Partially Coupled Stochastic Gradient Estimation for Multivariate Equation-Error Systems
Abstract
:1. Introduction
- This paper decomposes the multivariate equation-error autoregressive system into several sub-identification models according to the number of the system outputs.
- A multivariate partially coupled generalized stochastic gradient (M-PC-GSG) algorithm is proposed for the multivariate equation-error system by utilizing the coupling identification concept, which can reduce the computation amounts compared with the traditional stochastic gradient algorithm.
- A multivariate partially coupled multi-innovation generalized stochastic gradient (M-PC-MI-GSG) algorithm is proposed by using the multi-innovation identification theory, which has higher parameter estimation accuracy than the M-PC-GSG algorithm.
2. System Description and Identification Model
3. The Partially Coupled Stochastic Gradient Algorithm
4. The Partially Coupled Multi-Innovation Stochastic Gradient Algorithm
- Let , choose an innovation length p, set the initial values , , , , , , 1, ⋯, , , and set the data length K.
- Collect the observation data and , read from using (52).
- Construct using (53).
- Compute using (41).
- Compute using (48).
- Increase t by 1 if , and then go to Step 2. Otherwise, obtain parameter estimates and and stop computing.
5. The Simulation Examples
- From Table 1 and Table 2, Figure 2 and Figure 5, it can be shown that the parameter estimation errors of the M-PC-GSG and the M-PC-MI-GSG algorithms become smaller as the data length t increases, which means that the proposed algorithms are effective in parameter estimation for the multivariate autoregressive system.
- Figure 2 and Figure 5 show that the M-PC-MI-GSG algorithm has higher parameter estimation accuracy than the M-PC-GSG algorithm under the same noise variances and same data length. Introducing the innovation length p can effectively improve the parameter estimation accuracy for the M-PC-GSG algorithm, and the parameter estimates can be more stationary as the innovation length p increases.
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
Abbreviations | Explanations |
M-PC-GSG | the multivariate partially coupled generalized stochastic gradient algorithm |
M-PC-MI-GSG | the multivariate partially coupled multi-innovation generalized stochastic |
gradient algorithm | |
M-FF-GSG | the multivariate forgetting factor generalized stochastic gradient algorithm |
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Algorithms | t | 100 | 200 | 500 | 1000 | 2000 | 3000 | True Values |
---|---|---|---|---|---|---|---|---|
M-PC-GSG | 0.19022 | 0.28281 | 0.33788 | 0.36637 | 0.36787 | 0.36879 | 0.42000 | |
0.77045 | 0.84529 | 0.88400 | 0.91050 | 0.90956 | 0.90832 | 0.93000 | ||
0.42624 | 0.49137 | 0.54550 | 0.57360 | 0.57695 | 0.57314 | 0.56000 | ||
0.25045 | 0.25999 | 0.29032 | 0.31802 | 0.32379 | 0.32917 | 0.31000 | ||
−0.16279 | −0.13896 | −0.12959 | −0.13205 | −0.13608 | −0.13540 | −0.25000 | ||
−0.06264 | 0.01024 | 0.07393 | 0.10879 | 0.14383 | 0.16385 | 0.68000 | ||
−0.18783 | −0.21131 | −0.22760 | −0.22982 | −0.23502 | −0.23697 | −0.33000 | ||
0.06255 | 0.05905 | 0.07305 | 0.09172 | 0.10850 | 0.12021 | 0.44000 | ||
60.04604 | 53.54881 | 48.52141 | 45.66110 | 43.08714 | 41.59002 | |||
M-PC-MI-GSG | 0.24413 | 0.35242 | 0.38045 | 0.39205 | 0.38805 | 0.38809 | 0.42000 | |
0.80411 | 0.89433 | 0.91071 | 0.92629 | 0.92058 | 0.91797 | 0.93000 | ||
0.49437 | 0.54833 | 0.57802 | 0.58115 | 0.58224 | 0.57102 | 0.56000 | ||
0.30721 | 0.28679 | 0.29391 | 0.32301 | 0.32347 | 0.32964 | 0.31000 | ||
−0.05715 | −0.06016 | −0.08608 | −0.10747 | −0.13040 | −0.13492 | −0.25000 | ||
0.12099 | 0.22052 | 0.30492 | 0.34234 | 0.38260 | 0.40361 | 0.68000 | ||
−0.23138 | −0.26019 | −0.26756 | −0.26447 | −0.26951 | −0.27084 | −0.33000 | ||
0.11762 | 0.11798 | 0.15107 | 0.18454 | 0.20984 | 0.22701 | 0.44000 | ||
47.51695 | 39.87369 | 33.61530 | 30.01753 | 26.59647 | 24.80351 | |||
M-PC-MI-GSG | 0.26824 | 0.39655 | 0.39206 | 0.40527 | 0.40117 | 0.40344 | 0.42000 | |
0.81764 | 0.91822 | 0.91009 | 0.93027 | 0.92223 | 0.92506 | 0.93000 | ||
0.53011 | 0.56224 | 0.57291 | 0.58006 | 0.58163 | 0.56221 | 0.56000 | ||
0.33781 | 0.28934 | 0.27962 | 0.32670 | 0.31900 | 0.33021 | 0.31000 | ||
−0.02859 | −0.07525 | −0.14398 | −0.17985 | −0.21158 | −0.20979 | −0.25000 | ||
0.40063 | 0.49444 | 0.57042 | 0.57794 | 0.60699 | 0.61726 | 0.68000 | ||
−0.27689 | −0.30473 | −0.29946 | −0.29170 | −0.29725 | −0.29909 | −0.33000 | ||
0.22503 | 0.21310 | 0.25734 | 0.29682 | 0.32439 | 0.34271 | 0.44000 | ||
30.60589 | 22.77807 | 16.20218 | 12.94875 | 9.87303 | 8.55799 | |||
M-PC-MI-GSG | 0.22875 | 0.41123 | 0.39312 | 0.41067 | 0.40847 | 0.41377 | 0.42000 | |
0.82538 | 0.91675 | 0.89962 | 0.92660 | 0.91916 | 0.93461 | 0.93000 | ||
0.50062 | 0.56129 | 0.56967 | 0.58480 | 0.57905 | 0.55554 | 0.56000 | ||
0.34784 | 0.29455 | 0.27015 | 0.32493 | 0.31639 | 0.33324 | 0.31000 | ||
−0.08032 | −0.14448 | −0.22262 | −0.25289 | −0.27162 | −0.25101 | −0.25000 | ||
0.60874 | 0.66577 | 0.71758 | 0.67480 | 0.69550 | 0.69273 | 0.68000 | ||
−0.31148 | −0.33276 | −0.30958 | −0.29949 | −0.30552 | −0.31357 | −0.33000 | ||
0.32292 | 0.26569 | 0.31977 | 0.36227 | 0.39067 | 0.40749 | 0.44000 | ||
20.99172 | 13.61264 | 9.45336 | 5.90521 | 4.39162 | 3.04341 |
Algorithms | t | 100 | 200 | 500 | 1000 | 2000 | 3000 | True Values |
---|---|---|---|---|---|---|---|---|
M-PC-GSG | −0.38109 | −0.34290 | −0.35197 | −0.35358 | −0.35240 | −0.35140 | −0.36000 | |
0.17154 | 0.19202 | 0.22318 | 0.20571 | 0.20894 | 0.20290 | 0.22000 | ||
0.52049 | 0.46240 | 0.42063 | 0.40418 | 0.38885 | 0.38130 | 0.34000 | ||
0.36772 | 0.39041 | 0.40431 | 0.43059 | 0.44228 | 0.45084 | 0.45000 | ||
0.32582 | 0.28799 | 0.27265 | 0.25708 | 0.26433 | 0.26468 | 0.25000 | ||
−0.05956 | −0.02720 | −0.02478 | −0.02008 | −0.00934 | −0.00876 | 0.11000 | ||
−0.32562 | −0.33413 | −0.35093 | −0.36678 | −0.37198 | −0.37339 | −0.41000 | ||
−0.42387 | −0.42801 | −0.43950 | −0.44549 | −0.45522 | −0.45784 | −0.48000 | ||
−0.07449 | −0.04447 | 0.00357 | 0.05541 | 0.08891 | 0.10599 | 0.35000 | ||
0.13315 | 0.09440 | 0.04896 | 0.00007 | −0.03206 | −0.04982 | −0.31000 | ||
62.46630 | 55.62215 | 48.66825 | 41.81593 | 37.21977 | 34.99402 | |||
M-PC-MI-GSG | −0.34351 | −0.30662 | −0.36709 | −0.35952 | −0.35025 | −0.35053 | −0.36000 | |
0.13616 | 0.21845 | 0.24620 | 0.20140 | 0.21882 | 0.20575 | 0.22000 | ||
0.46447 | 0.34810 | 0.32286 | 0.33605 | 0.34131 | 0.33736 | 0.34000 | ||
0.53467 | 0.51841 | 0.49198 | 0.50896 | 0.49490 | 0.49143 | 0.45000 | ||
0.34052 | 0.24404 | 0.24150 | 0.23199 | 0.26443 | 0.26536 | 0.25000 | ||
−0.08458 | −0.00717 | −0.00412 | 0.01422 | 0.05155 | 0.05032 | 0.11000 | ||
−0.40747 | −0.39750 | −0.42113 | −0.42870 | −0.41996 | −0.41446 | −0.41000 | ||
−0.44953 | −0.45067 | −0.47122 | −0.47509 | −0.49160 | −0.49275 | −0.48000 | ||
−0.00019 | 0.05273 | 0.13849 | 0.22633 | 0.25865 | 0.27774 | 0.35000 | ||
−0.10619 | −0.14079 | −0.18170 | −0.23788 | −0.25384 | −0.26009 | −0.31000 | ||
45.05598 | 34.25524 | 25.50852 | 16.96953 | 12.12756 | 10.74341 | |||
M-PC-MI-GSG | −0.35745 | −0.30970 | −0.38425 | −0.36288 | −0.34050 | −0.34946 | −0.36000 | |
0.12136 | 0.23250 | 0.25401 | 0.20572 | 0.22745 | 0.20850 | 0.22000 | ||
0.40415 | 0.28647 | 0.30611 | 0.33618 | 0.34589 | 0.33781 | 0.34000 | ||
0.51978 | 0.48285 | 0.46617 | 0.48824 | 0.46462 | 0.46024 | 0.45000 | ||
0.38782 | 0.22572 | 0.22904 | 0.22473 | 0.27031 | 0.26832 | 0.25000 | ||
−0.02008 | 0.06621 | 0.05334 | 0.06901 | 0.10837 | 0.09507 | 0.11000 | ||
−0.41690 | −0.39157 | −0.43170 | −0.43710 | −0.42419 | −0.41720 | −0.41000 | ||
−0.46440 | −0.45497 | −0.48430 | −0.48122 | −0.50181 | −0.49994 | −0.48000 | ||
0.09086 | 0.14050 | 0.23086 | 0.31994 | 0.33401 | 0.34857 | 0.35000 | ||
−0.24541 | −0.23841 | −0.25163 | −0.30483 | −0.29672 | −0.29278 | −0.31000 | ||
32.60246 | 22.30989 | 14.47746 | 6.91170 | 4.31185 | 3.73785 |
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Ma, P.; Wang, L. Partially Coupled Stochastic Gradient Estimation for Multivariate Equation-Error Systems. Mathematics 2022, 10, 2955. https://doi.org/10.3390/math10162955
Ma P, Wang L. Partially Coupled Stochastic Gradient Estimation for Multivariate Equation-Error Systems. Mathematics. 2022; 10(16):2955. https://doi.org/10.3390/math10162955
Chicago/Turabian StyleMa, Ping, and Lei Wang. 2022. "Partially Coupled Stochastic Gradient Estimation for Multivariate Equation-Error Systems" Mathematics 10, no. 16: 2955. https://doi.org/10.3390/math10162955
APA StyleMa, P., & Wang, L. (2022). Partially Coupled Stochastic Gradient Estimation for Multivariate Equation-Error Systems. Mathematics, 10(16), 2955. https://doi.org/10.3390/math10162955