Principal component analysis (PCA) is a widely used descriptive multivariate technique in the analysis of quantitative data. When applying PCA to mixed quantitative and qualitative data, we utilize an optimal scaling ...
详细信息
Principal component analysis (PCA) is a widely used descriptive multivariate technique in the analysis of quantitative data. When applying PCA to mixed quantitative and qualitative data, we utilize an optimal scaling technique for quantifying qualitative data. PCA with optimal scaling is called nonlinear PCA. The alternating least squares (ALS) algorithm is used for computing nonlinear PCA. The ALS algorithm is stable in convergence and simple in implementation;however, the algorithm tends to converge slowly for large data matrices owing to its linear convergence. Then the v epsilon-ALS algorithm, which incorporates the vector s accelerator into the ALS algorithm, is used to accelerate the convergence of the ALS algorithm for nonlinear PCA. In this paper, we improve the v epsilon-ALS algorithm via a restarting procedure and further reduce its number of iterations and computation time. The restarting procedure is performed under simple restarting conditions, and it speeds up the convergence of the v epsilon-ALS algorithm. The v epsilon-ALS algorithm with a restarting procedure is referred to as the v epsilon R-ALS algorithm. Numerical experiments examine the performance of the v epsilon R-ALS algorithm by comparing its number of iterations and computation time with those of the ALS and v epsilon-ALS algorithms.
Principal components analysis (PCA) is a popular descriptive multivariate method for handling quantitative data and it can be extended to deal with qualitative data and mixed measurement level data. The existing algor...
详细信息
Principal components analysis (PCA) is a popular descriptive multivariate method for handling quantitative data and it can be extended to deal with qualitative data and mixed measurement level data. The existing algorithms for extended PCA are PRINCIPALS of Young et al.(1978) and PRINCALS of Gifi (1989) in which the alternating least squares algorithm is utilized. These algorithms based on the least squares estimation may require many iterations in their application to very large data sets and variable selection problems and may take a long time to converge. In this paper, we derive a new iterative algorithm for accelerating the convergence of PRINCIPALS and PRINCALS by using the vector epsilon algorithm of Wynn (1962). The proposed acceleration algorithm speeds up the convergence of the sequence of the parameter estimates obtained from PRINCIPALS or PRINCALS. Numerical experiments illustrate the potential of the proposed acceleration algorithm. (C) 2010 Elsevier B.V. All rights reserved.
The EM algorithm of Dempster, Laird and Rubin [1977. Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39, 1-22] is a very general and popular iterative computational algorithm...
详细信息
The EM algorithm of Dempster, Laird and Rubin [1977. Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39, 1-22] is a very general and popular iterative computational algorithm that is used to find maximum likelihood estimates from incomplete data and is widely used to perform statistical analysis with missing data, because of its stability, flexibility and simplicity. However, a common criticism is that the convergence of the EM algorithm is slow. Various algorithms to accelerate the convergence of the EM algorithm have been proposed. In this paper, we propose the "epsilon-accelerated EM algorithm" that speeds up the convergence of the EM sequence via the vector epsilon algorithm of Wynn [1962. Acceleration techniques for iterated vector and matrix problems. Math. Comp. 16, 304-322]. We also demonstrate its theoretical properties. The epsilon-accelerated EM algorithm has been successfully extended to the EM algorithm without affecting its stability, flexibility and simplicity. Numerical experiments illustrate the potential of the epsilon-accelerated EM algorithm. (c) 2006 Elsevier B.V. All rights reserved.
We present an alternative to the hector epsilon-algorithm based on hector continued fractions and which is applicable when the sequence to be accelerated is generated by a one-point iteration function. These fractions...
详细信息
We present an alternative to the hector epsilon-algorithm based on hector continued fractions and which is applicable when the sequence to be accelerated is generated by a one-point iteration function. These fractions are constructed in the language of Clifford algebras. which allow three-term recurrence relations. The new algorithm evidently has considerably greater numerical precision than the old one. Results from numerical experiments are reported.
Some open problems in vector Pade approximation are stated, and some recent de Montessus-type theorems governing convergence of rows of the vector Pade table are contrasted. We show how these results indicate when it ...
详细信息
Some open problems in vector Pade approximation are stated, and some recent de Montessus-type theorems governing convergence of rows of the vector Pade table are contrasted. We show how these results indicate when it is more appropriate to use generalised inverse vector Pade approximants or their hybridised form. We show how a straightforward analogue of the Berlekamp-Massey algorithm may be used to calculate generalised inverse vector Pade approximants. This algorithm is applied to the derivation of a new low-order hybrid vector approximant. Related results include the case of a row convergence theorem for a complex-valued power series using a Clifford inverse instead of the Moore-Penrose inverse.
The vector epsilon algorithm (VEA) has many advantages as a method for accelerating the convergence of a sequence of vectors. A vector Pade approximant P(z)/Q(z) of type [n/2k] can be associated with each entry of the...
详细信息
The vector epsilon algorithm (VEA) has many advantages as a method for accelerating the convergence of a sequence of vectors. A vector Pade approximant P(z)/Q(z) of type [n/2k] can be associated with each entry of the vectorepsilon table. In the scalar case, it reduces to the Pade approximant p(z)/q(z) of type [n - k/k]. It is thought that the disadvantages of VEA are (indirectly) attributable to the positivity property of Q(x), x epsilon R, recalling that in the scalar case, Q(z) proportional to q(z)(2). In this paper, a specification of a polynomial sigma(z) of degree k is given, such that sigma(z)(2) approximate to Q(z). The coefficients of sigma(z) specify an accelerator for a sequence of vectors which should avoid many of the numerical difficulties of VEA.
This is an expository paper that describes and compares five methods for extrapolating to the limit (or anti-limit) of a vector sequence without explicit knowledge of the sequence generator. The methods are the minima...
详细信息
This is an expository paper that describes and compares five methods for extrapolating to the limit (or anti-limit) of a vector sequence without explicit knowledge of the sequence generator. The methods are the minimal polynomial extrapolation (MPE) studied by Cabay and Jackson, Mešina, and Skelboe; the reduced rank extrapolation (RRE) of Eddy (which we show to be equivalent to Mešina’s version of MPE); the vector and scalar versions of the epsilonalgorithm (VEA, SEA) introduced by Wynn and extended by Brezinski and Gekeler; and the topological epsilonalgorithm (TEA) of Brezinski. We cover the derivation and error analysis of iterated versions of the algorithms, as applied to both linear and nonlinear problems, and we show why these versions tend to converge quadratically. We also present samples from extensive numerical testing that has led us to the following conclusions: (a) TEA, in spite of its role as a theoretical link between the polynomial-type and the epsilon-type methods, has no practical application; (b) MPE is at least as good as RRE, and VEA at least as good as SEA, in almost all situations (c) there are circumstances in which either MPE or VEA is superior to the other.
暂无评论