HW5 (Due 12/6)

Let's try to extrapolate with a Gaussian process. Consider a signal of length-\(50\) with \(x[n]=20 sin(2 \pi n/20)\) for \(n=0, \cdots, 49\) (x=20*sin(2*pi*(0:49)/20) in Matlab notation). Try to extend the signal with 20 more samples (a total of 70 samples). Assume the entire signal is zero-mean and \(\rho = 0.5\). And assume the variance is \(1\). (The complete covariance matrix should have diagonal elements of \(1\) and the \(j\)th off-diagonal elements equal to \(\rho^j\).)

1. Plot the extrapolated signal of total length of 70. Plot the lower and upper bound as well (mean \(\pm\) standard derviation)

2. Try to vary \(\rho\) and comment on your observation.

3. Check whether the Markov property of the final signal is preserved by verifying if sample 51 and sample 53 are conditionally independent given sample 52. Recall that for joint Gaussian variables \(X,Y\), and \(Z\), \(X\) and \(Y\) are conditionally independent given \(Z\) if and only if \(\rho_{XY} = \rho_{XZ} \rho_{YZ}\).

Please try to install Pyro and run this jupyter notebook. Note that you can download the notebook from the link “View page source”. You would like to save the file with an extension .ipynb.

1. Please modify the observation from 9.5 to 10.

2. What should be the analytical estimate of the true weight?

3. After modifying and running your notebook, please save your notebook as pdf to submit.