Last updated: 2021-09-30
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Let X be a column centered and scaled genotype matrix, X∈Rn×p. Each individual can be thought of as representing a point in p-dimensional space, where each dimension represents a single SNP. The goal of PCA is to find a new set of orthogonal axes (the principal components), each of which is made up from a linear combination of the original axes, such that the projection of the original data onto these new axes leads to an efficient summary of the structure of the data.
X has the following SVD: Xn×p=Un×nΣn×pVp×p⊺ The projection matrix is V. The transformed data matrix is Y_{n \times p} = X_{n \times p} V_{p \times p} = U_{n\times n} \Sigma_{n \times p}. The ith row of Y represents the individual’s position or projection on the ith principal component.
We can find U using eigenvectors of the similarity matrix between individuals, H_{n \times n} = \frac{1}{p} XX^\intercal.