Discriminant analysis in stationary time series based on robust cepstral coefficients

Abstract

Time series analysis is essential in fields such as finance, economics, environmental science, and biomedical engineering for understanding underlying mechanisms, forecasting, and identifying patterns. Traditional time-domain methods, which focus on trends, seasonality, and noise, often overlook periodicities and harmonic structures that are better captured in the frequency domain. Analyzing time series in the frequency domain enables the identification of these spectral properties, providing deeper insights into the underlying processes. These insights can help differentiate data-generating processes of different populations and assist in the discrimination and classification of time series. The literature commonly uses smoothed estimators like the smoothed periodogram to minimize bias, obtaining an average spectrum from individual replicates within a population to classify new time series. However, if there is spectral variability among replicates within each population, such methods become unfeasible. Moreover, abrupt values can significantly impact spectrum estimators, complicating practical discrimination and classification. There is a gap in the literature for methods that consider within-population spectral variability, separate white noise effects from autocorrelations, and use robust estimators in the presence of outliers. This paper addresses this gap by presenting a robust framework for classifying replicate groups of time series by transforming them into the frequency domain using the Fourier Transform to compute the power spectrum. Then, after taking the logarithm of the spectra, the inverse Fourier Transform is used to achieve the cepstrum. To mitigate the effects of outliers and improve the consistency of spectral estimates, we employ the multitaper periodogram alongside the M-periodogram. These spectral features are then utilized in Linear Discriminant Analysis (LDA) to enhance classification accuracy and interpretability. This integrated approach offers significant potential for applications requiring precise temporal pattern distinction and resilience to data anomalies.