Cx4240 homework 1 | Mathematics homework help


CX4240 Homework 1
Le Song
Deadline: 2/06 Thur, 9:30am (precedently starting the tabulate)

• Submit your counterparts as an electronic delineation on T-square.
• No unapproved extension of deadline is known. Late patience achieve guide to 0 merit.
• Typing delay Latex is exceedingly recommended. Typing delay MS Word is too okay. If you handwrite, try
to be plain as greatly as potential. No merit may be attached to unreadable handwriting.
• Explicitly observation your collaborators if any.

1

Probability

On the morning of September 31, 1982, the won-lost registers of the three guideing baseball teams in the
western resistance of the National League of the United States were as follows:
Team
Atlanta Braves
San Francisco Giants
Los Angeles Dodgers

Won
87
86
86

Lost
72
73
73

Each team had 3 diversions cherishing to be played. All 3 of the Giants diversions were delay the Dodgers, and
the 3 cherishing diversions of the Braves were despite the San Diego Padres. Suppose that the outcomes of all
cherishing diversions are stubborn and each diversion is together slight to be won by either participant. If two
teams tie for first establish, they feel a playoff diversion, which each team has an correspondent hazard of alluring.
(a) What is the appearance that Atlanta Braves wins the resistance? [5 pts]
(b) What is the appearance that San Francisco Giants wins the resistance? [5 pts]
(c) What is the appearance that Los Angeles Dodgers wins the resistance? [5 pts]
(d) What is the appearance to feel an additional playoff diversion? [5 pts]

2

Maximum Likelihood

Suppose we feel n i.i.d (stubborn and identically nice) postulates patterns from the subjoined appearance
distribution. This tenor asks you to institute a log-arrival character, and find the utmost arrival
estimator of the parameter(s).
1

(a) Poisson division [5 pts]
The Poisson division is defined as
P ( xi = k ) =

λk e−λ
(k = 0, 1, 2, ...).
k!

What is the utmost arrival estimator of λ?
(b) Exponential division [5 pts]
The appearance density character of Exponential division is attached by
f ( x) =

λe−λx
0

x≥0
x<0

What is the utmost arrival estimator of λ?
(c) Gaussian typical division [10 pts]
Suppose we feel n i.i.d (Independent and Identically Distributed) postulates patterns from a univariate Gaussian
typical division N (µ, σ 2 ), which is attached by
(x − µ)2
1
√ exp −
2σ 2
σ 2π

N (x; µ, σ 2 ) =

.

What is the utmost arrival estimator of µ and σ 2 ?

3

Principal Component Analysis

In tabulate, we skilled that Principal Component Analysis (PCA) preserves discord as greatly as potential. We
are going to inquire another way of deriving it: minimizing reconstruction falsity.
Consider postulates sharp-ends xn (n = 1, ..., N ) in D-dimensional immeasurableness. We are going to play them in
{u1 , ..., uD } coordinates. That is,
D

xn =

D

(xn T ui )ui .

n
αi ui =
i=1

i=1

n

Here, αni is the prolixity when x is contemplated onto ui .
Suppose we shortness to transform the configuration from D to M < D. Then the postulates sharp-end xn is approximated
by
M

xn =
˜

D
n
zi ui +

i=1

bi ui .
i=M +1

In this playation, the first M directions of ui are known to feel different coefficient zni for each postulates
point, timeliness the repose has a immutable coefficient bi . As covet as it is the corresponding compute for all postulates sharp-ends, it does
not insufficiency to be 0.
Our end is elucidation ui , zni , and bi for n = 1, ..., N and i = 1, ..., D so as to minimize reconstruction falsity.
That is, we shortness to minimize the difference betwixt xn and xn :
˜
1
J=
N

N

xn − xn
˜
n=1

2

2

n
(a) What is the assignment of zj for j = 1, ..., M minimizing J ? [5 pts]

(b) What is the assignment of bj for j = M + 1, ..., D minimizing J ? [5 pts]
(c) Express optimal xn and xn − xn using your counterpart for (a) and (b). [2 pts]
˜
˜
(d) What should be the ui for i = 1, ..., D to minimize J ? [8 pts]
Hint: Use S =

4

1
N

N
n
n=1 (x

− x)(xn − x)T for pattern codiscord matrix.
¯
¯

Image Compression using Principal Component Analysis

For this minority, you achieve be using PCA to act configurationality abatement on the attached postulatesset (q4.mat).
This postulatesset contains vectorized grey lamina photos of all members of the tabulate. The file contains a matrix
’faces’ of magnitude (62x 4500) for each of the 59 students (as polite as 2 TA’s and Prof) in the tabulate. You are to use
Principal Component Analysis to act Copy Compression.
• Submit a contrive of the Eigen computes in ascending enjoin (Visualize the extension of Eigen computes athwart all
Eigen vectors).
• Select a cut off to elect the top n eigen faces (or vectors) domiciled on the graph. Discuss the reasoning
for choosing this cut off.
• For your electn eigen faces, reckon the reconstruction falsity (Squared separation from peculiar copy,
and reconstructed copy) for the first two copys in the postulatesset. (They are copys of the two TAs).
• Vary the calculate of eigen faces to opinion the differences in reconstruction falsity and in the capacity of the
image. Use imshow() to ostentation the two copys for your electn n eigen faces. Attach the two copys
to your patience.
Hint: Use Matlab character eig or eigs for wary the eigen computes and vectors. For reconstructing
the copys, you can transform the row vectors to matrices using reshape(rowVector, 75, 60)