The following table contains output from a lasso fit to a linear model with d = 5 variables and n = 100 observations. Starting from the left, the columns are λ, and B1, ..., ß5, i.e. each row has λ and the transposed column vector B(λ). 0.00000 0.05470 0.13093 -0.04217 0.09980 -0.01947 1.39802 0.03968 0.11610 -0.01917 0.08656 0.00000 3.00093 0.02288 0.09856 0.00000 0.06971 0.00000 5.70455 0.00000 0.06926 0.00000 0.04054 0.00000 9.18968 0.00000 0.02941 0.00000 0.00000 0.00000 12.13018 0.00000 0.00000 0.00000 0.00000 0.00000 For each of the required computations below, briefly report your procedure and the required quantity. a) For each row in the table, compute s, the proportion of shrinkage defined as s = s(λ) = ||B(2)||1/ max 2 ||B(2)||1 · b) Consider λ = 7.447115. Note that λ' is the intermediate value between λ = 5.70455 and λ = 9.18968 of the 4th and 5th rows above. Using this value of 1', compute and report the shrunk estimator ß(λ′). c) Consider the vector of predictors x = (1.06931, -0.9703, -1.06931, 1.12871, 1.0297). Using your shrunk estimator B(λ') (as column vector) compute the predicted value ŷ = x²ß(X′). d) Repeat b) for the regularization parameter taking value λ" = 13, that is, determine ẞ(λ").
The following table contains output from a lasso fit to a linear model with d = 5 variables and n = 100 observations. Starting from the left, the columns are λ, and B1, ..., ß5, i.e. each row has λ and the transposed column vector B(λ). 0.00000 0.05470 0.13093 -0.04217 0.09980 -0.01947 1.39802 0.03968 0.11610 -0.01917 0.08656 0.00000 3.00093 0.02288 0.09856 0.00000 0.06971 0.00000 5.70455 0.00000 0.06926 0.00000 0.04054 0.00000 9.18968 0.00000 0.02941 0.00000 0.00000 0.00000 12.13018 0.00000 0.00000 0.00000 0.00000 0.00000 For each of the required computations below, briefly report your procedure and the required quantity. a) For each row in the table, compute s, the proportion of shrinkage defined as s = s(λ) = ||B(2)||1/ max 2 ||B(2)||1 · b) Consider λ = 7.447115. Note that λ' is the intermediate value between λ = 5.70455 and λ = 9.18968 of the 4th and 5th rows above. Using this value of 1', compute and report the shrunk estimator ß(λ′). c) Consider the vector of predictors x = (1.06931, -0.9703, -1.06931, 1.12871, 1.0297). Using your shrunk estimator B(λ') (as column vector) compute the predicted value ŷ = x²ß(X′). d) Repeat b) for the regularization parameter taking value λ" = 13, that is, determine ẞ(λ").
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Transcribed Image Text:The following table contains output from a lasso fit to a linear model with d = 5 variables and n = 100
observations. Starting from the left, the columns are λ, and B1, ..., ß5, i.e. each row has λ and the transposed
column vector B(λ).
0.00000 0.05470 0.13093 -0.04217 0.09980 -0.01947
1.39802 0.03968 0.11610 -0.01917 0.08656 0.00000
3.00093 0.02288 0.09856 0.00000 0.06971 0.00000
5.70455 0.00000 0.06926 0.00000 0.04054 0.00000
9.18968 0.00000 0.02941 0.00000 0.00000 0.00000
12.13018 0.00000 0.00000 0.00000 0.00000 0.00000
For each of the required computations below, briefly report your procedure and the required quantity.
a) For each row in the table, compute s, the proportion of shrinkage defined as
s = s(λ) = ||B(2)||1/ max 2 ||B(2)||1 ·
b) Consider λ = 7.447115. Note that λ' is the intermediate value between λ = 5.70455 and λ = 9.18968 of
the 4th and 5th rows above. Using this value of 1', compute and report the shrunk estimator ß(λ′).
c) Consider the vector of predictors x = (1.06931, -0.9703, -1.06931, 1.12871, 1.0297). Using your shrunk estimator
B(λ') (as column vector) compute the predicted value ŷ = x²ß(X′).
d) Repeat b) for the regularization parameter taking value λ"
=
13, that is, determine ẞ(λ").
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