Conformal prediction for high-dimensional functional time series: Applications to subnational mortality

TL;DR

This work takes a model-agnostic and distribution-free approach, namely conformal prediction, to construct prediction intervals in high-dimensional functional time series and compares the finite-sample forecast performance of these two conformal methods using empirical coverage probability and the mean interval score.

Abstract

In statistics, forecast uncertainty is often quantified using a specified statistical model, though such approaches may be vulnerable to model misspecification, selection bias, and limited finite-sample validity. While bootstrapping can potentially mitigate some of these concerns, it is often computationally demanding. Instead, we take a model-agnostic and distribution-free approach, namely conformal prediction, to construct prediction intervals in high-dimensional functional time series. Among a rich family of conformal prediction methods, we study split and sequential conformal prediction. In split conformal prediction, the data are divided into training, validation, and test sets, where the validation set is used to select optimal tuning parameters by calibrating empirical coverage probabilities to match nominal levels; after this, prediction intervals are constructed for the test set, and their accuracy is evaluated. In contrast, sequential conformal prediction removes the need for a validation set by updating predictive quantiles sequentially via an autoregressive process. Using subnational age-specific log-mortality data from Japan and Canada, we compare the finite-sample forecast performance of these two conformal methods using empirical coverage probability and the mean interval score.

Figures (5)

• Figure 1: A map of 47 Japanese prefectures ordered geographically from the most northern region (Hokkaido) to the most southern region (Okinawa).
• Figure 2: The raw (first row) and smoothed (second row) age-specific $\log$ mortality rates between 1975 and 2023 in Okinawa, Japan. Curves are ordered chronologically using a rainbow color palette; the oldest curves are shown in red, and the most recent in violet.
• Figure 3: A diagram of the expanding-window forecast scheme, for the observations in the test set. The same scheme is also applied to calibreate tuning parameters using a validation set in the split conformal prediction method. In the forecast accuracy evaluation, we consider forecast horizon $h=1, 2,\dots,10$.
• Figure 4: For forecast horizon $h=1, 2,\dots,10$, a comparison of the ECPs of the female and male data between the split and sequential conformal predictions at the nominal coverage probability of 95%. Each boxplot contains the ECP values from the 47 prefectures.
• Figure 5: For forecast horizon $h=1, 2,\dots,10$, a comparison of the mean interval scores of the female and male data between the split and sequential conformal predictions. Each boxplot contains the mean interval scores from the 47 prefectures. For the split conformal prediction, the relatively larger mean interval score is because there are only a few data curves for calibration in the validation set for a relatively large forecast horizon.