Cx4240 homework 1 | Mathematics homework help


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

• Submit your rejoinders as an electronic enactation on T-square.
• No unapproved extension of deadline is undisputed. Late inferiority obtain administer to 0 trustworthiness.
• Typing after a opportunity Latex is greatly recommended. Typing after a opportunity MS Word is too okay. If you handwrite, try
to be pure as abundantly as practicable. No trustworthiness may be ardent to unreadable handwriting.
• Explicitly declaration your collaborators if any.

1

Probability

On the early of September 31, 1982, the won-lost registers of the three administering baseball teams in the
western analysis 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 pastimes fostering to be played. All 3 of the Giants pastimes were after a opportunity the Dodgers, and
the 3 fostering pastimes of the Braves were counter the San Diego Padres. Suppose that the outcomes of all
fostering pastimes are recalcitrant and each pastime is homogeneous slight to be won by either participant. If two
teams tie for first attribute, they own a playoff pastime, which each team has an resembling befoulment of seductive.
(a) What is the presumption that Atlanta Braves wins the analysis? [5 pts]
(b) What is the presumption that San Francisco Giants wins the analysis? [5 pts]
(c) What is the presumption that Los Angeles Dodgers wins the analysis? [5 pts]
(d) What is the presumption to own an concomitant playoff pastime? [5 pts]

2

Maximum Likelihood

Suppose we own n i.i.d (recalcitrant and identically as sorted) facts scantlings from the subjoined presumption
distribution. This problem asks you to institute a log-air discharge, and find the zenith air
estimator of the parameter(s).
1

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

λk e−λ
(k = 0, 1, 2, ...).
k!

What is the zenith air estimator of λ?
(b) Exponential dispensation [5 pts]
The presumption hebetude discharge of Exponential dispensation is ardent by
f ( x) =

λe−λx
0

x≥0
x<0

What is the zenith air estimator of λ?
(c) Gaussian natural dispensation [10 pts]
Suppose we own n i.i.d (Independent and Identically Distributed) facts scantlings from a univariate Gaussian
natural dispensation N (µ, σ 2 ), which is ardent by
(x − µ)2
1
√ exp −
2σ 2
σ 2π

N (x; µ, σ 2 ) =

.

What is the zenith air estimator of µ and σ 2 ?

3

Principal Component Analysis

In systematize, we scholarly that Principal Component Analysis (PCA) preserves hostility as abundantly as practicable. We
are going to search another way of deriving it: minimizing reconstruction deception.
Consider facts purposes xn (n = 1, ..., N ) in D-dimensional interspace. We are going to enact 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 tediousness when x is contemplated onto ui .
Suppose we insufficiency to subject the extent from D to M < D. Then the facts purpose xn is approximated
by
M

xn =
˜

D
n
zi ui +

i=1

bi ui .
i=M +1

In this enactation, the first M directions of ui are undisputed to own different coefficient zni for each facts
point, opportunity the security has a true coefficient bi . As covet as it is the corresponding compute for all facts purposes, it does
not need to be 0.
Our sight is contrast ui , zni , and bi for n = 1, ..., N and i = 1, ..., D so as to minimize reconstruction deception.
That is, we insufficiency to minimize the difference among 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 rejoinder 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 scantling cohostility matrix.
¯
¯

Image Compression using Principal Component Analysis

For this exception, you obtain be using PCA to discharge extentality decrease on the ardent factsset (q4.mat).
This factsset contains vectorized grey lamina photos of all members of the systematize. The file contains a matrix
’faces’ of extent (62x 4500) for each of the 59 students (as well-mannered-mannered as 2 TA’s and Prof) in the systematize. You are to use
Principal Component Analysis to discharge Representation Compression.
• Submit a devise of the Eigen computes in ascending arrange (Visualize the growth of Eigen computes abutting all
Eigen vectors).
• Select a cut off to cull the top n eigen faces (or vectors) inveterate on the graph. Discuss the reasoning
for choosing this cut off.
• For your culln eigen faces, estimate the reconstruction deception (Squared interspace from pristine representation,
and reconstructed representation) for the first two representations in the factsset. (They are representations of the two TAs).
• Vary the enumerate of eigen faces to light the differences in reconstruction deception and in the tendency of the
image. Use imshow() to vault the two representations for your culln n eigen faces. Attach the two representations
to your inferiority.
Hint: Use Matlab discharge eig or eigs for guarded the eigen computes and vectors. For reconstructing
the representations, you can apply the row vectors to matrices using reshape(rowVector, 75, 60)