Principal component analysis of 1/fα noise

Principal component analysis (PCA) is a popular data analysis method. One of the motivations for using PCA in practice is to reduce the dimension of the original data by projecting the raw data onto a few dominant eigenvectors with large variance (energy). Due to the ubiquity of 1/fα noise in science and engineering, in this Letter we study the prototypical stochastic model for 1/fα processes--the fractional Brownian motion (fBm) processes using PCA, and find that the eigenvalues from PCA of fBm processes follow a power-law, with the exponent being the key parameter defining the fBm processes. We also study random-walk-type processes constructed from DNA sequences, and find that the eigenvalue spectrum from PCA of those random-walk processes also follow power-law relations, with the exponent characterizing the correlation structures of the DNA sequence. In fact, it is observed that PCA can automatically remove linear trends induced by patchiness in the DNA sequence, hence, PCA has a similar capability to the detrended fluctuation analysis. Implications of the power-law distributed eigenvalue spectrum are discussed