We propose a new autocorrelation measure for functional time series that we term “spherical autocorrelation”. It is based on measuring the average angle between lagged pairs of series after having been projected onto the unit sphere. This new measure enjoys several complimentary advantages compared to existing autocorrelation measures for functional data, since it both 1) describes a notion of “sign” or “direction” of serial dependence in the series, and 2) is more robust to outliers. The asymptotic properties of estimators of the spherical autocorrelation are established, and are used to construct confidence intervals and portmanteau white noise tests. These confidence intervals and tests are shown to be effective in simulation experiments, and demonstrated in applications to model selection for daily electricity price curves, and measuring the volatility in densely observed asset price data.
Chi-Kuang Yeh. Gregory Rice. Joel A. Dubin. "Functional spherical autocorrelation: A robust estimate of the autocorrelation of a functional time series." Electron. J. Statist. 17 (1) 650 - 687, 2023. https://doi.org/10.1214/23-EJS2112