Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements
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1 Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall 06 Patter Classfcato Chapter 4 (Part ): No-Parametrc Classfcato (Sectos ) 4.3) All materals these sldes were tae from Patter Classfcato (2d ed) by. O. Duda, P. E. Hart ad D. G. Stor, Joh Wley & Sos, 2000 wth the permsso of the authors ad the publsher Itroducto Desty Estmato Parze Wdows Itroducto 4 Desty Estmato 5 All Parametrc destes are umodal (have a sgle local maxmum), whereas may practcal problems volve mult- modal destes Noparametrc procedures ca be used wth arbtrary dstrbutos ad wthout the assumpto that the forms of the uderlyg destes are ow There are two types of oparametrc methods: Estmatg P P(x ω j ) Bypass probablty ad go drectly to a-posteror a probablty estmato Basc dea: Probablty that a vector x wll fall rego s: P = p( x' )dx' () P s a smoothed (or averaged) verso of the desty fucto p(x) f we have a sample of sze ; therefore, the probablty that pots fall s the: P = P ( P ) (2) ad the expected value for s: E() = P (3)
2 ML estmato of P = θ Max( P θ ) s reached for θ ˆ θ = P 6 Combg equato (), (3) ad (4) yelds: p( x ) / V 7 Therefore, the rato / s a good estmate for the probablty P ad hece for the desty fucto p. p(x) s cotuous ad that the rego s so small that p does ot vary sgfcatly wth t, we ca wrte: p( x' )dx' p( x )V (4) where s a pot wth ad V the volume eclosed by. Desty Estmato (cot.) Justfcato of equato (4) p( x' )dx' p( x )V (4) We assume that p(x) s cotuous ad that rego s so small that p does ot vary sgfcatly wth.. Sce p(x) = costat, t s ot a part of the sum. 8 p ( x' )dx' p( x' ) dx' = p( x' ) = ( x )dx' = p( x' ) μ( ) Where: μ() s: a surface the Eucldea space 2 a volume the Eucldea space 3 a hypervolume the Eucldea space Sce p(x) p(x ) = costat, therefore the Eucldea space 3 : ad p( x' )dx' p( x ) p( x ). V V 9 Codto for covergece The fracto /(V) s a space averaged value of p(x). p(x) s obtaed oly f V approaches zero. lm V 0, = 0 p( x ) = 0 (f = fxed) Ths s the case where o samples are cluded : t s a uterestg case! lm V 0, 0 p( x ) = I ths case, the estmate dverges: t s a uterestg case! 0 The volume V eeds to approach 0 ayway f we wat to use ths estmato Practcally, V caot be allowed to become small sce the umber r of samples s always lmted Oe wll have to accept a certa amout of varace the rato o / Theoretcally, f a ulmted umber of samples s avalable, we ca crcumvet ths dffculty To estmate the desty of x, we form a sequece of regos, 2, cotag x: the frst rego cotas oe sample, the secod two samples ad so o. Let V be the volume of, the umber of samples fallg ad p (x) be the th estmate for p(x): p (x) = ( /)/V (7) 2
3 2 3 Three ecessary codtos should apply f we wat p (x) to coverge to p(x) ) lmv = 0 2 ) lm 3 ) lm / = 0 There are two dfferet ways of obtag sequeces of regos that t satsfy these codtos: (a) Shr a tal rego where V = / ad show that Ths s called the Parze-wdow estmato method p ( x ) = p( x ) (b) Specfy as some fucto of, such as = ; ; the volume V s grow utl t ecloses eghbors of x. Ths s called the -earest eghbor estmato method Parze Wdows Parze-wdow approach to estmate destes assume that the rego s a d-dmesoal d dmesoal hypercube 4 The umber of samples ths hypercube s: = = = x x ϕ h 5 d V = h (h : legth of the edge of ) Let ϕ(u) be the followg wdow fucto : u j j =,...,d ϕ(u) = 2 0 otherwse ϕ(( ((x-x )/h ) s equal to uty f x falls wth the hypercube of volume V cetered at x ad equal to zero otherwse. By substtutg equato (7), we obta the followg estmate: = = x x p(x) ϕ = V h P (x) estmates p(x) as a average of fuctos of x ad the samples (x ) ( =,,). These fuctos ϕ ca be geeral! Parze Wdow Example Draw samples from a Normal dstrbuto, N(0,) Let ϕ(u) = (/ (2π) exp(-u 2 /2) h = h / (>) Thus: = = x x p ( x ) ϕ = h h s a average of ormal destes cetered at the samples x. 6 Numercal results: For = ad h = p ( x ) ( x x ) e / 2 2 = ϕ = ( x x ) N ( x, ) 2π For = 0 ad h = 0.,, the cotrbutos of the dvdual samples are clearly observable! 7 3
4 Aalogous results are also obtaed two dmesos as llustrated: 2 Case where p(x) = λ.u(a,b) + λ 2.T(c,d) (uow desty) (mxture of a uform ad a tragle desty)
5 K - Nearest eghbor estmato Goal: a soluto for the problem of the uow best wdow fucto Let the cell volume be a fucto of the trag data Ceter a cell about x ad let t grows utl t captures samples ( = f()) are called the earest-eghbors eghbors of x 2 possbltes ca occur: Desty s hgh ear x; ; therefore the cell wll be small whch provdes a good resoluto Desty s low; therefore the cell wll grow large ad stop utl l hgher desty regos are reached We ca obta a famly of estmates by settg = 24 Illustrato For = ad = = ; the estmate becomes: P (x) = /.V = / V = / 2 x- Yes! Well ot so good as the probablty goes to fty at x but at least we do ot have holes the desty! Thgs get better as gets bgger! Ad we stll do t t have holes the desty eve for hgher dmesos! Estmato of a-posteror a probabltes Goal: estmate P(ω x) from a set of labeled samples Let s s place a cell of volume V aroud x ad capture samples samples amogst tured out to be labeled ω the: p (x, ω ) = /.V A estmate for p (ω x) s: p( x, ω ) p ( ω x ) = = j c p ( x, ω ) = j= j / s the fracto of the samples wth the cell that are labeled ω For mmum error rate, the most frequetly represeted category wth the cell s selected If s large ad the cell suffcetly small, the performace wll approach the best possble So whether we use Parze wdows (or K-thK earest eghbors to determe our wdow sze V, we ca drectly get the a posteror probabltes. 5
6 30 3 The earest eghbor rule Let D = {x, x 2,, x } be a set of labeled prototypes Let x D be the closest prototype to a test pot x the the earest-eghbor eghbor rule for classfyg x s to assg t the label assocated wth x The earest-eghbor eghbor rule leads to a error rate greater tha the mmum possble: the Bayes rate If the umber of prototype s large (ulmted), the error rate of the earest-eghbor eghbor classfer s ever worse tha twce the Bayes rate (t ca be demostrated!) If,, t s always possble to fd x suffcetly close so that: P(ω x ) P(ω x) The earest-eghbor eghbor rule Goal: Classfy x by assgg t the label most frequetly represeted amog the earest samples ad use a votg scheme 6
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