STAT215 - Assignment 4 Solved

 

Problem 1: Consider a Gaussian linear dynamical system (LDS),

 [ ~T                                                                                                          ] [ ii T                                                                               ]

p(x1:T , y1:T ) = jV (x1 | 0, q2)     jV (xt | axt_1 + b, q2)    jV (yt | xt, r2) ,

 t=2                                                   t=1

for xt, yt E R for all t, and parameters a, b E R and q2, r2 E R+. Compute the forward filtered distribution p(xt | y1:t) in terms of the model parameters and the filtered distribution p(xt_1 | y1:t_1). Solve for the base case p(x1 | y1). For reference, consult the state space modeling chapters of either the Bishop or the Murphy textbook.

Your solution here.

 Problem 2: Sample a time series of length T = 30 from the Gaussian LDS in Problem 1 with param­eters a = 1, b = 0,q = 0.1, r = 0.3. Plot the sample of x1:T as a solid line, and plot the observed y1:T as +’s. Write code to compute the filtered distribution p(xt | y1:t) you derived in Problem 1. Then plot the mean of the filtered distribution E[xt | y1:t] over time as a solid line, and plot a shaded region encompassing the mean ±2 standard deviations of the filtered distribution. All plots should be on the same axis. Include a legend.

Write your code in a Colab notebook and include a PDF printout of your notebook as well as the raw .ipynb file.

 Problem 3: Reproduce Figure 2.5 of Rasmussen and Williams, Gaussian Processes for Machine Learn­ing, available at http://www.gaussianprocess.org/gpml/chapters/RW2.pdf. Use a randomly generated dataset as described in the figure caption and surrounding text.

Write your code in a Colab notebook and include a PDF printout of your notebook as well as the raw .ipynb file.