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HW2 (Due 9/26)
(Gaussian process) Load the data file from here. It contains two variables, pos and observed. They are observed locations and corresponding observations from a multivariate normal vector variables of length 200.
Assuming the random variable is zero mean and a covariance between \(i\)th and \(j\)th variables equal to \(\exp(-\lambda (i-j)^2)\) (the model is sometimes known as a Gaussian process with a squared exponential kernel).
Let \(\lambda = 0.001\). Predict the rest of the unknown locations (compute the conditional mean and conditional variance) given what have been observed.
Plot the result (the mean) with error bar (mean +/- standard deviation).
Try again with only the first 10 observations
Try with different \(\lambda\) (0.0001 and 0.01). Explain what you observed.
(Similarity measure) Try to verify that \(Sim( \mathcal{N}({\boldsymbol \mu}_p,\Sigma_p),\mathcal{N}({\boldsymbol \mu}_q,\Sigma_q)) = \frac{\mathcal{N}({\boldsymbol \mu}_p; {\boldsymbol \mu}_q, \Sigma_p + \Sigma_q)}{\sqrt{\mathcal{N}(0; 0, 2 \Sigma_p)
\mathcal{N}(0; 0, 2 \Sigma_q)}}\) from Slide 14 of Lecture 5.
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