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

This paper proposes a model-agnostic, distribution-free approach using split and sequential conformal prediction to construct prediction intervals for high-dimensional functional time series, demonstrating their effectiveness in forecasting subnational mortality rates for Japan and Canada compared to traditional methods.

The Gist

Imagine you are a weather forecaster. You don't just want to say, "It will rain tomorrow." You want to say, "It will rain, and I'm 95% sure the amount will be between 1 and 3 inches." That "between 1 and 3 inches" part is your prediction interval. It's your safety net.

This paper is about building better, safer nets for predicting complex, changing patterns—specifically, how death rates change over time across different regions in Japan and Canada.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Problem: The "Crystal Ball" is Cracked

Traditionally, statisticians build complex mathematical models (like a crystal ball) to predict the future. They assume the world follows specific rules.

• The Risk: If the rules you assumed are slightly wrong (model misspecification), your crystal ball gives you a false sense of security. Your "safety net" might be too small, and you'll get wet when it rains.
• The Old Fix: Some people use "bootstrapping" (resampling data over and over), but that's like trying to count every grain of sand on a beach by picking them up one by one. It's accurate but takes forever and requires a supercomputer.

2. The Solution: The "Conformal Prediction" Safety Net

The author, Han Lin Shang, suggests a smarter way called Conformal Prediction. Think of this not as a crystal ball, but as a tailor.

• Instead of guessing the rules of the universe, the tailor looks at how much the fabric (the data) actually stretches and shrinks in the past.
• It doesn't care what kind of fabric you have (it's "model-agnostic"). It just measures the real-world wiggles and creates a net that fits those wiggles perfectly.
• The Goal: To create a prediction interval that is guaranteed to catch the future outcome 95% of the time, no matter how messy the data is.

3. The Challenge: Too Many Threads (High-Dimensional Data)

The data here isn't just one line; it's a massive tapestry.

• The Data: Death rates for every age group (0 to 100+) in 47 different Japanese prefectures, tracked over 49 years.
• The Analogy: Imagine trying to predict the weather for 47 different cities simultaneously, where the temperature in each city changes every day in a unique curve. That is a "High-Dimensional Functional Time Series." It's a lot of moving parts.

4. The Two Methods: The "Split" vs. The "Live Update"

The paper compares two ways to make this tailor work:

Method A: Split Conformal Prediction (The "Rehearsal" Approach)

• How it works: You take your data and cut it into three pieces:
  1. Training: To learn the pattern.
  2. Validation (Rehearsal): To test how wide your net should be.
  3. Test: The real future prediction.
• The Flaw: It's like a chef tasting a soup before the main event to decide how much salt to add. But once the main event starts, the chef can't taste it again. If the soup changes flavor later (new data arrives), the chef is stuck with the old salt level.
• Result: In the paper, this method often made the net too small (underestimated risk), especially for long-term predictions.

Method B: Sequential Conformal Prediction (The "Live Update" Approach)

• How it works: This method doesn't need a separate "rehearsal" phase. As soon as new data arrives (e.g., this year's death rates), it instantly updates the net.
• The Analogy: Imagine a smart thermostat that doesn't just guess the temperature; it constantly feels the room and adjusts the heating right now. If the room gets colder, the net widens immediately.
• Result: This method was conservative. It made the net slightly larger than necessary (overestimated risk), but that's a good thing! It meant the actual outcomes were almost always caught inside the net.

5. The Verdict: Better Safe Than Sorry

The author tested these methods on real data from Japan and Canada.

• The Winner: Sequential Conformal Prediction.
• Why? It didn't waste data on a "rehearsal" set. It learned on the fly. While it created slightly wider nets (which might look like "less precise" at first glance), those nets were more reliable. They caught the truth more often than the "Split" method.

Summary in One Sentence

Instead of guessing the future based on rigid rules, this paper teaches us to build a flexible, self-updating safety net that gets smarter every time a new piece of data arrives, ensuring we are rarely caught off guard by the unexpected.

Technical Summary

Here is a detailed technical summary of the paper "Conformal prediction for high-dimensional functional time series: Applications to subnational mortality" by Han Lin Shang.

1. Problem Statement

The paper addresses the challenge of quantifying forecast uncertainty in High-Dimensional Functional Time Series (HDFTS).

• Context: Traditional time series forecasting often relies on specific statistical models (e.g., ARIMA, factor models) to generate prediction intervals. These approaches are vulnerable to model misspecification, selection bias, and often lack valid finite-sample coverage guarantees.
• The HDFTS Challenge: In many fields (climatology, finance, demography), data consists of a large number of functional curves ($N$) observed over time ($T$), where the number of cross-sections often exceeds the number of time points ($N > T$). Existing literature on functional time series largely focuses on low-dimensional settings ($N$ fixed, $T \to \infty$), leaving a gap in methods for HDFTS.
• Goal: To develop a model-agnostic and distribution-free approach to construct prediction intervals for HDFTS that maintains valid coverage probabilities in finite samples without relying on strict distributional assumptions.

2. Methodology

The paper proposes a framework combining Functional Data Analysis (FDA) with Conformal Prediction.

A. Data Decomposition

To handle the HDFTS structure (specifically subnational mortality rates), the authors employ two decomposition strategies that allow for exact recovery of the original data:

1. One-way Functional ANOVA: Decomposes the log-mortality rates into a functional grand effect, a functional row effect (representing regional/prefecture differences), and a time-varying residual process. Robust estimation is achieved via functional median polish.
2. Functional Factor Model: Based on Leng et al. (2026), this models the data using functional factor loadings and real-valued factors. It utilizes an information criterion to estimate the number of factors ($q$) and employs eigenanalysis of the covariance matrix to estimate latent factors.

B. Conformal Prediction Variants

The core contribution is the application of two conformal prediction methods tailored for functional time series:

1. Split Conformal Prediction:

   • Procedure: The data is split into training, validation, and test sets.
   • Calibration: The validation set is used to calculate non-conformity scores (residuals). A tuning parameter ($\xi_\alpha$) is selected to ensure that a specific proportion (e.g., 95%) of validation residuals fall within the bounds.
   • Forecasting: Prediction intervals for the test set are constructed using the calibrated parameter.
   • Limitation: Requires a dedicated validation set, which reduces the sample size available for training and calibration, potentially leading to suboptimal performance in long-horizon forecasts.

2. Sequential Conformal Prediction:

   • Procedure: This method eliminates the need for a separate validation set.
   • Mechanism: It updates predictive quantiles sequentially as new data arrives. It models the temporal dependence of absolute residuals using an autoregressive quantile regression process.
   • Advantage: It is "tuning-parameter-free" regarding sample splitting and adapts dynamically to distribution shifts, making it more efficient for finite samples.

C. Evaluation Metrics

The accuracy of the interval forecasts is evaluated using:

• Empirical Coverage Probability (ECP): The proportion of times the true value falls within the predicted interval.
• Coverage Probability Difference (CPD): The absolute difference between ECP and the nominal level (e.g., 0.95).
• Mean Interval Score (MIS): A scoring rule that penalizes both lack of coverage and excessive width (sharpness). Lower scores indicate better performance.

3. Key Contributions

• First Application to HDFTS: This is the first study to apply conformal prediction specifically to the High-Dimensional Functional Time Series setting.
• Model Agnosticism: The proposed framework does not rely on the correctness of the underlying forecasting model (e.g., ARIMA or ETS used for point forecasts), making it robust to model misspecification.
• Comparative Analysis: The paper rigorously compares Split vs. Sequential conformal prediction, demonstrating that the sequential approach is superior for finite-sample HDFTS applications because it avoids the data loss associated with sample splitting.
• Reproducibility: The authors provide open-source code (R/Python) for computing interval forecast errors using Japanese and Canadian mortality data.

4. Results

The study utilizes Japanese subnational age- and sex-specific log-mortality rates (1975–2023) and validates findings with Canadian data (1950–2016).

• Coverage Performance:
  • Split Conformal: Tends to underestimate coverage probabilities (ECP < 0.95), particularly at longer forecast horizons ($h=3$ to $7$). This is attributed to the calibration on a limited validation set not being optimal for the test set.
  • Sequential Conformal: Tends to overestimate coverage (ECP > 0.95), resulting in conservative but reliable intervals.
• Interval Score (Sharpness vs. Coverage):
  • Despite being conservative, the Sequential Conformal Prediction consistently achieves lower (better) Mean Interval Scores compared to the Split method.
  • The Split method suffers from wider intervals or poor coverage at longer horizons due to the reduced training sample size after splitting.
• Forecast Horizon: As the forecast horizon ($h$) increases, the performance gap widens, with the Sequential method maintaining stability while the Split method degrades.
• Robustness: Results were consistent across different forecasting models (ARIMA vs. ETS) for the principal component scores and across both sexes and two different countries (Japan and Canada).

5. Significance and Conclusion

• Practical Utility: For demographic and actuarial applications (like mortality forecasting), having reliable uncertainty quantification is crucial for policy planning and risk management. The sequential conformal approach provides a robust tool that does not require complex model re-specification when data distributions shift.
• Methodological Shift: The paper advocates moving away from rigid, model-dependent uncertainty quantification toward distribution-free, conformal methods, especially in high-dimensional settings where $N > T$.
• Recommendation: The authors recommend Sequential Conformal Prediction as the preferred method for HDFTS due to its ability to update quantiles dynamically without sacrificing data for validation, thereby offering a better trade-off between coverage reliability and interval sharpness.

Future Directions: The authors suggest extending the sequential method to model temporal dependence using other time-series models beyond quantile regression and exploring joint modeling of male and female data to handle higher-dimensional arrays using two-way functional ANOVA.