MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Part f only. It should be one figure with two lines. 

## Problem 3. [SW 14.10] You have a sample of size \( n = 1 \) with data \( y_1 = 2 \) and \( x_1 = 1 \). You are interested in the value of \( \beta \) in the regression \( Y = X \beta + u \). (Note there is no intercept.) (a) Plot the sum of squared residuals \( (y_1 - b x_1)^2 \) as a function of \( b \). You can choose your own range for \( b \), one reasonable choice is \( b \in [-2, 5] \). *(Use any software you prefer. Excel is one option.)* (b) Show that the least squares estimate of \( \beta \) is \( \hat{\beta}^{OLS} = 2 \). (c) Using \( \lambda_{Lasso} = 1 \), plot the Lasso penalty term \( \lambda_{Lasso} |b| \) as a function of \( b \). (d) Using \( \lambda_{Lasso} = 1 \), plot the Lasso penalized sum of squared residuals \( (y_1 - b x_1)^2 + \lambda_{Lasso} |b| \). Please put all three lines in one plot. (e) Find the value of \( \hat{\beta}^{Lasso} \). (f) Using \( \lambda_{Lasso} = 0.5 \), repeat (c) and (d) (put both lines in one plot). Find the value of \( \hat{\beta}^{Lasso} \).
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Transcribed Image Text:## Problem 3. [SW 14.10] You have a sample of size \( n = 1 \) with data \( y_1 = 2 \) and \( x_1 = 1 \). You are interested in the value of \( \beta \) in the regression \( Y = X \beta + u \). (Note there is no intercept.) (a) Plot the sum of squared residuals \( (y_1 - b x_1)^2 \) as a function of \( b \). You can choose your own range for \( b \), one reasonable choice is \( b \in [-2, 5] \). *(Use any software you prefer. Excel is one option.)* (b) Show that the least squares estimate of \( \beta \) is \( \hat{\beta}^{OLS} = 2 \). (c) Using \( \lambda_{Lasso} = 1 \), plot the Lasso penalty term \( \lambda_{Lasso} |b| \) as a function of \( b \). (d) Using \( \lambda_{Lasso} = 1 \), plot the Lasso penalized sum of squared residuals \( (y_1 - b x_1)^2 + \lambda_{Lasso} |b| \). Please put all three lines in one plot. (e) Find the value of \( \hat{\beta}^{Lasso} \). (f) Using \( \lambda_{Lasso} = 0.5 \), repeat (c) and (d) (put both lines in one plot). Find the value of \( \hat{\beta}^{Lasso} \).
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