Multitaper Transfer Function Estimates and the Hunt for Coherent Signals Trespassing on Time Series Regression Models

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Authors
Griffith, Skye
Keyword
Time Series , Complex Regression , Multitaper , Spectrum , Transfer Function , Signal Detection , Coherence , Stationary
Abstract
This work provides a test statistic for the detection of coherent harmonic signals underlying time series regression models. Time series regression observes a response series modelled as a function of one or more predictor series, relating predictors and response over time. Unfortunately, standard time-domain regression models don’t account for what temporal factors may be present, operating under assumptions of independence commonly unsatisfied by time series data. Linear filter model coefficients provide an unintuitive alternative, being not necessarily analogous to standard regression coefficients. Analysing the filter model via complex regression in the frequency domain by way of a transfer function, however, provides insight into the frequency structures of the original time series. Identifying the distribution of the transfer function's modulus, phase, and variance allows for further interpretation of the model. Multitaper spectrum estimation expresses the transfer function in terms of the eigencoefficients of predictor and response signals. Advantages of the multitaper method over alternative estimation procedures include that it controls for broadband bias, is an approximately maximum likelihood estimator of the spectrum, and its eigenspectra form a collection of uncorrelated direct spectrum estimators. The multitaper transfer function is estimated, by frequency, as the coefficient obtained from complex regression on response and predictor eigencoefficients. The limiting distributions of eigencoefficents and transfer functions are known if it's assumed that both series are white noise processes, but less is known about their distributions given harmonic elements embedded in more general, stationary noise. We wish to relate structural temporal components constituting a time series regression model's coherency, by isolating the frequency bands in which those structural components exist. This work explores the distributional behaviour of the multitaper transfer function estimator in bands featuring common structure between response and predictor, versus bands without. We discuss the joint distributions of complex eigencoefficients, along with the multitaper transfer function estimator, itself. We investigate the transfer function's potential to detect frequency bands containing coherent line components, and develop a statistical test based on its differential distributional properties across frequency.
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