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MahalanobisDistance

Create the MahalanobisDistance function that we created in the lecture. It takes as input a single class from the pca model mdl created using the 'my_fitpca' function and an [1xN] feature vector v, containing N feature measurements for a sample. It outputs the Mahalanobis distance from the feature vector to the pca model. It also outputs b, the coordinates within the pca model space for the feature vector, and std_per_mode, which contains the standard deviation distances from the mean for each mode of variation in the pca model.

The steps to perform in the function are as follows:

• compute b, our PCA coordinates for v, by subtracting pcamdl.mu from v and then projecting the result onto the eigenvectors by pre-multiplying by the eigenvectors matrix
• compute std_per_mode, the standard deviation distance of v from the mean for each mode of variation. This is the magnitude of b, normalized by the square root of the eigenvalues
• compute md, the Mahalanobis distance, as the square root of the sum of the squared entries in std_per_mode

Function:

function [md,b,std_per_mode] = MahalanobisDistance(pcamdl,v)


% compute b, our PCA coordinates for v
b = v - pcamdl.mu;

% compute std_per_mode, the standard deviation distance from the mean for v for each mode of variation
std_per_mode = sqrt(sum(b .^ 2)) ./ sqrt(pcamdl.eigvals);

% compute md, the Mahalanobis distance
md = sqrt(sum(std_per_mode));

end

Code:

mdl.class(1).eigvects = [0.7071 0.7071]; 
mdl.class(1).eigvals = 4; 
mdl.class(1).mu = [2 2];
v = [2,2];
md = MahalanobisDistance(mdl.class(1),v) % md = 0 because v = mu
v = [2,2] + 2*[0.7071 0.7071];
md = MahalanobisDistance(mdl.class(1),v) % md = 1 because v is 2/sqrt(eigval)=2/2=1 standard deviations from mean along eigenvector
v = [2,2] - 6*[0.7071 0.7071];
md = MahalanobisDistance(mdl.class(1),v) % md = 3 because v is 6/sqrt(eigval)=6/2=3 standard deviations from mean along eigenvector
v = [2,2] + [-0.7071 0.7071];
md = MahalanobisDistance(mdl.class(1),v) % md = 0 because v lies in a direction from mu that is orthogonal to the eigenvector

0 of 3 Passed

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• Assessment result: incorrectIs b correct?Variable size_of_b has an incorrect value. size(b) = [1 4] but should be size [4 1]
• Assessment result: incorrectIs std_per_mode correct?Variable std_per_mode has an incorrect value. Function found std_per_mode = [1.3699 1.5086 1.8488 2.0059]' but should have returned [0.086622 0.83919 0.25637 1.6388]'
• Assessment result: incorrectIs md correct?Variable md has an incorrect value. Function found md = 2.5948 but should have returned 1.861