# CSE/ISYE 6740 Homework 1 Probability solution

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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
−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

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.
· ⃗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,
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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