Part G 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 \( (y_1 - bx_1)^2 \) 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 \( (y_1 - bx_1)^2 + \lambda_{Lasso} |b| \). 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 \( \hat{\beta}^{Lasso} \).(g) Using \( \lambda_{Lasso} = 5 \), repeat (c) and (d) (put both lines in one plot). Find the value of \( \hat{\beta}^{Lasso} \).

Given Information: Consider the given regression model without intercept: y=Xβ+u Sample size n=1 with data y1=2 and x1=1The SSE is given as: SSE=y1-bx12 =2-b2 Consider the given information in which it is mentioned that b∈-2,5. Thus, we have: g) i. λLasso=5, The lasso penalty term is given by λLassob. Thus, we have: b Lasso penaltyλLassob -2 10 -1 5 0 0 1 5 2 10 3 15 4 20 5 25 ii. The lasso penalized sum of squared residuals is given as: y1-bx12+λLassob Using λLasso=5 b Lasso penalize sum of squared residualsy1-bx12+λLassob -2 26 -1 14 0 4 1 6 2 10 3 16 4 24 5 34 Thus, the data we have: b Lasso penalty Lasso penalize sum of squared residuals -2 10 26 -1 5 14 0 0 4 1 5 6 2 10 10 3 15 16 4 20 24 5 25 34 The step-by-step procedure to plot the sum of square residuals using the excel function is given by: Enter the data into excel. go to the tab Insert> recommended chart > All chart > X-Y scatter > Scatter with smooth lines. Click ok. The chart is shown below: The value of β^Lasso can be obtained as: SSE=y1-bx12+λbSSE=2-b2+5bdSSEdb=022-b-1+5=0 -4+2b+5=0 2b+1=0 b=-12=-0.5 β^Lasso=-0.5

Math

Statistics

Problem 3. [SW 14.10] You have a sample of size n = 1 with data y₁ = 2 and ₁= 1. You are interested in the value of 3 in the regression Y = XB+ u. (Note there is no intercept.) (a) Plot the sum of squared residuals (y₁ - bx₁)² as function of b. You can choose your own range for b, one reasonable choice is b € [-2,5]. (Use any software you prefer. Excel is one option.) OLS (b) Show that the least squares estimate of 3 is 3 = 2. (c) Using Lasso = 1, plot the Lasso penalty term XLasso b as a function of b. 2 (d) Using Lasso = 1, plot the Lasso penalized sum of squared residuals (y₁ - br₁)² + XLasso |b|. Please put all three lines in one plot. Lasso (e) Find the value of 3 (f) Using Lasso = 0.5, repeat (c) and (d) (put both lines in one plot). Find the value of (g) Using ALasso = 5, repeat (c) and (d) (put both lines in one plot). Find the value of B Lasso Lasso

Problem 3. [SW 14.10] You have a sample of size n = 1 with data y₁ = 2 and ₁= 1. You are interested in the value of 3 in the regression Y = XB+ u. (Note there is no intercept.) (a) Plot the sum of squared residuals (y₁ - bx₁)² as function of b. You can choose your own range for b, one reasonable choice is b € [-2,5]. (Use any software you prefer. Excel is one option.) OLS (b) Show that the least squares estimate of 3 is 3 = 2. (c) Using Lasso = 1, plot the Lasso penalty term XLasso b as a function of b. 2 (d) Using Lasso = 1, plot the Lasso penalized sum of squared residuals (y₁ - br₁)² + XLasso |b|. Please put all three lines in one plot. Lasso (e) Find the value of 3 (f) Using Lasso = 0.5, repeat (c) and (d) (put both lines in one plot). Find the value of (g) Using ALasso = 5, repeat (c) and (d) (put both lines in one plot). Find the value of B Lasso Lasso

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Part G 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 \( (y_1 - bx_1)^2 \) 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 \( (y_1 - bx_1)^2 + \lambda_{Lasso} |b| \). 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 \( \hat{\beta}^{Lasso} \).

(g) Using \( \lambda_{Lasso} = 5 \), repeat (c) and (d) (put both lines in one plot). Find the value of \( \hat{\beta}^{Lasso} \).

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