Statistics Seminar Series
Tuesday, December 2, 4:10pm, MSB 1147 (Colloquium Room)
Speaker: Shahin Tavakoli (University of Cambridge)
Title: “Fourier Analysis of Functional Time Series, with Applications to DNA Dynamics”
Abstract: The fundamental tool of Functional Data Analysis, the Karhunen--Loeve decomposition, gives means for understanding the randomness of a random function: it decomposes it in a sum of uncorrelated random functions living in 1-dimensional subspace, and is the basis for technology transfer from multivariate analysis, by providing optimal linear finite dimensional approximations of the random function. Though it is arguably the best tool in an i.i.d. setting, its use in the context of dependent functional data, such as a stationary functional time series, is not appropriate, since it is based only on the lag-0 autocovariance operator.
In the first part of this talk, I will present a canonical extension of the Karhunen--Loeve decomposition for stationary functional time series, based on a combination of a Cramer representation of functional time series, together with Karhunen--Loeve decompositions. The resulting expansion is shown to extend the properties of the Karhunen--Loeve expansion, providing a decomposition of the functional time series into a sum of rank-1 random functions that are uncorrelated accross all time lags, and optimal linear finite dimensional representations that dominate fPCA.
The construction is based on the (unknown) eigenstructure of the spectral density operator (indexed by frequency and curvelength), which is the functional extension of the spectral density matrix of multivariate time series. The second part of this talk presents the basic ingredients for such a frequency domain approach to functional time series. In particular, we propose estimators of the spectral density operator, and study their asymptotics under functional cumulant-type mixing conditions.
The last part of the talk deals with an applications to DNA dynamics. Using the tools previously introduced, we compare the dynamics of two closed DNA strands (also called DNA minicircles) through the lens of the spectral density operator. We develop methods for comparing the spectral density operator at a hierarchy of stages. First, we compare the spectral density operators at the level of frequencies, and localize their differences along frequencies. Second, we compare the spectral density operators along the length of the curve, within each selected frequencies, and localize their differences, using a hierarchical multiple testing approach.