1. The goal of this problem is to design a classi er that will predict if a person, represented by measurements of their face, is happy or angry. A key to any classi cation task is to use good features that discriminate between the two categories.
Consider
is happy
person
clues.
The image below depicts a set of landmarks that can be automatically measured in a face image. These include points corresponding to the eyes, the brows, the nose, and the mouth. We will use n = 9 distances between pairs of these points to classify whether the image represents someone who is happy or angry.
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Features extracted from m = 128 face images (like the two shown above) provided in the m-by-n matrix X in the le face_emotion_data.mat. This le also includes the m 1 vector of labels y. Here happy faces are labeled +1 and angry faces are labeled
1. Your task is to nd the weights for a linear classi er that will use the features to predict whether the emotion displayed on a face image is happy or angry.
De ne a feature vector xiT
=
x1i
x2ix9i
T
and classi er weights w =
w1 w2
w9
T
so that the label, yi xi
w.
a) Use the training data X and y and a least squares problem to train your classi er weights.
b) Explain how to use the weights you found to classify a new face image as happy or angry?
c) Which features seem to be most important? Justify your answer. Note that the nine columns of the training data feature matrix X have been normalized to have the same two-norm.
d) Design a classi er based on three of the nine features. Which three should you choose? Describe the procedure for designing your classi er.
e) What percent of the training labels are incorrectly classi ed using all nine fea-tures? What percent of the training labels are incorrectly classi ed using your reduced set of three features?
f) Now use cross validation to assess your classi er performance. Divide the available
data in to eight subsets of sixteen samples (e.g., examples 1 16; 17 32; : : : ; 113 128). Use seven sets to design your classi er weights, then use the remaining hold-out set to evaluate the classi er performance. Compute the number of mis-classi cations made on this hold-out set and divide that number by 16 (the size of the set) to estimate the error rate for that hold-out set. Repeat this process eight times using the eight di erent possible divisions between training and hold-out sets and average the error rates to obtain a nal performance estimate.
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