14.5: Principal Component Analysis

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PCA breaks n-dimensional data into n vectors so that each data point can be represented by a linear combination of the n vectors. These n vectors have two interesting properties: first, they are ordered by their variance so that the first vector is representative of the data with the highest variation in the data, and second, they are orthogonal. These vectors are therefore called principal components.

This approach has a strong geometrical interpretation: given data such as two-dimensional points, say in the shape of a rectangle, the points along the long axis of the rectangle have higher variance than those along the the short axis. Every point in this point cloud can then be reconstructed by a linear combination of the principal component along the long axis and the principal component along the short axis. Finding these vectors is therefore akin finding the principal axes of the rectangle regardless of its orientation.