Nonparametric collective spectral density estimation with an application to clustering the brain signals

Abstract

In this paper, we develop a method for the simultaneous estimation of spectral density functions (SDFs) for a collection of stationary time series that share some common features. Due to the similarities among the SDFs, the log-SDF can be represented using a common set of basis functions. The basis shared by the collection of the log-SDFs is estimated as a low-dimensional manifold of a large space spanned by a prespecified rich basis. A collective estimation approach pools information and borrows strength across the SDFs to achieve better estimation efficiency. Moreover, each estimated spectral density has a concise representation using the coefficients of the basis expansion, and these coefficients can be used for visualization, clustering, and classification purposes.The Whittle pseudo-maximum likelihood approach is used to fit the model and an alternating blockwise Newton-type algorithm is developed for the computation. A web-based shiny App found at "https://ncsde.shinyapps.io/NCSDE" is developed for visualization, training, and learning the SDFs collectively using the proposed technique. Finally, we apply our method to cluster similar brain signals recorded by the for identifying synchronized brain regions according to their spectral densities. KEYWORDS roughness penalty, time series clustering, Whittle likelihood INTRODUCTIONNonparametric techniques for estimating functional structures have been developed in a variety of settings including regression, density estimation, and survival analysis. In time series analysis, the spectral density function plays an important role in characterizing the frequency content of a signal. Mainly, the estimated spectral density can be used to detect the periodicities of the signals in the frequency domain.In practice, it is common to utilize a discrete Fourier transform (DFT) of the input signal and provide a mathematical approximation of the full integral solution of the Fourier transformation. The squared-magnitude of a DFT of the data is called periodogram. However, the raw periodogram is not a consistent estimator for the spectral density of a stationary random process. One classical method to obtain a consistent estimator is to smooth the periodogram across frequencies. Yuen 1 analyzed the performance of three methods of periodogram smoothing for spectrum estimation. Wahba 2 developed an objective optimum smoothing procedure for estimating the log-spectral density using the spline to smooth the log-periodogram. A discrete spectral average estimator and lag window estimators were introduced in the work of Brockwell and Davis. 3 Both of the two methods are consistent. Brillinger 4 introduced periodogram kernel smoothing.Statistics in Medicine. 2018;37:4789-4806.wileyonlinelibrary.com/journal/sim

Collective spectral density estimation and clustering for spatially-correlated data

“…In this paper, to cluster spatial data that share similar spectral features, we extend the methodology of collective spectral density functions estimation as proposed by Maadooliat et al (2018) to two-dimensional case, and take the spatial dependence of the subregions into account to produce homogeneous spatial clusters. To begin, we use a framework similar to principal component analysis (PCA) to construct a low-dimensional basis expansion that explains the similar features of the 2D-SDFs.…”