Dynamical AutoCorrelation for Functional Time Series with the Application to fMRI Data

Abstract The concept of dynamical correlation is extended to functional time series, FTS. The dynamical autocorrelation is a scalar, not a functional, measure of cor- relation for FTS; i.e., the expectation of linear relationship of coecients/weights of time and/or space dependent functional bases. This introduces the dynamical autocorrelation as a sensitive measure compared to the functional autocorre- lation. The proposed measure of autocorrelation can be obtained for true, i.e., continuously measured, functional data or possibly to approximated functional data. Estimators of the dynamical autocorrelation are presented based on the Karhunen-Lo'eve expression for observed regular FTS. The Central Limit theo- rem is applied to obtain the asymptotic distribution of proposed estimators of the dynamical autocorrelation. Simulation studies are performed to evaluate the ac- curacy of proposed estimators of the dynamical autocorrelation in the functional autoregressive models with a linear autocovariance operator of order one and two as well as their corresponding standard deviation. Finally, the methodology is applied to a functional Magnetic Resonance Imaging (fMRI) dataset in order to measure the tempo-spatial dynamical autocorrelation among brain images. The dynamical autocorrelation can capture the dierence among runs and participants using the ANOVA test. The results confirm the length of time a brain reacts to a stimulus.