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Polynomial Representations of High-Dimensional Observations of Random Processes

Author

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  • Pavel Loskot

    (Zhejiang University/University of Illinois at Urbana-Champaign Institute, Haining 314400, China)

Abstract

The paper investigates the problem of performing a correlation analysis when the number of observations is large. In such a case, it is often necessary to combine random observations to achieve dimensionality reduction of the problem. A novel class of statistical measures is obtained by approximating the Taylor expansion of a general multivariate scalar symmetric function by a univariate polynomial in the variable given as a simple sum of the original random variables. The mean value of the polynomial is then a weighted sum of statistical central sum-moments with the weights being application dependent. Computing the sum-moments is computationally efficient and amenable to mathematical analysis, provided that the distribution of the sum of random variables can be obtained. Among several auxiliary results also obtained, the first order sum-moments corresponding to sample means are used to reduce the numerical complexity of linear regression by partitioning the data into disjoint subsets. Illustrative examples provided assume the first and the second order Markov processes.

Suggested Citation

  • Pavel Loskot, 2021. "Polynomial Representations of High-Dimensional Observations of Random Processes," Mathematics, MDPI, vol. 9(2), pages 1-24, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:123-:d:476416
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    References listed on IDEAS

    as
    1. Drezner, Zvi, 1995. "Multirelation -- a correlation among more than two variables," Computational Statistics & Data Analysis, Elsevier, vol. 19(3), pages 283-292, March.
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    Cited by:

    1. Pavel Loskot, 2021. "A Generative Model for Correlated Graph Signals," Mathematics, MDPI, vol. 9(23), pages 1-12, November.

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