State Space Models

Matthieu Bloch

Thursday September 29, 2022

Today in ECE 6555

Don't forget
- Midterm Exam coming up on Thursday October 6, 2022
  - In class - 75 minutes
  - Open notes (your notes, slides, problem set solutions - no textbook)
  - DL students: schedule within one week (official instruction through GTPE)
- Problem Set 3 due Thursday October 13, 2022 on Gradescope
  - Only 3 problems
- Make sure you start homework early and don't spend 30 hours
Last time
- Computing innovation process through LDL decomposition and Gaussian elimination
- This is messy!
Today's plan
- State space model: often a more natural way to derive innovations
Questions?

The exponentially correlated process is a wide sense stationary process defined by an autocorrelation function of the form \[ \dotp{y_i}{y_j} \eqdef a^{\abs{i-j}} \qquad\forall i,j\qquad \textsf{for some } a\in(0;1) \]
The covariance matrix of the process is a Toeplitz matrix of the form \[ \matR_y =\left[\begin{array}{ccccc} 1&a&a^2&a^3&a^4\\ a&1&a&a^2&a^3\\ a^2&a&1&a&a^2\\ a^3&a^2&a&1&a\\ a^4&a^3&a^2&a&1\\ \end{array}\right] \]
Computing the innovation is tedious if we try to obtain the LDL decomposition of $\matR_y$
Is there a simpler way?

Consider the linear stochastic differential equations $\vecy_{i+1}-a \vecy_i=\vecu_i$ for $i\geq 0$, $a\in\bbR$
- Initial conditions $\vecy_0$
- $\vecu_i$ such that $\dotp{\vecu_i}{\vecu_j}=\matQ_i\delta_{ij}$, $\norm{\vecy_0}^2=\Pi_0$, $\dotp{\vecu_i}{\vecy_0}=0$
- The process may be non-stationary
$\forall i\geq j$ we have $\dotp{\vecu_i}{\vecy_j}=0$. For $i\geq 0$, if $\Pi_i\eqdef \norm{\vecy_i}^2$ then $\Pi_{i+1}=a^2\Pi_i+\matQ_i$
In general the process is not stationary
If $\matQ_i=\matQ$, $\Pi_0 = \frac{\matQ}{1-a^2}$ with $\abs{a}<1$, the process is stationary.
- What happens in the scalar case for $\matQ=1-a^2$?
Innovations can be computed regardless of stationarity
The innovation of the process defined by the stochastic differential equation above is $\vece_i=\vecu_i$ for $i\geq 0$
How general is such a approach?

An autoregressive process is defined by the stochastic equation \[ y_{i+1} = a_{0,i}y_i + a_{1,i} y_{i-1}+\cdots + a_{n-1,i}y_{i-n+1} + u_i\qquad \forall i\geq 0 \] where:
- $u_i$ is zero mean with $\dotp{u_i}{u_j}=Q_i\delta_{i,j}$ $\forall i \geq 0$
- $y_{0},\cdots,y_{-n+1}$ zero mean with know covariance matrix $\Pi_0$
The autoregressive model can be represented as a simple order 1 recursion of the form \[ \vecx_{i+1} = \matF_i\vecx_i + \matG_i\vecu_i\qquad y_i =\matH_i\vecx_i \]

An autoregressive moving-average process is defined by the stochastic equation \[ y_{i+1} = a_{0}y_i + a_{1} y_{i-1}+\cdots + a_{n-1}y_{i-n+1} + b_0u_i + \cdots+b_{n-1}u_{i-n+1}\qquad \forall i\geq 0 \] where:
- $u_i$ is zero mean with $\dotp{u_i}{u_j}=Q_i\delta_{i,j}$ $\forall i \geq 0$
- $y_{0},\cdots,y_{-n+1}$ zero mean with known covariance matrix $\Pi_0$
The autoregressive moving-average model can be represented as an order 1 recursion

The standard state space model is of the form \[ \vecx_{i+1} = \matF_i\vecx_i + \matG_i\vecu_i\qquad \vecy_i = \matH_i \vecx_i + \vecv_i \] with known matrices $\set{\matF_i,\matG_i,\matH_i}$ and \[ \dotp{\left[\begin{array}{c}\vecx_0\\\vecu_i\\\vecv_i\end{array}\right]}{\left[\begin{array}{c}\vecx_0\\\vecu_j\\\vecv_j\\1\end{array}\right]} \eqdef \left[\begin{array}{cccc}\Pi_0&0&0&0\\0&\matQ_i\delta_{ij}&\matS_i\delta_{ij}&0\\0&\matS_i^T\delta_{ij}&\matR_i\delta_{ij}&0\end{array}\right] \]

- For $i\geq j$ $\dotp{\vecu_i}{\vecx_j}=0$ and $\dotp{\vecv_i}{\vecx_j}=0$
- For $i> j$ $\dotp{\vecu_i}{\vecy_j}=0$ and $\dotp{\vecv_i}{\vecy_j}=0$
- For $i= $ $\dotp{\vecu_i}{\vecy_j}=\matS_i$ and $\dotp{\vecv_i}{\vecy_j}=\matR_i$
- If $\matF_i$ non singular, $\dotp{\vecu_i}{\vecx_0}=0$ if and only if $\forall i\geq 0$ $\dotp{\vecu_i}{\vecx_i}=0$
Let $\dotp{\vecx_i}{\vecx_i}=\Pi_i$. Then \[ \forall i\geq 0\qquad \Pi_{i+1} = \matF_i\Pi_i\matF_i^T + \matG_i\matQ_i\matG_i^T \] Let $\Phi(i,j)\eqdef \prod_{\ell=i-1}^j\matF_\ell$ for $i>j$ and $\Phi(i,i)=\matI$. Then \[ \dotp{\vecx_i}{\vecx_j} = \begin{cases}\Phi(i,j)\Pi_j\text{ for }i\geq j\\\Pi_i\Phi(j,i)^T\text{ for }i\leq j\end{cases} \] \[ \dotp{\vecy_i}{\vecy_j} = \begin{cases}\matH_i\Phi(i,j+1)\matN_j\text{ for }i> j\text{ with }\matN_i=\matF_i\Pi_i\matH_i^T+\matG_i\matS_i\\ \matR_i+\matH_i\Pi_i\matH_i^T\text{ for }i= j\\ \matN_i^T\Phi(j,i+1)^T\matH)j^T\text{ for }i< j\end{cases} \]