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- CSE/ISYE 6740 Homework 1 Probability

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Category: CSE/ISYE 6740

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1 Probability [15 pts]

1. We select a positive integer I with P{I = n} =

1

2n . If I = n, we toss a coin with probability

of heads p = e

−n

. What is the probability that the result is heads? [3.5 pts]

2. In a network of computers, 15% of the computers are infected by a virus V . An anti-virus

scan has the property that if a computer is infected with V , the scan will detect the infection

to be positive 95% of the time. However, if the computer is not infected, the scan will still

detect the infection to be positive 10% of the time. All the computers which are detected to be

infected, are applied with a corrective software-patch, which causes corruption of computers’

files 20% of the time. Given that a computer picked at random has corrupted files, what is

the probability that it was actually infected with the virus V to begin with? [3.5 pts]

3. Charlie has a choice to take a bus or walk to attend CSE6740 lecture. If he walks, he gets late

with a probability of 1

2

. However, if he takes a bus, he gets late only with a probability of 1

6

.

Further, if he gets on time, he always keeps the same mode of travel the day after, whereas

he always changes when he gets late. Let p be the probability that Charlie walks on the first

day.

(a) What is the probability that Charlie walks on the n

th day? [4 pts]

(b) What is the probability that Charlie gets late on the n

th day? [4 pts]

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2 Maximum Likelihood [15 pts]

Suppose we have n i.i.d (independent and identically distributed) data samples from the following

probability distribution. This problem asks you to build a log-likelihood function, and find the

maximum likelihood estimator of the parameter(s).

(a) Exponential distribution [5 pts]

The exponential distribution is defined as

P(x|β) = 1

β

e

− x

β , with 0 ≤ x < ∞

Please find the MLE of the parameter β

(b) Pareto distribution [5 pts]

The Pareto distribution has been used in economics for a density function with a slowly decaying

tail:

f(x|x0, θ) = θx0

θx

−θ−1

, x ≥ x0, θ > 1

assume that x0 > 0 is given. Find the MLE of θ.

(c) Normal linear regression model [5 pts]

The regression equations can be written in matrix form as

y = Xβ + ε

where y is the N × 1 vector of observations of the dependent variable, X is the N × K matrix

of regressors, and ε is the N × 1 error terms. With the i.i.d assumption, multivariate normal

distribution of ε on X, and full rank X, we can construct that the likelihood function of the linear

regression model is

L(β, σ2

; y, X) =

2πσ2

−N/2

exp

−

1

2σ

2

X

N

i=1

(yi − xiβ)

2

!

Show that the MLE of the regression coefficients β and the variance of the error terms σ

2 are

βˆN = (XTX)

−1XTy

σˆ

2

N =

1

N

X

N

i=1

yi − xiβˆN

3 PCA [20 pts]

Suppose that we use q directions, given by q orthogonal length-one vectors ⃗w1, … ⃗wq. Please prove

that minimizing the mean squared error is equivalent to maximizing the sum of the variances of

the scores along these directions.

1. Write w for the matrix forms by stacking the ⃗wi

. Prove that wT w = Iq. [4 pts]

2. Find the matrix of p-dimensional approximations based on these scores in terms of x and w.

Hint: your answer should reduce to ( ⃗xi

· ⃗w1) ⃗w1 when q = 1. [4 pts]

3. Using the conclusion from question 3.1, show that the MSE(mean squared error) of using the

vectors ⃗w1, … ⃗wq is the sum of two terms, one of which depends only on x and not w, and

the other depends only on the scores along those directions (and not otherwise on what those

directions are). [10 pts]

4. Explain in what sense minimizing projection residuals is equivalent to maximizing the sum

of variances along the different directions. [2 pts]

4 Clustering [20 pts]

Given N data points x

n

(n = 1, …, N), K-means clustering algorithm groups them into K clusters.

With respect to K-means clustering answer the following question:

1. Consider the given single dimensional data with 4 data points x1 = 1, x2 = 3, x3 = 6, x4 = 7.

Let’s consider k = 3 for this situation. What is the optimal clustering for this data? [4 pts]

2. For the above part (1), show that by changing the center initialization we get a suboptimal

cluster assignment that cannot be further improved. [4 pts]

3. Prove that the K-means algorithm converges to a local optimum in finite steps. [8 pts]

4. Original K-means algorithm uses Euclidian distance as the metric to compute the distance

between data points. What is the disadvantage of using this distance function and suggest a

solution to overcome this? [4 pts]

5 Programming: Image Compression [Report 10 pts + Code 20

pts]

In this programming assignment, you are going to apply clustering algorithms for image compression. Before starting this assignment, we strongly recommend reading PRML Section 9.1.1, page

428 – 430.

To ease your implementation, we provide a skeleton code containing image processing part.

homework1.m is designed to read an RGB bitmap image file, then cluster pixels with the given number of clusters K. It shows converted image only using K colors, each of them with the representative color of centroid. To see what it looks like, you are encouraged to run homework1(‘beach.bmp’,

3) or homework1(‘football.bmp’, 2), for example.

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Your task is implementing the clustering parts with two algorithms: K-means and K-medoids.

We learned and demonstrated K-means in class, so you may start from the sample code we distributed.

The file you need to edit is mykmeans.m and mykmedoids.m, provided with this homework.

In the files, you can see it calls Matlab function kmeans initially. Comment this line out, and

implement your own in the files. You would expect to see similar result with your implementation

of K-means, instead of kmeans function in Matlab.

K-medoids

In class, we learned that the basic K-means works in Euclidean space for computing distance

between data points as well as for updating centroids by arithmetic mean. Sometimes, however,

the dataset may work better with other distance measures. It is sometimes even impossible to

compute arithmetic mean if a feature is categorical, e.g, gender or nationality of a person. With

K-medoids, you choose a representative data point for each cluster instead of computing their

average.

Given N data points xn

(n = 1, …, N), K-medoids clustering algorithm groups them into K

clusters by minimizing the distortion function J =

PN

n=1

PK

k=1 r

nkD(xn

, µk

), where D(x, y) is a

distance measure between two vectors x and y in same size (in case of K-means, D(x, y) = ∥x−y∥

2

),

µ

k

is the center of k-th cluster; and r

nk = 1 if xn belongs to the k-th cluster and r

nk = 0 otherwise.

In this exercise, we will use the following iterative procedure:

• Initialize the cluster center µ

k

, k = 1, …, K.

• Iterate until convergence:

– Update the cluster assignments for every data point xn

: r

nk = 1 if k =j D(xn

, µj

), and

r

nk = 0 otherwise.

– Update the center for each cluster k: choosing another representative if necessary.

There can be many options to implement the procedure; for example, you can try many distance

measures in addition to Euclidean distance, and also you can be creative for deciding a better

representative of each cluster. We will not restrict these choices in this assignment. You are

encouraged to try many distance measures as well as way of choosing representatives.

Formatting instruction

Both mykmeans.m and mykmedoids.m take input and output format as follows. You should not

alter this definition, otherwise your submission will print an error, which leads to zero credit.

Input

• pixels: the input image representation. Each row contains one data point (pixel). For

image dataset, it contains 3 columns, each column corresponding to Red, Green, and Blue

component. Each component has an integer value between 0 and 255.

• K: the number of desired clusters. Too high value of K may result in empty cluster error.

Then, you need to reduce it.

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Output

• class: cluster assignment of each data point in pixels. The assignment should be 1, 2, 3, etc.

For K = 5, for example, each cell of class should be either 1, 2, 3, 4, or 5. The output should

be a column vector with size(pixels, 1) elements. Start from 0 if you are using python.

• centroid: location of K centroids (or representatives) in your result. With images, each

centroid corresponds to the representative color of each cluster. The output should be a

matrix with K rows and 3 columns. The range of values should be [0, 255], possibly floating

point numbers.

Hand-in

Both of your code and report will be evaluated. Submit mykmeans.m and mykmedoids.m files as a

zip to Homework 1 Programming (submit homework1.py if you are using python). In your report,

answer to the following questions:

1. Within the K-medoids framework, you have several choices for detailed implementation.

Explain how you designed and implemented details of your K-medoids algorithm, including

(but not limited to) how you chose representatives of each cluster, what distance measures

you tried and chose one, or when you stopped iteration.

2. Attach a picture of your own. We recommend size of 320 × 240 or smaller.

3. Run your K-medoids implementation with the picture you chose above, with several different

K. (e.g, small values like 2 or 3, large values like 16 or 32) What did you observe with

different K? How long does it take to converge for each K?

4. Run your K-medoids implementation with different initial centroids/representatives. Does

it affect final result? Do you see same or different result for each trial with different initial

assignments? (We usually randomize initial location of centroids in general. To answer this

question, an intentional poor assignment may be useful.)

5. Repeat question 3 and 4 with K-means. Do you see significant difference between K-medoids

and K-means, in terms of output quality, robustness, or running time?

Note

• You may see some error message about empty clusters even with Matlab implementation,

when you use too large K. Your implementation should treat this exception as well. That

is, do not terminate even if you have an empty cluster, but use smaller number of clusters in

that case.

• We will grade using test pictures which are not provided. We recommend you to test your

code with several different pictures so that you can detect some problems that might happen

occasionally.

• If we detect copy from any other student’s code or from the web, you will not be eligible for

any credit for the entire homework, not just for the programming part. Also, directly calling

Matlab function kmeans or other clustering functions is not allowed.

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