Detailed Question:

The homework is for computational statistics course (Optimization, R coding, regression).

This is an advanced course. THE R CODES ARE IMPORTANT AND SHOULD BE INCLUDED IN THE ANSWER.

Instructions:

A scientist studied the effects of a very strong storm in trees in a forest in Minnesota as a function of the tree diameters and the severity of winds in the areas where the trees were located. The data for n = 659 trees are given at tree.txt file

The data set consists of the following variables:

D = Diameter of the tree.
S = Severity of the storm at the location of the tree in the scale of 0 to 1, with 1 being most severe.
y = 0 if the tree survived and 1 if the tree is blown down.

Here, y₁, ... , y_n is a set of n observations where yi ∼ Bernouli(Π_i). We are to estimate Π_i = P(y_i = 1), the probability that a tree is blow out, as a function of D and S. In particular we parameterize this probability by the logit function

Π_i(β) = exp(x_i^Tβ)/1 + exp(x_i^Tβ) for i = 1, ......n (1)

where x_i^T = (x_i1, x_i2, x_i3), where x_i1 = 1 is the intercept, x_i2 = log(D), and x_i3 = S, and β = (β₁, β₂, β₃)^T is a set of p parameters to be estimated.

(a) Write the likelihood function l(β) as a function of y_i's and Π_i(β).

(b) Obtain the first differential dΠ_i(dβ) and the second differential ddΠ_i(dβ, dβ).

(c) Obtain the first differential dl(dβ) and use it to derive formulas for ∂l(β)/∂β)_i for B = 1, 2, 3.

(d) Write an R code to implement steepest ascent method with step-halving to obtain the maximum likelihood estimates for β.

(e) Obtain the second differential ddl(dβ, dβ).

(f) Obtain the expected value dd[dl(dβ, dβ)] and use it derive formulas for the elements of the Fisher information matrix.

(g) Write an R code to implement the Fisher-scoring algorithm to obtain the maximum likelihood estimates for β. [Do not use iteratively reweighted least squares to run Fisher scoring.]

(h) Use the glm function in R to obtain maximum likelihood estimates for β.

(i) Plot the three-dimensional graph of Π(β), using your maximum likelihood estimates and as a function of X₁ = S and X₂ = log(D), For your plot, X₁ should range in the interval (0,1) and X₂ should range in the range of observed values for log(D). Moreover, draw the constant value contours and give an interpretation of the values.

(j) What is the prediction of your model for the probability that a tree with diameter 10 would blow down as a function of the severity of wind, as it ranges from 0 to 1. Show the probability using a graph.

(h) What is the prediction of your model for the probability that a tree would be blown down with wind severity 0.4 and diameters ranging from 5 to 30.

Attachment:- tree.txt