MAST30027 Assignment 3

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MAST30027: Modern Applied Statistics

Assignment 3, 2024.

Due: 11:59pm Monday September 23rd

  • This assignment is worth 12% of your total mark.
  • To get full marks, show your working including 1) R commands and outputs you use, 2) mathematics derivation, and 3) rigorous explanation why you reach conclusions or answers. If you just provide final answers, you will get zero mark.
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1. The file assignment3_2024_prob1.txt contains 300 observations. We can read the observations and make a histogram as follows:

“`R

X <- scan(file=”assignment3_2024_prob1.txt”, what=double())
Read 300 items
length(X)
[1] 300
hist(X, breaks=0:max(X))
“`
We will model the observed data using a mixture of three negative binomial distributions. Specifically, we assume that the observations $X_1,…, X_{300}$ are independent of each other, and each $X_i$ follows the mixture model
$$
Z_i \sim Categorical(\pi_1, \pi_2, 1 – \pi_1 – \pi_2),
$$
$$
X_i|Z_i = 1 \sim NegBinomial(r, p_1),
$$
$$
X_i|Z_i = 2 \sim NegBinomial(r, p_2),
$$
$$
X_i|Z_i = 3 \sim NegBinomial(r, p_3),
$$
where $r$ is some known constant. The negative binomial distribution has the probability mass function
$$
f(x; r, p) = \binom{x + r – 1}{r – 1}p^r(1 – p)^x.
$$
We aim to obtain MLE of the parameters $\theta = (\pi_1, \pi_2, p_1, p_2, p_3)$ using the EM algorithm.

(a) (5 marks) Let $X = (X_1,…, X_{300})$ and $Z = (Z_1,…, Z_{300})$. Derive the expectation of the complete log – likelihood, $Q(\theta, \theta^0) = E_{Z|X, \theta^0}[\log(P(X, Z|\theta))]$.
(b) (3 marks) Derive E – step of the EM algorithm.
(c) (5 marks) Derive M – step of the EM algorithm.
(d) (5 marks) Note: Your answer for this problem should be typed. Answers including screen – captured R codes or figures won’t be marked.
Implement the EM algorithm and obtain MLE of the parameters by applying the implemented algorithm to the observed data, $X_1,…, X_{300}$, assuming a value of $r = 10$. Terminate the algorithm when the number of EM iterations reaches 500, or when the incomplete log – likelihood has changed by less than $10^{-5}$ ($\epsilon = 10^{-5}$). Run the EM algorithm twice with the following two different initial values, and report the parameter estimates corresponding to the highest incomplete log – likelihood.
|$\pi_1$|$\pi_2$|$p_1$|$p_2$|$p_3$|
|—|—|—|—|—|
|1st initial values|0.1|0.3|0.1|0.8|0.5|
|2nd initial values|0.2|0.4|0.3|0.2|0.7|
For each EM run, check that the incomplete log – likelihoods increase at each EM – iteration by plotting them.

2. The file assignment3_2024_prob2.txt contains 100 additional observations. We can read the 300 obser – vations from the Problem 1 and the new 100 observations and make histograms as follows:

“`R

X <- scan(file=”assignment3_2024_prob1.txt”, what=double())
Read 300 items
X.more <- scan(file=”assignment3_2024_prob2.txt”, what=double())
Read 100 items
length(X)
[1] 300
length(X.more)
[1] 100
xgrid <- 0:max(c(X, X.more))
par(mfrow=c(2, 2))
hist(X, breaks=xgrid, ylim=c(0, 21), main=”Histogram of X”, xlab=”X”)
hist(X.more, breaks=xgrid, ylim=c(0, 21), main=”Histogram of X.more”, xlab=”X.more”)
hist(c(X, X.more), breaks=xgrid, ylim=c(0, 21), main=”Histogram of X & X.more”, xlab=”X & X.more”)
``
Let $X_1,..., X_{300}$ and $X_{301},..., X_{400}$ denote the 300 observations fromassignment3_2024_prob1.txtand the 100 observations fromassignment3_2024_prob2.txt`, respectively. We assume that the observations $X_1,…, X_{400}$ are independent to each other. We model $X_1,…, X_{300}$ using a mixture of three negative binomial distributions (as we did in Problem 1), but we model $X_{301},…, X_{400}$ using the first of those three negative binomial distributions. Specifically, for $i = 1,…, 300$, $X_i$ follows the mixture model
$$
Z_i \sim Categorical(\pi_1, \pi_2, 1 – \pi_1 – \pi_2),
$$
$$
X_i|Z_i = 1 \sim NegBinomial(r, p_1),
$$
$$
X_i|Z_i = 2 \sim NegBinomial(r, p_2),
$$
$$
X_i|Z_i = 3 \sim NegBinomial(r, p_3),
$$
whereas for $i = 301,…, 400$, $X_i$ follows
$$
X_i \sim NegBinomial(r, p_1),
$$
where $r$ is some known constant. We aim to obtain MLE of the parameters $\theta = (\pi_1, \pi_2, p_1, p_2, p_3)$ using the EM algorithm.

(a) (5 marks) Let $X = (X_1,…, X_{400})$ and $Z = (Z_1,…, Z_{300})$. Derive the expectation of the com – plete log – likelihood, $Q(\theta, \theta^0) = E_{Z|X, \theta^0}[\log(P(X, Z|\theta))]$.
(b) (5 marks) Derive E – step and M – step of the EM algorithm.
(c) (5 marks) Note: Your answer for this problem should be typed. Answers including screen – captured R codes or figures won’t be marked.
Implement the EM algorithm and obtain MLE of the parameters by applying the implemented algorithm to the observed data, $X_1,…, X_{400}$, assuming a value of $r = 10$. Terminate the algorithm when the number of EM iterations reaches 500, or when the incomplete log – likelihood has changed by less than $10^{-5}$ ($\epsilon = 10^{-5}$). Run the EM algorithm twice with the following two different initial values, and report the parameter estimates corresponding to the highest incomplete log – likelihood.
|$\pi_1$|$\pi_2$|$p_1$|$p_2$|$p_3$|
|—|—|—|—|—|
|1st initial values|0.1|0.3|0.1|0.8|0.5|
|2nd initial values|0.2|0.4|0.3|0.2|0.7|
For each EM run, check that the incomplete log – likelihoods increase at each EM – iteration by plotting them.