5. (20 pts) Poisson regression. The Poisson distribution is commonly used to model count data, e.g. the number of failures of disk drives in a data center over some period of time. If Y is the random variable that represents the number of events in the time interval, then the probability that exactly y events occur is P(Y = y) = Hex where p is the single parameter (the mean) for the Poisson distribution. Suppose we have some predictor (or explanatory) variables that we think might influence the number of events. For the example of the data center, these explanatory variables could be temperature, age of drives, manufacturer, etc. We have the count data y1, 32, -...I'm and for each y; we have a vector of explanatory variables a;. The Poisson model is Hi = mi = explomoro + auri + .. .+ dinIn) =eliz where / is the fitted value for observation i (i = 1, ..., m), a; is the vector of predictor variables for observation i, and r ( R" is the vector of parameters to be estimated (i.e. the regression coefficients). Here, the vector r includes the intercept term, so a; = 1. That is to say, there are n parameters to be estimated, including the intercept. A c Rx" is the model matrix, the first column of which is a column of ones for the intercept. In the Poisson regression model, the function that "links" the linear predictor to the response is log (/) = afr In the terminology of Generalized Linear Models, this is the link function (for Poisson data). Your assignment is to obtain the maximum likelihood estimates for the model parameters r for a data set of ship damage incidents (using the Poisson model). Follow the very same steps that were demonstrated in class for logistic regression. (a) Write the likelihood function. (b) Write the log-likelihood function I(x). (c) Write an expression for the gradient VI(x). (d) Write an expression for the Hessian V-1(x). (e) Write Vi(x) and V-I(x) in matrix notation. (f) Write the step for using Newton's method, but express the step as solving a weighted least squares problem where the response is a nonlinear function of r