Homework 4 in STAT630: Advanced Statistical Data Analysis @ CSU
[12 pts] (Boos and Stephanski 2.48) Suppose that the data \(Y_1, \dots, Y_n\) are assumed to come from a mixture of two binomial distributions. Thus, \[ f(y; p, \theta_1, \theta_2) = p {n \choose y} \theta_1^y(1 - \theta_1)^{n-y} + (1 - p) {n \choose y} \theta_2^y(1 - \theta_2)^{n-y}. \] Find an expression for \(Q(p, \theta_1, \theta_2, p^v, \theta_1^v, \theta_2^v)\) and the updating formulas. (Hint: to get the updating formulas, take partial derivatives of \(Q\) with respect to \(p, \theta_1\), and \(\theta_2\) and proceed as usual.)
Synthetic data are available here. This is synthetic data which comes from a mixture of three bivariate Gaussian distributions.
[20 pts] Write your own code to perform the EM algorithm to fit a model to this data. As we learned in class, a “useful” solution is found by finding a local maximum, as the global maximum occurs at the boundary of the parameter space. Turn in your plan (psuedocode), including all derived quantities, as well as your actual code, and report parameter estimates.
[10 pts] Use the R package mclust
to find a “useful” solution. You should download the package, and
familiarize yourself with what it does. The package will likely fit more
than one model. Briefly explain the best fitting model for 3 mixture
components and report parameter estimates.
[5 pts] Using either your results from a) or b), produce a plot which shows to your boss (who doesn’t know about the EM algorithm or coding) that your solution is useful.
[3 pts] Use the mclust package to
choose the number of mixture components (pretending that you don’t know
that the number of mixture components is three). What number of
components does it choose?