Bolla, Marianna BME
Spectra of Spectra
Asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of autocovariances and the union of spectra of the spectral density matrices at the Fourier frequencies. This result has a lot of important consequences as for the relations between time and frequency domain inferences. As a consequence, the principal component transformation of the multivariate real time series results in the complex increment process in the frequency domain. Based on inverse Fourier transformation, low rank approximation of high dimensional time series and dimension reduction is also possible. A dynamic PCA algorithm is illustrated on multivariate financial and gait data.
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