Abstract

1
Introduction
In the analysis of data with large numbers of variables, a common objective is to reduce the dimension- ality of the data set. Principal components analysis (PCA) is a popular dimension-reducing tool that replaces the variables in the data set by a smaller number of derived variables. However, for example, in PCA of a data set with a large number of variables, the result may not be easy to interpret. One way to give a simple interpretation of principal components is to select a subset of variables that best approximates all the variables. For other several situations, we meet variable selection problems in the context of PCA. Various variable selection criteria in PCA has been proposed by Jolliffe (1972, 1973), McCabe (1984), Robert and Escoufier (1976) and Krzanowski (1987). Al-Kandari et al. (2001, 2005) gave guidelines as to the types of data for which each variable selection criteria is useful. Cadima et al. (2004) reported computational experiments carried out with several heuristic algorithms for the optimization problems resulting from the variable selection criteria in PCA found in the above literature. Tanaka and Mori (1997) proposed modified PCA (M.PCA) for deriving principal components which are computed by using only a selected subset of variables but which represent all the variables including those not selected. Since M.PCA includes variable selection procedures in the analysis, its criteria can be used directly to find a reasonable subset of variables. Mori et al. (1997) extended M.PCA to qualitative data and provided variable selection procedures, in which the alternating least squares (ASL) algorithm of Young et al. (1978) is utilized. Then the ALS algorithm iterates between optimal scaling for quantifying qualitative data and estimating parameters in M.PCA. In this situation, the computation time of the ALS algorithm has been a big issue so far, because the total number of iterations of the algorithm is much larger for quantification using the ALS algorithm, and it takes a long computational time until its convergence for searching a reasonable subset even though a cost- saving selection procedure such as Backward elimination or Forward selection is employed. For such situations, to accelerate the convergence of the ALS algorithm in PCA of qualitative data, Kuroda et al. (2011) derived a new iterative algorithm using the vector ε (νε) algorithm by Wynn (1962). In this paper, we investigate how much the proposed ve acceleration algorithm improves the computational efficiency for the variable selection problem in M.PCA of qualitative data. Section 2 introduces the formulation of M.PCA, and Backward elimination and Forward selection procedures used in variable selection. In Section 3, we describe the ALS algorithm for quantifying qualitative data in PCA and, in Section 4, present the νε accelerated ALS algorithm for speeding up the convergence. Numerical experiments in Section 5 illustrate the performance of the vs accelerated ALS algorithm. In Section 6, we present our concluding remarks.
2
Modified PCA for variable selection
2.1
Formulation of modified PCA
Let X = (X 1 X 2 · · · X n ) be an n × p matrix of n observations on p variables and be columnwise standardized. In PCA, we postulate that X is approximated by the following bilinear form: (1) Z = XA, where Z = ( Z 1 Z 2 ･ ･ ･ Z r ) is an n × r matrix of n component scores on r (1 ≤ r ≤ p ) components, and A = ( A 1 A 2 · · · A r ) is a p × r matrix consisting of the eigenvectors of X T X /n and A T A = I r . M.PCA derives principal components which are computed as linear combinations of a subset of variables but which can reproduce all the variables very well. Let X be decomposed into an n × q submatrix X V 1 and an n × ( p − q ) remaining submatrix X V 2 . Then M.PCA finds r linear combinations Z = X V i A. The matrix A consists of the eigenvectors associated with the largest r eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ r and is obtained by solving the eigenvalue problem: [(S i 1 + S 12S 21 ) − DS 11 ] A = 0, S11 S12 where S = is the covariance matrix of X = ( X v 1 , X v 2 ) and D is a q × q diagonal matrix S21 S 22 of eigenvalues. A best subset of q variables has the largest value of the proportion P = C j =1 λ j /tr(S) or the RV-coefficient RV = { ∑ j =1 λ j /tr( S 2 ) } 1/2 . Here we use P as variable selection criteria.
2.2
Variable selection procedures
In order to find a subset of q variables, we employ Backward elimination and Forward selection of Mori et al. (1998, 2006) as cost-saving stepwise selection procedures in which only one variable is removed or added sequentially.
[Backward elimination]
Stage A: Initial fixed-variables stage A-1 Assign q variables to subset XV1, usually q := p. A-2 Solve the eigenvalue problem (2). A-3 Look carefully at the eigenvalues, determine the number r of principal components to be used. A-4 Specify kernel variables which should be involved in X V1 , if necessary. The number of kernel variables is less than q . Stage B:  Variable selection stage (Backward) B-1 Remove one variable from among q variables in XV1, make a temporary subset of size q − 1, and compute P based on the subset. Repeat this for each variable in XVi, then obtain q values on P . Find the best subset of size q − 1 which provides the largest P among these q values and remove the corresponding variable from the present X V 1 . Put q := q − 1. B-2 If P or q is larger than preassigned values, go to B-1. Otherwise stop.
[Forward selection]
Stage A:  Initial fixed-variables stage A-1 ~ A-3 Same as A-1 to A-3 in Backward elimination. A-4 Redefine q as the number of kernel variables (here, q ≥ r ). If you have kernel variables, assign them to X V 1 . If not, put q := r , find the best subset of q variables which provides the largest P among all possible subsets of size q and assign it to XV1. Stage B:  Variable selection stage (Forward) Basically the opposites of Stage B in Backward elimination. In Backward elimination, to find the best subset of q − 1 variables, we perform M.PCA for each of q possible subsets of the q − 1 variables among q variables selected in the previous selection step. The total number of estimations for M.PCA from q = p − 1 to q = r is therefore large, i.e., p + (p − 1) +۰۰+ (r + 1) = (p − r)(p + r + 1)/2. In Forward selection, the total number of estimations for M.PCA from q = r to q = p − 1 is pCr + (p − r) + (p − (r+1))+ · · · +2 = pCr + (p − r − 1)(p − r+2)/2.
3
Alternating least squares algorithm for M.PCA of qualitative data
When observed data are qualitative, ordinary PCA can not be directly applied to such data. In such situations, optimal scaling is used to quantify the observed qualitative data and then ordinary PCA can be applied. To quantify X j of qualitative variable j with K j categories, the vector is coded by using an n × K j indicator matrix G j with entries g(j) ik = 1 if object i belongs to category k, andייגו(j) = 0 if object i belongs to some other category k ′ ( ≠ k ), i = 1, . . . , n and k = 1, . . . , K j . Then the optimally scaled vector X ξ of X j is given by X ξ = G jα j , where α j is a K j × 1 score vector for categories of X j . Let X* = (X i X z  . . . X z ) be an n × p matrix of optimally scaled observations to satisfy restrictions
where 1 n and O p are vectors of ones and zeros of length n and p respectively. To apply PCA to qualitative data, we determine the optimal scaling parameter X*, in addition to estimating Z and A. A possible computational algorithm for estimating simultaneously Z , A and X* is the ALS algorithm. The existing ALS algorithms for PCA of qualitative data are PRINCIPALS of Young et al. (1978) and PRINCALS of Giff (1989).  Mori et al. (1996) used PRINCIPALS in applying M.PCA to qualitative data. PRINCIPALS alternates between ordinary PCA and optimal scaling, and minimizes (4)
under the restriction (3). Then θ is to be determined by model parameters Z and A and optimal scaling parameter X*, by updating each of the parameters in turn, keeping the others fixed. For the initialization of PRINCIPALS, we determine initial data X*(0) . The observed data X may be used as X*( 0 ) after it is standardized to satisfy the restriction (3). For given initial data X*( 0 ) with the restriction (3), PRINCIPALS iterates the following two steps: ● Model parameter estimation step : Obtain A(t) by solving
where A T A = I r and D r is an r × r diagonal matrix of eigenvalues, and the superscript ( t ) indicates the t-th iteration. Compute Z(t) from Z(t) = X*(t)A(t) ● Optimal scaling step : Calculate Î ( t +1) = Z ( t ) A ( t ) T from Equation (1). Find X *( t +1) such that X*( t+1) = arg min tr(X* − X(t+1) T (X* − X(t+1) for fixed Å ( t +1) under measurement restrictions on each of the variables. Scale X*( t +1) by columnwise centering and normalizing.
4
The vector ε acceleration of the ALS algorithm
We briefly introduce the νε algorithm of Wynn (1962). The νε algorithm is utilized to speed up the convergence of a slowly convergent vector sequence and is very effective for linearly converging sequences. Kuroda et al. (2011) provided the νε accelerated ALS algorithm for PCA of mixed mea- surement level data. The acceleration algorithm speeds up the convergence of the ALS sequence via the νε algorithm and demonstrated that its speed of convergence is significantly faster than that of the ALS algorithm. Let {Y(t)}t>0 = {Y(0), Y(1), Y(2), . . .} be a linear convergent sequence generated by an iterative computational procedure and let {Y(t)} t ≥ 0 = {Y(0), Y(1), Y(2), . . .} be the accelerated sequence of {Y(f)}t>0. Then the v ε algorithm generates {Y(f)}t>0 by using
where [Y]−1 = Y /||Y||² and ||Y|| is the Euclidean norm of Y.  For the detailed derivation of Equation (6), see Kuroda et al. (2011). When {Y(t)}t>0 converges to a limit point Y( ∞ ) of {Y(t)}t>0, it is known that, in most cases, { Y ( f )} t≥ 0 generated by the νε algorithm converges to Y ( ∞ ) faster than {Y(t)}t>0. We assume that { X *( t )} t >0 generated by PRINCIPALS converges to a limit point X*( ∞ ) . Then the νε accelerated PRINCIPALS (vε-PRINCIPALS) produces a faster convergent sequence { X * ( t ) t >o of {X*( t )} t>0 and enables the acceleration of convergence of PRINCIPALS. The general procedure of vε-PRINCIPALS iterates the following two steps: ● PRINCIPALS step : Compute model parameters A ( t ) and Z ( t ) and determine optimal scaling parameter X*(t+1) . ● Acceleration step : Calculate X*(t-1) using {X*(t-1), X*(t), X*(t+1)} from the vs algorithm:
where vecX* = (Xi T Xi T ... Xi T ) T , and check the convergence by
where δ is a desired accuracy. Before starting the iteration, we determine initial data X*( 0 ) satisfying the restriction (3) and execute the PRINCIPALS step twice to generate {X*(0), X*(1), X*(2)}. v ε -PRINCIPALS is designed to generate { X*( t )} t>0 converging to X*( ∞ ). Thus the estimate of X* can be obtained from the final value of {X*( ) } t ≥ 0 when v ε -PRINCIPALS terminates. The estimates of Z and A can then be calculated immediately from the estimate of X* in the Model parameter estimation step of PRINCIPALS. Note that X*(t-1) obtained at the t-th iteration of the Acceleration step is not used as the estimate X*( t+1 ) at the ( t + 1)-th iteration of the PRINCIPALS step. Thus v ε -PRINCIPALS speeds up the convergence of { X*( t )} t≥0 without affecting the convergence properties of ordinary PRINCIPALS.
5
Numerical experiments
In this section, we study how much faster v ε -PRINCIPALS converges than ordinary PRINCIPALS when applying to variable selection in M.PCA of qualitative data.  Backward elimination and Forward selection procedures are used in the problem. All computations are performed with the statistical package R. CPU times taken are measured by the function proc.time 1 . For all experiments, δ for convergence of ε -PRINCIPALS is set to 10~ 8 and PRINCIPALS terminates when | θ(t+1) - θ(t)| < 10~8, where θ(t) is the t-th update of θ calculated from Equation (4).
Simulated data
We apply PRINCIPALS and νε-PRINCIPALS to variable selection in M.PCA of qualitative data using simulated data that consist of 100 observations on 10 variables with 5 levels. Table 1 shows the number of iterations and CPU time taken by two algorithms for finding a subset of q variables based on 3 (= r ) principal components. We compute the iteration and CPU time speed-ups for comparing the speed of convergence of PRINCIPALS with that of vs-PRINCIPALS. The iteration speed-up is defined as the number of iterations required for PRINCIPALS divided by the number of iterations required for v ε -PRINCIPALS. The CPU time speed-up is calculated similarly to the iteration speed-up. The values of the second to fifth columns in the table indicate that the number of iterations of PRINCIPALS is very large and a long time is taken for convergence, while vc-PRINCIPALS converges considerably faster than PRINCIPALS. We can see from the sixth and seventh columns in the table that vε-PRINCIPALS requires the number of iterations 3 to 5 times smaller and CPU time 2 to 5 times shorter than vε -PRINCIPALS. In particular, vε -PRINCIPALS speeds up well the convergence for the larger numbers of iterations. The last row in the table shows the total number of iterations and total CPU time for selecting 8 subsets for q = 3, . . . , 10. When searching the best subset for each q , PRINCIPALS requires 64491 iterations in Backward elimination and 178249 iterations in Forward selection, while v ε -PRINCIPALS finds the subsets after 17530 and 32405 iterations, respectively. These values show that the compu- tational times by νε-PRINCIPALS are reduced to only 28%(= 1/3.52) and 19% = (1/5.16) of those of ordinary PRINCIPALS. The iteration and CPU time speed-ups given in the sixth and seventh columns of the table demonstrate that the νε acceleration algorithm works well to accelerate the convergence of {X*( t )} t>0 and consequently results in greatly reduced computation times in variable selection problems.
5.2
Real data
We consider the variable selection problems in M.PCA of qualitative data to mild distribution of consciousness (MDOC) data from Sano et al. (1977).  The data consist of 87 individuals and 23 categorical variables with 4 levels. In the variable selection problem, we select a suitable subset based on 2 (= r) principal components. Table 2 summarizes the results of variable selection using Backward elimination and Forward selection procedures for finding a subset of q variables. The proportion P in the eighth column of the table indicates the variation explained by the first 2 principal components for the selected q variables. Iizuka et al. (2003) selected the subset of 6 variables found by either procedures as a best subset, since P slightly changes until q = 6 in Backward elimination and after q = 6 in Forward selection. The values of the second to fifth columns in the table show that ε-PRINCIPALS finds a subset of q variables by taking the smaller numbers of iterations and shorter CPU times than PRINCIPALS. We see from the last row in the table that the computational time of vε-PRINCIPALS is less than one-half of that of PRINCIPALS when finding 22 subsets using two selection procedures.
6
Concluding remarks
In this paper, we examine the performance of the νε accelerated ALS algorithm by applying to variable selection problems in M.PCA of qualitative data. The νε algorithm in itself is a fairly simple computational procedure and speeds up the convergence of the ALS algorithm without losing its stable convergence property. Numerical experiments employing simulated and real data demonstrated that the νε accelerated ALS algorithm improves the speed of convergence of the ordinary ALS algorithm and enables greatly
5.1
the reduction of computation times in the variable selection problems for finding a suitable variable set using Backward elimination and Forward selection. The results indicate that the vs acceleration works effectively in saving the computational time in variable selection problems. The computations of variable selection in M.PCA of qualitative data are partially performed by the statistical package VASpca(VAriable Selection in principal component analysis) that was developed by Mori, Iizuka, Tarumi and Tanaka in 1999 and can be obtained from Mori's website in Appendix. We will provide VASpca using vε -PRINCIPALS as the iterative algorithm for PCA and M.PCA of qualitative data.
Acknowledgment
This research is supported by the Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C), No 20500263 and No 22500265.
Appendix
URL of VASpca http://mo161.soci.ous.ac.jp/vaspca/indexE.html