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Category: E 6690

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P1. It is well known that ridge regression tends to give similar coefficient values to correlated variables, whereas

the lasso may give quite different coefficient values to correlated variables. We will now explore this property in

a very simple setting.

Suppose that n = 2, p = 2, x11 = x12, x21 = x22. Furthermore, suppose that y1 + y2 = 0 and x11 + x21 = 0

and x12 + x22 = 0, so that the estimate for the intercept in a least squares, ridge regression, or lasso model is

zero: βˆ

0 = 0.

(a) (2pt) Write out the ridge regression optimization problem in this setting.

(b) (2pt) Argue that in this setting, the ridge coefficient estimates satisfy βˆ

1 = βˆ

2.

(c) (2pt) Write out the lasso optimization problem in this setting.

(d) (4pt) Argue that in this setting, the lasso coefficients βˆ

1 and βˆ

2 are not unique-in other words, there are

many possible solutions to the optimization problem in (c). Describe these solutions.

P2. Consider a simple regression with n = p, and X a diagonal matrix with 1’s on the diagonal and 0’s in all

off-diagonal elements. To simplify the problem further, assume that we are performing regression without an

intercept. In this case, the usual least squares problem simplifies to finding β1, . . . , βp that minimize

X

p

j=1

(yj − βj )

2

.

The least squares solution is given by βˆ

j = yj .

And in this setting, ridge regression amounts to finding β1, . . . , βp such that

X

p

j=1

(yj − βj )

2 + λ

X

p

j=1

β

2

j

(1)

is minimized, and the lasso amounts to finding the coefficients such that

X

p

j=1

(yj − βj )

2 + λ

X

p

j=1

|βj | (2)

is minimized. One can show that in this setting, the ridge regression estimates take the form

βˆR

j = yj/(1 + λ), (3)

and the lasso estimates take the form

βˆL

j =

yj − λ/2 if yj > λ/2;

yj + λ/2 if yj < −λ/2;

0 if |yj | ≤ λ/2.

(4)

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(a) (5pt) Consider (1) with p = 1. For some choice of y1 and λ > 0, plot (1) as a function of β1. Your plot

should confirm that (1) is solved by (3).

(b) (5pt) Consider (2) with p = 1. For some choice of y1 and λ > 0, plot (2) as a function of β1. Your plot

should confirm that (2) is solved by (4).

P3. In this problem, we will derive the Bayesian connection to the lasso and ridge regression.

(a) (2pt) Suppose that yi = β0 +

Pp

j=1 xijβj +i where 1, . . . , n are independent and identically distributed

from a N (0, σ2

) distribution. Write out the likelihood for the data.

(b) (3pt) Assume the following prior for β : β1, . . . , βp are independent and identically distributed according to

a double-exponential distribution with mean 0 and common scale parameter b; i.e, p(β) = 1

2b

exp(−|β|/b)

and |β| =

Pp

j=1 |βj |. Write out the posterior for β in this setting.

(c) (5pt) Argue that the lasso estimate is the mode for β under this posterior distribution.

(d) (5pt) Now assume that the following prior for β : β1, . . . , βp are independent and identically distributed

according to a normal distribution with mean 0 and variance c. write out the posterior for β in this setting.

(e) (5pt) Argue that the ridge regression estimate is both the mode and the mean for β under this posterior

distribution.

P4. In this exercise, we will generate simulated data, and will then use this data to perform best subset

selection.

(a) (2pt) Use the rnorm() function to generate a predictor X of length n = 100, as well as a noise vector

of length n = 100.

(b) (3pt) Generate a response vector Y of length n = 100 according to the model

Y = β0 + β1X + β2X2 + β3X3 + ,

where β0, β1, β2, and β3 are constants of your choice.

(c) (5pt) Use the regsubsets() function to perform best subset selection in order to choose the best model

containing the predictors X, X2

, . . . , X10. What is the best model obtained according to Cp, BIC, and

adjusted R2

? Show some plots to provide evidence for your answer, and report the coefficients of the

best model obtained. Note you will need to use the data.frame() function to create a single data set

containing both X and Y .

(d) (5pt) Repeat (c), using forward stepwise selection and also using backwards stepwise selection. How does

your answer compare to the result in (c)?

(e) (5pt) Now fit a lasso model to the simulated data, again using X, X2

, . . . , X10 as predictors. Use crossvalidation to select the optimal value of λ. Create plots of the cross-validation error as a function of λ.

Report the resulting coefficient estimates, and discuss the results obtained.

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(f) (5pt) Now generate a response vector Y according to the model

Y = β0 + β7X7 + ,

and perform best subset selection and the lasso. Discuss the results obtained.

P5. In this exercise, we will predict the number of applications received using the other variables in the college

data set.

(a) (2pt) Split the data set into a training set and a test set.

(b) (3pt) Fit a linear model using least squares on the training set, and report the test error obtained.

(c) (5pt) Fit a ridge regression model on the training set, with λ chosen by cross-validation. Report the test

error obtained.

(d) (5pt) Fit a lasso model on the training set, with λ chosen by cross-validation. Report the test error

obtained, along with the number of non-zero coefficient estimates.

P6. We have seen that as the number of features used in a model increases, the training error will necessarily

decrease, but the test error may not. We will now explore this in a simulated data set.

(a) (2pt) Generate a data set with p = 20 features, n = 1000 observations, and an associated quantitative

response vector generated according to the model

Y = Xβ + ,

where β has some elements that are exactly equal to zero.

(b) (2pt) Split your data set into a training set containing 100 observations and a test set containing 900

observations.

(c) (2pt) Perform best subset selection on the training set, and plot the training set MSE associated with the

best model of each size.

(d) (2pt) Plot the test set MSE associated with the best model of each size.

(e) (2pt) For which model size does the test set MSE take on its minimum value? Comment on your results.

If it takes on its minimum value for a model containing only an intercept or a model containing all of the

features, then play around with the way that you are generating the data in (a) until you come up with a

scenario in which the test set MSE is minimized for an intermediate model size.

(f) (2pt) How does the model at which the test set MSE is minimized compare to the true model used to

generate the data? Comment on the coefficient values.

(g) (3pt) Create a plot displaying qPp

j=1(βj − βˆr

j

)

2 for a range of values of r, where βˆr

j

is the jth coefficient

estimate for the best model containing r coefficients. Comment on what you observe. How does this

compare to the test MSE plot from (d)?

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