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Fig. 3.8. Principal components of data distributions. In principal component analysis, the data originally represented in (x, y) coordinates are transformed into the principal component coordinate system: The first principal component (PC₁) aligns with the direction of maximum variance in the data, and the second (PC₂) is orthogonal to it. The lengths of the axes reflect the variance along each coordinate dimension. (a) The two-dimensional distribution has a linear structure, and the first component alone is a good representation. However, with a nonlinear distribution (b), PCA does not result in a good lower dimensional representation, even though the distribution lies on a one-dimensional curve.