (of)g(the)g(w)o(eigh)o(t)150 2129 y(matrix)10 b(\(e.g.,)g Fj(U)16 b Fu(for)10 b(the)g(exp)q(ert)h(net)o(w)o(orks\))e(is)i(a)f (separate)g(parameter)g(v)o(ector)g(corresp)q(onding)150 2186 y(to)16 b(the)i(v)o(ector)e Fi(\014)j Fu(of)d(this)i(section.)26 b(These)17 b(ro)o(w)g(v)o(ectors)f(are)h(up)q(dated)h(indep)q(enden)o (tly)i(and)150 2242 y(in)c(parallel.)p 150 2282 660 2 v 187 2309 a Fh(10)219 2324 y Fg(As)d(McCullagh)i(and)e(Nelder)h (\(1983\))f(note,)g Fc(z)g Fg(has)g(the)g(in)o(terpretation)j(as)d(the) g(linearization)j(of)d(the)g(link)150 2370 y(function)h(around)h(the)e (curren)o(t)g(v)n(alue)h(of)f(the)g(mean.)952 2775 y Fu(34)p eop %%Page: 35 35 35 34 bop 150 192 a Fp(App)r(endix)18 b(B)g({)h(Multinomial)d(logit)i (mo)r(dels)150 277 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y(App)q(endix)g(A,)e(using)h(Equation)g(51)e(and)i (Equation)f(52)g(for)f(the)i(link)g(function)g(and)g(the)150 868 y(mean,)15 b(resp)q(ectiv)o(ely)l(.)952 2775 y(36)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF