Variable selection in principal components analysis of qualitative data using the accelerated ALS algorithm

Transcription

1 Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm Masahiro Kuroda Yuichi Mori Masaya Iizuka Michio Sakakihara (Okayama Uiversity of Sciece) (Okayama Uiversity of Sciece) (Okayama Uiversity) (Okayama Uiversity of Sciece) * supported by the Japa Society for the Promotio of Sciece (JSPS), Grat-i-Aid for Scietific Research (C), No Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 1/29

2 Motivatio Variable sectio i PCA of qualitative data The ALS algorithm for quatifyig qualitative data PCA.ALS PRINCIPALS: Youg, Takae & de Leeuw (1978) (SAS) PRINCALS: Gifi (1990) (SPSS) Acceleratio of PCA.ALS usig the vector ε (vε) algorithm vε-pca.als: Kuroda, Mori, Iizuka & Sakakihara (2011) i CSDA. Modified PCA (M.PCA) Formulatio of M.PCA: Taaka & Mori (1997) Backward elimiatio & Forward selectio procedures: Mori et al. (1998, 2006) Applicatio of vε-pca.als to variable selectio i M.PCA of qualitative data Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 2/29

3 PCA with variables measured by mixed scaled levels X : p matrix ( observatios o p variables; columwise stadardized) I PCA, X is postulated to be approximated by a biliear structure of the form: where ˆX = ZA, Z is a r matrix of compoet scores o r compoets (1 r p), A is a p r matrix cosistig of the eigevectors of X X/ ad A A = I r. We fid Z ad A such that θ = tr(x ˆX) (X ˆX) = tr(x ZA ) (X ZA ) is miimized for the prescribed umber of compoets r. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 3/29

4 PCA with variables measured by mixed scaled levels For oly quatitative variables (iterval ad ratio scales) We ca fid Z ad A (or ˆX = ZA ) miimizig θ = tr(x ˆX) (X ˆX) = tr(x ZA ) (X ZA ). For mixed scaled variables (omial, ordial, iterval ad ratio scales) Optimal scalig is ecessary to quatify the observed qualitative data, i.e., we eed to fid a optimally scaled observatio X miimizig θ = tr(x ˆX) (X ˆX) = tr(x ZA ) (X ZA ), where [ X X X ] 1 = 0 p ad diag = I p, i additio to Z ad A, simultaeously. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 4/29

5 Alteratig least squares algorithm to fid the optimal scaled observatio X To fid model parameters (Z, A) ad optimal scalig parameter X, the Alteratig Least Squares (ALS) algorithm ca be utilized: PCA.ALS (PRINCIPALS, PRINCALS) Model parameter estimatio step : estimate Z ad A coditioally o fixed X. Optimal scalig step : fid X for miimizig θ coditioally o fixed Z ad A. X p = Model parameter estimatio = Optimal scalig Z r p A r Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 5/29

6 Alteratig least squares algorithm to fid the optimal scaled observatio X [ PCA.ALS algorithm ] PRINCIPALS (Youg et al, 1978) Superscript (t) idicates the t-th iteratio. Model parameter estimatio step: Obtai A (t) by solvig [ X (t) X (t) ] A = AD r, where A A = I r ad D r is a r r diagoal eigevalue matrix. Compute Z (t) from Z (t) = X (t) A (t). Optimal scalig step: Calculate ˆX (t+1) = Z (t) A (t). Fid X (t+1) such that X (t+1) = arg mi X tr(x ˆX (t+1) ) (X ˆX (t+1) ) for fixed ˆX (t+1) uder measuremet restrictios o each variables. Scale X (t+1) by columwise ormalizig ad ceterig. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 7/29

7 Acceleratio of PCA.ALS by the vector ε accelerator To accelerate the computatio, we ca use vector ε accelerator (vε accelerator) by Wy (1962), which speeds up the covergece of a slowly coverget vector sequece, is very effective for liearly covergig sequeces, geerates a sequece {Ẏ(t) } t 0 from the iterative sequece {Y (t) } t 0. Covergece: The accelerated sequece {Ẏ(t) } t 0 coverges to the statioary poit Y of {Y (t) } t 0 faster tha {Y (t) } t 0. Computatioal cost: At each iteratio, the vε algorithm requires oly O(d) arithmetic operatios while the Newto-Raphso ad quasi-newto algorithms are achieved at O(d 3 ) ad O(d 2 ) where d is the dimesio of Y. Covergece speed: The best speed of covergece is superliear. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 8/29

8 Acceleratio of PCA.ALS by the vector ε accelerator The vε accelerator is give by Ẏ (t 1) = Y (t) + [ [ Y (t 1) Y (t)] 1 + [Y (t+1) Y (t)] ] 1 1, where [Y] 1 = Y/ Y 2 ad Y is the Euclidea orm of Y. The accelerated vector Ẏ (t 1) is obtaied by the origial sequece (Y (t 1), Y (t), Y (t+1) ) The vε accelerator does ot deped o the statistical model {Y (t) } t 0. Therefore, whe the vε algorithm is applied to ALS, it guaratees the covergece properties of the ALS. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 9/29

9 Acceleratio of PCA.ALS by the vector ε accelerator To accelerate PCA.ALS, we itroduce the vε algorithm ito PCA.ALS, i.e., From a sequece {X (t) } t 0 = {X (0), X (1),, X ( ) } i PCA.ALS, make a accelerated sequece {Ẋ (t) } t 0 = {Ẋ (0), Ẋ (1),, X ( ) }. [ Geeral procedure of vε-pca.als ] Alterate the followig two steps util the algorithm is coverged: PCA.ALS step: Compute model parameters A (t) ad Z (t) ad determie optimal scalig parameter X (t+1). Acceleratio step: Calculate Ẋ (t 1) usig {X (t 1), X (t), X (t+1) } from the vε algorithm: [ [ ] 1 [ 1 ] 1 vecẋ (t 1) = vecx (t) + vec(x (t 1) X (t) ) + vec(x (t+1) X )] (t), where vecx stads for the vectors of colums of X, ad check the covergece by vec(ẋ (t 1) Ẋ (t 2) ) 2 < δ, where δ is a desired accuracy. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 10/29

10 Acceleratio of PCA.ALS by the vector ε accelerator X p = Model parameter estimatio = Optimal scalig Z r p A r p PCA.ALS step: {X (0),..., X (t) } Ẋ Acceleratio step: {Ẋ (0),..., Ẋ (s) } Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 11/29

11 Acceleratio of PCA.ALS by the vector ε accelerator Sice vε-pca.als is desiged to geerate {Ẋ (t) } t 0 covergig to X ( ), the estimate of X ca be obtaied from the fial value of {Ẋ (t) } t 0 whe vε-pca.als termiates, the estimates of Z ad A ca the be calculated immediately from the estimate of X i the Model parameter estimatio step of PCA.ALS. Note that Ẋ (t 1) obtaied at the t-th iteratio of the Acceleratio step is ot used as the estimate X (t+1) at the (t + 1)-th iteratio of the PCA.ALS step. Thus vε-pca.als speeds up the covergece of {X (t) } t 0 without affectig the covergece properties of PCA.ALS procedure. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 12/29

12 Modified PCA (M.PCA) of Taaka & Mori (1997) Modified PCA (M.PCA) derives pricipal compoets that are computed as liear combiatios of a subset of variables but ca reproduce all the variables very well. Let X be decomposed ito a q submatrix X V1 ad a (p q) remaiig submatrix X V2. The M.PCA fids r liear combiatios Z = X V1 A. X p = Variable selectio q X V1 p X V2 = M.PCA Z r Approximate: Z PCA Z Z PCA = XA = A from [X X/]A = AD r Z = X V1 A = A from [(S S 12 S 21 ) DS 11 ]A = 0 Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 13/29

13 Formulatio of Modified PCA (M.PCA) The matrix A cosists of the eigevectors associated with the largest r eigevalues λ 1 λ 2 λ r ad is obtaied by solvig the eigevalue problem: [(S S 12 S 21 ) DS 11 ]A = 0, ( ) S11 S where S = 12 S 21 S is the covariace matrix of X = (X V1, X V2 ) ad D is 22 a q q diagoal matrix of eigevalues. A best subset of q variables has the largest value of the proportio P = r j=1 λ j/tr(s) = we use P as variable selectio criteria. the RV -coefficiet RV = { r j=1 λ2 j /tr(s2 )} 1/2. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 14/29

14 Variable selectio procedures i M.PCA I order to fid a subset of q variables, we employ two variable selectio procedures of Mori et al. (1998, 2006) Backward elimiatio procedure ad Forward selectio procedure. These procedures are cost-savig stepwise selectio procedures ad are remove or add oly oe variable sequetially. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 15/29

15 Backward elimiatio [Backward elimiatio] Stage A: Iitial fixed-variables stage A-1 Assig q variables to subset X V1, usually q := p. A-2 Solve the eigevalue problem. A-3 Look carefully at the eigevalues, determie the umber r of pricipal compoets to be used. A-4 Specify kerel variables which should be ivolved i X V1, if ecessary. The umber of kerel variables is less tha q. Stage B: Variable selectio stage (Backward) B-1 Remove oe variable from amog q variables i X V1, make a temporary subset of size q 1, ad compute P based o the subset. Repeat this for each variable i X V1, the obtai q values o P. Fid the best subset of size q 1 which provides the largest P amog these q values ad remove the correspodig variable from the preset X V1. Put q := q 1. B-2 If P or q is larger tha preassiged values, go to B-1. Otherwise stop. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 16/29

16 Forward selectio [Forward selectio] Stage A: Iitial fixed-variables stage A-1 A-3 Same as A-1 to A-3 i Backward elimiatio. A-4 Redefie q as the umber of kerel variables (here, q r). If you have kerel variables, assig them to X V1. If ot, put q := r, fid the best subset of q variables which provides the largest P amog all possible subsets of size q ad assig it to X V1. Stage B: Variable selectio stage (Forward) B-1 Addig oe of the p q variables i X V2 to X V1, make a temporary subset of size q + 1 ad obtai P. Repeat this for each variable i X V2, the obtai p q P s. Fid the best subset of size q + 1 which provides the largest (or smallest) P amog the p q P s ad add the correspodig variable to the preset subset of X V1. Put q := q + 1. B-2 If the P or q are smaller (or larger) tha preassiged values, go back to B-1. Otherwise stop. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 17/29

17 Variable selectio i M.PCA Variable selectio is to fid a subset of q variables that best approximates all the variables. Fid X V1 usig Backward elimiatio or Forward selectio procedures. X p = Variable selectio q X V1 p X V2 = M.PCA Z r Approximate: Z PCA Z Z PCA = XA = A from [X X/]A = AD r Z = X V1 A = A from [(S S 12 S 21 ) DS 11 ]A = 0 Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 18/29

18 Variable selectio i M.PCA of qualitative data M.PCA of qualitative data: Iteratio betwee Variable selectio ad PCA.ALS X p = Variable selectio q p X V 1 X V 2 M.PCA usig PCA.ALS Z r Steps of variable selectio ad PCA.ALS Variable selectio: Select X V1 usig Backward elimiatio or Forward selectio. PCA.ALS: Quatify X V 1 ad compute A ad Z usig the ALS algorithm. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 19/29

19 vε-pca.als for variable selectio i M.PCA of qualitative data X p = Variable selectio q p X V 1 X V 2 M.PCA usig vε-pca.als Acceleratio Z r Steps of variable selectio ad PCA.ALS Variable selectio: Select X V1 usig Backward elimiatio or Forward selectio. vε-pca.als: PCA.ALS: Quatify X V 1 ad compute A ad Z usig the ALS algorithm vε acceleratio: Geerate a accelerated sequece of X V 1 usig the vε algorithm Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 21/29

20 Numerical experimets: Variable selectio i M.PCA of qualitative data Computatio Algorithm: PRINCIPALS ad vε-principals Covergece criteria: δ = 10 8 The umber of pricipal compoets (PCs): r = 3 Compare The umber of iteratios ad CPU time Iteratio ad CPU time speed-ups Iteratio (CPU time) speed-ups = The umber of iteratios (CPU time) of PRINCIPALS The umber of iteratios (CPU time) of vε-principals [Simulatio 1]: Computatio of PCs The size of sample () 100 ** The umber of items (p) 40 items with 20 levels (Data 1) ** The umber of items (p) 20 items with 10 levels (Data 2) ++ Replicatio: 50 times [Simulatio 2]: Variable selectio i M.PCA The size of sample (): 100 The umber of items (p): 10 items with 5 levels (Data 3) Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 22/29

21 Numerical experimets: Data 1 & 2 Table 1: Summary of statistics ad speed-ups (a) Data 1 ( with 20 levels) PRINCIPALS vε-principals Speed-ups Iteratio CPU time Iteratio CPU time Iteratio CPU time Mi st Qu Media Mea rd Qu Max (b) Data 2 ( with 10 levels) PRINCIPALS vε-principals Speed-ups Iteratio CPU time Iteratio CPU time Iteratio CPU time Mi st Qu Media Mea rd Qu Max Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 23/29

22 Numerical experimets: Boxplots of iteratio ad CPU time speed-ups Data 1 ( with 20 levels) Data 2 ( with 10 levels) Iteratio CPU time Iteratio CPU time Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 24/29

23 Numerical experimets: Data 3 Table 2(a): The umbers of iteratios ad CPU times of PRINCIPALS ad vε-principals ad their speed-ups i applicatio to variable selectio for fidig a subset of q variables usig simulated data. (a) Backward elimiatio PRINCIPALS vε-principals Speed-up q Comb. Iteratio CPU time Iteratio CPU time Iteratio CPU time Total Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 25/29

24 Numerical experimets: Data 3 Table 2(b): The umbers of iteratios ad CPU times of PRINCIPALS ad vε-principals ad their speed-ups i applicatio to variable selectio for fidig a subset of q variables usig simulated data. (b) Forward selectio PRINCIPALS vε-principals Speed-up q Comb. Iteratio CPU time Iteratio CPU time Iteratio CPU time Total Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 26/29

25 Numerical experimets: Data 3 The umber of iteratios CPU time Backward elimiatio Forward selectio Backward elimiatio Forward selectio The umber of iteratios The umber of iteratios CPU time CPU time q q q q Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 27/29

26 Coclusio [Numerical experimets] The vε acceleratio works well to reduce the computatioal time of PRINCIPALS. [Future works] Meas of iteratio & CPU time speed-ups: Computatio of PCs Iteratio CPU time Data 1 ( with 20 levels) (36%) (38%) Data 2 ( with 10 levels) (31%) (34%) Iteratio & CPU time speed-ups: Variable selectio Data 3 Iteratio CPU time Backward elimiatio 3.68 (27%) 3.52 (28%) Forward selectio 5.50 (18%) 5.16 (19%) We apply vε-principals with a re-start procedure to variable selectio i PCA of qualitative data. Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 28/29

27 Refereces Refereces GIFI, A. (1989): Algorithm descriptios for ANACOR, HOMALS, PRINCALS, ad OVERALS. Report RR Leide: Departmet of Data Theory, Uiversity of Leide. KURODA, M. ad SAKAKIHARA, M. (2006): Acceleratig the covergece of the EM algorithm usig the vector ε algorithm. Computatioal Statistics ad Data Aalysis 51, Kuroda, M., Mori, Y., Iizuka, M. ad Sakakihara, M. (2011). Acceleratio of the alteratig least squares algorithm for pricipal compoets aalysis. Computatioal Statistics ad Data Aalysis, 55, MICHAILIDIS, G. ad DE LEEUW, J. (1998): The Gifi system of descriptive multivariate aalysis. Statistical Sciece 13, MORI, Y., TANAKA, Y. ad TARUMI, T. (1997): Pricipal compoet aalysis based o a subset of variables for qualitative data. I: C. Hayashi, K. Yajima, H. Bock, N. Ohsumi, Y. Taaka, Y. Baba (Eds.): Data Sciece, Classificatio, ad Related Methods (Proceedigs of IFCS-96). Spriger-Verlag, YOUNG, F.W., TAKANE, Y., ad DE LEEUW, J. (1978): Pricipal compoets of mixed measuremet level multivariate data: A alteratig least squares method with optimal scalig features. Psychometrika 43, WANG, M., KURODA, M., SAKAKIHARA, M. ad GENG, Z. (2008): Acceleratio of the EM algorithm usig the vector epsilo algorithm. Computatioal Statistics 23, WYNN, P. (1962): Acceleratio techiques for iterated vector ad matrix problems. Mathematics of Computatio 16, Variable selectio i pricipal compoets aalysis of qualitative data usig the accelerated ALS algorithm 29/29