Computation of the State Bias and Initial States for Stochastic State Space Systems in the General 2-D Roesser Model Form

TL;DR

This paper analyzes the bias in state estimation for 2-D stochastic systems in the Roesser model, proving it is negligible under certain conditions and proposing an improved iterative identification algorithm.

Contribution

It demonstrates that the bias in state estimation is insignificant when states are uncorrelated and introduces a second iteration to enhance initial state computation.

Findings

Bias is negligible with large enough i and uncorrelated states.

The proposed second iteration improves state estimates.

The revised algorithm provides a complete 2-D stochastic system identification method.

Abstract

Recently \cite{Ramos2017a} presented a subspace system identification algorithm for 2-D purely stochastic state space models in the general Roesser form. However, since the exact problem requires an oblique projection of $Y_f^h$ projected onto $W_p^h$ along $\widehat{X}_f^{vh}$, where $W_p^h= \begin{bmatrix}\widehat{X}_p^{vh} \\ Y_p^h \end{bmatrix}$, this presents a problem since $\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$ are unknown. In the above mentioned paper, the authors found that by doing an orthogonal projection $Y_f^h/Y_p^h$, one can identify the future horizontal state matrix $\widehat{X}_f^{h}$ with a small bias due to the initial conditions that depend on $\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$. Nevertheless, the results on modeling 2-D images were very good despite lack of knowledge of $\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$. In this note we delve into the bias term and prove that it is insignificant, provided $i$ is chosen large enough and the vertical and horizontal states are uncorrelated. That is, the cross covariance of the state estimates $x_{r,s}^{h}$ and $x_{r,s}^{v}$ is zero, or $P_{hv}=0_{n_x\times n_x}$ and $P_{vh}=0_{n_x\times n_x}$. Our simulations use $i=30$. We also present a second iteration to improve the state estimates by including the vertical states computed from a vertical data processing step, i.e., by doing an orthogonal projection $Y_f^v/Y_p^v$. In this revised algorithm we include a step to compute the initial states. This new portion, in addition to the algorithm presented in \cite{Ramos2017a}, forms a complete 2-D stochastic subspace system identification algorithm.