%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: wahba.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %X-Note: roottr.dvi/wahba.dvi merged manually %DVIPSCommandLine: dvips roottr.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1998.07.02:1125 %DVIPSCommandLine: dvips wahba.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1998.07.01:2011 %%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 round sub abs 0.00001 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Fy(\))29 b(with)g Ft(t)3498 4466 y Fs(`)3556 4454 y Fx(2)c(T)714 4562 y Fy(and)i(limits)h(of)g(suc)n(h)f (functions)h(in)g(the)g(norm)f(induced)h(b)n(y)f(the)h(inner)f(pro)r (duct)p Black Black 714 4728 a Ft(E)5 b(X)i Fy(\()p Ft(s)p Fy(\))p Ft(X)g Fy(\()p Ft(t)p Fy(\))22 b(=)h Ft(K)6 b Fy(\()p Ft(s;)14 b(t)p Fy(\))p Ft(:)1997 b Fm(\(1.14\))797 4894 y Fy(Then)27 b(w)n(e)h(ha)n(v)n(e)p Black Black eop %%Page: 7 8 7 7 bop Black Black -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V Black -450 5700 a FB(Generic)20 b(author)h(design)g (sample)g(pages)62 b(1998/07/01)35 b(20:11)198 100 y Fv(1.2)78 b(F)-6 b(r)l(om)28 b(Soft)g(Classi\014c)l(ation)g(to)g(Har)l (d)h(Classi\014c)l(ation)f(to)g(SVM's)1611 b(7)p Black 945 349 a Fy(The)27 b(Isometric)g(Isomorphism)g(Theorem)f(\(P)n(arzen)g (1962,)g(1970\))1028 565 y Fp(To)43 b(every)e(RKHS)h Fx(H)1708 577 y Fs(K)1816 565 y Fp(there)f(corresponds)e(a)k(zero)f (mean)g(Gaussian)f(stochastic)945 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Fm(\(1.18\))945 4129 y Fy(Let)27 b Ft(f)9 b Fy(\()p Ft(t)p Fy(\))23 b(=)g(ln\()p Ft(p)p Fy(\()p Ft(t)p Fy(\))p Ft(=)p Fy(\(1)17 b Fx(\000)h Ft(p)p Fy(\()p Ft(t)p Fy(\)\))27 b(b)r(e)h(the)f(logit)g([also)f(called)h(the) g(log)g(o)r(dds)g(ratio].)f(Assuming)945 4237 y(that)c(the)g(cost)f(of) h(misclassi\014cation)e(is)i(the)g(same)f(for)g(b)r(oth)h(kinds)g(of)g (misclassi\014cation,)e(then)945 4345 y(the)k(optimal)f(strategy)f(for) h(generalization,)e([minimization)j(of)f(exp)r(ected)h(loss],)f(if)h (one)f(knew)945 4453 y Ft(f)9 b Fy(,)28 b(w)n(ould)g(b)r(e)h(to)f (classify)g(as)f Fx(A)i Fy(if)g Ft(f)9 b Fy(\()p Ft(t)p Fy(\))25 b Ft(>)f Fy(0)k(and)g Fx(B)i Fy(if)f Ft(f)9 b Fy(\()p Ft(t)p Fy(\))25 b Ft(<)f Fy(0.)k(Th)n(us,)g(letting)h([)p Ft(f)9 b Fy(])3664 4465 y Fn(\003)3726 4453 y Fy(=)24 b(1)k(if)945 4560 y Ft(f)36 b(>)27 b Fy(0)j(and)g(0)g(otherwise,)g(one) g(really)f(w)n(an)n(ts)g(to)i(kno)n(w)e(sign)h Ft(f)9 b Fy(,)30 b(equiv)-5 b(alen)n(tly)30 b(it)h(is)f(desired)945 4668 y(to)40 b(estimate)g([)p Fx(\000)p Ft(f)9 b Fy(\()p Ft(t)p Fy(\)])1659 4680 y Fn(\003)1738 4668 y Fy(from)40 b(the)h(training)e(set)i Fx(f)p Ft(y)2655 4680 y Fs(i)2681 4668 y Ft(;)14 b(t)2748 4680 y Fs(i)2776 4668 y Fx(g)p Ft(;)g(i)43 b Fy(=)h(1)p Ft(;)14 b Fx(\001)g(\001)g(\001)f Ft(;)h(n)p Fy(.)41 b(This)f(particular)945 4776 y(form)n(ulation)27 b(is)h(con)n(v)n(enien)n(t,)f(b)r(ecause)h(w)n(e)g(note)g(that)g(if) 2802 4754 y(^)2784 4776 y Ft(f)37 b Fy(is)28 b(used)g(for)f (classi\014cation,)g(then)945 4884 y(the)h(n)n(um)n(b)r(er)f(of)h (misclassi\014cations)e(on)h(the)h(training)f(set)g(will)h(just)g(b)r (e)3263 4822 y Fq(P)3351 4843 y Fs(n)3351 4909 y(i)p Fr(=1)3463 4884 y Fy([)p Fx(\000)p Ft(y)3592 4896 y Fs(i)3637 4862 y Fy(^)3619 4884 y Ft(f)8 b Fy(\()p Ft(t)3730 4896 y Fs(i)3758 4884 y Fy(\)])3813 4896 y Fn(\003)3852 4884 y Fy(.)p Black Black eop %%Page: 8 9 8 8 bop Black Black -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V Black -450 5700 a FB(Generic)20 b(author)h(design)g (sample)g(pages)62 b(1998/07/01)35 b(20:11)-34 100 y Fv(8)709 b(Supp)l(ort)29 b(V)-6 b(e)l(ctor)29 b(Machines,)f(R)l(epr)l (o)l(ducing)h(Kernel)f(Hilb)l(ert)g(Sp)l(ac)l(es)h(and)f(the)h(R)l (andomize)l(d)f(GA)n(CV)p Black 714 349 a Fu(1.2.2)98 b(Soft)31 b(Classi\014cation)714 565 y Fy(If,)39 b(on)g(the)h(other)f (hand)g(one's)f(goal)g(is)h(not)h(simply)f(classi\014cation,)f(but)i (to)f(understand)714 673 y(ho)n(w)34 b(the)h(relativ)n(e)e(risk)h([)p Ft(e)1575 643 y Fs(f)7 b Fr(\()p Fs(t)p Fr(\))1694 673 y Fy(])35 b(of)g Fx(A)g Fy(to)f Fx(B)s Fy(,)g(v)-5 b(aries)33 b(with)i Ft(t)p Fy(,)g(as)f(is)g(frequen)n(tly)h(the)g(case)e(in)714 781 y(demographic)g(and)h(en)n(vironmen)n(tal)g(studies,)g(then)i(one)e (is)g(in)n(terested)h(in)g(estimating)f(the)714 889 y(actual)e(v)-5 b(alue)32 b(of)h Ft(f)41 b Fy(for)32 b(all)g Ft(t)h Fy(in)g(a)f(region) f(of)i Fx(T)21 b Fy(,)33 b(for)f(whic)n(h)g(one)h(is)f(lik)n(ely)g(to)h (ha)n(v)n(e)e(future)714 997 y(observ)-5 b(ations.)20 b(See,)h(for)g(example,)g(WW)n(GKK)h(\(1994,)e(1995\).)g(In)i(this)f (latter)h(case)e(one)i(migh)n(t)714 1105 y(estimate)31 b Ft(f)41 b Fy(from)31 b(the)h(training)g(set)f(b)n(y)h(the)g(metho)r (ds)g(of)g(p)r(enalized)g(log)f(lik)n(eliho)r(o)r(d,)g(that)714 1213 y(is,)c(one)g(\014nds)h Ft(f)36 b Fy(in)28 b Fx(f)p Ft(span)f(\036)1638 1225 y Fs(\027)1680 1213 y Fx(g)18 b(\010)g(H)1893 1225 y Fs(K)1984 1213 y Fy(to)28 b(minimize)p Black Black 728 1373 a(1)p 724 1410 50 4 v 724 1486 a Ft(n)837 1325 y Fs(n)797 1350 y Fq(X)803 1527 y Fs(i)p Fr(=1)931 1429 y Fx(L)p Fy(\()p Ft(y)1061 1441 y Fs(i)1089 1429 y Ft(;)14 b(f)1167 1441 y Fs(i)1194 1429 y Fy(\))19 b(+)f Ft(\025)p Fx(k)p Ft(h)p Fx(k)1508 1394 y Fr(2)1508 1449 y Fn(H)1565 1457 y Fo(K)1622 1429 y Ft(:)1861 b Fm(\(1.19\))714 1669 y Fy(Here,)18 b Ft(f)965 1681 y Fs(i)1036 1622 y Ft(:)1015 1669 y Fy(=)23 b Ft(f)9 b Fy(\()p Ft(t)1215 1681 y Fs(i)1242 1669 y Fy(\))19 b(and)g Fx(L)p Fy(\()p Ft(y)1576 1681 y Fs(i)1603 1669 y Ft(;)14 b(f)1681 1681 y Fs(i)1709 1669 y Fy(\))k(is)h(the)g(negativ)n(e)e(log)h (lik)n(eliho)r(o)r(d)g(function)3077 1641 y Fl(3)3116 1669 y Fy(.)g(In)h(this)g(example)714 1777 y(the)32 b(lik)n(eliho)r(o)r (d)g(that)g Ft(y)1469 1789 y Fs(i)1527 1777 y Fy(=)e(1)h(is)h Ft(p)p Fy(\()p Ft(t)1887 1789 y Fs(i)1915 1777 y Fy(\),)g(and)g(the)g (lik)n(eliho)r(o)r(d)g(that)g Ft(y)2923 1789 y Fs(i)2981 1777 y Fy(=)e Fx(\000)p Fy(1)h(is)h(\(1)21 b Fx(\000)g Ft(p)p Fy(\()p Ft(t)3587 1789 y Fs(i)3615 1777 y Fy(\)\).)714 1885 y(Th)n(us)32 b Fx(L)p Fy(\()p Ft(y)1059 1897 y Fs(i)1087 1885 y Ft(;)14 b(f)1165 1897 y Fs(i)1192 1885 y Fy(\))1276 1838 y Ft(:)1255 1885 y Fy(=)31 b Ft(l)r Fy(\()p Ft(y)1451 1897 y Fs(i)1478 1885 y Ft(f)1519 1897 y Fs(i)1546 1885 y Fy(\))i(where)e Ft(l)r Fy(\()p Ft(\034)9 b Fy(\))32 b(=)f Ft(l)r(n)p Fy(\(1)20 b(+)h Ft(e)2415 1855 y Fn(\000)p Fs(\034)2508 1885 y Fy(\).)33 b(T)-7 b(o)32 b(see)g(this,)h(let)f Ft(p)3218 1897 y Fs(i)3298 1838 y Ft(:)3277 1885 y Fy(=)e Ft(p)p Fy(\()p Ft(t)3476 1897 y Fs(i)3504 1885 y Fy(\))j(and)714 1993 y(note)27 b(that)p Black Black 820 2142 a Fx(L)p Fy(\(1)p Ft(;)14 b(f)1029 2154 y Fs(i)1056 2142 y Fy(\))83 b(=)g Fx(\000)14 b Fy(ln)o(\()1551 2109 y Fs(e)1582 2084 y Fo(f)1610 2097 y(i)p 1509 2123 175 4 v 1509 2175 a Fr(1+)p Fs(e)1624 2155 y Fo(f)1652 2168 y(i)1693 2142 y Fy(\))83 b(=)g Fx(\000)14 b Fy(ln)f Ft(p)2159 2154 y Fs(i)755 2272 y Fx(L)p Fy(\()p Fx(\000)p Fy(1)p Ft(;)h(f)1029 2284 y Fs(i)1056 2272 y Fy(\))83 b(=)g Fx(\000)14 b Fy(ln)o(\()1579 2240 y Fr(1)p 1509 2254 V 1509 2305 a(1+)p Fs(e)1624 2285 y Fo(f)1652 2298 y(i)1693 2272 y Fy(\))83 b(=)g Fx(\000)14 b Fy(ln\(1)k Fx(\000)g Ft(p)2321 2284 y Fs(i)2348 2272 y Fy(\))3506 2202 y Fm(\(1.20\))714 2432 y Fy(Th)n(us,)27 b(w)n(e)g(ma)n(y)g(rewrite)g(\(1.19\))g(as)p Black Black 728 2592 a(1)p 724 2629 50 4 v 724 2705 a Ft(n)837 2544 y Fs(n)797 2569 y Fq(X)803 2746 y Fs(i)p Fr(=1)931 2648 y Ft(l)r Fy(\()p Ft(y)1031 2660 y Fs(i)1058 2648 y Ft(f)1099 2660 y 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Fy(=)38 b Ft(K)6 b(c)24 b Fy(+)h Ft(T)12 b(d)36 b Fy(and,)h(if)g Ft(K)43 b Fy(is)36 b(of)h(full)h(rank,)e Ft(T)3087 3064 y Fn(0)3109 3095 y Ft(c)j Fy(=)f(0,)f(with)g(the)714 3202 y(mo)r(di\014cations)f(noted)g(if)h Ft(K)42 b Fy(is)36 b(not)h(of)f(full)h(rank.)f Ft(c)g Fy(and)h Ft(d)f Fy(are)g (substituted)h(in)n(to)f(\(1.21\))714 3310 y(and,)31 b(if)h Ft(K)37 b Fy(is)32 b(of)f(full)i(rank)d(or)h(the)h(dimension)g (of)f Ft(c)h Fy(is)f(reduced)h(appropriately)-7 b(,)30 b(a)h(strictly)714 3418 y(con)n(v)n(ex)d(optimization)i(problem)f(with) i(readily)e(accessible)g(gradien)n(t)g(and)h(Hessian)f(results.)714 3526 y(A)38 b(natural)g(target)f(for)h(c)n(ho)r(osing)f Ft(\025)i Fy(is)f(then)h(to)g(minimize)g(the)f(comparativ)n(e)f (Kullbac)n(k-)714 3634 y(Liebler)j(distance)g Ft(C)6 b(K)g(L)p Fy(\()p Ft(\025)p Fy(\))1718 3587 y Ft(:)1697 3634 y Fy(=)44 b Ft(C)6 b(K)g(L)p Fy(\()p Ft(f)2078 3646 y Fs(tr)r(ue)2210 3634 y Ft(;)14 b(f)2288 3646 y Fs(\025)2331 3634 y Fy(\))41 b(b)r(et)n(w)n(een)g Ft(f)2779 3646 y Fs(tr)r(ue)2952 3634 y Fy(and)f Ft(f)3167 3646 y Fs(\025)3210 3606 y Fl(4)3249 3634 y Fy(.)g(Here)h Ft(f)3563 3646 y Fs(\025)3647 3634 y Fy(is)714 3742 y(the)29 b(minimizer)g(of)h (\(1.21\))e(and)h Ft(f)1789 3754 y Fs(tr)r(ue)1950 3742 y Fy(is)h(the)f(logit)g(of)g(the)g(`true')h(distribution)f Ft(p)3331 3754 y Fs(tr)r(ue)3493 3742 y Fy(whic)n(h)714 3850 y(generated)17 b(the)i(data.)g Ft(C)6 b(K)g(L)p Fy(\()p Ft(\025)p Fy(\))19 b(in)g(this)g(case)f(b)r(ecomes)g Ft(E)2541 3862 y Fs(tr)r(ue)2688 3788 y Fq(P)2776 3808 y Fs(n)2776 3875 y(i)p Fr(=1)2901 3850 y Ft(l)r Fy(\()p Ft(y)3001 3862 y Fs(i)3028 3850 y Ft(f)3069 3862 y Fs(\025i)3136 3850 y Fy(\),)h(see)f(Xiang)h(and)714 3958 y(W)-7 b(ah)n(ba)19 b(\(1996\))g(for)g(more)h(details.)g(Later)f(w)n(e)h(will)g(turn)g(to)g (the)g(randomized)g(GA)n(CV)g(metho)r(d)714 4066 y(for)27 b(estimating)g(a)g(computable)g(pro)n(xy)f(for)h Ft(C)6 b(K)g(L)p Fy(\()p Ft(\025)p Fy(\))27 b(\(Xiang)h(and)f(W)-7 b(ah)n(ba)27 b(1996,1997,)c(Lin)714 4174 y(and)k(W)-7 b(ah)n(ba,)27 b(in)h(preparation\).)p Black 714 4355 798 3 v 714 4456 a Fl(3.)39 b(In)23 b(the)h(statistics)h(literature)g (the)f(usual)g(log)h(lik)n(eliho)r(o)r(d)g(functional)g(is)f(form)n (ulated)g(for)h Fj(y)f Fl(=)d(1)j(or)714 4547 y(0.)714 4630 y(4.)39 b(Recall)26 b(that)g(the)f(Kullbac)n(k-Liebler)g(distance) i(is)f(not)f(really)i(a)f(distance.)p Black Black Black eop %%Page: 9 10 9 9 bop Black Black -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V Black -450 5700 a FB(Generic)20 b(author)h(design)g (sample)g(pages)62 b(1998/07/01)35 b(20:11)198 100 y Fv(1.2)78 b(F)-6 b(r)l(om)28 b(Soft)g(Classi\014c)l(ation)g(to)g(Har)l (d)h(Classi\014c)l(ation)f(to)g(SVM's)1611 b(9)p Black Black 945 2671 a @beginspecial 72 @llx 208 @lly 541 @urx 584 @ury 3586 @rwi @setspecial %%BeginDocument: wahba-fig1.ps 150 dict begin gsave /bd{bind def}def /PVer 3200 def % drawing commands /I{Coord SetPage 1 setlinecap 1 setlinejoin LineTypes{RastersPerPoint ScaleArray}forall /Helvetica findfont PointSize RastersPerPoint mul Cex mul scalefont setfont /Colors GColors def }bd /BWOp{ bw{currentrgbcolor add add 2.9 gt{1}{0}ifelse setgray}if exec bw{CurSc Sc}if }bd /IBu{}def /B{newpath}bd /E{{stroke}BWOp}bd /C{currentpoint Penup{newpath}{E}ifelse moveto}bd /F{closepath eofill}bd /M{moveto}bd /L{lineto}bd /Pu{load /XEQ exch def /Penup true def}bd /Pd{/XEQ{}def /Penup false def}bd /Px{currentdict /XEQ known{XEQ}if}bd /h{Penup{rmoveto Px}{rlineto}ifelse}bd /j{neg Penup{rmoveto Px}{rlineto}ifelse}bd /k{exch neg exch Penup{rmoveto Px}{rlineto}ifelse}bd /l{neg exch neg exch Penup{rmoveto Px}{rlineto}ifelse}bd /m{moveto Penup{Px}if}bd /x{0 Penup{rmoveto Px}{rlineto}ifelse}bd /y{0 exch Penup{rmoveto Px}{rlineto}ifelse}bd /w{neg x}bd /z{neg y}bd /a{B M dup y exch x neg y closepath}bd /b{a eofill}bd /c{a E}bd /A{PageBegin BackCol 0 ge {CurSc gsave 0 SC Page aload pop exch 4 -1 roll dup 5 1 roll sub 3 1 roll exch dup 4 1 roll sub 4 2 roll b grestore SC}if }bd /Ph{gsave Pch-x Pch-y rmoveto Show grestore}bd /Pc{Pch Ph}bd /T{/Adjust exch def gsave translate StringRot rotate 0 0 moveto dup stringwidth pop neg Adjust mul 0 rmoveto currentpoint translate TextShow grestore}bd /X{erasepage InPage{PageEnd}if}bd /Z{gsave showpage grestore PageEnd}bd /W{grestore end}bd % parameter setting commands /saveAt{Attr begin /Atlinewidth currentlinewidth def /Atfont currentfont def /Atdash currentdash 2 array astore def /AtSc CurSc def end}bd /restAt{Attr begin Atlinewidth Atfont Atdash AtSc end SC aload pop setdash setfont setlinewidth }bd /SC{dup dup 0 eq {pop BackIdx {BackCol dup 0 lt {pop 1 setgray} {Sc}ifelse} {/Colors BColor def 1 Sc}ifelse} {/Colors GColors def Sc}ifelse /CurSc exch def }bd /St{1 sub LineTypes dup 3 1 roll length Rem floor get 0 setdash}bd /Sw{abs 2 div RastersPerPoint mul setlinewidth SetClip}bd /Sp{dup 42 eq{pop IBu}{Pch exch 0 exch put SetPchSize}ifelse}bd /Sx{dup Cex div /Ratio exch def /Cex exch def currentfont Ratio scalefont setfont /Pch-x Pch-x Ratio mul def /Pch-y Pch-y Ratio mul def /Text-y Text-y Ratio mul def}bd /So{4 1 roll exch 4 -1 roll Plot astore pop SetClip}bd /Sg{4 1 roll exch 4 -1 roll Figure astore pop SetClip}bd /Sr{/StringRot exch def}bd /Sh{/CharRot exch def}bd /Sd{0 eq /ClipToPlot exch def SetClip}bd /Iv{/InPage false def/Clip 4 array def/Page 4 array def /Figure [0 0 1 1] def/Plot [0 0 1 1] def/ClipToPlot true def /Cex 1 def/Outline false def /Pch 1 string def/Pch-x 0 def/Pch-y 0 def/Text-y 0 def /LineTypes [ % in default units [] [1 2] [4 4] [8 4] [13 3] [16 2 2 2] [8 2 2 2] [1 13] [6 5] [12 4] ] def /Attr 5 dict def/CurSc 1 def/Penup false def/XEQ{}def }bd /Rem{2 copy div floor mul sub floor cvi}bd /RastersPerPoint{RastersPerInch 72 div}bd /ScaleArray{/Factor exch def /Array exch def 0 1 Array length 1 sub {dup Array exch get Factor mul Array 3 1 roll put}for}bd /Coord{Region aload pop /uy exch def /ux exch def /ly exch def /lx exch def uy ly sub ux lx sub Landscape{exch}if /Width exch def /Height exch def lx ly translate Landscape{90 rotate 0 Height neg translate}if 1 RastersPerPoint div dup scale}bd /SetPchSize{gsave newpath 0 0 moveto Pch false charpath flattenpath pathbbox exch 3 1 roll add 2 div neg /Pch-y exch def add 2 div neg /Pch-x exch def grestore}bd /Show{Outline{false charpath E}{{show}BWOp}ifelse}bd /SimpleShow{0 Text-y 2 div neg rmoveto Show}bd /FancyShow{ /RotDiff exch def /Cos RotDiff cos abs def /Sin RotDiff sin abs def { ( ) dup 0 4 -1 roll put dup stringwidth pop /CharWidth exch def Cos 0 eq{ Text-y Sin div }{ Sin 0 eq{ CharWidth Cos div }{ Text-y Sin div dup CharWidth Cos div dup 4 1 roll lt{exch pop}{pop}ifelse }ifelse }ifelse 2 div /CharDist exch def CharDist 0 translate 0 0 moveto gsave RotDiff rotate CharWidth 2 div neg Text-y 2 div neg rmoveto Show grestore CharDist 0 translate 0 0 moveto }forall }bd /TextShow{ CharRot StringRot sub dup 0 eq{pop SimpleShow}{FancyShow}ifelse }bd /BoxClip{/CLW currentlinewidth def 2{CLW add 4 1 roll}repeat 2{CLW sub 4 1 roll}repeat % initclip saveAt grestore gsave restAt newpath 2 index exch 2 index exch dup 6 index exch moveto 3{lineto}repeat closepath clip newpath}bd /Subregion{ 7 dict begin/A exch def/Uy exch def/Ux exch def/Ly exch def/Lx exch def Ux Lx sub A 0 get mul Lx add Uy Ly sub A 1 get mul Ly add Ux Lx sub A 2 get mul Lx add Uy Ly sub A 3 get mul Ly add end}bd /SetFigure{Page aload pop Figure Subregion}bd /SetPlot{SetFigure Plot Subregion}bd /SetClip{ClipToPlot{SetPlot}{SetFigure}ifelse BoxClip}bd /SetPage{0 0 Width Height Page astore RastersPerPoint ScaleArray}bd /PageBegin{save /PageContext exch def /InPage true def gsave}bd /PageEnd{grestore PageContext restore /InPage false def}bd /SImage{ dup /Bits exch def 8 exch idiv /PerByte exch def /Ht exch def /Wid exch def 1 Wid div 1 Ht div scale /saveSc CurSc def /Colors IColors def /curstr Wid PerByte add 1 sub PerByte idiv string def /MaskByte 16#ff Bits 8 sub bitshift def 1 1 Ht{ currentfile curstr readhexstring pop pop /CurPos 0 def /Left PerByte def 1 1 Wid{ Left PerByte eq{ /CurByte curstr CurPos get def /MaskMove -8 Bits add def /CurPos CurPos 1 add def }if CurByte MaskMove bitshift MaskByte and dup 0 gt{ Sc exch dup 3 1 roll B M 1 z 1 w 1 y F }{pop pop}ifelse Left 1 sub dup /Left exch def 0 le {/Left PerByte def} {/MaskMove MaskMove Bits add def}ifelse }for pop }for /Colors GColors def saveSc Sc }bd /Ic {} bd /Sc { dup dup 1 lt exch 0 ge and {1 exch sub setgray} % explicit gray level [0,1) {1 sub Colors dup 3 1 roll length Rem floor get dup type /arraytype eq {aload pop setrgbcolor} {setgray} ifelse } ifelse } bd /If {} bd /Sf {dup 0 lt /Outline exch def abs 1 sub Fonts dup 3 1 roll length Rem floor get findfont PointSize Cex mul RastersPerPoint mul scalefont dup setfont dup /FontMatrix get /Matrix exch def /FontBBox get aload pop Matrix transform 4 2 roll Matrix transform exch pop add /Text-y exch def pop SetPchSize} bd % fixed controlling parameters /Landscape false def /Region [72 208.8 540 583.2] def /bw statusdict begin /processcolors where {pop processcolors} {1} ifelse end 1 eq def /RastersPerInch 300 def /PointSize 14 def /Fonts [ /Helvetica /Courier /Times-Roman /Helvetica-Oblique /Helvetica-Bold /Helvetica-BoldOblique /Courier-Oblique /Courier-Bold /Courier-BoldOblique /Times-Italic /Times-Bold /Times-BoldItalic /Symbol /AvantGarde-Book /AvantGarde-BookOblique /AvantGarde-Demi /AvantGarde-DemiOblique /Bookman-Demi /Bookman-DemiItalic /Bookman-Light /Bookman-LightItalic /Helvetica-Narrow /Helvetica-Narrow-Bold /Helvetica-Narrow-BoldOblique /Helvetica-Narrow-Oblique /NewCenturySchlbk-Roman /NewCenturySchlbk-Bold /NewCenturySchlbk-Italic /NewCenturySchlbk-BoldItalic /Palatino-Roman /Palatino-Bold /Palatino-Italic /Palatino-BoldItalic /ZapfChancery-MediumItalic /ZapfDingbats ] def /GColors [ 0 0.6 0.3 0.9 0.4 0.7 0.1 0.5 0.8 0.2 ] def /BackIdx true def /BackCol -1 def /IColors [ 0.933333 0.866667 0.8 0.733333 0.666667 0.6 0.533333 0.466667 0.4 0.333333 0.266667 0.2 0.133333 0.0666667 0 ] def /Colors GColors def /IBu {Pch 0 8#267 put SetPchSize} def /Bu {Pc} def % all initialization action here Iv I currentdict /Ic known {Ic} if currentdict /If known {If} if A 1 St 1 Sw 1 SC 0 Sr 42 Sp 1 Sx 0.147179 0.924615 0.228846 0.816026 So 0 1 0 1 Sg 0 Sh 1 Sd 1 Sf (tau) 1045 196 0.5 T 90 Sr 90 Sh 287 391 m 18 w 287 561 m 18 w 287 730 m 18 w 287 900 m 18 w 287 1069 m 18 w 287 1239 m 18 w E 287 391 m 848 y E (0.0) 217 391 0.5 T (0.5) 217 561 0.5 T (1.0) 217 730 0.5 T (1.5) 217 900 0.5 T (2.0) 217 1069 0.5 T (2.5) 217 1239 0.5 T 0 Sr 0 Sh 0 Sd 1516 916 287 357 c 1 Sd 652 357 m 18 z 1045 357 m 18 z 1270 357 m 18 z E 652 357 m 618 x E (e-) 652 252 0.5 T (0) 1045 252 0.5 T (e+) 1270 252 0.5 T 3 St 0 Sd B 343 1266 m 3 3 j 3 3 j 3 4 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 4 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 4 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 4 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 3 2 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 2 j 3 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 2 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 2 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 2 j 3 3 j 3 3 j 3 3 j 2 3 j 3 2 j 3 3 j 3 3 j 3 3 j 3 2 j 2 3 j 3 3 j 3 3 j 3 2 j 3 3 j 2 3 j 3 3 j 3 2 j 3 3 j 3 3 j 2 2 j 3 3 j 3 3 j 3 3 j 3 2 j 2 3 j 3 3 j 3 2 j 3 3 j 3 3 j 2 2 j 3 3 j 3 3 j 3 2 j 3 3 j 2 2 j 3 3 j 3 3 j 3 2 j 3 3 j 3 2 j 2 3 j 3 3 j 3 2 j 3 3 j 3 2 j 2 3 j 3 2 j 3 3 j 3 2 j 3 3 j 2 3 j 3 2 j 3 3 j 3 2 j 3 3 j 2 2 j 3 3 j 3 2 j 3 2 j 3 3 j 2 2 j 3 3 j 3 2 j 3 3 j 3 2 j 3 3 j 2 2 j 3 2 j 3 3 j 3 2 j 3 3 j 2 2 j 3 2 j 3 3 j 3 2 j 3 2 j 2 3 j 3 2 j 3 2 j 3 3 j 3 2 j 2 2 j 3 3 j 3 2 j 3 2 j 3 3 j 2 2 j 3 2 j 3 2 j 3 3 j 3 2 j 3 2 j 2 2 j 3 2 j 3 3 j 3 2 j 3 2 j 2 2 j 3 2 j 3 3 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 3 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 1 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 1 j 3 2 j 2 2 j 3 2 j 3 2 j 3 1 j 3 2 j 2 2 j 3 2 j 3 1 j 3 2 j 3 2 j 2 2 j 3 1 j 3 2 j 3 2 j 3 1 j 2 2 j 3 2 j 3 1 j 3 2 j 3 2 j 2 1 j 3 2 j 3 1 j 3 2 j 3 2 j 2 1 j 3 2 j 3 1 j 3 2 j 3 2 j 3 1 j 2 2 j 3 1 j 3 2 j 3 1 j 3 2 j 2 1 j 3 2 j 3 1 j 3 2 j 3 1 j 2 2 j 3 1 j 3 1 j 3 2 j 3 1 j 2 2 j 3 1 j 3 2 j 3 1 j 3 1 j 2 2 j 3 1 j 3 1 j 3 2 j 3 1 j 3 1 j 2 2 j 3 1 j 3 1 j 3 2 j 3 1 j 2 1 j 3 2 j 3 1 j 3 1 j 3 1 j 2 2 j 3 1 j 3 1 j 3 1 j 3 1 j 2 2 j 3 1 j 3 1 j 3 1 j 3 1 j 2 2 j 3 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 2 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 x 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 x 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 x 3 1 j 3 1 j 3 1 j 3 1 j 3 x 2 1 j 3 1 j 3 1 j 3 x 3 1 j 2 1 j 3 1 j 3 x 3 1 j 3 1 j 2 x 3 1 j 3 1 j 3 1 j 3 x 2 1 j 3 1 j 3 x 3 1 j 3 1 j 2 x 3 1 j 3 1 j 3 x 3 1 j 2 x 3 1 j 3 1 j 3 x 3 1 j 3 1 j 2 x 3 1 j 3 x 3 1 j 3 1 j 2 x 3 1 j 3 x 3 1 j 3 x 2 1 j 3 x 3 1 j 3 1 j 3 x 2 1 j 3 x 3 1 j 3 x 3 1 j 2 x 3 1 j 3 x 3 1 j 3 x 3 1 j 2 x 3 1 j 3 x 3 1 j 3 x 2 1 j 3 x 3 x 3 1 j 3 x 2 1 j 3 x 3 1 j 3 x 3 1 j 2 x 3 x 3 1 j 3 x 3 1 j 2 x 3 x 3 1 j 3 x 3 1 j 3 x 2 x 3 1 j 3 x 3 1 j 3 x 2 x 3 1 j 3 x 3 x 3 1 j 2 x 3 x 3 1 j 3 x 3 x 2 1 j 3 x 3 x 3 1 j 3 x 2 x 3 1 j 3 x 3 x 3 1 j 3 x 2 x 3 1 j 3 x 3 x 3 x 2 1 j 3 x 3 x 3 1 j 3 x 2 x 3 x 3 1 j 3 x 3 x 2 1 j 3 x 3 x 3 x E 2 St B 343 1239 m 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x E 1 St B 1048 391 m 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x E B 343 730 m 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x E 4 St B 343 730 m 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 1 j 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 1 j 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 1 j 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 1 j 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 1 j 3 x 3 x 3 x 3 x 2 x 3 x 3 1 j 3 x 3 x 2 x 3 x 3 1 j 3 x 3 x 2 x 3 x 3 1 j 3 x 3 x 2 x 3 1 j 3 x 3 x 3 x 3 1 j 2 x 3 x 3 1 j 3 x 3 x 2 1 j 3 x 3 x 3 1 j 3 x 2 x 3 1 j 3 x 3 1 j 3 x 2 x 3 1 j 3 x 3 1 j 3 x 2 1 j 3 x 3 1 j 3 x 3 1 j 3 1 j 2 x 3 1 j 3 x 3 1 j 3 1 j 2 x 3 1 j 3 1 j 3 1 j 3 x 2 1 j 3 1 j 3 1 j 3 x 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 2 j 3 1 j 3 1 j 2 1 j 3 1 j 3 2 j 3 1 j 3 1 j 2 2 j 3 1 j 3 2 j 3 1 j 3 2 j 2 1 j 3 2 j 3 1 j 3 2 j 3 2 j 2 1 j 3 2 j 3 2 j 3 2 j 3 1 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 3 j 3 2 j 2 2 j 3 2 j 3 3 j 3 2 j 3 2 j 2 3 j 3 2 j 3 2 j 3 3 j 3 2 j 3 3 j 2 2 j 3 3 j 3 2 j 3 3 j 3 2 j 2 3 j 3 2 j 3 3 j 3 2 j 3 3 j 2 2 j 3 3 j 3 2 j 3 3 j 3 3 j 2 2 j 3 3 j 3 2 j 3 3 j 3 2 j 2 3 j 3 2 j 3 3 j 3 2 j 3 3 j 2 2 j 3 3 j 3 2 j 3 2 j 3 3 j 3 2 j 2 3 j 3 2 j 3 2 j 3 2 j 3 3 j 2 2 j 3 2 j 3 2 j 3 3 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 1 j 3 2 j 3 2 j 3 2 j 2 1 j 3 2 j 3 1 j 3 2 j 3 2 j 2 1 j 3 2 j 3 1 j 3 2 j 3 1 j 2 1 j 3 2 j 3 1 j 3 1 j 3 2 j 2 1 j 3 1 j 3 1 j 3 2 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 x 3 1 j 3 1 j 3 1 j 2 1 j 3 x 3 1 j 3 1 j 3 1 j 2 x 3 1 j 3 x 3 1 j 3 1 j 2 x 3 1 j 3 x 3 1 j 3 1 j 3 x 2 1 j 3 x 3 1 j 3 x 3 x 2 1 j 3 x 3 1 j 3 x 3 1 j 2 x 3 x 3 1 j 3 x 3 x 2 1 j 3 x 3 x 3 1 j 3 x 2 x 3 1 j 3 x 3 x 3 x 3 1 j 2 x 3 x 3 x 3 1 j 3 x 2 x 3 x 3 x 3 1 j 3 x 2 x 3 x 3 x 3 x 3 1 j 2 x 3 x 3 x 3 x 3 x 2 x 3 1 j 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 1 j 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 1 j 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 1 j 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 1 j 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x E 5 St B 343 730 m 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 1 j 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 1 j 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 1 j 3 x 3 x 3 x 3 x 3 x 2 1 j 3 x 3 x 3 x 3 x 2 1 j 3 x 3 x 3 x 3 1 j 2 x 3 x 3 1 j 3 x 3 x 2 1 j 3 x 3 1 j 3 x 3 1 j 2 x 3 1 j 3 x 3 1 j 3 x 3 1 j 2 1 j 3 x 3 1 j 3 1 j 3 1 j 2 1 j 3 x 3 1 j 3 1 j 3 2 j 2 1 j 3 1 j 3 1 j 3 1 j 3 2 j 2 1 j 3 2 j 3 1 j 3 2 j 3 2 j 2 1 j 3 2 j 3 2 j 3 2 j 3 3 j 3 2 j 2 3 j 3 2 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 4 j 3 4 j 3 3 j 2 4 j 3 5 j 3 4 j 3 5 j 3 5 j 2 5 j 3 6 j 3 6 j 3 6 j 3 6 j 2 7 j 3 7 j 3 7 j 3 8 j 3 8 j 3 9 j 2 9 j 3 10 j 3 10 j 3 10 j 3 11 j 2 12 j 3 12 j 3 13 j 3 14 j 3 14 j 2 15 j 3 16 j 3 16 j 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x E 6 St B 343 1137 m 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 2 3 j 3 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 2 4 j 3 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 4 j 3 3 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 4 j 2 3 j 3 3 j 3 4 j 3 3 j 3 3 j 2 4 j 3 3 j 3 3 j 3 3 j 3 3 j 2 4 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 4 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 3 j 3 3 j 3 3 j 3 3 j 3 2 j 2 3 j 3 3 j 3 3 j 3 3 j 3 3 j 2 2 j 3 3 j 3 3 j 3 3 j 3 3 j 2 2 j 3 3 j 3 3 j 3 2 j 3 3 j 3 3 j 2 2 j 3 3 j 3 2 j 3 3 j 3 3 j 2 2 j 3 3 j 3 2 j 3 3 j 3 2 j 2 3 j 3 2 j 3 3 j 3 2 j 3 2 j 2 3 j 3 2 j 3 3 j 3 2 j 3 2 j 2 3 j 3 2 j 3 2 j 3 2 j 3 3 j 3 2 j 2 2 j 3 2 j 3 3 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 3 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 1 j 3 2 j 3 2 j 3 2 j 3 2 j 2 2 j 3 2 j 3 1 j 3 2 j 3 2 j 2 2 j 3 1 j 3 2 j 3 2 j 3 1 j 2 2 j 3 2 j 3 1 j 3 2 j 3 2 j 2 1 j 3 2 j 3 1 j 3 2 j 3 2 j 2 1 j 3 2 j 3 1 j 3 1 j 3 2 j 3 1 j 2 2 j 3 1 j 3 2 j 3 1 j 3 1 j 2 2 j 3 1 j 3 1 j 3 2 j 3 1 j 2 1 j 3 1 j 3 2 j 3 1 j 3 1 j 2 1 j 3 2 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 1 j 3 1 j 2 1 j 3 1 j 3 1 j 3 x 3 1 j 2 1 j 3 1 j 3 1 j 3 x 3 1 j 2 1 j 3 x 3 1 j 3 1 j 3 x 2 1 j 3 1 j 3 x 3 1 j 3 1 j 3 x 2 1 j 3 x 3 1 j 3 x 3 1 j 2 x 3 1 j 3 x 3 x 3 1 j 2 x 3 1 j 3 x 3 x 3 1 j 2 x 3 x 3 1 j 3 x 3 x 2 x 3 1 j 3 x 3 x 3 x 3 x 2 x 3 1 j 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x E 1 St 0 SC 0.7 Sx 1 Sd 690 377 1101 828 b 1 SC 1101 828 m 377 y 1791 828 m 377 y E 1101 828 m 690 x 1101 1205 m 690 x E 3 St B 1154 1156 m 106 x E 1 St (ln\(1+exp\(-tau\)\)) 1313 1156 0 T 2 St B 1154 1102 m 106 x E 1 St ([-tau]+) 1313 1102 0 T B 1154 1048 m 106 x E ([-tau]*) 1313 1048 0 T 4 St B 1154 994 m 106 x E 1 St (1/\(1+exp\(tau\)\)) 1313 994 0 T 5 St B 1154 941 m 106 x E 1 St ([1-exp\(tau\)]+) 1313 941 0 T 6 St B 1154 887 m 106 x E 1 St (v\(e-,e+,tau\)) 1313 887 0 T Z W %%EndDocument @endspecial 103 x Ff(Figure)30 b(1.1)83 b Fy(Pseudo-distance)26 b(functions)i(of)f Fm(tau)g Fy(\()p Fj(\034)9 b Fy(\))28 b(men)n(tioned)f(in)h(the)g(text.)p Black 945 3019 a Fu(1.2.3)98 b(Bac)m(k)33 b(to)f(Hard)g(Classi\014cation)945 3235 y Fy(Section)26 b(1.2.1)f(suggests)f(that)j(w)n(e)e(c)n(ho)r(ose)g Ft(f)32 b Fx(2)23 b(f)p Ft(span)i(\036)2756 3247 y Fs(\027)2798 3235 y Fx(g)15 b Fy(+)g Ft(h)26 b Fy(with)h Ft(h)22 b Fx(2)i(H)3416 3247 y Fs(K)3506 3235 y Fy(to)i(minimize)p Black Black 959 3395 a(1)p 955 3432 50 4 v 955 3508 a Ft(n)1068 3347 y Fs(n)1028 3372 y Fq(X)1035 3549 y Fs(i)p Fr(=1)1148 3451 y Fy([)p Ft(\017)18 b Fx(\000)h Ft(y)1348 3463 y Fs(i)1375 3451 y Ft(f)1416 3463 y Fs(i)1443 3451 y Fy(])1466 3463 y Fn(\003)1523 3451 y Fy(+)f Ft(\025)p Fx(k)p Ft(h)p Fx(k)1786 3417 y Fr(2)1786 3472 y Fn(H)1843 3480 y Fo(K)1900 3451 y Ft(;)1814 b Fm(\(1.22\))945 3673 y Fy(for)30 b(some)f(\014xed)i Ft(\017)c(>)g Fy(0,)j(thereb)n(y)g(p)r (enalizing)g(the)h(misclassi\014cation)e(rate)h(rather)f(than)h(the)945 3781 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b Fx(\001)g(\001)g(\001)f Ft(;)h(d)p Fy(.)20 b(Bradley)-7 b(,)18 b(Mangasarian)e(and)j(Street)g (\(1997\))f(recen)n(tly)714 1004 y(considered)29 b(problems)h(where)h (the)g(rather)f(nast)n(y)g(function)h([)p Fx(\000)p Ft(\034)9 b Fy(])2834 1016 y Fn(\003)2903 1004 y Fy(is)31 b(replaced)f(with)h (other)714 1112 y(more)21 b(tractable)h(functions)h(including)g(the)g (sigmoidal)e(appro)n(ximation)g(1)p Ft(=)p Fy(\(1)8 b(+)g Ft(e)3299 1081 y Fs(a\034)3376 1112 y Fy(\))23 b(and)f(the)714 1219 y(function)33 b([1)21 b Fx(\000)g Ft(e)1255 1189 y Fs(a\034)1332 1219 y Fy(])1355 1231 y Fr(+)1410 1219 y Fy(,)33 b(conca)n(v)n(e)d(for)i Ft(\034)41 b(<)30 b Fy(0.)i(Here,)g([)p Ft(x)p Fy(])2495 1231 y Fr(+)2582 1219 y Fy(=)e Ft(x;)14 b(x)32 b(>)f Fy(0)p Ft(;)14 b Fy(=)30 b(0)i(otherwise.)f(F)-7 b(or)714 1327 y(comparison,)25 b(these)j(t)n(w)n(o)e(functions)i(are)e(also)h(plotted)g(in)h(Figure)f (1.1)f(with)i Ft(a)23 b Fy(=)g(1.)k(Bradley)714 1435 y Fz(et)j(al)g Fy(considered)e(examples)h(with)g(a)g(large)e(n)n(um)n (b)r(er)i(of)g(v)-5 b(ariables,)28 b(where)g(the)i(goal)e(w)n(as)g(to) 714 1543 y(screen)c(out)h(some)f(non-informativ)n(e)g(v)-5 b(ariables)24 b(for)g(deletion.)h(They)g(p)r(enalized)g(the)h(n)n(um)n (b)r(er)714 1651 y(of)g(v)-5 b(ariables)26 b(included)h(and)g(used)g (10-fold)e(cross)h(v)-5 b(alidation)26 b(on)h(the)g (misclassi\014cation)e(rate)714 1759 y(to)33 b(c)n(ho)r(ose)f(a)g(p)r (enalt)n(y)i(parameter)d(on)i(the)h(n)n(um)n(b)r(er)f(of)g(v)-5 b(ariables.)32 b(See)h(also)f(Bennett)i(and)714 1867 y(Blue)27 b(\(1997\).)714 2023 y Fu(1.2.4)98 b(Con)m(v)m(ex)32 b(Compromises)c(with)j(SVM's)714 2239 y Fy(Let)c Ft(v)s Fy(\()p Ft(\034)9 b Fy(\))1060 2192 y Ft(:)1039 2239 y Fy(=)23 b Ft(v)1167 2251 y Fs(\017)1195 2259 y Fh(+)1241 2251 y Fs(;\017)1289 2259 y Fe(\000)1342 2239 y Fy(\()p Ft(\034)9 b Fy(\))29 b(b)r(e)f(de\014ned)g(b)n(y)p Black Black 755 2415 a Ft(v)795 2427 y Fs(\017)823 2435 y Fh(+)870 2427 y Fs(;\017)918 2435 y Fe(\000)971 2415 y Fy(\()p Ft(\034)9 b Fy(\))84 b(=)e([)p Fx(\000)p Fy(\()p Ft(\034)28 b Fx(\000)1588 2374 y Fs(\017)1616 2382 y Fh(+)1663 2374 y Fr(+)p Fs(\017)1742 2382 y Fe(\000)p 1588 2396 203 4 v 1673 2444 a Fr(2)1801 2415 y Fy(\)])1856 2427 y Fr(+)2273 2415 y Ft(\034)93 b(<)82 b(\017)2583 2427 y Fn(\000)1164 2547 y Fy(=)1341 2506 y Fr(\()p Fs(\034)7 b Fn(\000)p Fs(\017)1485 2514 y Fh(+)1531 2506 y Fr(\))1557 2481 y Fh(2)p 1321 2528 289 4 v 1321 2575 a Fr(2\()p Fs(\017)1408 2583 y Fh(+)1455 2575 y Fn(\000)p Fs(\017)1535 2583 y Fe(\000)1584 2575 y Fr(\))1994 2547 y Ft(\017)2028 2559 y Fn(\000)2107 2547 y Fx(\024)101 b Ft(\034)93 b Fx(\024)82 b Ft(\017)2583 2559 y Fr(+)1164 2681 y Fy(=)g(0)642 b Ft(\017)2029 2693 y Fr(+)2107 2681 y Fx(\024)83 b Ft(\034)5 b(:)3506 2543 y Fm(\(1.23\))714 2841 y Fy(F)-7 b(or)30 b(\014xed)g Ft(\017)1103 2853 y Fn(\000)1187 2841 y Ft(<)d(\017)1313 2853 y Fr(+)1368 2841 y Fy(,)k Ft(v)1462 2853 y Fs(\017)1490 2861 y Fh(+)1536 2853 y Fs(;\017)1584 2861 y Fe(\000)1637 2841 y Fy(\()p Ft(\034)9 b Fy(\))32 b(is)e(con)n(v)n(ex)f(and)h(p)r (ossesses)g(a)g(con)n(tin)n(uous)f(\014rst)h(deriv)-5 b(ativ)n(e,)714 2948 y(and)31 b(a)g(non-negativ)n(e)e(second)i(deriv)-5 b(ativ)n(e)30 b(ev)n(erywhere)f(except)j(at)f Ft(\017)2939 2960 y Fn(\000)3026 2948 y Fy(and)g Ft(\017)p Fy(+,)f(where)h(the)714 3056 y(second)g(deriv)-5 b(ativ)n(e)31 b(could)h(b)r(e)g(de\014ned)g(b) n(y)g(assigning)e(it)i(to)g(b)r(e)g(con)n(tin)n(uous)f(from)h(the)g (left,)714 3164 y(sa)n(y)-7 b(.)29 b Ft(v)916 3176 y Fs(\017)944 3184 y Fh(+)991 3176 y Fs(;\017)1039 3184 y Fe(\000)1091 3164 y Fy(\()p Ft(\034)9 b Fy(\))32 b(is)e(plotted)g(in) h(Figure)e(1.1)h(as)f Fg(v\(e+,e-,tau\))p Fy(,)h(along)f(with)i Ft(v)3148 3176 y Fr(0)p Fs(;)p Fr(0)3238 3164 y Fy(\()p Ft(\034)9 b Fy(\))3397 3117 y Ft(:)3376 3164 y Fy(=)27 b([)p Fx(\000)p Ft(\034)9 b Fy(])3624 3176 y Fr(+)3679 3164 y Fy(.)714 3272 y Ft(v)754 3284 y Fs(\017)782 3292 y Fh(+)828 3284 y Fs(;\017)876 3292 y Fh(+)927 3272 y Fy(\()p Ft(\034)g Fy(\))1112 3225 y Ft(:)1092 3272 y Fy(=)54 b([)p Ft(\017)31 b Fx(\000)f Ft(\034)9 b Fy(])1462 3284 y Fr(+)1518 3272 y Fy(.)47 b(The)f Ft(v)s Fy('s)h(ma)n(y)f(b)r(e)h (though)n(t)f(of)h(as)f(\(in)h(some)f(sense\))g(con)n(v)n(ex)714 3380 y(appro)n(ximations)26 b(to)i([)p Ft(\017)18 b Fx(\000)h Ft(\034)9 b Fy(])1630 3392 y Fn(\003)1669 3380 y Fy(,)28 b(whic)n(h)g(for)g Ft(\017)2120 3392 y Fn(\000)2200 3380 y Ft(<)23 b(\017)2322 3392 y Fr(+)2405 3380 y Fy(p)r(ossess)k(a)h(con)n (tin)n(uous)f(\014rst)h(deriv)-5 b(ativ)n(e)714 3488 y(and)27 b(non-negativ)n(e)f(second)h(deriv)-5 b(ativ)n(e)27 b(whic)n(h)g(could)h(b)r(e)g(de\014ned)g(ev)n(erywhere.)-34 3836 y Fw(1.3)101 b(The)34 b(Randomized)h(GA)m(CV)g(for)g(Cho)s(osing)h Fd(\025)p -34 3703 3736 3 v 714 4052 a Fy(So)d(far)h(our)f(discussion)g (has)h(b)r(een)g(a)g(relativ)n(ely)f(straigh)n(tforw)n(ard)e (description)i(of)h(bridges)714 4160 y(b)r(et)n(w)n(een)20 b(w)n(ell)h(kno)n(wn)f(results)g(in)i(optimization)e(in)h(RKHS,)g (Gaussian)f(pro)r(cesses,)f(p)r(enalized)714 4268 y(lik)n(eliho)r(o)r (d)43 b(metho)r(ds)h(in)g(soft)f(classi\014cation)g(\(more)g(commonly)g (kno)n(wn)g(as)f(risk)h(factor)714 4376 y(estimation\))34 b(and)h(SVM)g(metho)r(ds.)h(This)e(section)h(is)f(more)g(heuristic)h (and)g(in)g(the)g(nature)714 4484 y(of)29 b(w)n(ork)f(in)i(progress.)d (The)j(goal)e(is)h(to)h(explore)e(to)i(what)f(exten)n(t)h(the)g (randomized)e(GA)n(CV)714 4592 y(metho)r(d)49 b(in)g(Xiang)f(and)g(W)-7 b(ah)n(ba)48 b(\(1996,1997\))d(for)j(c)n(ho)r(osing)g Ft(\025)h Fy(n)f(the)h(case)f Ft(g)s Fy(\()p Ft(\034)9 b Fy(\))59 b(=)714 4700 y Ft(l)r(n)p Fy(\(1)26 b(+)h Ft(e)1022 4670 y Fn(\000)p Fs(\034)1115 4700 y Fy(\))42 b(ma)n(y)e(b)r(e)i(extended)f(to)g(apply)g(in)g(the)h(con)n(text)f(of)g (SVM's.)g(Minimization)714 4808 y(of)g(the)i(generalized)d(comparativ)n (e)g(Kullbac)n(k-Liebler)g(distance)h(\(GCKL\))i(of)e Ft(f)3386 4820 y Fs(\025)3471 4808 y Fy(to)h(the)714 4916 y(`)p Ft(tr)r(ue)893 4885 y Fn(0)943 4916 y Ft(f)35 b Fy(as)27 b(a)f(function)i(of)e Ft(\025)i Fy(is)f(the)g(target)f(of)h (the)g(GA)n(CV.)g(W)-7 b(e)27 b(\014rst)g(describ)r(e)g(the)g(GCKL)p Black Black eop %%Page: 11 12 11 11 bop Black Black -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V Black -450 5700 a FB(Generic)20 b(author)h(design)g (sample)g(pages)62 b(1998/07/01)35 b(20:11)198 100 y Fv(1.3)78 b(The)28 b(R)l(andomize)l(d)g(GA)n(CV)g(for)f(Cho)l(osing)i Fj(\025)2141 b Fv(11)p Black 945 349 a Fy(and)34 b(ho)n(w)g(it)h (relates)e(in)i(some)f(cases)f(to)h(the)h(exp)r(ected)g (misclassi\014cation)e(rate.)g(Then)i(w)n(e)945 457 y(describ)r(e)26 b(ho)n(w)g(the)g(\(computable\))h(minimizer)g(of)f(the)h(GA)n(CV)g (should)f(b)r(e)h(a)f(go)r(o)r(d)g(estimate)945 565 y(of)d(the)g (minimizer)h(of)f(the)g(\(not)h(computable\))f(GCKL.)g(The)g (randomized)f(trace)h(metho)r(d)g(for)945 673 y(computing)k(the)i(GA)n 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Fy(.)g(Giv)n(en)945 1644 y Ft(f)986 1656 y Fs(\025)1029 1644 y Fy(,)j(de\014ne)h(the)g(generalized)e(comparativ)n(e)f(Kullbac)n (k-Liebler)g(distance)i(\(GCKL)h(distance\))945 1752 y(with)h(resp)r(ect)f(to)h Ft(g)i Fy(as)p Black Black 945 1964 a Ft(GC)6 b(K)g(L)p Fy(\()p Ft(f)1282 1976 y Fs(tr)r(ue)1414 1964 y Ft(;)14 b(f)1492 1976 y Fs(\025)1535 1964 y Fy(\))1611 1917 y Ft(:)1590 1964 y Fy(=)23 b Ft(GC)6 b(K)g(L)p Fy(\()p Ft(\025)p Fy(\))23 b(=)g Ft(E)2226 1976 y Fs(tr)r(ue)2373 1908 y Fy(1)p 2369 1945 50 4 v 2369 2021 a Ft(n)2482 1860 y Fs(n)2443 1885 y Fq(X)2445 2062 y Fs(j)s Fr(=1)2576 1964 y Ft(g)s Fy(\()p Ft(y)2692 1976 y Fs(j)2727 1964 y Ft(f)2768 1976 y Fs(\025j)2842 1964 y Fy(\))p Ft(:)840 b Fm(\(1.24\))945 2216 y Fy(If)28 b Ft(g)s Fy(\()p Ft(\034)9 b Fy(\))25 b(=)f Ft(l)r(n)p Fy(\(1)18 b(+)g Ft(e)1585 2186 y Fn(\000)p Fs(\034)1678 2216 y Fy(\),)29 b(then)g Ft(GC)6 b(K)g(L)p Fy(\()p Ft(\025)p Fy(\))28 b(reduces)g(to)g(the)g(usual)g(CKL)3321 2188 y Fl(5)3388 2216 y Fy(,)g(a)n(v)n(eraged)d(o)n(v)n(er)945 2324 y(the)j(attribute)g(v)n(ectors)e(of)h(the)h(training)f(set.)h(If)g Ft(g)s Fy(\()p Ft(\034)9 b Fy(\))24 b(=)e([)p Ft(\017)c Fx(\000)g Ft(\034)9 b Fy(])2997 2336 y Fn(\003)3036 2324 y Fy(,)28 b(then)945 2490 y Ft(E)1006 2502 y Fs(tr)r(ue)1139 2490 y Fy([)p Ft(\017)18 b Fx(\000)g Ft(y)1338 2502 y Fs(j)1373 2490 y Ft(f)1414 2502 y Fs(\025j)1488 2490 y Fy(])1511 2502 y Fn(\003)1572 2490 y Fy(=)23 b Ft(p)1702 2505 y Fr([)p Fs(tr)r(ue)p Fr(])p Fs(j)1902 2490 y Fy([)p Ft(\017)18 b Fx(\000)h Ft(f)2102 2502 y Fs(\025j)2176 2490 y Fy(])2199 2502 y Fn(\003)2255 2490 y Fy(+)f(\(1)g Fx(\000)g Ft(p)2555 2505 y Fr([)p Fs(tr)r(ue)p Fr(])p Fs(j)2756 2490 y Fy(\)[)p Ft(\017)h Fy(+)f Ft(f)2988 2502 y Fs(\025j)3062 2490 y Fy(])3085 2502 y Fn(\003)3737 2490 y Fm(\(1.25\))1572 2623 y Fy(=)23 b Ft(p)1702 2638 y Fr([)p Fs(tr)r(ue)p Fr(])p Fs(j)1902 2623 y Ft(;)70 b(f)2036 2635 y Fs(\025j)2133 2623 y Ft(<)22 b Fx(\000)p Ft(\017)1418 b Fm(\(1.26\))1572 2755 y Fy(=)23 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Fy(\(1)18 b(+)g Ft(\017)p Fy(\))23 b(=)g(+1.)1028 2625 y(Letting)42 b Ft(\026)1384 2637 y Fs(\025i)1499 2625 y Fy(=)47 b Ft(\026)p Fy(\()p Ft(f)1734 2637 y Fs(\025i)1801 2625 y Fy(\))42 b(and)g Ft(\026)2101 2582 y Fr([)p Fn(\000)p Fs(i)p Fr(])2101 2650 y Fs(\025i)2266 2625 y Fy(=)47 b Ft(\026)p Fy(\()p Ft(f)2510 2582 y Fr([)p Fn(\000)p Fs(i)p Fr(])2501 2650 y Fs(\025i)2627 2625 y Fy(\),)c(w)n(e)f(ma)n(y)f (write)h(\(ignoring,)g(for)f(the)945 2733 y(momen)n(t,)27 b(the)h(p)r(ossibilit)n(y)g(of)f(dividing)h(b)n(y)f(0\),)945 2956 y Ft(D)r Fy(\()p Ft(\025)p Fy(\))d Fx(\031)f(\000)1318 2900 y Fy(1)p 1315 2937 50 4 v 1315 3013 a Ft(n)1427 2852 y Fs(n)1388 2877 y Fq(X)1394 3054 y Fs(i)p Fr(=1)1564 2900 y Ft(@)5 b(g)p 1531 2937 157 4 v 1531 3013 a(@)g(f)1621 3025 y Fs(\025i)1708 2898 y Fy(\()p Ft(f)1781 2910 y Fs(\025i)1866 2898 y Fx(\000)18 b Ft(f)1999 2855 y Fr([)p Fn(\000)p Fs(i)p Fr(])1990 2924 y Fs(\025i)2116 2898 y Fy(\))p 1708 2937 441 4 v 1727 3033 a(\()p Ft(y)1800 3045 y Fs(i)1846 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3293 y Fo(\025i)2455 3284 y Fn(\000)p Fs(\026)2547 3253 y Fh([)p Fe(\000)p Fo(i)p Fh(])2547 3306 y Fo(\025i)p 2353 3312 299 4 v 2374 3377 a Fs(y)2408 3385 y Fo(i)2434 3377 y Fn(\000)p Fs(\026)2526 3347 y Fh([)p Fe(\000)p Fo(i)p Fh(])2526 3399 y Fo(\025i)2662 3331 y Fy(\))2704 3231 y Ft(:)1010 b Fm(\(1.41\))945 3544 y Fy(Next,)28 b(appro)n(ximate)e Ft(\026)1701 3556 y Fs(\025i)1786 3544 y Fx(\000)18 b Ft(\026)1919 3501 y Fr([)p Fn(\000)p Fs(i)p Fr(])1919 3569 y Fs(\025i)2064 3544 y Fy(as)p Black Black 945 3741 a Ft(\026)995 3753 y Fs(\025i)1080 3741 y Fx(\000)g Ft(\026)1213 3698 y Fr([)p Fn(\000)p Fs(i)p Fr(])1213 3766 y Fs(\025i)1353 3741 y Fy(=)23 b Ft(\026)p Fy(\()p Ft(f)1564 3753 y Fs(\025i)1631 3741 y Fy(\))c Fx(\000)f Ft(\026)p Fy(\()p Ft(f)1897 3698 y Fr([)p Fn(\000)p Fs(i)p Fr(])1888 3766 y Fs(\025i)2014 3741 y Fy(\))23 b Fx(\031)2196 3685 y Ft(@)5 b(\026)p 2167 3722 157 4 v 2167 3798 a(@)g(f)2257 3810 y Fs(\025i)2333 3741 y Fy(\()p Ft(f)2406 3753 y Fs(\025i)2492 3741 y Fx(\000)18 b Ft(f)2625 3698 y Fr([)p Fn(\000)p Fs(i)p Fr(])2616 3766 y Fs(\025i)2741 3741 y Fy(\))p Ft(:)941 b Fm(\(1.42\))945 3930 y Fy(Making)27 b(the)h(de\014nitions)976 4118 y Ft(g)1019 4084 y Fn(0)1016 4139 y Fs(i)1067 4118 y Fy(=)1197 4062 y Ft(@)5 b(g)p 1164 4099 V 1164 4175 a(@)g(f)1254 4187 y Fs(\025i)3737 4118 y Fm(\(1.43\))966 4330 y Ft(\026)1016 4296 y Fn(0)1016 4350 y Fs(i)1067 4330 y Fy(=)1193 4274 y Ft(@)g(\026)p 1164 4311 V 1164 4387 a(@)g(f)1254 4399 y Fs(\025i)3737 4330 y Fm(\(1.44\))945 4572 y Ft(h)993 4584 y Fs(ii)1067 4572 y Fy(=)1164 4515 y Ft(f)1205 4527 y Fs(\025i)1291 4515 y Fx(\000)18 b Ft(f)1424 4471 y Fr([)p Fn(\000)p Fs(i)p Fr(])1415 4540 y Fs(\025i)p 1164 4553 376 4 v 1184 4650 a Ft(y)1225 4662 y Fs(i)1271 4650 y Fx(\000)g Ft(\026)1404 4606 y Fr([)p Fn(\000)p Fs(i)p Fr(])1404 4675 y Fs(\025i)3737 4572 y Fm(\(1.45\))p Black Black eop %%Page: 14 15 14 14 bop Black Black -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V Black -450 5700 a FB(Generic)20 b(author)h(design)g (sample)g(pages)62 b(1998/07/01)35 b(20:11)-34 100 y Fv(14)670 b(Supp)l(ort)29 b(V)-6 b(e)l(ctor)29 b(Machines,)f(R)l(epr)l (o)l(ducing)h(Kernel)f(Hilb)l(ert)g(Sp)l(ac)l(es)h(and)f(the)h(R)l (andomize)l(d)f(GA)n(CV)p Black Black 714 2671 a @beginspecial 72 @llx 208 @lly 541 @urx 584 @ury 3586 @rwi @setspecial %%BeginDocument: wahba-fig2.ps 150 dict begin gsave /bd{bind def}def /PVer 3200 def % drawing commands /I{Coord SetPage 1 setlinecap 1 setlinejoin LineTypes{RastersPerPoint ScaleArray}forall /Helvetica findfont PointSize RastersPerPoint mul Cex mul scalefont setfont /Colors GColors def }bd /BWOp{ bw{currentrgbcolor add add 2.9 gt{1}{0}ifelse setgray}if exec bw{CurSc Sc}if }bd /IBu{}def /B{newpath}bd /E{{stroke}BWOp}bd /C{currentpoint Penup{newpath}{E}ifelse moveto}bd /F{closepath eofill}bd /M{moveto}bd /L{lineto}bd /Pu{load /XEQ exch def /Penup true def}bd /Pd{/XEQ{}def /Penup false def}bd /Px{currentdict /XEQ known{XEQ}if}bd /h{Penup{rmoveto Px}{rlineto}ifelse}bd /j{neg Penup{rmoveto Px}{rlineto}ifelse}bd /k{exch neg exch Penup{rmoveto Px}{rlineto}ifelse}bd /l{neg exch neg exch Penup{rmoveto Px}{rlineto}ifelse}bd /m{moveto Penup{Px}if}bd /x{0 Penup{rmoveto Px}{rlineto}ifelse}bd /y{0 exch Penup{rmoveto Px}{rlineto}ifelse}bd /w{neg x}bd /z{neg y}bd /a{B M dup y exch x neg y closepath}bd /b{a eofill}bd /c{a E}bd /A{PageBegin BackCol 0 ge {CurSc gsave 0 SC Page aload pop exch 4 -1 roll dup 5 1 roll sub 3 1 roll exch dup 4 1 roll sub 4 2 roll b grestore SC}if }bd /Ph{gsave Pch-x Pch-y rmoveto Show grestore}bd /Pc{Pch Ph}bd /T{/Adjust exch def gsave translate StringRot rotate 0 0 moveto dup stringwidth pop neg Adjust mul 0 rmoveto currentpoint translate TextShow grestore}bd /X{erasepage InPage{PageEnd}if}bd /Z{gsave showpage grestore PageEnd}bd /W{grestore end}bd % parameter setting commands /saveAt{Attr begin /Atlinewidth currentlinewidth def /Atfont currentfont def /Atdash currentdash 2 array astore def /AtSc CurSc def end}bd /restAt{Attr begin Atlinewidth Atfont Atdash AtSc end SC aload pop setdash setfont setlinewidth }bd /SC{dup dup 0 eq {pop BackIdx {BackCol dup 0 lt {pop 1 setgray} {Sc}ifelse} {/Colors BColor def 1 Sc}ifelse} {/Colors GColors def Sc}ifelse /CurSc exch def }bd /St{1 sub LineTypes dup 3 1 roll length Rem floor get 0 setdash}bd /Sw{abs 2 div RastersPerPoint mul setlinewidth SetClip}bd /Sp{dup 42 eq{pop IBu}{Pch exch 0 exch put SetPchSize}ifelse}bd /Sx{dup Cex div /Ratio exch def /Cex exch def currentfont Ratio scalefont setfont /Pch-x Pch-x Ratio mul def /Pch-y Pch-y Ratio mul def /Text-y Text-y Ratio mul def}bd /So{4 1 roll exch 4 -1 roll Plot astore pop SetClip}bd /Sg{4 1 roll exch 4 -1 roll Figure astore pop SetClip}bd /Sr{/StringRot exch def}bd /Sh{/CharRot exch def}bd /Sd{0 eq /ClipToPlot exch def SetClip}bd /Iv{/InPage false def/Clip 4 array def/Page 4 array def /Figure [0 0 1 1] def/Plot [0 0 1 1] def/ClipToPlot true def /Cex 1 def/Outline false def /Pch 1 string def/Pch-x 0 def/Pch-y 0 def/Text-y 0 def /LineTypes [ % in default units [] [1 2] [4 4] [8 4] [13 3] [16 2 2 2] [8 2 2 2] [1 13] [6 5] [12 4] ] def /Attr 5 dict def/CurSc 1 def/Penup false def/XEQ{}def }bd /Rem{2 copy div floor mul sub floor cvi}bd /RastersPerPoint{RastersPerInch 72 div}bd /ScaleArray{/Factor exch def /Array exch def 0 1 Array length 1 sub {dup Array exch get Factor mul Array 3 1 roll put}for}bd /Coord{Region aload pop /uy exch def /ux exch def /ly exch def /lx exch def uy ly sub ux lx sub Landscape{exch}if /Width exch def /Height exch def lx ly translate Landscape{90 rotate 0 Height neg translate}if 1 RastersPerPoint div dup scale}bd /SetPchSize{gsave newpath 0 0 moveto Pch false charpath flattenpath pathbbox exch 3 1 roll add 2 div neg /Pch-y exch def add 2 div neg /Pch-x exch def grestore}bd /Show{Outline{false charpath E}{{show}BWOp}ifelse}bd /SimpleShow{0 Text-y 2 div neg rmoveto Show}bd /FancyShow{ /RotDiff exch def /Cos RotDiff cos abs def /Sin RotDiff sin abs def { ( ) dup 0 4 -1 roll put dup stringwidth pop /CharWidth exch def Cos 0 eq{ Text-y Sin div }{ Sin 0 eq{ CharWidth Cos div }{ Text-y Sin div dup CharWidth Cos div dup 4 1 roll lt{exch pop}{pop}ifelse }ifelse }ifelse 2 div /CharDist exch def CharDist 0 translate 0 0 moveto gsave RotDiff rotate CharWidth 2 div neg Text-y 2 div neg rmoveto Show grestore CharDist 0 translate 0 0 moveto }forall }bd /TextShow{ CharRot StringRot sub dup 0 eq{pop SimpleShow}{FancyShow}ifelse }bd /BoxClip{/CLW currentlinewidth def 2{CLW add 4 1 roll}repeat 2{CLW sub 4 1 roll}repeat % initclip saveAt grestore gsave restAt newpath 2 index exch 2 index exch dup 6 index exch moveto 3{lineto}repeat closepath clip newpath}bd /Subregion{ 7 dict begin/A exch def/Uy exch def/Ux exch def/Ly exch def/Lx exch def Ux Lx sub A 0 get mul Lx add Uy Ly sub A 1 get mul Ly add Ux Lx sub A 2 get mul Lx add Uy Ly sub A 3 get mul Ly add end}bd /SetFigure{Page aload pop Figure Subregion}bd /SetPlot{SetFigure Plot Subregion}bd /SetClip{ClipToPlot{SetPlot}{SetFigure}ifelse BoxClip}bd /SetPage{0 0 Width Height Page astore RastersPerPoint ScaleArray}bd /PageBegin{save /PageContext exch def /InPage true def gsave}bd /PageEnd{grestore PageContext restore /InPage false def}bd /SImage{ dup /Bits exch def 8 exch idiv /PerByte exch def /Ht exch def /Wid exch def 1 Wid div 1 Ht div scale /saveSc CurSc def /Colors IColors def /curstr Wid PerByte add 1 sub PerByte idiv string def /MaskByte 16#ff Bits 8 sub bitshift def 1 1 Ht{ currentfile curstr readhexstring pop pop /CurPos 0 def /Left PerByte def 1 1 Wid{ Left PerByte eq{ /CurByte curstr CurPos get def /MaskMove -8 Bits add def /CurPos CurPos 1 add def }if CurByte MaskMove bitshift MaskByte and dup 0 gt{ Sc exch dup 3 1 roll B M 1 z 1 w 1 y F }{pop pop}ifelse Left 1 sub dup /Left exch def 0 le {/Left PerByte def} {/MaskMove MaskMove Bits add def}ifelse }for pop }for /Colors GColors def saveSc Sc }bd /Ic {} bd /Sc { dup dup 1 lt exch 0 ge and {1 exch sub setgray} % explicit gray level [0,1) {1 sub Colors dup 3 1 roll length Rem floor get dup type /arraytype eq {aload pop setrgbcolor} {setgray} ifelse } ifelse } bd /If {} bd /Sf {dup 0 lt /Outline exch def abs 1 sub Fonts dup 3 1 roll length Rem floor get findfont PointSize Cex mul RastersPerPoint mul scalefont dup setfont dup /FontMatrix get /Matrix exch def /FontBBox get aload pop Matrix transform 4 2 roll Matrix transform exch pop add /Text-y exch def pop SetPchSize} bd % fixed controlling parameters /Landscape false def /Region [72 208.8 540 583.2] def /bw statusdict begin /processcolors where {pop processcolors} {1} ifelse end 1 eq def /RastersPerInch 300 def /PointSize 14 def /Fonts [ /Helvetica /Courier /Times-Roman /Helvetica-Oblique /Helvetica-Bold /Helvetica-BoldOblique /Courier-Oblique /Courier-Bold /Courier-BoldOblique /Times-Italic /Times-Bold /Times-BoldItalic /Symbol /AvantGarde-Book /AvantGarde-BookOblique /AvantGarde-Demi /AvantGarde-DemiOblique /Bookman-Demi /Bookman-DemiItalic /Bookman-Light /Bookman-LightItalic /Helvetica-Narrow /Helvetica-Narrow-Bold /Helvetica-Narrow-BoldOblique /Helvetica-Narrow-Oblique /NewCenturySchlbk-Roman /NewCenturySchlbk-Bold /NewCenturySchlbk-Italic /NewCenturySchlbk-BoldItalic /Palatino-Roman /Palatino-Bold /Palatino-Italic /Palatino-BoldItalic /ZapfChancery-MediumItalic /ZapfDingbats ] def /GColors [ 0 0.6 0.3 0.9 0.4 0.7 0.1 0.5 0.8 0.2 ] def /BackIdx true def /BackCol -1 def /IColors [ 0.933333 0.866667 0.8 0.733333 0.666667 0.6 0.533333 0.466667 0.4 0.333333 0.266667 0.2 0.133333 0.0666667 0 ] def /Colors GColors def /IBu {Pch 0 8#267 put SetPchSize} def /Bu {Pc} def % all initialization action here Iv I currentdict /Ic known {Ic} if currentdict /If known {If} if A 1 St 1 Sw 1 SC 0 Sr 42 Sp 1 Sx 0.147179 0.924615 0.228846 0.816026 So 0 1 0 1 Sg 0 Sh 1 Sd 1 Sf (f) 1045 196 0.5 T 90 Sr 90 Sh (mu\(f\)) 126 815 0.5 T 287 391 m 18 w 287 561 m 18 w 287 730 m 18 w 287 900 m 18 w 287 1069 m 18 w 287 1239 m 18 w E 287 391 m 848 y E (-1.0) 217 391 0.5 T (0.0) 217 730 0.5 T (0.5) 217 900 0.5 T (1.0) 217 1069 0.5 T (1.5) 217 1239 0.5 T 0 Sr 0 Sh 0 Sd 1516 916 287 357 c 1 Sd 652 357 m 18 z 820 357 m 18 z 1045 357 m 18 z 1270 357 m 18 z 1438 357 m 18 z E 652 357 m 786 x E (e-) 652 252 0.5 T (0) 1045 252 0.5 T (e+) 1270 252 0.5 T 3 St 0 Sd B 343 442 m 3 1 h 3 x 3 1 h 2 x 3 1 h 3 x 3 1 h 3 x 2 1 h 3 x 3 1 h 3 x 3 1 h 2 x 3 1 h 3 1 h 3 x 3 1 h 2 x 3 1 h 3 x 3 1 h 3 x 3 1 h 2 1 h 3 x 3 1 h 3 x 3 1 h 2 1 h 3 x 3 1 h 3 x 3 1 h 2 1 h 3 x 3 1 h 3 1 h 3 x 2 1 h 3 1 h 3 x 3 1 h 3 1 h 2 x 3 1 h 3 1 h 3 x 3 1 h 3 1 h 2 1 h 3 x 3 1 h 3 1 h 3 x 2 1 h 3 1 h 3 1 h 3 x 3 1 h 2 1 h 3 1 h 3 1 h 3 x 3 1 h 2 1 h 3 1 h 3 1 h 3 x 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 x 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 x 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 2 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 2 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 2 h 3 1 h 3 1 h 3 1 h 3 1 h 3 2 h 2 1 h 3 1 h 3 1 h 3 2 h 3 1 h 2 1 h 3 1 h 3 2 h 3 1 h 3 1 h 2 2 h 3 1 h 3 1 h 3 1 h 3 2 h 2 1 h 3 1 h 3 2 h 3 1 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 1 h 3 2 h 2 1 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 2 2 h 3 2 h 3 1 h 3 2 h 3 2 h 2 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 2 h 3 2 h 3 1 h 3 2 h 3 2 h 2 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 2 h 3 1 h 3 2 h 3 2 h 3 2 h 2 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 2 h 3 1 h 3 2 h 3 2 h 3 2 h 2 1 h 3 2 h 3 2 h 3 1 h 3 2 h 3 2 h 2 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 2 2 h 3 2 h 3 1 h 3 2 h 3 2 h 2 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 1 h 2 2 h 3 1 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 1 h 3 2 h 3 1 h 3 1 h 2 2 h 3 1 h 3 1 h 3 2 h 3 1 h 2 1 h 3 1 h 3 2 h 3 1 h 3 1 h 2 2 h 3 1 h 3 1 h 3 1 h 3 2 h 2 1 h 3 1 h 3 1 h 3 1 h 3 2 h 3 1 h 2 1 h 3 1 h 3 1 h 3 2 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 2 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 2 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 x 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 2 1 h 3 1 h 3 1 h 3 x 3 1 h 2 1 h 3 1 h 3 1 h 3 1 h 3 1 h 3 x 2 1 h 3 1 h 3 1 h 3 1 h 3 x 2 1 h 3 1 h 3 1 h 3 1 h 3 x 2 1 h 3 1 h 3 1 h 3 x 3 1 h 2 1 h 3 1 h 3 x 3 1 h 3 1 h 2 1 h 3 x 3 1 h 3 1 h 3 x 3 1 h 2 1 h 3 x 3 1 h 3 1 h 3 x 2 1 h 3 1 h 3 x 3 1 h 3 1 h 2 x 3 1 h 3 1 h 3 x 3 1 h 2 x 3 1 h 3 1 h 3 x 3 1 h 2 x 3 1 h 3 x 3 1 h 3 1 h 3 x 2 1 h 3 x 3 1 h 3 x 3 1 h 2 x 3 1 h 3 x 3 1 h 3 1 h 2 x 3 1 h 3 x 3 1 h 3 x 2 1 h 3 x 3 x 3 1 h E 6 St B 343 391 m 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 1 h 3 2 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 4 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 4 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 4 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 4 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 4 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 4 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 4 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 4 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 4 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 4 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 4 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 4 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 4 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 3 h 3 3 h 3 3 h 3 3 h 3 3 h 2 4 h 3 3 h 3 3 h 3 3 h 3 3 h 2 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 2 h 3 1 h 2 2 h 3 1 h 3 2 h 3 1 h 3 2 h 2 1 h 3 2 h 3 1 h 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x 3 x 2 x 3 x 3 x 3 x E 1 St 0 SC 0.7 Sx 1 Sd 690 162 371 1043 b 1 SC 371 1043 m 162 y 1061 1043 m 162 y E 371 1043 m 690 x 371 1205 m 690 x E 3 St B 424 1156 m 106 x E 1 St (ln\(1+exp\(-tau\)\)) 584 1156 0 T 6 St B 424 1102 m 106 x E 1 St (v\(e-,e+,tau\)) 584 1102 0 T Z W %%EndDocument @endspecial 105 x Ff(Figure)e(1.2)83 b Fy(Prediction)24 b(functions)h Fj(\026)p Fl(\()p Fj(f)8 b Fl(\))25 b Fy(de\014ned)f(in)h (the)g(text)g(for)f Fj(g)s Fl(\()p Fj(\034)9 b Fl(\))20 b(=)h Fj(l)q(n)p Fl(\(1)11 b(+)g Fj(e)3427 2744 y Fb(\000)p Fi(\034)3514 2776 y Fl(\))25 b Fy(and)714 2884 y Fj(g)s Fl(\()p Fj(\034)9 b Fl(\))19 b(=)i Fj(v)994 2892 y Fi(\017)1020 2903 y Fh(+)1068 2892 y Fi(;\017)1113 2903 y Fe(\000)1166 2884 y Fl(\()p Fj(\034)9 b Fl(\))p Fy(.)p Black 714 3134 a(in)27 b(\(1.41\))g(giv)n(es,)g(\014nally)p Black Black 714 3350 a Ft(D)r Fy(\()p Ft(\025)p Fy(\))d Fx(\031)e(\000)1087 3293 y Fy(1)p 1083 3330 50 4 v 1083 3407 a Ft(n)1196 3246 y Fs(n)1157 3271 y Fq(X)1163 3448 y Fs(i)p Fr(=1)1290 3350 y Ft(g)1333 3315 y Fn(0)1330 3370 y Fs(i)1358 3350 y Ft(h)1406 3362 y Fs(ii)1483 3293 y Ft(y)1524 3305 y Fs(i)1570 3293 y Fx(\000)c Ft(\026)1703 3305 y Fs(\025i)p 1466 3330 320 4 v 1466 3407 a Fy(1)g Fx(\000)g 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b(an)n(y)g Ft(f)9 b Fx(\003)25 b Fy(for)h(whic)n(h)g Ft(\026)p Fy(\()p Ft(f)9 b Fx(\003)p Fy(\))22 b Fx(6)p Fy(=)h Ft(\026)p Fy(\()p Ft(f)9 b Fy(\).)26 b(It)h(can)e(b)r(e)h(sho)n(wn)g(that)g Ft(g)f Fy(=)e Ft(l)28 b Fy(and)d Ft(g)h Fy(=)d Ft(v)3340 4892 y Fs(\017)3368 4900 y Fh(+)3414 4892 y Fs(;\017)3462 4900 y Fe(\000)3541 4880 y Fy(with)714 4988 y(the)34 b(`prediction)f(functions')i Ft(\026)e Fy(as)h(in)g(Figure)f(1.2)g(ha)n (v)n(e)f(prop)r(ert)n(y)h(\(1.47\).)g(Then)h(w)n(e)g(ha)n(v)n(e)p Black Black eop %%Page: 15 16 15 15 bop Black Black -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V Black -450 5700 a FB(Generic)20 b(author)h(design)g (sample)g(pages)62 b(1998/07/01)35 b(20:11)198 100 y Fv(1.3)78 b(The)28 b(R)l(andomize)l(d)g(GA)n(CV)g(for)f(Cho)l(osing)i Fj(\025)2141 b Fv(15)p Black 945 349 a Fy(the)945 565 y(Lea)n(ving-Out-One)25 b(Lemma)i(\(Cra)n(v)n(en)f(and)i(W)-7 b(ah)n(ba)27 b(1979,)f(Xiang)h(and)g(W)-7 b(ah)n(ba)27 b(1996\))1028 781 y Fp(Let)42 b Ft(f)1252 738 y Fr([)p Fn(\000)p Fs(i)p Fr(])1243 806 y Fs(\025)1412 781 y Fp(be)h(the)g (minimizer)c(of)k(\(1.33\))e(as)i(before.)e(Then)h Ft(f)3337 738 y Fr([)p Fn(\000)p Fs(i)p Fr(])3328 806 y Fs(\025)3497 781 y Fp(is)h(also)f(the)945 889 y(minimizer)e(of)1525 856 y Fr(1)p 1521 870 42 4 v 1521 918 a Fs(n)1586 827 y Fq(P)1674 847 y Fs(n)1674 914 y(i)p Fr(=1)1799 889 y Ft(g)s Fy(\()p Ft(y)1915 901 y Fs(j)1950 889 y Ft(f)1991 901 y Fs(j)2026 889 y Fy(\))18 b(+)g Ft(J)2205 901 y Fs(\025)2249 889 y Fy(\()p Ft(f)9 b Fy(\))44 b Fp(given)d(the)h(data)g Ft(Y)3127 859 y Fr([)p Fn(\000)p Fs(i)p Fr(])3267 889 y Fy(=)945 1004 y Fx(f)p Ft(y)1028 1016 y Fr(1)1064 1004 y Ft(;)14 b Fx(\001)g(\001)g(\001)g Ft(;)g(y)1290 1016 y Fs(i)p Fn(\000)p Fr(1)1402 1004 y Ft(;)g(\026)p Fy(\()p Ft(f)1571 961 y Fr([)p Fn(\000)p Fs(i)p Fr(])1562 1029 y Fs(\025i)1688 1004 y Fy(\))p Ft(;)g(y)1798 1016 y Fs(i)p Fr(+1)1910 1004 y Ft(;)g Fx(\001)g(\001)g(\001)f Ft(;)h(y)2135 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b(NY.)p Black 718 1990 a(3.)p Black 72 w(Bradley)-6 b(,)26 b(P)-6 b(.,)26 b(Mangasarian,)j(O.) c(&)h(Street,)f(N.)h(\(1997\),)h(`F)-6 b(eature)26 b(selection)h(via) 849 2082 y(mathematical)e(programming',)h Fv(INF)n(ORMS)h(J.)g (Complexity,)g(to)i(app)l(e)l(ar)p Fl(.)p Black 718 2206 a(4.)p Black 72 w(Co)n(x,)d(D.)g(&)f(O'Sulliv)l(an,)h(F.)g(\(1990\),)h (`Asymptotic)e(analysis)i(of)f(p)r(enalized)g(lik)n(eliho)r(o)r(d)h (and)849 2298 y(related)f(estimators',)h Fv(A)n(nn.)h(Statist.)e Ff(18)p Fl(,)h(1676{1695.)p Black 718 2422 a(5.)p Black 72 w(Cra)n(v)n(en,)f(P)-6 b(.)26 b(&)f(W)-6 b(ah)n(ba,)26 b(G.)g(\(1979\),)h(`Smo)r(othing)f(noisy)f(data)h(with)g(spline)g (functions:)849 2513 y(estimating)g(the)f(correct)i(degree)f(of)g(smo)r (othing)g(b)n(y)f(the)g(metho)r(d)g(of)h(generalized)849 2605 y(cross-v)l(alidation',)i Fv(Numer.)f(Math.)f Ff(31)p Fl(,)h(377{403.)p Black 718 2729 a(6.)p Black 72 w(Girard,)g(D.)e (\(1991\),)j(`Asymptotic)c(optimalit)n(y)h(of)i(the)e(fast)i 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Ff(6)p Fl(,)f(1{34.)p Black 714 3684 a(10.)p Black 72 w(Gong,)h(J.,)g(W)-6 b(ah)n(ba,)25 b(G.,)i(Johnson,)f(D.)g(&)f(T)-6 b(ribbia,)26 b(J.)h(\(1998\),)g(`Adaptiv)n(e)e(tuning)g(of)849 3775 y(n)n(umerical)g(w)n(eather)i(prediction)e(mo)r(dels:)h(sim)n (ultaneous)g(estimation)g(of)g(w)n(eigh)n(ting,)849 3867 y(smo)r(othing)g(and)f(ph)n(ysical)h(parameters',)g Fv(Monthly)i(We)l (ather)h(R)l(eview)d Ff(125)p Fl(,)h(210{231.)p Black 714 3991 a(11.)p Black 72 w(Gu,)f(C.)h(&)e(W)-6 b(ah)n(ba,)25 b(G.)i(\(1993\),)g(`Semiparametric)e(analysis)i(of)f(v)l(ariance)g (with)g(tensor)849 4082 y(pro)r(duct)f(thin)h(plate)g(splines',)h Fv(J.)g(R)l(oyal)g(Statistic)l(al)i(So)l(c.)f(Ser.)g(B)d Ff(55)p Fl(,)i(353{368.)p Black 714 4207 a(12.)p Black 72 w(Hastie,)g(T.)f(&)g(Tibshirani,)h(R.)e(\(1990\),)i Fv(Gener)l(alize)l(d)j(Additive)e(Mo)l(dels)p Fl(,)e(Chapman)f(and)849 4298 y(Hall,)i(335pp.)p Black 714 4423 a(13.)p Black 72 w(Hutc)n(hinson,)e(M.)i(\(1989\),)g(`A)f(sto)r(c)n(hastic)g 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y Fv(REFERENCES)3132 b(21)p Black Black 945 349 a Fl(16.)p Black 73 w(Li,)26 b(K.)f(C.)i(\(1986\),)g(`Asymptotic)e(optimalit)n(y)g (of)h Fj(C)2649 357 y Fi(L)2721 349 y Fl(and)f(generalized)i(cross)g(v) l(alidation)f(in)1080 441 y(ridge)h(regression)g(with)f(application)h (to)e(spline)i(smo)r(othing',)f Fv(A)n(nn.)h(Statist.)g Ff(14)p Fl(,)f(1101{1112.)p Black 945 565 a(17.)p Black 73 w(Mangasarian,)i(O.)d(\(1994\),)j(`Misclassi\014cation)h (minimization',)d Fv(J.)h(Glob)l(al)h(Optimization)1080 656 y Ff(5)p Fl(,)f(309{323.)p Black 945 781 a(18.)p Black 73 w(Micc)n(helli,)g(C.)f(\(1986\),)i(`In)n(terp)r(olation)e(of)h (scattered)f(data:)g(distance)g(matrices)g(and)1080 872 y(conditionally)h(p)r(ositiv)n(e)f(de\014nite)f(functions',)i Fv(Constructive)j(Appr)l(oximation)d Ff(2)p Fl(,)f(11{22.)p Black 945 997 a(19.)p Black 73 w(P)n(arzen,)g(E.)g(\(1962\),)h(`An)f (approac)n(h)f(to)h(time)f(series)i(analysis',)h Fv(A)n(nn.)f(Math.)g (Statist.)1080 1088 y Ff(32)p Fl(,)g(951{989.)p Black 945 1213 a(20.)p Black 73 w(P)n(arzen,)f(E.)g(\(1970\),)h(Statistical)g (inference)g(on)e(time)g(series)i(b)n(y)e(rkhs)g(metho)r(ds,)g Fv(in)g Fl(R.)h(Pyk)n(e,)1080 1304 y(ed.,)g(`Pro)r(ceedings)i(12th)e (Biennial)h(Seminar',)e(Canadian)i(Mathematical)f(Congress,)1080 1395 y(Mon)n(treal.)53 b(1-37.)p Black 945 1520 a(21.)p Black 73 w(Sc)n(ho)r(en)n(b)r(erg,)25 b(I.)h(\(1942\),)h(`P)n(ositiv)n (e)g(de\014nite)e(functions)h(on)g(spheres',)g Fv(Duke)j(Math.)e(J.) 1080 1611 y Ff(9)p Fl(,)g(96{108.)p Black 945 1736 a(22.)p Black 73 w(Sc)n(h\177)-38 b(olk)n(opf,)25 b(B.,)i(Sung,)e(K.,)h (Burges,)h(C.,)g(Girosi,)h(F.,)e(Niy)n(ogi,)g(P)-6 b(.,)26 b(P)n(oggio,)i(T.)f(&)e(V)-6 b(apnik,)25 b(V.)1080 1827 y(\(1997\),)j(Comparing)e(supp)r(ort)f(v)n(ector)g(mac)n(hines)h(with)f (gaussian)j(k)n(ernels)d(to)h(radial)h(basis)1080 1918 y(function)f(classi\014ers,)i(T)-6 b(ec)n(hnical)26 b(Rep)r(ort)g (1599,)h(Cen)n(ter)f(for)h(biological)h(and)d(Computational)1080 2010 y(Learning,)i(MIT,)f(Boston)h(MA.)p Black 945 2134 a(23.)p Black 73 w(Sk)n(orokho)r(d,)e(A.)g(&)h(Y)-6 b(adrenk)n(o,)24 b(M.)j(\(1973\),)g(`On)e(absolute)i(con)n(tin)n(uit)n(y)d(of)j (measures)1080 2226 y(corresp)r(onding)g(to)f(homogeneous)g(Gaussian)g (\014elds',)h Fv(The)l(ory)i(of)e(Pr)l(ob)l(ability)h(and)g(its)1080 2317 y(Applic)l(ations)f Ff(XVI)r(I)r(I)p Fl(,)g(27{40.)p Black 945 2441 a(24.)p Black 73 w(Smola,)e(A.)h(&)f(Sc)n(h\177)-38 b(olk)n(opf,)26 b(B.)g(\(1997\),)i(`On)d(a)h(k)n(ernel-based)f(metho)r (d)f(for)j(pattern)1080 2533 y(recognition,)h(regression,)f(appro)n (ximation,)f(and)f(op)r(erator)i(in)n(v)n(ersion',)f(ms.)p Black 945 2657 a(25.)p Black 73 w(V)-6 b(apnik,)24 b(V.)i(\(1995\),)h Fv(The)h(Natur)l(e)h(of)e(Statistic)l(al)i(L)l(e)l(arning)f(The)l(ory)p Fl(,)g(Springer.)p Black 945 2782 a(26.)p Black 73 w(Villalob)r(os,)g (M.)e(&)f(W)-6 b(ah)n(ba,)26 b(G.)g(\(1987\),)h(`Inequalit)n(y)e (constrained)h(m)n(ultiv)l(ariate)f(smo)r(othing)1080 2873 y(splines)i(with)f(application)g(to)g(the)g(estimation)f(of)i(p)r (osterior)g(probabilities',)h Fv(J.)f(A)n(m.)g(Statist.)1080 2964 y(Asso)l(c.)g Ff(82)p Fl(,)f(239{248.)p Black 945 3089 a(27.)p Black 73 w(W)-6 b(ah)n(ba,)25 b(G.)h(\(1978\),)h(`Improp)r (er)e(priors,)i(spline)f(smo)r(othing)g(and)f(the)g(problem)g(of)i (guarding)1080 3180 y(against)g(mo)r(del)f(errors)g(in)g(regression',)i Fv(J.)f(R)l(oy.)g(Stat.)i(So)l(c.)e(Ser.)h(B)e Ff(40)p Fl(,)g(364{372.)p Black 945 3305 a(28.)p Black 73 w(W)-6 b(ah)n(ba,)25 b(G.)h(\(1981\),)h(`Spline)f(in)n(terp)r(olation)h(and)e (smo)r(othing)h(on)g(the)f(sphere',)h Fv(SIAM)h(J.)1080 3396 y(Sci.)g(Stat.)i(Comput.)c Ff(2)p Fl(,)h(5{16.)p Black 945 3521 a(29.)p Black 73 w(W)-6 b(ah)n(ba,)25 b(G.)h(\(1982)p Fv(a)p Fl(\),)i(Constrained)e(regularization)i(for)e (ill)h(p)r(osed)f(linear)g(op)r(erator)1080 3612 y(equations,)h(with)f (applications)h(in)e(meteorology)h(and)g(medicine,)f Fv(in)g Fl(S.)h(Gupta)f(&)h(J.)g(Berger,)1080 3703 y(eds,)g 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b(&)g(D.)h(dePriest,)g(eds,)g (`Statistical)1080 4534 y(Image)g(Pro)r(cessing)h(and)f(Graphics',)h (Marcel)g(Dekk)n(er,)e(pp.)g(275{290.)p Black 945 4658 a(33.)p Black 73 w(W)-6 b(ah)n(ba,)25 b(G.)h(\(1990\),)h Fv(Spline)h(Mo)l(dels)g(for)f(Observational)i(Data)p Fl(,)e(SIAM.)51 b(CBMS-NSF)1080 4749 y(Regional)27 b(Conference)g (Series)f(in)g(Applied)f(Mathematics,)h(v.)g(59.)p Black 945 4874 a(34.)p Black 73 w(W)-6 b(ah)n(ba,)25 b(G.)h(\(1992\),)h (Multiv)l(ariate)g(function)f(and)f(op)r(erator)i(estimation,)f(based)g (on)1080 4965 y(smo)r(othing)g(splines)g(and)g(repro)r(ducing)g(k)n (ernels,)g Fv(in)f Fl(M.)h(Casdagli)i(&)d(S.)h(Eubank,)f(eds,)p Black Black eop %%Page: 22 23 22 22 bop Black Black -452 -350 3 150 v -452 5899 V 4350 -350 V 4350 5899 V -649 -300 150 3 v 4400 -300 V -649 5702 V 4400 5702 V Black -450 5700 a FB(Generic)20 b(author)h(design)g (sample)g(pages)62 b(1998/07/01)35 b(20:11)-34 100 y Fv(22)670 b(Supp)l(ort)29 b(V)-6 b(e)l(ctor)29 b(Machines,)f(R)l(epr)l (o)l(ducing)h(Kernel)f(Hilb)l(ert)g(Sp)l(ac)l(es)h(and)f(the)h(R)l (andomize)l(d)f(GA)n(CV)p Black 849 349 a Fl(`Nonlinear)f(Mo)r(deling)g (and)e(F)-6 b(orecasting,)27 b(SFI)f(Studies)f(in)g(the)h(Sciences)g (of)g(Complexit)n(y)-6 b(,)849 441 y(Pro)r(c.)27 b(V)-6 b(ol)26 b(XI)r(I',)f(Addison-W)-6 b(esley)g(,)25 b(pp.)g(95{112.)p Black 714 565 a(35.)p Black 72 w(W)-6 b(ah)n(ba,)26 b(G.,)g(Johnson,)h (D.,)f(Gao,)h(F.)f(&)f(Gong,)i(J.)f(\(1995)p Fv(a)p Fl(\),)i(`Adaptiv)n (e)d(tuning)g(of)849 656 y(n)n(umerical)g(w)n(eather)i(prediction)e(mo) r(dels:)h(randomized)f(GCV)h(in)g(three)f(and)h(four)849 748 y(dimensional)g(data)g(assimilation',)i Fv(Mon.)f(We)l(a.)g(R)l (ev.)f Ff(123)p Fl(,)g(3358{3369.)p Black 714 872 a(36.)p Black 72 w(W)-6 b(ah)n(ba,)26 b(G.,)g(W)-6 b(ang,)26 b(Y.,)g(Gu,)g(C.,)h(Klein,)f(R.)f(&)h(Klein,)g(B.)g(\(1994\),)h (Structured)e(mac)n(hine)849 964 y(learning)i(for)f(`soft')i (classi\014cation)f(with)f(smo)r(othing)g(spline)g(ANO)n(V)-9 b(A)24 b(and)h(stac)n(k)n(ed)h(tuning,)849 1055 y(testing)g(and)g(ev)l (aluation,)g Fv(in)f Fl(J.)i(Co)n(w)n(an,)g(G.)f(T)-6 b(esauro)27 b(&)e(J.)h(Alsp)r(ector,)h(eds,)f(`Adv)l(ances)f(in)849 1146 y(Neural)h(Information)g(Pro)r(cessing)h(Systems)e(6',)h(Morgan)h (Kau\013man,)e(pp.)g(415{422.)p Black 714 1271 a(37.)p Black 72 w(W)-6 b(ah)n(ba,)26 b(G.,)g(W)-6 b(ang,)26 b(Y.,)g(Gu,)g(C.,)h(Klein,)f(R.)f(&)h(Klein,)g(B.)g(\(1995)p Fv(b)p Fl(\),)i(`Smo)r(othing)d(spline)849 1362 y(ANO)n(V)-9 b(A)24 b(for)j(exp)r(onen)n(tial)e(families,)j(with)e(application)g(to) g(the)g(Wisconsin)g(Epidemiological)849 1453 y(Study)e(of)j(Diab)r (etic)f(Retinopath)n(y',)f Fv(A)n(nn.)i(Statist.)g Ff(23)p Fl(,)g(1865{1895.)p Black 714 1578 a(38.)p Black 72 w(Xiang,)g(D.)e (\(1996\),)j(Mo)r(del)e(Fitting)g(and)g(T)-6 b(esting)26 b(for)h(Non-Gaussian)f(Data)g(with)g(a)g(Large)849 1669 y(Data)g(Set,)g(PhD)f(thesis,)h(T)-6 b(ec)n(hnical)27 b(Rep)r(ort)e(957,)i(Univ)n(ersit)n(y)e(of)h(Wisconsin-Madison,)849 1761 y(Madison)h(WI.)p Black 714 1885 a(39.)p Black 72 w(Xiang,)g(D.)e(&)h(W)-6 b(ah)n(ba,)25 b(G.)h(\(1996\),)h(`A)f (generalized)h(appro)n(ximate)e(cross)i(v)l(alidation)f(for)849 1977 y(smo)r(othing)g(splines)g(with)g(non-Gaussian)g(data',)h Fv(Statistic)l(a)i(Sinic)l(a)d Ff(6)p Fl(,)g(675{692.)p Black 714 2101 a(40.)p Black 72 w(Xiang,)h(D.)e(&)h(W)-6 b(ah)n(ba,)25 b(G.)h(\(1997\),)h(Appro)n(ximate)e(smo)r(othing)g (spline)h(metho)r(ds)f(for)i(large)849 2192 y(data)f(sets)g(in)g(the)f (binary)g(case,)i(T)-6 b(ec)n(hnical)27 b(Rep)r(ort)e(982,)i(Departmen) n(t)d(of)j(Statistics,)849 2284 y(Univ)n(ersit)n(y)e(of)h(Wisconsin,)h (Madison)g(WI.)51 b(T)-6 b(o)26 b(app)r(ear)g(in)g(the)f(Pro)r (ceedings)i(of)g(the)e(1997)849 2375 y(ASA)g(Join)n(t)h(Statistical)h (Meetings,)g(Biometrics)g(Section,)f(pp)f(94-98)h(\(1998\).)p Black Black eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF