The near-perfect case of dimensionality reduction

Applying principal component analysis (PCA) to a dataset of four populations sampled evenly: the three primary colors (Red, Green, and Blue) and Black illustrate a near-ideal dimension reduction example. PCA condensed the dataset of these four samples from a 3D Euclidean space (Fig. 1B) into three principal components (PCs), the first two of which explained 88% of the variation and can be visualized in a 2D scatterplot (Fig. 1C). Here, and in all other color-based analyses, the colors represent the true 3D structure, whereas their positions on the 2D plots are the outcome of PCA. Although PCA correctly positioned the primary colors at even distances from each other and Black, it distorted the distances between the primary colors and Black (from 1 in 3D space to 0.82 in 2D space). Thereby, even in this limited and near-perfect demonstration of data reduction, the observed distances do not reflect the actual distances between the samples (which are impossible to recreate in a 2D dataset). In other words, distances between samples in a reduced dimensionality plot do not and cannot be expected to represent actual genetic distances. Evenly increasing all the sample sizes yields identical results irrespective of the sample size (Fig. 1D,E).

When analyzing human populations, which harbor most of the genomic variation between continental populations (12%) with only 1% of the genetic variation distributed within continental…