模式识别作业Homework#2

Homework #2

Note:In some problem (this is true for the entire quarter) you will need to make some assumptions since the problem statement may not fully specify the problem space. Make sure that you make reasonable assumptions and clearly state them.

Work alone: You are expected to do your own work on all assignments; there are no group assignments in this course. You may (and are encouraged to) engage in general discussions with your classmates regarding the assignments, but specific details of a solution, including the solution itself, must always be your own work.

Problem:

In this problem we will investigate the importance of having the correct model for classification. Load file hw2.mat and open it in Matlab using command load hw2. Using command whos, you should see six array c1, c2, c3 and t1, t2, t3, each has size 500 by 2. Arrays c1, c2, c3 hold the training data, and arrays t1, t2, t3 hold the testing data. That is arrays c1, c2, c3 should be used to train your classifier, and arrays t1, t2, t3 should be used to test how the classifier performs on the data it hasn’t seen. Arrays c1 holds training data for the first class, c2 for the second class, c3 for the third class. Arrays t1, t2, t3 hold the test data, where the true class of data in t1, t2, t3 comes from the first, second, third classed respectively. Of course, array ci and ti were drawn from the same distribution for each i. Each training and testing example has 2 features. Thus all arrays are two dimensional, the number of rows is equal to the number of examples, and there are 2 columns, column 1 has the first feature, column 2 has the second feature.

(a)Visualize the examples by using Matlab scatter command a plotting each class in

different color. For example, for class 1 use scatter(c1(:,1),c1(:,2),’r’);. Other possible colors can be found by typing help plot.

(b)From the scatter plot in (a), for which classes the multivariate normal distribution looks

like a possible model, and for which classes it is grossly wrong? If you are not sure how to answer this part, do parts (c-d) first.

(c)Suppose we make an erroneous assumption that all classed have multivariate normal

Nμ. Compute the Maximum Likelihood estimates for the means and distributions()∑,

covariance matrices (remember you have to do it separately for each class). Make sure you use only the training data; this is the data in arrays c1, c2, and c3.

(d)You can visualize what the estimated distributions look like using Matlab contour().

Recall that the data should be denser along the smaller ellipse, because these are closer to the estimated mean.

(e)Use the ML estimates from the step (c) to design the ML classifier (this is the Bayes

classifier under zero-one loss function with equal priors). Thus we are assuming that priors are the same for each class. Now classify the test example (that is only those