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Optimal subspaces and constrained principal component analysis. (English) Zbl 1043.15002
Let \(D\) be a linear \(d\)-dimensional subspace of the Euclidean vector space \(\mathbb R^p\) (its elements are written as columns), and let \(z_1,z_2,\ldots,z_n\) be points of \(\mathbb R^p\). The author determines a linear \(d+s\)-dimensional subspace \(H\) of \(\mathbb R^p\) containing \(D\) and minimizing the sum of the squared distances of the \(z_i\)’s to \(H\). The solution is given elegantly in terms of eigenvectors which belong to the largest \(s\) eigenvalues of the symmetric matrix \(M_DZ'ZM_D\). Here \(M_D\) is the matrix of the orthogonal projection of \(\mathbb R^p\) onto \(D^\perp\), \(Z\) is the matrix with rows \(z_1',z_2',\ldots,z_n'\), and a prime denotes transposition.
MSC:
15A03 Vector spaces, linear dependence, rank, lineability
62H25 Factor analysis and principal components; correspondence analysis
15A18 Eigenvalues, singular values, and eigenvectors
Software:
Flury
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