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线性判别分析(LDA)入门.pdf

发布时间：2022-06-14 发布人：admin 分类：说明书 资料大小：2.11M 资料格式：pdf 举报 版权申诉
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    预备知识
    分类问题的描述
    拉格朗日乘子法
    Two-classes 情形的数学推导
    基本思想
    目标函数
    极值求解
    阀值选取
    推广到 Multi-classes 情形
    降维问题的描述
    目标函数与极值求解
    降维幅度
    其他几个相关问题
    后记
    <=978; (LDA):; 2013 9 26 z peghoty 1 A6 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2j| f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Two-classes 37 D 2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4e? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 [F Multi-classes 3 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2w= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 eVY 5 _g 3 3 3 4 4 6 7 11 12 12 12 16 17 18 1
    Fisher’s Linear Discriminant (FLD ∗ ) Linear Discriminant Analysis (LDA) e4SUR kf,wÆ4V LDA FLD_yY9x :k FLD LDA,`F~ LDA ( \ S, the two terms are often used interchangeably, [2]). LDA6 ^jk45R ,W ^ B7jk^YRE#.6Z!Æ /U2hV h RVGv℄^2}yjfj ([1]). Æ 1 a#a\b (Sir Ronald Aylmer Fisher) V U LDA (Ce FLD)E.. ∗FLDd Ronald Fisher ( 1) 1936  ([4]), Ronald Fisher (1890-1962)dX=℄ 3 9SY9v[59.|d? 3v?SSX. 2
    X = {x1, x2, · · · , xN } Xi = {xj | Lj = li}, i = 1, 2, · · · , c §1 B7 §1.1 R ~ ℄:8 -9 (W9wd) {(xi, Li)}N i=1,0! xi ∈ Rd (d ≥ 2) i }:8 (~^fY} dkF6x), Li xi^f :9,XU! c (c ≥ 2)},5 l1, l2, · · · , lc,. Li ∈ {l1, l2, · · · , lc}.DVY}?U x (V=~UYZ1)Z,kCV0^f5 y"? FV U LDAWe^SkfY.k1A,e a{UY,^{k iUY,5 |Xi| = Ni > 0 (i = 1, 2, · · · , c).D, §1.2 U LDAE!jFU (Lagrange),3}>B U,Z ;#b{ 1 . := z = f (x)C φi(x) = 0 (i = 1, 2, · · · , m)  ,0! j Lagrange f, FSCC C kRE=, Z= : 1. F (x) = f (x) + λiφi(x),0! λi (i = 1, 2, · · · , m) Lagrange. 2.=k.0 mPi=1 0! dF )Ta F xBE. = (1.1)  x1, x2, · · · , xn; λ1, λ2, · · · , λm,0! (x1, x2, · · · , xn)We 5.  x = (x1, x2, · · · , xn) ∈ Rn. i = 1, 2, · · · , m = ( ∂F ∂x1 , ∂F ∂x2 , · · · , ∂F ∂xn dx cPi=1 Ni = N. dF dx = 0, φi(x) = 0, (1.1) 3
    §2 Two-classes48!E ^ two-classes (. c = 2):D,2 LDASE. §2.1 a8 LDA{:F{UYhF18?6xf=,^=[17 #[,. f=!18 F. Sf`,WF_x.^b, jY" k` k U LDA3: Y}0F T ,F Rd!UYiVF R1 (9 dF 1),8# FkYUYiV~S~ (.17#[),`FkU YiV~|~ (.1>=[).ZT _UYA" (preserve as much of the class discriminatory information as possible).U,F Y}feW ^FUYw. ,0F T?"?F0AVD` 0! w ∈ RdAV6x. j (2.2),e ^ 0! zi = wT xi, i = 1, 2, · · · , N.  2.1b (2.2)#ÆX6 wT x= w x ( ),ib,{D  θw w x. d = 2w9, 2~`> . Z1 = {zj | xj ∈ X1}, Z2 = {zj | xj ∈ X2}, Z = {z1, z2, · · · , zN }, y = wT x = w · x = kwk · kxk · cos θ, (2.3) y = T (x) = wT x, (2.2) 4
    Æ 2#0ab  2., kxk · cos θS x# wX>^, (2.3).,U kwk = 1, yw x# wX>^.  S3,Æ 32Y}>B (0! d = 2). Æ 3>B Æ 3!R\ Rkhw}UY,`9{ 2d V0k6.6x wk6.^2wÆ_d ,Æ2Æ8ÆV, ~ ^w}UY.  LDA3!,F, wCV0F Tf /:kYU Y T4j[S,`kUY T4j[|.a, 0\eY"f,#2Y}Vx.2wf!,3 fe:iEY}^ w x J(w),D=Y}1f (> 7),F w∗.{ w. k, J(w)E. 5
    §2.2 9^ 5 x1, x2aUY X1 X2_ (! ),. ~U0F T4j z1, z2,. j TVb (2.2), UY Xi! h. Xi!~}zvh_. zi = T (xi) = wT xi = wT 1 Ni Xx∈Xi Ni Xx∈Xi Ni Xx∈Xi Ni Xz∈Zi zi = T (xi), i = 1, 2. x, i = 1, 2, wT x = xi = x = 1 1 1 z, i = 1, 2, (2.4) Æ 4 x1, x2^ z1, z2aaÆ Æ 42y x1, x2^ z1, z2aaÆ.Y63,# (1) z1w z2~|~; (2) Zi!zv~! zi (i = 1, 2)S~.  (1)^f jvÆ (Between-class scatter) , x kREd;` (2)^f vÆ (Within-class scatter), x 2,0! s2 kREd,0! s2  X1 X2^fÆy (scatter),~D`S s2 =,C3^zv}. i =Xz∈Zi JB = |z1 − z2|2 (z − zi)2, JW = s2 1 + s2 i = 1, 2 1 2 (2.5) (2.6) 6
     2.2w\C[ A jvÆ vÆ? S" A45V V A45V,\Cz? [10]~`> Æ|vx91.S 54j , X1 X2zT!zW#W3W v7, x1 x2zTw X1 X2. wzT w x1/ x2/#,,h w x1/[, JB (= |z1 − z2|2) d,ge,j x1/X.Yz,aV z5; h w x2/[, JB,gj x2/X.Y,x # z5. Æ 5jh6x8Æ b|91.,{D [ A jvÆ vÆ<|h. 5~ `>[ A<|hkj. ,y JB JWw}x,63 : JB~>~,` JW~7~ ..bw ,F9^Vb D,^ J(w).Fw RE: (1)F J(w)v< w`a; (2)j Lagrange f=v<>. •v< |z1 − z2|2 1 + s2 s2 2 JB JW JW J(w) = = , (2.7) 7
    JB = |z1 − z2|2 JB = wT SBw. SB = (x1 − x2)(x1 − x2)T , = wT (x1 − x2)wT (x1 − x2) = (wT x1 − wT x2)2 = (wT (x1 − x2))2 1. JBv< = wT (x1 − x2)(x1 − x2)T w (j#  F,. aT b = bT a) e jvÆE (Between-class scatter matrix)  JB >< a,X SBe6x x1 − x2w06x4DxbF,~YE Y{Y1Y}, 80 1. 2. JWv
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