Q2: Gaussian linear state-space model estimation and Kalman filtering

Consider the state-space model

where the{t}and{t}are i.i.d. respectively.

Xt=AXt1+tYt=BXt+t,

t N(0,Q)andt N(0,R),

(a)In the simple case of supervised state-space model estimation, i.e. when both the stateXtRpand observationYtRqvectors are available in some training data, the parameters{A,B,Q,R}can be estimated using standard regression methods.

Load the provided fileGaussianLDS_XY_data.matwhich contains the training datasetXtrain(dim,time)for the state vector (dim=1, . . . ,p=2) andYtrain(dim,time)for the observation vector (dim=1,2,...,q=10). Estimate the parameters{A,B,Q,R}based on these training data. To solve for{A,B}use the least-square solutions given in Lecture 19. To solve for the noise covariance matrices use the formulas given in the same lecture. Report the estimated parameters.

(b)Apply the estimated parameters to the Kalman filter equations to infer from the test dataYtest(also available in the same file) the true states based on the posterior means. Plot the observations from the test data (one component in each subplot) for the timest=2500, . . . , 3000. On a different figure, plot the true states{Xt}from the test data and the inferred posterior means. Organize this second figure in two subplots, one for each state dimension, and plot for timest=2500, . . . , 3000.