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DATA DRIVEN COMPUTING

Creation date：2024-04-24 17:36:50

Belonging category：经济Economics

Tag COM 2004

DATA DRIVEN COMPUTING

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DATA DRIVEN COMPUTING
Answer THREE questions.
All questions carry equal weight. Figures in square brackets indicate the percentage of avail-
able marks allocated to each part of a question.
COM 2004 1

1. This question concerns probability theory.
a) The discrete random variable X represents the outcome of a biased coin toss. X has
the probability distribution given in the table below,
x H T
P(X = x) θ 1−θ
where H represents a head and T represents a tail.
(i) Write an expression in terms of θ for the probability of observing the sequence
H, T, H, H. [5%]
ANSWER:
θ× (1−θ)×θ×θ = θ3(1−θ)
(ii) A sequence of coin tosses is observed that happens to contain NH heads and
NT tails. Write an expression in terms of θ for the probability of observing this
specific sequence. [5%]
ANSWER:
θNH (1−θ)NT
(iii) Show that having observed a sequence of coin tosses containing NH heads and
NT tails, the maximum likelihood estimate of the parameter θ is given by
NH
NH +NT
[20%]
ANSWER:
Write down the expression for p(x) in terms of the parameter θ,
P(x;θ) = θNH (1−θ)NT
Find maximum by differentiating w.r.t. the parameter θ and setting to 0,
dP(x;θ)
dθ
= NHθ
NH−1(1−θ)NT −θNH NT (1−θ)
NT−1 = 0
Solve for θ,
NHθ
NH−1(1−θ)NT −θNH NT (1−θ)
NT−1 = 0
NH(1−θ)−θNT = 0
NH = NHθ+NT θ
θ =
NH
NH +NT

b) The discrete random variables X1 and X2 represent the outcome of a pair of independent
but biased coin tosses. Their joint distribution P(X1,X2) is given by the probabilities in
the table below,
X1 = H X1 = T
X2 = H λ 3λ
X2 = T 2λ ρ
(i) Write down the probability P(X1 = H,X2 = H). [5%]
ANSWER:
λ
(ii) Calculate the probability P(X1 = H) in terms of λ. [5%]
ANSWER:
λ+2λ = 3λ
(iii) Calculate the probability P(X2 = H) in terms of λ. [5%]
ANSWER:
λ+3λ = 4λ
(iv) Given that the coin tosses are independent and that λ is greater than 0, use your
previous answers to calculate the value of λ. [15%]
ANSWER:
Independence implies that
P(X1 = H,X2 = H) = P(X1 = H)×P(X2 = H)
So
λ = 3λ×4λ
λ =
1
12
COM 2004 3
UNIVERSITY OF SHEFFIELD COM 2004
(v) Calculate the value of ρ. [5%]
ANSWER:
Also entries in the table must sum to one so
λ+3λ+2λ+ρ = 1
Therefore
ρ =
1
2

c) Consider the distribution sketched int the figure below.
0 1
2λ
λ
b
p(x)
x
p(x) =


2λ if 0 <= x < b
λ if b <= x <= 1
0 otherwise
(i) Write an expression for λ in terms of the parameter b. [15%]
ANSWER:
2λb+(1−b)λ = 1
λ =
1
1+b
(ii) Two independent samples, x1 and x2, are observed. x1 has the value 0.25 and
x2 has the value 0.75. Sketch p(x1,x2;b) as a function of b as b varies between
0 and 1. Using your sketch, calculate the maximum likelihood estimate of the
parameter b given the observed samples. [20%]
ANSWER:
• For b in the range 0 to 0.25, p(x1,x2) = λ×λ =
1
(1+b)2
, i.e., 1 to 16/25
• For b in the range 0.25 to 0.75, p(x1,x2) = λ× 2λ =
2
(1+b)2
, i.e., 32/25 to
32/49
• For b in the range 0.75 to 1.0, p(x1,x2) = 2λ×2λ =
4
(1+b)2
, 64/49 to 1
The p(x1,x2) has a maximum value of 64/49 when b = 0.75.

2. This question concerns the multivariate normal distribution.
a) Consider the data in the following table showing the height (x1) and arm span (x2) of a
sample of 8 adults.
x1 151.1 152.4 152.9 156.8 161.8 158.6 157.4 158.8
x2 154.5 162.2 151.5 158.2 165.3 165.6 159.8 162.0
The joint distribution of the two variables is to be modeled using a multivariate Gaussian
with mean vector, µ and covariance matrix, Σ.
(i) Calculate an appropriate value for the mean vector, µ. [5%]
ANSWER:
(156.2,159.9)
(ii) Write down the formula for sample variance. Use it to calculate the unbiased
variance estimate for both height and arm span. [10%]
ANSWER:
Var(x) =
1
N−1
N
∑
i=1
(xi−µx)
2
The variance of height and arm space are 13.9 and 24.9, respectively.
(iii) Write down the formula for sample covariance. Use it to calculate the unbiased
estimate of the covariance between height and arm span. [10%]
ANSWER:
Cov(x,y) =
1
N−1
N
∑
i=1
(xi−µx)(yi−µy)
The covariance between height and arm span is 13.5.
(iv) Write down the covariance matrix, Σ. [5%]
ANSWER:
(
13.9 13.5
13.5 24.9
)

(v) Compute the inverse covariance matrix, Σ−1. [15%]
ANSWER:
(
0.154 −0.0840
−0.0840 0.0860
)
b) Remember that the pdf of a multivariate Gaussian is given by
p(x) =Ce−
1
2 (x−µ)
T Σ−1(x−µ)
where C is a scaling constant that does not depend on x.
Using the answer to 2 (a) and the equation above, answer the following questions.
(i) Who should be considered more unusual:
• Ginny who is 162.1 cm tall and has arms 164.2 cm long, or
• Cho who is 156.0 cm tall and has arms 153.1 cm long?
Show your reasoning. [20%]
ANSWER:
Compute −(x− µ)T Σ−1(x− µ) for each person and see which is smaller. For
Ginny it is -2.67 and for Cho it is -3.71. Therefore Cho’s proportions are more
unusual.
(ii) A large sample of women is taken and it is found that 120 have measurements
similar to those of Ginny. How many women in the same sample would be ex-
pected to have measurements similar to those of Cho? [15%]
ANSWER:
The samples should be in the ratio
exp(−(x1−µ)
T Σ−1(x1−µ)) : exp(−(x2−µ)
T Σ−1(x2−µ))
Plugging in the numbers from the previous section
120∗ exp(−3.71)/exp(−2.67) = 42
COM 2004 7

c) A person’s ‘ape index’ is defined as their arm span minus their height.
(i) Use the data in 2 (a) to estimate a mean and variance for ape index. [10%]
ANSWER:
µ = 3.66
σ2 = 11.6
σ = 3.41
(ii) The figure below shows a standard normal distribution, i.e., X ∼ N(0,1). The
percentages indicate the proportion of the total area under the curve for each
segment.
−3 −2 −1 0 1 2 3
19.1%19.1%
15.0%15.0%
9.2%9.2%
4.4%4.4%
1.7%1.7%
0.5%0.5%
Using the diagram estimate the proportion of the population who will have an ape
index greater than 10.5? [5%]
ANSWER:
10.5 is 2 σ greater than the mean. Therefore about 2.2% of the population will
be expected to have an ape index of 10.5 or greater.
(iii) Using the figure above estimate the mean-centred range of ape indexes that
would include 99% of the population. [5%]
ANSWER:
This would be about + or - 2.5 sigma about the mean. So
• smallest: 3.66−3.41×2.5 =−4.87
• biggest: 3.66+3.41×2.5 = 12.19

i.e., the range [−4.87,12.19].

3. This question concerns classifiers.
a) Consider a Bayesian classification system based on a pair of univariate normal distribu-
tions. The distributions have equal variance and equal priors. The mean of class 1 is
less than the mean of class 2. For each case below say whether the decision threshold
increases, decreases, remains unchanged or can move in either direction.
(i) The mean of class 2 is increased. [5%]

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LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 FINC5090 MATH1052 SciComp Elec4622 FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 MAT 237 BISM2201 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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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