Polynomial Representations of High-Dimensional Observations of Random Processes
Abstract
:1. Introduction
2. Background
2.1. Random Processes
2.2. Estimation Methods
2.3. Generating Random Processes
2.4. Polynomials and Multivariate Functions
3. Background Extensions
3.1. Linear LS Estimation
3.2. Generating Pairwise-Correlated Gaussian Processes
4. Polynomial Statistics and Sum-Moments for Vectors of Random Variables
4.1. Related Concepts
4.2. Multiple Random Processes
5. Illustrative Examples
5.1. Linear Regression
5.2. Comparison of Central Moments
5.3. Signal Processing Problems for the 1st Order Markov Process
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
1MP | 1st order Markov process |
2MP | 2nd order Markov process |
AR | autoregressive |
LMMSE | linear minimum mean square error |
LS | least squares |
MMSE | minimum mean square error |
MSE | mean square error |
MA | moving average |
TV | total variance |
absolute value, set cardinality, sum of vector elements | |
expectation | |
correlation | |
covariance | |
variance | |
matrix inverse | |
matrix/vector transpose | |
distribution of a random variable X | |
f, , | function f, and its first and second derivatives |
positive non-zero integers | |
, | real numbers, positive real numbers |
mean value of random variable X | |
j-th sample of process i | |
Lambert function |
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Loskot, P. Polynomial Representations of High-Dimensional Observations of Random Processes. Mathematics 2021, 9, 123. https://doi.org/10.3390/math9020123
Loskot P. Polynomial Representations of High-Dimensional Observations of Random Processes. Mathematics. 2021; 9(2):123. https://doi.org/10.3390/math9020123
Chicago/Turabian StyleLoskot, Pavel. 2021. "Polynomial Representations of High-Dimensional Observations of Random Processes" Mathematics 9, no. 2: 123. https://doi.org/10.3390/math9020123