Correspondence Analysis & Related Methods

Transcription

1 Coespondence Analyss & Related Methods Ineta contbutons n weghted PCA PCA s a method of data vsualzaton whch epesents the tue postons of ponts n a map whch comes closest to all the ponts, closest n sense of weghted least-squaes. Mchael Geenace SESSIN 13 Dagnostcs, contbutons n weghted PCA and Coespondence Analyss 1% 71% The neta (weghted vaance) explaned n the map apples to all the ponts f we say 83% of the neta s explaned n the map, 71% on the fst dmenson and 1% on the second, ths s a fgue calculated fo all ow (o column) ponts togethe. Ineta contbutons n weghted PCA Geomety of neta contbutons Ths type of neta-explaned-by-axes calculaton can be made fo ndvdual ponts. These moe detaled esults ae ads to ntepetaton n the fom of numecal dagnostcs, called contbutons. Especally when thee s not a hgh pecentage of neta explaned by the map, these contbutons wll help us to dentfy ponts whch ae epesented naccuately. The netas and the pecentages tell us how much of the vaance n the table s explaned by the pncpal axes. The contbutons do the same, but fo each pont ndvdually, and help us to see (a) whch ponts ae beng explaned bette than othes; (b) whch ponts ae contbutng to the soluton moe than othes. centod c d (pncpal coodnate) -th pont a wth mass m poecton on Total neta of the cloud of ponts =µ m d = µ m µ k = µ k λ k Ineta of -th pont = m d = m µ k Ineta contbuton of -th pont to k-th = m k-th pncpal

2 1 3 n Axes 1... p m 1 f 11 m 1 f 1... m 1 f 1p m f 1 m f... m f p m 3 f 31 m 3 f 3... m 3 f 3p m n f n1 m n f n... m n f np λ 1 λ... Decomposton of neta λ p centod c m 1 d 1 m d m 3 d 3 d -th pont a wth mass m poecton on Total neta of the cloud of ponts =µ m d = µ m µ k = µ k λ k Ineta of -th pont = m d = m µ k m n d n Ineta contbuton of -th pont to k-th = m k-th pncpal 1 3 n Axes 1... p m 1 f 11 m 1 f 1... m 1 f 1p m f 1 m f... m f p m 3 f 31 m 3 f 3... m 3 f 3p m n f n1 m n f n... m n f np λ 1 λ... Ineta contbutons λ p centod c m 1 d 1 m d m 3 d 3 m n d n θ k d -th pont a wth mass m poecton on m / λ k amount of neta of k explaned by pont (contbuton, CTR) k-th pncpal m / m d amount of neta of pont explaned by k (squaed coelaton, CR) m / m d = / d,.e. the squae of / d = cos(θ k ), whee θ k s the angle pont- Ineta contbutons fo CA of autho autho col name mass qlt n k=1 co ct k= co ct 1 a b c d e f g h k l m n o p q s t u v w x y z Summay Contbutons to neta Each pncpal neta can decomposed nto pats due to each pont, ethe ow ponts o column ponts. These contbutons explan how each pncpal has been constucted (hence the nfluence of each pont n defnng the dmenson). The neta of a pont s smlaly decomposed ove all the axes, thanks to usng Eucldean-type dstance and Pythagoas theoem. Each component on an can be expessed elatve to the pont neta and ths s the same as the squaed cosne (.e., squaed coelaton) between the pont and the. These values can be added ove axes and tell you how well the pont s epesented n the soluton space.

3 R mplementaton of CA (epeat) Computaton of contbutons # ead n data nto data-fame data_set # the next 14 commands ae all you need to compute CA esults data.p <- data_set/sum(data_set) data. <- apply(data.p,1,sum) data.c <- apply(data.p,,sum) data.d <- dag(data.) data.dc <- dag(data.c) data.dmh <- dag(1/sqt(data.)) data.dcmh <- dag(1/sqt(data.c)) data.p data.s data.svd data.sc data.csc data.pc data.cpc # the symmetc map <- as.matx(data.p) <- data.dmh %*% (data.p-data.%o%data.c) %*% data.dcmh <- svd(data.s) <- data.dmh%*%data.svd$u <- data.dcmh%*%data.svd$v <- data.sc%*%dag(data.svd$d) <- data.csc%*%dag(data.svd$d) plot(data.pc[,1],data.pc[,],type="n",pty="s") text(data.pc[,1],data.pc[,],label=ownames(data)) # now do t n one shot usng ca package (fst nstall fom CRAN) lbay(ca) plot(ca(data_set)) # compute matx of contbutons fo ows and netas data.con <- data.pc^ * data. apply(data.con, 1, sum) # compute contbutons and squaed coelatons data.ct <- t( t(data.con) / apply(data.con,, sum) ) data.co <- data.con / apply(data.con, 1, sum) # compute qualtes n -d soluton apply(data.co[,1], 1, sum) # compute matx of contbutons fo columns and netas data.ccon <- data.cpc^ * data.c apply(data.ccon, 1, sum) # compute contbutons and squaed coelatons data.cct <- t( t(data.ccon) / apply(data.ccon,, sum) ) data.cco <- data.ccon / apply(data.ccon, 1, sum) # compute qualtes n -d soluton apply(data.cco[,1], 1, sum)

4 Coespondence Analyss & Related Methods Mchael Geenace SESSIN CRRESPNDENCE ANALYSIS & CLUSTER ANALYSIS. CRRESPNDENCE ANALYSIS & BIPLT Coespondence analyss (CA) s a method of data vsualzaton that eveals contnuous stuctues (the dmensons) But n ou seach fo stuctue n the table we can also consde clusteng the ows and columns, to eveal dscete stuctues (the clustes, o classes) A smple example 988 students, males and females classfed each accodng to the paents havng been o not to unvesty, coss-tabulated wth the choce of studes at hgh school A smple example 988 students, males and females classfed each accodng to the paents havng been o not to unvesty, coss-tabulated wth the choce of studes at hgh school F_no F_un M_un Ineta = Ch-squae = Whch two ows can we mege so that neta (o ch-squae) s educed the least? F_no and F_un educes neta by {F_no, F_un} M_un Ineta = Whch two ows can we mege so that neta s educed the least? and M_un educes neta by F_no female, paents no unvesty F_un female, paents unvesty male, paents no unvesty M_un male, paents unvesty NS non-scence MA mathematcs LS lfe scences PS physcal scences F_no female, paents no unvesty F_un female, paents unvesty male, paents no unvesty M_un male, paents unvesty NS non-scence MA mathematcs LS lfe scences PS physcal scences

5 A smple example 988 students, males and females classfed each accodng to the paents havng been o not to unvesty, coss-tabulated wth the choce of studes at hgh school A smple example 988 students, males and females classfed each accodng to the paents havng been o not to unvesty, coss-tabulated wth the choce of studes {F_no, F_un} {, M_un} Ineta = Whch two ows can we mege so that neta (o ch-squae) s educed the least? nly two ows left to mege and ths educes neta by F_no F_un M_un F_no F_un F_no female, paents no unvesty F_un female, paents unvesty male, paents no unvesty M_un male, paents unvesty NS non-scence MA mathematcs LS lfe scences PS physcal scences Fom Geenace( ), the ctcal pont fo the ch-squae s 13.11, that s fo the neta 13.11/988 = Ths gves multple compason test fo dffeences between ows. M_un Masses of clustes G 1 and G Wad clusteng The type of clusteng pefomed by ths pocedue of mnmzng the educton of neta at each step s called Wad clusteng (see ou eale classes on cluste analyss) Wad clusteng s a heachcal clusteng analyss whch needs (a) descpton vectos of obects to be clusteed (b) weghts fo each obect If you pefe to have a dstance cteon fo clusteng, ths s t d 1 ( G1, G) = pof1 pof c 1 + We want to pefom Wad clusteng on the pofles, wth weghts equal to the masses. Ch-squae dstance Pofles of clustes G 1 and G Snce Wad clusteng calculates Eucldean dstances between vectos, we would need to pepae the pofles so that the Eucldean dstances wll be ch-squaed that s, we have to dvde the pofle elements by the squae oots of the aveage (expected) values. But we need to weght the ponts use XLSTAT o Fonn Mutagh s R code http//asto.u-stasbg.f/~fmutagh/mda-sw/coespondances Bplot Coespondence analyss s based on the SVD of Pncpal coodnates D 1/ T 1/ ( P c ) Dc F = D 1/ UD α G = D 1/ c VD α = UD Standad coodnates 1 T 1 1/ 1/ T D ( P c ) Dc = D UDα ( Dc V) = α V T 1/ Φ = D U 1/ Γ D V = c We want the ght hand sde n the fom of scala poducts between the coodnate matces In full space In educed space (e.g., -d) ow pofle element p c c p = f γ p c c + f γ 1 1 +L f γ + f γ 1 1 c / c f 1γ 1 + fγ aveage pofle element FΓ T scala poduct between ow pofle and column vetex

6 Bplot vaatons by Gabel & Geenace (epeat) ow pofle element p Gabel s modfcaton p devaton of pofle fom aveage c / c f 1γ 1 + fγ aveage pofle element c f c γ + f c γ 1 1 Geenace s modfcaton (the standad bplot) p c 1/ 1/ / c 1/ f 1c γ 1 + fc γ standadzed devaton of pofle fom aveage (Relatve) column contbuton c g 1/ k / λ k = c ( λk γ k ) / λk = c γ k scala poduct between ow pofle and column vetex vetces (standad coodnates) shunk by the espectve masses vetces shunk by squae oots of the espectve masses; squaes of these escaled column coodnates ae exactly the (elatve) contbutons of the column to the espectve dmenson