Statistical Methods in the Frequency Domain
In Chapters 3 and 4, we saw many applied time series problems that involved relating series to each other or to evaluating the effects of treatments or design parameters that arise when time-varying phenomena are subjected to periodic stimuli. In many cases, the nature of the physical or biological phenomena under study are best described by their Fourier components rather than by the difference equations involved in ARIMA or state-space models. The fundamental tools we use in studying periodic phenomena are the discrete Fourier transforms (DFTs) of the processes and their statistical properties. Hence, in Section 5.2, we review the properties of the DFT of a multivariate time series and discuss various approximations to the likelihood function based on the large-sample properties and the properties of the complex multivariate normal distribution. This enables extension of the classical techniques discussed in the following paragraphs to the multivariate time series case.
KeywordsDiscrete Wavelet Transform Multivariate Time Series Spectral Envelope Frequency Domain Method Spectral Matrix
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