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- HOMEWORK 1 – V2 Sta414/Sta2104 Probability and Calculus

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1. Probability and Calculus.

1.1. Variance and covariance – 15 pts. Let X, Y be two independent random vectors in R

m.

(a) Show that their covariance is zero.

(b) For a constant matrix A ∈ R

m×m, show the following two properties:

E(X + AY ) = E(X) + AE(Y )

Var(X + AY ) = Var(X) + AVar(Y )A

T

(c) Using part (b), show that if X ∼ N (µ, Σ), then AX ∼ N (Aµ, AΣAT

). Here, you may use

the fact that linear transformation of a Gaussian random vector is again Gaussian.

1.2. Densities – 10 pts. Answer the following questions:

(a) Can a probability density function (pdf) ever take values greater than 1?

(b) Let X be a univariate normally distributed random variable with mean 0 and variance

1/100. What is the pdf of X?

(c) What is the value of this pdf at 0?

(d) What is the probability that X = 0?

1.3. Calculus – 10 pts. Let x, y ∈ R

m and A ∈ R

m×m. In vector notation, what is

(a) the gradient with respect to x of x

T y?

(b) the gradient with respect to x of x

T x?

(c) the gradient with respect to x of x

T Ax?

(d) the gradient with respect to x of Ax?

2. Regression.

2.1. Linear regression – 15 pts. Suppose that X ∈ R

n×m with n ≥ m and Y ∈ R

n

, and that

Y |X, β ∼ N (Xβ, σ2

I). We know that the maximum likelihood estimate βˆ of β is given by

βˆ = (XT X)

−1XT Y.

(a) Find the distribution of βˆ, its expectation and covariance matrix.

(b) Write the log-likelihood implied by the model above, and compute its gradient w.r.t. β.

(c) Assuming that σ

2

is known, what is the probability that an individual parameter βˆ

i

is in

the -neighborhood of the corresponding entry of the true parameter βi

, i.e. P(|βˆ

i −βi

| ≤ )?

(Hint: Use Gaussian CDF Φ(t).)

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2.2. Ridge regression and MAP – 20 pts. Suppose that we have Y |X, β ∼ N (Xβ, σ2

I) and we

place a normal prior on β, i.e., β ∼ N (0, τ 2

I).

(a) Show that the MAP estimate of β given Y in this context is

βˆMAP = (XT X + λI)

−1XT Y

where λ = σ

2/τ 2

.

(b) Show that ridge regression is equivalent to adding m additional rows to X where the j-th

additional row has its j-th entry equal to √

λ and all other entries equal to zero, adding m

corresponding additional entries to Y that are all 0, and and then computing the maximum

likelihood estimate of β using the modified X and Y .

2.3. Cross validation – 30 pts. In this problem, you will write a function that performs

K-fold cross validation procedure to tune the penalty parameter λ in Ridge regression. Your

cross_validation function will rely on 6 short functions which are defined below along with

their variables.

• data is a variable and refers to a (y, X) pair (can be test, training, or validation) where y

is the target (response) vector, and X is the feature matrix.

• model is a variable and refers to the coefficients of the trained model, i.e. βˆ

λ.

• data_shf = shuffle_data(data) is a function and takes data as an argument and returns

its randomly permuted version along the samples. Here, we are considering a uniformly

random permutation of the training data. Note that y and X need to be permuted the same

way preserving the target-feature pairs.

• data_fold, data_rest = split_data(data, num_folds, fold) is a function that takes

data, number of partitions as num_folds and the selected partition fold as its arguments

and returns the selected partition (block) fold as data_fold, and the remaining data as

data_rest. If we consider 5-fold cross validation, num_folds=5, and your function splits

the data into 5 blocks and returns the block fold (∈ {1, 2, 3, 4, 5}) as the validation fold

and the remaining 4 blocks as data_rest. Note that data_rest ∪ data_fold = data, and

data_rest ∩ data_fold = ∅.

• model = train_model(data, lambd) is a function that takes data and lambd as its arguments, and returns the coefficients of ridge regression with penalty level λ. For simplicity,

you may ignore the intercept and use the expression in question 2.2.

• predictions = predict(data, model) is a function that takes data and model as its

arguments, and returns the predictions based on data and model.

• error = loss(data, model) is a function which takes data and model as its arguments

and returns the average squared error loss based on model. This means if data is composed

of y ∈ R

n and X ∈ R

n×p

, and model is βˆ, then the return value is ky − Xβˆk

2/n.

• cv_error = cross_validation(data, num_folds, lambd_seq) is a function that takes

the training data, number of folds num_folds, and a sequence of λ’s as lambd_seq as its

arguments and returns the cross validation error across all λ’s. Take lambd_seq as evenly

spaced 50 numbers over the interval (0.02, 1.5). This means cv_error will be a vector of 50

errors corresponding to the values of lambd_seq. Your function will look like:

data = shuffle_data(data)

for i = 1,2,…,length(lambd_seq)

2

lambd = lambd_seq(i)

cv_loss_lmd = 0.

for fold = 1,2, …,num_folds

val_cv, train_cv = split_data(data, num_folds, fold)

model = train_model(train_cv, lambd)

cv_loss_lmd += loss(val_cv, model)

cv_error(i) = cv_loss_lmd / num_folds

return cv_error

(a) Download the dataset from the course webpage dataset.mat and place it in your working directory, or note its location file_path. For example, file path could be /Users/yourname/Desktop/

• In R:

library(R.matlab)

dataset = readMat(’file_path/dataset.mat’)

data.train.X = dataset$data.train.X

data.train.y = dataset$data.train.y[1,]

data.test.X = dataset$data.test.X

data.test.y = dataset$data.test.y[1,]

• In Python:

import scipy.io as sio

dataset = sio.loadmat(’file_path/dataset.mat’)

data_train_X = dataset[’data_train_X’]

data_train_y = dataset[’data_train_y’][0]

data_test_X = dataset[’data_test_X’]

data_test_y = dataset[’data_test_y’][0]

(b) Write the above 6 functions, and identify the correct order and arguments to do cross

validation.

(c) Find the training and test errors corresponding to each λ in lambd_seq. This part does not

use the cross_validation function but you may find the other functions helpful.

(d) Plot training error, test error, and 5-fold and 10-fold cross validation errors on the same

plot for each value in lambd_seq. What is the value of λ proposed by your cross validation

procedure? Comment on the shapes of the error curves.

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