2. Fit auto-regressive (AR) models with regular and stochastic gradient de- scents. Refer to the two-tap...

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2. Fit auto-regressive (AR) models with regular and stochastic gradient de- scents. Refer to the two-tap predictor example in the slides of chapter 5a, consider a AR(2) model x(n) = -wx(n-1) - wx(n - 2) + (n), where x(n) is a time series data, x(n 1) and x(n 2) are the lag 1 and lag 2 series of x(n), and e(n) is a Gaussian noise with zero mean. Dateset description: Two datasets ("Casel.csv", "Case2.csv") are given in this question, each dataset contains a length-1000 series x(n). To fit lin- ear regression model above, you will need to produce lag 1 series x(n-1) and lag 2 series x(n - 2) by yourself. To make x(n-1) and x(n 2) have the same length 1000, filling up the lag by zeros. For example, x(n 1) = [0, x(1), x(2), ..., x (999)]. (a) Estimate w and w for the two datasets using regular gradient de- scent. (b) Estimate w and w for the two datasets using stochastic gradient descent, and compare the result with part (a). (c) For the two datasets, find their correlation matrices R, and the eigen- value spread X(R) = Aman. Hint: To find Amax and Amin, conduct eigendecomposition to R using build-in functions. 2. Fit auto-regressive (AR) models with regular and stochastic gradient de- scents. Refer to the two-tap predictor example in the slides of chapter 5a, consider a AR(2) model x(n) = -wx(n-1) - wx(n - 2) + (n), where x(n) is a time series data, x(n 1) and x(n 2) are the lag 1 and lag 2 series of x(n), and e(n) is a Gaussian noise with zero mean. Dateset description: Two datasets ("Casel.csv", "Case2.csv") are given in this question, each dataset contains a length-1000 series x(n). To fit lin- ear regression model above, you will need to produce lag 1 series x(n-1) and lag 2 series x(n - 2) by yourself. To make x(n-1) and x(n 2) have the same length 1000, filling up the lag by zeros. For example, x(n 1) = [0, x(1), x(2), ..., x (999)]. (a) Estimate w and w for the two datasets using regular gradient de- scent. (b) Estimate w and w for the two datasets using stochastic gradient descent, and compare the result with part (a). (c) For the two datasets, find their correlation matrices R, and the eigen- value spread X(R) = Aman. Hint: To find Amax and Amin, conduct eigendecomposition to R using build-in functions.

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Solutions Step 1 Question Fit autoregressive AR models with regular and stochastic gradient descents Refer to the twotap predictor example in the slides of chapter 5a consider a AR2 model xnw1xn1w2xn2... View the full answer