Application of PCA in the Frequency Domain using the Covariance Spectrum

In this paper we will consider a re-interpretation of principal component analysis (PCA) for the case of time series of correlated data where the principal components are partly covered in noise, so that only in a part of the considered frequency band, the principal components are visible. In this case it might be useful to consider a representation of the spectral density matrix that is a real and one-sided function of frequency, in this paper denoted the covariance spectrum, which is directly representing the covariance matrix as a function of frequency. This means that if a principal component is dominating in a narrow frequency band, it might not be visible on the time domain, but be visible in the narrow band. The covariance spectrum can then be added in the considered band to represent the covariance matrix of the principal component, and classical PCA can be used to illustrate the properties of the principal component. Basic theory is introduced, and the principle is illustrated on a case with two harmonics in white noise acting on an arbitrary mechanical system.