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STA 5208 Homework 1

1 Simulation example

Construct a numerical simulation example to illustrate the diﬀerence between the following two procedures for comparing OLS (ordinary least squares), PLS (partial least squares), or other estimators. In this simulation example, please ﬁrst generate the data under the multiple linear regression model.

1.1 Procedure 1

1. Center the data. Xi = Xi - , i = Yi - for i = 1, . . . , n, where = n~ 1 Xi and = n~ 1 Yi .

2. Data splitting. Splitting the data into 50%-50% training-testing groups with index sets labelled as [Tr] and [Te] so that they can form a partition of {1, 2, . . . , n}.

3. Estimation. Obtain tr from the training set.

4. Evaluation. Compute PE = i×[T e](yi - i )2 , where the predictor i = tTr xi .

5. Comparison. Apply steps 3 and 4 to diﬀerent methods (OLS, PLS, LASSO, etc.) and compare the prediction errors.

1.2 Procedure 2

1. Data splitting. Splitting the data into 50%-50% training-testing groups with index sets labelled as [Tr] and [Te] so that they can form a partition of {1, 2, . . . , n}.

2. Center the partitioned data. x,tr = ntr(~)1 i×[Tr]xi , y,tr = ntr(~)1 i×[Tr]yi . Center both the training and

4. Prediction error. Compute PE = i×[T e] ╱yi - y,tr - tr (xi - x,tr )、2 , where the predictor i = tTr xi .

5. Comparison. Apply steps 3 and 4 to diﬀerent methods and compare the prediction errors.

2 Invariance of OLS

Assume that XT X is non-degenerate and Γ is a p × p orthogonal matrix. Deﬁne = XΓ . From the OLS ﬁt of Y

= Γ, = , =

3 Multivariate Linear Model

For each unit i = 1, . . . , n, we have multiple responses yi = (yi1 , . . . , yiq )T e Rq and multiple covariates xi = (xi1 , . . . , xip )T e Rp . Please carefully deﬁne the centered data matrices Y e Rn—q and X e Rn—p, and the OLS

objective function L(B) to be minimized. Express the OLS estimator = ╱1 , . . . , q 、in terms of the observed

4 Matrix derivatives

Under the same MLM as in Problem 3, let L(B) denote the objective function to be minimized.

4.1 Derivation

Calculate the derivative D = L(B), which should be a p × q matrix.

4.2 Simulation

Simulate a data set under this MLM and numerically compute the derivative as Dij = {L(B + δ1ij ) - L(B)}/δ for i = 1, . . . , p and j = 1, . . . , q, for a small δ, say 0.0001. Compare this answer to your analytical solution in the previous part of the problem.