A C-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains

Wen Li; Rihuan Ke; Wai Ki Ching; Michael K. Ng

doi:10.1016/j.camwa.2019.03.016

A C-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains

Wen Li, Rihuan Ke, Wai Ki Ching, Michael K. Ng^*

^*Corresponding author for this work

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

7 Citations (Scopus)

Abstract

In this paper, we study a new tensor eigenvalue problem, which involves E- and S-eigenvalues as its special cases. Some theoretical results such as existence of an eigenvalue and the number of eigenvalues are given. For an application of the proposed eigenvalue problem, we establish a tensor model for a higher-order multivariate Markov chain. The core issue of this problem is to study a stationary probability distribution of a higher-order multivariate Markov chain. A sufficient condition of the unique stationary positive distribution is given. An algorithm for computing stationary probability distribution is also developed. Numerical examples of applications in stock market modeling, sales demand prediction and biological sequence analysis are given to illustrate the proposed tensor model and the computed stationary probability distribution can provide a better prediction in these Markov chain applications.

Original language	English
Pages (from-to)	1008-1025
Number of pages	18
Journal	Computers and Mathematics with Applications
Volume	78
Issue number	3
DOIs	https://doi.org/10.1016/j.camwa.2019.03.016
Publication status	Published - 1 Aug 2019

Bibliographical note

Funding Information:
This author is supported by National Natural Science Foundation of China (Grant Nos. 11671158, 11771159 and U181164), and Major Project (Grant No. 2016KZDXM025) and Innovation Team Project (Grant No. 2015KCXTD007) of Guangdong Provincial Universities.Research supported in part by HKRGC GRF15210815, National Natural Science Foundation of China (Grant No. 11671158), and IMR and RAE research fund, Faculty of Science, The University of Hong Kong.Research supported in part by HKRGC GRF12302715, 12306616, 12200317 and 12300218.

Publisher Copyright:
© 2019 Elsevier Ltd

Keywords

Eigenpair
Higher-order
Markov chain
Multivariate
Stationary probability distribution
Tensor

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title = "A C-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains",

abstract = "In this paper, we study a new tensor eigenvalue problem, which involves E- and S-eigenvalues as its special cases. Some theoretical results such as existence of an eigenvalue and the number of eigenvalues are given. For an application of the proposed eigenvalue problem, we establish a tensor model for a higher-order multivariate Markov chain. The core issue of this problem is to study a stationary probability distribution of a higher-order multivariate Markov chain. A sufficient condition of the unique stationary positive distribution is given. An algorithm for computing stationary probability distribution is also developed. Numerical examples of applications in stock market modeling, sales demand prediction and biological sequence analysis are given to illustrate the proposed tensor model and the computed stationary probability distribution can provide a better prediction in these Markov chain applications.",

keywords = "Eigenpair, Higher-order, Markov chain, Multivariate, Stationary probability distribution, Tensor",

author = "Wen Li and Rihuan Ke and Ching, \{Wai Ki\} and Ng, \{Michael K.\}",

note = "Funding Information: This author is supported by National Natural Science Foundation of China (Grant Nos. 11671158, 11771159 and U181164), and Major Project (Grant No. 2016KZDXM025) and Innovation Team Project (Grant No. 2015KCXTD007) of Guangdong Provincial Universities.Research supported in part by HKRGC GRF15210815, National Natural Science Foundation of China (Grant No. 11671158), and IMR and RAE research fund, Faculty of Science, The University of Hong Kong.Research supported in part by HKRGC GRF12302715, 12306616, 12200317 and 12300218. Publisher Copyright: {\textcopyright} 2019 Elsevier Ltd",

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doi = "10.1016/j.camwa.2019.03.016",

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T1 - A C-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains

AU - Li, Wen

AU - Ke, Rihuan

AU - Ching, Wai Ki

AU - Ng, Michael K.

N1 - Funding Information: This author is supported by National Natural Science Foundation of China (Grant Nos. 11671158, 11771159 and U181164), and Major Project (Grant No. 2016KZDXM025) and Innovation Team Project (Grant No. 2015KCXTD007) of Guangdong Provincial Universities.Research supported in part by HKRGC GRF15210815, National Natural Science Foundation of China (Grant No. 11671158), and IMR and RAE research fund, Faculty of Science, The University of Hong Kong.Research supported in part by HKRGC GRF12302715, 12306616, 12200317 and 12300218. Publisher Copyright: © 2019 Elsevier Ltd

PY - 2019/8/1

Y1 - 2019/8/1

N2 - In this paper, we study a new tensor eigenvalue problem, which involves E- and S-eigenvalues as its special cases. Some theoretical results such as existence of an eigenvalue and the number of eigenvalues are given. For an application of the proposed eigenvalue problem, we establish a tensor model for a higher-order multivariate Markov chain. The core issue of this problem is to study a stationary probability distribution of a higher-order multivariate Markov chain. A sufficient condition of the unique stationary positive distribution is given. An algorithm for computing stationary probability distribution is also developed. Numerical examples of applications in stock market modeling, sales demand prediction and biological sequence analysis are given to illustrate the proposed tensor model and the computed stationary probability distribution can provide a better prediction in these Markov chain applications.

AB - In this paper, we study a new tensor eigenvalue problem, which involves E- and S-eigenvalues as its special cases. Some theoretical results such as existence of an eigenvalue and the number of eigenvalues are given. For an application of the proposed eigenvalue problem, we establish a tensor model for a higher-order multivariate Markov chain. The core issue of this problem is to study a stationary probability distribution of a higher-order multivariate Markov chain. A sufficient condition of the unique stationary positive distribution is given. An algorithm for computing stationary probability distribution is also developed. Numerical examples of applications in stock market modeling, sales demand prediction and biological sequence analysis are given to illustrate the proposed tensor model and the computed stationary probability distribution can provide a better prediction in these Markov chain applications.

KW - Eigenpair

KW - Higher-order

KW - Markov chain

KW - Multivariate

KW - Stationary probability distribution

KW - Tensor

UR - https://www.scopus.com/pages/publications/85063494932

U2 - 10.1016/j.camwa.2019.03.016

DO - 10.1016/j.camwa.2019.03.016

M3 - Article (Academic Journal)

AN - SCOPUS:85063494932

SN - 0898-1221

VL - 78

SP - 1008

EP - 1025

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 3

ER -

A C-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains

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