Statistical Computing

Question 1: Maximum Likelihood Estimation

a)Defining the model

The logistic regression is performed using train cancellation data. The response variable cancellation is a dummy variable with 0 representing train cancellation and 1 indicating that the train was cancelled. The day’s temperature and rainfall are the predictor variables used to model the probability of a train being cancelled in a particular day. The model has 3 coefficient parameters which are; beta 0, beta 1 – which is associated with temperature and beta 2 – associated with amount of rain.  These coefficients indicate the amount of change in the response variable for any unit change in the predictor variables. The exponential value of Beta 0 shows that odds of a train being cancelled assuming that rain and temperature were recorded as zero. The exponent value of beta 1 explains the odds of train being cancelled having controlled for rain in the model. In the same manner, exponent value of beta 2 indicates the odds of a train being cancelled in a particular day having controlled for temperature.

b)The modelled Probability pi

i)The log-likelihood function

QUESTION 3: RIDGE REGRESSION (15 marks)

Procedure:

(i)Loading the train cancellation information

(ii)The predictor variables are temperature and rainfall, and the response variable is train cancellation

(iii)The predictor variables are plotted against each other.

(iv)The correlation between the predictor variables is obtained with respect to each other. The coefficient estimates for a multilinear model with interaction terms are computed for a range of ridge parameters.

(v) The plots for ridge regression are obtained as shown in the output plots.

% Ridge regression

tc=xlsread(‘TrainCancellation.xlsx’);

temp=tc(:,2);

rainfall=tc(:,3);

cancellation=tc(:,4);

weather=[temp rainfall];

D=x2fx(weather, ‘interaction’)

D(:,1)=[];  %No constant term

k=1:1:365;

b=ridge(cancellation,D,k)

%% plotting the ridge trace

figure(1)

plot(k,b,’LineWidth’,2)

ylim([-100 100])

grid on

xlabel(‘Ridge Parameter’)

ylabel(‘Standardized Coefficient’)

title(‘{bf Ridge Trace}’)

legend(‘Temperature’,’Rainfall’,’Temperature-Rainfall’,’Cancellations’)

References

[1] Dobson, A. J., and A. G. Barnett. An Introduction to Generalized Linear Models. Chapman and Hall/CRC. Taylor & Francis Group, 2008.

[2] K. Sigmon and T. A. Davis, MATLAB Primer, Sixth Edition, Chapman and

Hall/CRC, Boca Raton, FL, 2002

[3] Marquardt, D.W. “Generalized Inverses, Ridge Regression, Biased Linear Estimation, and Nonlinear Estimation.” Technometrics. Vol. 12, No. 3, 1970, pp. 591–612.