Phase Distributions of Complex Multitaper Transfer Function Estimates

– In time series analysis, trends are often studied via sequences of observations taken with respect to, as the name suggests, time. Time series regression models are a classic way of viewing a response time series as a function of one or multiple predictor time series. However, the default assumptions of these models fail to account for the data’s temporal trends. By instead building a complex regression model in the frequency domain, these assumptions can be relaxed, providing insight into what coherency is present in the model. The coefficient of the resulting complex regression model is known as a transfer function of frequency. There are several techniques used to estimate transfer functions, but all are subject to the bias-variance trade-off occurring as a byproduct of transformation to the frequency domain. Multitaper spectrum estimation has been shown to minimize spectral leakage (broad-band bias) while providing flexibility in terms of bandwidth selection (variance). Thus, exploration of Multitaper Transfer Function Estimators (MTFEs) is an alluring topic of research. Previous work has explored distribution theory for the modulus of MTFEs, in addition to MTFE variance across simulations, and has revealed effective methods of signal detection more robust to frequency modulation than classic alternatives. However, the distribution of an MTFE's phase has not been explored. This paper demonstrates that for models whose underlying noise is stationary and Gaussian, the MTFE at a given frequency is distributed as a complex Gaussian random variable. From the perspective of the time domain, one can infer parameters of the phase distribution of the MTFE by way of estimating the autocovariance function of the response. This phase distribution provides information which may be useful for signal detection.