模式识别作业Homework#2

Homework #2Note: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 indifferent 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 lookslike 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 normalNμ. 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 Bayesclassifier 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 thoseexamples which are in arrays t1, t2, t3). Compute confusion array which has size 3 by 3, and in ith row and jth column contains the number of examples fro which the true class isi while the class your classifier gives is j. Note that all the off-diagonal elements in theconfusion array are errors. Compute the total classification error, easiest way to do it is to use Matlab function sum() and trace().(f)Inspect the off diagonal elements to see if which types of error are more common thanothers. That should give you an idea of where the decision boundaries lie. Now plot the decision regions experimentally (select a fine 2D grid, classify each point on this grid, and plot the class with distinct color). If you love solving quadratic systems of equations, you can find the decision boundaries analytically. Using your decision boundaries, explain why some errors are more common than others.(g)If the model assumed for the data is wrong, than the ML estimate of the parameters arenot even the best parameters to use for classification with that wrong model. That is because the multivariate normal is the wrong distribution to use with out data, the MLE parameters we computed in part (c) are not the ones which will give us the best classification with our wrong model. To confirm this, find parameters for the means and variances (you can change as many as you like, from one to all) which will give better classification rate than the one you have gotten in part (e). Hint: it pays to try to change covariance matrices, rather than the means.(h)Now let’s try to find a better model for our data. Notice that to determine the class of apoint; it is sufficient to consider the distance of that point from the origin. The distance from the origin is very important for classifying our data, while the direction is totally irrelevant. Convert all the training and testing arrays to polar coordinates using Matlib function cart2pol(). Ignore the first coordinate, which is the angle, and only use the second coordinate, which is the radius (or distance from the origin). Assume now that all classes come from normal distribution with unknown mean and variance. Estimate these unknown parameters using ML estimation again using only the training data (the arrays ci’s). Test how this new classifier works using the testing data (the arrays ti’s) by computing the confusion matrix and the total classification error. How does this classifier compare with the one using the multivariate normal assumption and why is there a difference?(i)Experimentally try to find better parameters than those found by ML method for classifierin (h). If you do find better parameters, do they lead to a significantly better classification error? How does it compare to part (g)? Why can’t you find significantly better parameters than MLE for the classifier in (h)?。