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.
Highlights
The interest in developing and improving statistical methods and models is driven by the ever increasing volumes and variety of data
The following concepts are briefly summarized: stationarity of random processes, estimation of general and central moments and of correlation and covariance using the method of moments, definition of cosine similarity and Minkowski distance, parameter estimation via linear regression, generation of the 1st and the 2nd order Markov processes, and selected properties of polynomials and multivariate functions are given
The main objective of this section is to define a universal function to effectively measure the statistics of random vectors and random processes observed at multiple discrete time instances
Summary
The interest in developing and improving statistical methods and models is driven by the ever increasing volumes and variety of data. An univariate correlation measure for multiple random variables is defined in [3] to be a sum of elements on the main diagonal of the covariance matrix. The mean value of each polynomial element corresponds to a general or central moment of the sum of random variables. 4 including defining a class of polynomial statistical measures and sum-moments for multiple random variables and random processes.
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