Interpretable models for forecasting high-dimensional functional time series

Abstract

We study the modeling and forecasting of high-dimensional functional time series, which can be temporally dependent and cross-sectionally correlated. We implement a functional analysis of variance (FANOVA) to decompose high-dimensional functional time series, such as subnational age- and sex-specific mortality observed over years, into two distinct components: a deterministic mean structure and a residual process varying over time. Unlike purely statistical dimensionality-reduction techniques, the FANOVA decomposition provides a direct and interpretable framework by partitioning the series into effects attributable to data-specific factors, such as regional and sex-level variations, and a grand functional mean. From the residual process, we implement a functional factor model to capture the remaining stochastic trends. By combining the forecasts of the residual component with the estimated deterministic structure, we obtain the forecasted curves for high-dimensional functional time series. Illustrated by the age-specific Japanese subnational mortality rates from 1975 to 2023, we evaluate and compare the accuracy of the point and interval forecasts across various forecast horizons. The results demonstrate that leveraging these interpretable components not only clarifies the underlying drivers of the data but also improves forecast accuracy, providing more transparent insights for evidence-based policy decisions.

Figures (5)

• Figure 1: Age-specific raw and smoothed $\log$ mortality rates from ages 0 to 100+ between 1975 and 2023 in Okinawa. Curves are ordered chronologically according to the colors of the rainbow. The left vertical axis measures log mortality rates. Mortality rates dip in early childhood, climb in the teen years, stabilize in the early 20s, and then steadily increase with age.
• Figure 2: Two-way functional analysis of variance decomposition: Functional grand effect, capturing the overall age profile; functional row effect, capturing the means across prefectures; and functional column effect, capturing the means across sexes.
• Figure 3: From the interaction term and time-varying residuals, we implement the one-way functional ANOVA for each gender to capture the functional grand effect and row effect, respectively.
• Figure 4: Via the functional factor model, we display the first set of factor loadings and factors, based on the estimated variance of the HDFTS residuals.
• Figure 5: A diagram of the expanding-window forecast scheme. The data begin in 1975 and end in 2023.