**7. Acknowledgment**

12 Principal Component Analysis

that the iteration speed-ups are 3.18 in Backward elimination and 3.99 in Forward selection

speed-ups are 2.18 in Backward elimination and 2.35 in Forward selection, and are not as large as the iteration speed-ups. The computation time per iteration of v*ε*-PRINCIPALS is greater than that of PRINCIPALS due to computation of the *Acceleration step*. Therefore, for the smaller number of iterations, the CPU time of v*ε*-PRINCIPALS is almost same as or may be longer than that of PRINCIPALS. For example, in Forward selection for *q* = 2, PRINCIPALS converges in almost cases after less than 15 iterations and then the CPU time speed-up is 1.48. 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. [Iizuka et al., 2003] selected the subset of 6 variables found by either procedures as a best subset, since *P* slightly changes

In this paper, we presented v*ε*-PRINCIPALS that accelerates the convergence of PRINCIPALS by using the v*<sup>ε</sup>* algorithm. The algorithm generates the v*<sup>ε</sup>* accelerated sequence {**X**˙ <sup>∗</sup>(*t*)} using {**X**∗(*t*)}*t*≥<sup>0</sup> but it does not modify the estimation equations in PRINCIPALS. Therefore the algorithm enables an acceleration of the convergence of PRINCIPALS, while still preserving the stable convergence property of PRINCIPALS. The v*ε* algorithm in itself is a fairly simple computational procedure and, at each iteration, it requires only *O*(*np*) arithmetic operations. For each iteration, the computational complexity of the v*ε* algorithm may be less expensive than that for computing a matrix inversion and for solving the eigenvalue problem in

The most appealing points of the v*ε* algorithm are that, if an original sequence converges to a limit point then the accelerated sequence converges to the same limit point as the original sequence and its speed of convergence is faster than the original sequence. In all the numerical

The numerical experiments employing simulated data in Section 4 demonstrated that v*ε* acceleration for PRINCIPALS significantly speeds up the convergence of {**X**∗(*t*)}*t*≥<sup>0</sup> in terms of the number of iterations and the computation time. In particular, the v*ε* acceleration effectively works to speed up the convergence for the larger number of iterations of PRINCIPALS. Furthermore, we evaluate the performance of the v*ε* acceleration for PRINCIPALS by applying to variable selection in M.PCA of qualitative data. Numerical experiments using simulated and real data showed that v*ε*-PRINCIPALS improves the speed of convergence of ordinary PRINCIPALS and enables greatly the reduction of computation times in the variable selection for finding a suitable variable set using Backward elimination and Forward selection procedures. The results indicate that the v*ε* acceleration well works in saving the

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 C. We will provide VASpca using v*ε*-PRINCIPALS as the iterative

}*t*≥<sup>0</sup> converges to the final value of {**X**∗(*t*)

}*t*≥0

}*t*≥0. The CPU time

and thus v*ε*-PRINCIPALS well accelerates the convergence of {**X**∗(*t*)

until *q* = 6 in Backward elimination and after *q* = 6 in Forward selection.

after the significantly fewer number of iterations than that of PRINCIPALS.

**6. Concluding remarks**

experiments, the v*<sup>ε</sup>* accelerated sequence {**X**˙ <sup>∗</sup>(*t*)

computational time in variable selection problems.

algorithm for PCA and M.PCA of qualitative data.

PRINCIPALS.

The authors would like to thank the editor and two referees whose valuable comments and kind suggestions that led to an improvement of this paper. This research is supported by the Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C), No 20500263.
