Detecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics

Abstract

Motivated by the problem of inferring the molecular dynamics of DNA in solution, and linking them with its base-pair composition, we consider the problem of comparing the dynamics of functional time series (FTS), and of localizing any inferred differences in frequency and along curvelength. The approach we take is one of Fourier analysis, where the complete second-order structure of the FTS is encoded by its spectral density operator, indexed by frequency and curvelength. The comparison is broken down to a hierarchy of stages: at a global level, we compare the spectral density operators of the two FTS, across frequencies and curvelength, based on a Hilbert-Schmidt criterion; then, we localize any differences to specific frequencies; and, finally, we further localize any differences along the length of the random curves, that is, in physical space. A hierarchical multiple testing approach guarantees control of the averaged false discovery rate over the selected frequencies. In this sense, we are able to attribute any differences to distinct dynamic (frequency) and spatial (curvelength) contributions. Our approach is presented and illustrated by means of a case study in molecular biophysics: how can one use molecular dynamics simulations of short strands of DNA to infer their temporal dynamics at the scaling limit, and probe whether these depend on the sequence encoded in these strands? Supplementary materials for this article are available online.

Wavelet spectral testing: Application to nonstationary circadian rhythms

“…The closest methodology for spectral comparison is framed as a (consistent) classification method of Fryzlewicz and Ombao (2009), further improved by Krzemieniewska et al (2014). Tavakoli and Panaretos (2016) compare pairs of stationary functional time series, by developing t -tests for the equality of their (Fourier) spectral density operators. Shumway (1988) compares groups of curves (with stationary stochastic errors) by testing whether the mean curves have the same Fourier spectrum at each given frequency.…”

Two-Sample Tests for Relevant Differences in the Eigenfunctions of Covariance Operators

“…Monographs detailing many of the available statistical procedures for functional data are Ramsay and Silverman (2005) and Horváth and Kokoszka (2012). This type of data naturally arises in various contexts such as environmental data (Aue et al, 2015), molecular biophysics (Tavakoli and Panaretos, 2016), climate science (Zhang et al, 2011;Aue et al, 2018), and economics (Kowal et al, 2019). Most of these examples intrinsically contain a time series component as successive curves are expected to depend on each other.…”

Functional lagged regression with sparse noisy observations

“…Indeed the abstraction of an infinite dimensional (Hilbert) space captures the scenario where the value of a series at each time is a square-integrable function, for example, a curve. Examples include time series of DNA minicircles evolving in solution (seen as a time series of closed curves in 3D indexed by discrete time Tavakoli and Panaretos, 2016) or the data constructed by dividing a continuously observed scalar time series into segments of an obvious periodicity, usually days. Examples of the latter form are particularly prominent in environmental applications, for example the analysis of particulate matter atmospheric pollution (Hörmann and Kokoszka, 2010;Hörmann et al, 2015Hörmann et al, , 2018Aue et al, 2015), traffic data modelling (Klepsch et al, 2017) or financial applications of intra-day trading (Müller et al, 2011;Kokoszka et al, 2017).…”