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Showing papers on "Approximation algorithm published in 1969"


Proceedings ArticleDOI
01 Nov 1969
TL;DR: A new algorithm for identification from input-output measurements of a canonical form for discrete-time linear systems with disturbances having rational spectral densities is obtained by a formal application of the recursive least squares formula.
Abstract: A new algorithm for identification from input-output measurements of a canonical form for discrete-time linear systems with disturbances having rational spectral densities is obtained by a formal application of the recursive least squares formula. Although in this case the assumptions of the least squares method are violated, the algorithm is shown to converge in mean square using a stochastic approximation proof. The proposed algorithm is computationally more expensive than the corresponding stochastic approximation formula [1], but converges much faster and there are no problems with choice of the gain constant. The complexity of the algorithm still compares favourably with other methods [2], [3], owing to its on-line structure.

81 citations


Journal ArticleDOI
TL;DR: It is shown that the least-square-error fit of the measured output signals of the systems offers a recursive formula which is a special case of the proposed algorithm, and the rate of convergence is computed.
Abstract: A stochastic approximation procedure that minimizes a mean-square-error criterion is proposed in this paper It is applied first to derive an algorithm for recursive estimation of the mean-square-error approximation of the function which relates the input signals and the responses of a memoryless system The input signals are assumed to be generated at random with an unknown probability density function, and the response is measured with an error which has zero mean and finite variance A performance index for evaluating the rate of convergence of the algorithm is defined and then the optimal form of the algorithm is derived It is shown that the least-square-error fit of the measured output signals of the systems offers a recursive formula which is a special case of the proposed algorithm A recursive formula for estimation of a priori probabilities of the pattern classes using unclassified samples is then presented The rate of convergence is computed A minimum square-error estimate of a continuous probability density function is also obtained by the same algorithm

47 citations



Journal ArticleDOI
TL;DR: Dynamic stochastic approximation algorithms are proposed to estimate the unknown time-varying parameters in a recursive fashion and both supervised and nonsupervised learning schemes are discussed and their convergence properties are investigated.
Abstract: The problem of learning in nonstationary environment is formulated as that of estimating time-varying parameters of a probability distribution which characterizes the process under study. Dynamic stochastic approximation algorithms are proposed to estimate the unknown time-varying parameters in a recursive fashion. Both supervised and nonsupervised learning schemes are discussed and their convergence properties are investigated. An accelerated scheme for the possible improvement of the dynamic algorithm is given. Numerical examples and an application of the proposed algorithm to a problem in weather forecasting are presented.

20 citations


Journal ArticleDOI
TL;DR: The theorems on the convergence of approximation methods in linear and non-linear problems which are proved here are more general that those in the papers mentioned, and at the same time are convenient for applications.
Abstract: THE topics dealt with in the present paper are closely related to Kantorovich's general theory of approximation methods (see [l, 2], and also [3–6]). The theorems on the convergence of approximation methods in linear and non-linear problems which are proved here are more general that those in the papers mentioned, and at the same time are convenient for applications. The closeness of the exact and approximation problems is characterized in a new way, using the concept of the compact approximation of operators. The large number of theoretical topics discussed has compelled applications to be relegated to separate papers. The most important applications are to finite-difference methods for solving boundary value problems.

9 citations


Journal ArticleDOI
TL;DR: An algorithm for determining an approximate solution of a large class of discrete linear programming problems, and an upper bound on the profit loss due to the approximation is computed and a geometrical interpretation of the algorithm is given.
Abstract: An algorithm is presented for determining an approximate solution of a large class of discrete linear programming problems, and an upper bound on the profit loss due to the approximation is computed. A subregion of the original polyhedron of feasible solutions is also defined; such a subregion certainly contains the optimal solution of the discrete linear programming problem considered. A geometrical interpretation of the algorithm is given.

5 citations


Proceedings ArticleDOI
01 Nov 1969
TL;DR: A stochastic approximation algorithm is developed for estimating a mixture of normal density functions with unknown means and unknown variances and it minimizes an information criterion which has interesting properties for density approximations.
Abstract: A stochastic approximation algorithm is developed for estimating a mixture of normal density functions with unknown means and unknown variances. The algorithm minimizes an information criterion which has interesting properties for density approximations. The question of the completeness of normal density functions for the approximation of the class of continuous probability density functions is analyzed.

4 citations



Journal ArticleDOI
TL;DR: In this article, the authors consider the problem of approximating random variables generated by sums of random variables and show that in many cases the corresponding c.f. is either unknown or too complicated for practical use so even when the d.f is completely known, some sort of approximation is needed.
Abstract: In many cases, especially in connection with random variables, generated by sums of random variables the fr.f. or d.f. is either unknown or too complicated for practical use so even when the d.f, is completely known, some sort of approximation is needed. In such a situation very often the corresponding c.f. is rather simple. Very often the normal approximation or the asymptotic expansion derived from it gives accurate results. The properties of such approximations are well known, and exact upper limits for the errors in these approximation formulae are easily available.

1 citations