The Wold decomposition of Hilbertian periodically correlated processes

2021 ◽  
Vol 101 ◽  
pp. 119-127
Author(s):  
A. Zamani ◽  
Z. Sajjadnia ◽  
M. Hashemi
Keyword(s):  
Wold Decomposition ◽  
Periodically Correlated Processes

On the spectral coherence between two periodically correlated processes

2021 ◽  
Author(s):  
Mahnaz Khalafi ◽  
Ahmad Reza Soltani ◽  
Masoud Golalipour ◽  
Farzad Najafiamiri
Keyword(s):  
Spectral Coherence ◽  
Periodically Correlated Processes

Time Series Models

2021 ◽  
pp. 255-270
Author(s):  
James Davidson
Keyword(s):  
Time Series ◽  
Random Walk ◽  
Time Series Analysis ◽  
Poisson Process ◽  
Final Section ◽  
Reflection Principle ◽  
Time Series Models ◽  
Wold Decomposition ◽  
Linear Processes ◽  
Series Analysis

This chapter reviews some important ideas from time series analysis. The concepts of stationarity, independence, and exchangeability are defined and illustrated with examples. The Poisson process is examined in detail and then the class of linear processes, noting the implications of the Wold decomposition. The final section studies the random walk and the reflection principle.

scholarly journals Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 717 ◽  
Author(s):  
Maël Dugast ◽  
Guillaume Bouleux ◽  
Eric Marcon
Keyword(s):  
Lie Group ◽  
Correlation Coefficients ◽  
Shape Space ◽  
Rotation Matrices ◽  
Dilation Matrix ◽  
Naimark Dilation ◽  
Periodically Correlated Processes ◽  
New Vision

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes; this enables a comparison between curves’ shapes and allows then the classification of random processes’ measures.

Sign in / Sign up

Export Citation Format

Share Document