Estimation of Stochastic Processes

Today in ECE 6555

• Don't forget
  • Problem set 2 due Thursday September 22, 2022 on Gradescope
  • Mathematics of ECE workshops website
  • Recordings available on mediaspace
• Last time
  • Gauss-Markov theorem
  • Be careful when specializing the stoachatic model to deterministic parameters
• Today's plan
  • Sensor fusion
  • Estimation for Stochastic processes
• Questions?

Combining estimators

• Consider a zero mean random variable \(\vecx\in\bbR^n\) (\(\matR_\vecx\succ 0\)) observed through \(\vecy_1\) and \(\vecy_2\) according to \[ \vecy_1 = \matH_1+\vecv_1\qquad \vecy_2 = \matH_2+\vecv_2 \] with \((\vecx,\vecv_1,\vecv_2)\) mutually independent and zero mean.

  • Let \(\hat{\vecx}_1\) be the LLSE of \(\vecx\) from \(\vecy_1\) with error covariance matrix \(\matP_1\)
  • Let \(\hat{\vecx}_2\) be the LLSE of \(\vecx\) from \(\vecy_2\) with error covariance matrix \(\matP_2\)

  Then \(\matP^{-1}\vecx = \matP_1^{-1}\hat{\vecx}_1+\matP_2^{-1}\hat{\vecx}_2\) with \(\matP^{-1} = \matP_1^{-1}+\matP_2^{-1}-\matR_\vecx^{-1}\)

Correspondence with deterministic least squares

• Consider the deterministic least square optimization \[ \min_{\vecx} (\vecx-\vecx_0)^T\Pi_0^{-1}(\vecx-\vecx_0) + \norm[W]{\vecy-\matH\vecx}^2 \]

• Consider the linear stochastic least square optimization for the linear model \(\vecy=\matH\vecx+\vecv\)

• Can we draw parallels between the two problems?

Estimation of Stochastic Processes

• Objective: extend our analysis to stochastic processes and introduce a temporal aspect
  • Idea is to understand how we can perform estimation when time enters the picture
  • We will eventually use this to study the Kalman filter

• A stochastic process is a collection of random variables \(\set{x_i}_{i\geq 0}\) characterized by the joint pdfs \(p_{X_{i_1}\cdots X_{i_k}}\) for all \(k\geq 1\) and all \(\set{i_j}_{j=1}^k\).

• Estimation model:
  • Signal process \(\set{x_i}_{i\geq 0}\) not observed, zero mean
  • Measurement process \(\set{y_i}_{i\geq 0}\) obverved, zero mean
  • Covariance and correlation matrices known: \(\matR_{xy}(i,\ell)\eqdef \E{s_i y_\ell^T}\), \(\matR_{y}(i,\ell)\eqdef \E{y_i y_\ell^T}\), \(\forall i,\ell\)
• Estimation goal: need to specify form of estimation and which observations to use

Smoothing, causal filtering, prediction

• Smoothing: estimate \(x_i\) from \(\set{y_j}_{j=0}^m\), \(m>i\) (using past, present and future observations) as \[ \hat{x}_{i|m} \eqdef \sum_{j=0}^{m} k_{i,j}y_j \]

• Causal filtering: estimate \(x_i\) from \(\set{y_j}_{j=0}^{i}\) (using past, and present observations) as \[ \hat{x}_{i|i} \eqdef \sum_{j=0}^{i} k_{i,j}y_j \]

• Prediction: estimate \(x_{i+\ell}\) from \(\set{y_j}_{j=0}^{i}\), \(\ell>1\) (using past observations) as \[ \hat{x}_{i+\ell|i} \eqdef \sum_{j=0}^{i} k_{i,j}y_j \]

• In all cases we want the estimation to be optimal (minimize error covariance matrix)

Smoothing

• Let's put what we've learned to work: geometry!

• Smoothing reduces to solving the normal equations and for \(\matR_y\succ 0\) \[ \hat{\vecx}_{s} = \matR_{\vecx\vecy}\matR_{\vecy}^{-1}\vecy \] where \[ \hat{\vecx}_{s}\eqdef\left[\begin{array}{c}\hat{x}_{0|m}\\\vdots\\\hat{x}_{m|m}\end{array}\right]\quad \matR_{\vecy}\eqdef\left[\matR_y(i,j)\right]\quad \matR_{\vecx\vecy}\eqdef\left[\matR_{xy}(i,j)\right] \]

Causal filtering

• Geometry strikes back…

• For \(\matR_\vecy\succ 0\) decomposed as \(\matR_\vecy=\matL\matD\matL^T\) (\(\matL\) lower triangular) \[ \hat{\vecx}_{f} = \mathcal{L}\left[\matR_{\vecx\vecy}\matL^T\matD^{-1}\right]\matL^{-1}\vecy \] where \[ \hat{\vecx}_{f}\eqdef\left[\begin{array}{c}\hat{x}_{0|0}\\\hat{x}_{1|1}\\\vdots\\\hat{x}_{m|m}\end{array}\right]\quad \matR_{\vecy}\eqdef\left[\matR_y(i,j)\right]\quad \matR_{\vecx\vecy}\eqdef\left[\matR_{xy}(i,j)\right] \] and \(\mathcal{L}[\cdot]\) is the operator that makes a matrix lower triangular.