Shahin Tavakoli

Fourier analysis of stationary time series in function space

We develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second-order structure of a stationary sequence of functional data. The key element in such a context is the spectral density operator, which generalises the notion of a spectral density matrix to the functional setting, and characterises the second-order dynamics of the process. Our main tool is the functional Discrete Fourier Transform (fDFT). We derive an asymptotic Gaussian representation of the fDFT, thus allowing the transformation of the original collection of dependent random functions into a collection of approximately independent complex-valued Gaussian random functions. Our results are then employed in order to construct estimators of the spectral density operator based on smoothed versions of the periodogram kernel, the functional generalisation of the periodogram matrix. The consistency and asymptotic law of these estimators are studied in detail. As immediate consequences, we obtain central limit theorems for the mean and the long-run covariance operator of a stationary functional time series. Our results do not depend on structural modelling assumptions, but only functional versions of classical cumulant mixing conditions, and are shown to be stable under discrete observation of the individual curves.

Tests for separability in nonparametric covariance operators of random surfaces

The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional {data analysis} settings, where this assumption is most relevant. We propose here to test separability by focusing on

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

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Research Data Supporting "Tests for separability in nonparametric covariance operators of random surfaces"

-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full non-separable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset.

Cramer-Karhunen-Loeve representation and harmonic principal component analysis of functional time series

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.

Spatial modeling of Object Data: Analysing dialect sound variations across the UK

#### Software - covsep-1.0.1.tar.gz : R package implementing the methodology of the paper. #### Data for reproducing the Numerical simulations: - reproduce-sims.R: R script to reproduce the simulation studies; WARNING: the simulations take a lot of time. - SIM2016a-SIM12feb.RData: results of the simulation studies, and parameters to reproduce them. - grid-sims2.RData: results of the simulation studies (Figure 5 and Figures S2 & S3 of the supplement), and parameters to reproduce them. - sim_function.R: function performing the simulation studies #### Data application: - Acoustic_Data_part1.RData: R workspace containing the preprocessed log-spectrograms considered in the phonetic application. (part 1) - Acoustic_Data_part2.RData: R workspace containing the preprocessed log-spectrograms considered in the phonetic application. (part 2) - Info_Acoustic_Data.txt: text file with information about the phonetic sounds. This includes the language, the word being pronounced and the gender of the speaker. - separability_Acoustic_Data.R: script to replicate the test for separability for the phonetic data as described in the manuscript.