[Paper Review] On projection methods for functional time series forecasting

Antonio Elías, Raúl Jiménez+1 more|arXiv (Cornell University)|May 10, 2021

Forecasting Techniques and Applications81 references4 citations

TL;DR

This paper proposes two model-free, nonparametric forecasting methods for functional time series (FTS) using k-nearest neighbors and a curve envelope approach to project future curves. The methods enable one-step-ahead forecasting and dynamic updating, outperforming benchmark models like FPCF in simulations and real-world data (electricity demand, NOx emissions), particularly under complex, nonlinear dynamics.

ABSTRACT

Two nonparametric methods are presented for forecasting functional time series (FTS). The FTS we observe is a curve at a discrete-time point. We address both one-step-ahead forecasting and dynamic updating. Dynamic updating is a forward prediction of the unobserved segment of the most recent curve. Among the two proposed methods, the first one is a straightforward adaptation to FTS of the $k$-nearest neighbors methods for univariate time series forecasting. The second one is based on a selection of curves, termed \emph{the curve envelope}, that aims to be representative in shape and magnitude of the most recent functional observation, either a whole curve or the observed part of a partially observed curve. In a similar fashion to $k$-nearest neighbors and other projection methods successfully used for time series forecasting, we ``project'' the $k$-nearest neighbors and the curves in the envelope for forecasting. In doing so, we keep track of the next period evolution of the curves. The methods are applied to simulated data, daily electricity demand, and NOx emissions and provide competitive results with and often superior to several benchmark predictions. The approach offers a model-free alternative to statistical methods based on FTS modeling to study the cyclic or seasonal behavior of many FTS.

Motivation & Objective

• Address the need for model-free, nonparametric forecasting methods in functional time series (FTS) that do not rely on parametric assumptions like ARFIMA or FPC-based models.
• Develop a unified framework for both one-step-ahead forecasting and dynamic updating—forward prediction of partially observed curves—using nonparametric projection techniques.
• Provide a flexible alternative to traditional FTS models by leveraging functional depth and nearest-neighbor principles to capture complex, nonlinear, and seasonal behaviors in FTS data.
• Evaluate the proposed methods on simulated data and real-world datasets (electricity demand, NOx emissions) to demonstrate robustness and competitiveness against established benchmarks.

Proposed method

• Adapt the k-nearest neighbors (k-NN) method from univariate time series to FTS by defining a distance metric between functional curves to identify the k most similar historical curves.
• Project the future segments of the k-nearest neighbors using a convex combination of their future curve segments to forecast the next curve in the series.
• Introduce the curve envelope method, which selects a set of representative curves based on shape and magnitude similarity to the most recent observed curve (or its partial segment), forming a local neighborhood for projection.
• Use functional depth measures to guide the selection of the curve envelope, ensuring that the selected curves are representative of the most recent functional observation in terms of location and shape.
• Apply dynamic updating by projecting the unobserved portion of a partially observed curve using the k-NN or envelope-based method, enabling real-time forecasting during curve observation.
• Optimize hyperparameters (k, embedding dimension E, envelope size m) via cross-validation or data-driven selection, with m set to 100 (small n) and 500 (large n) in simulations.

Experimental results

Research questions

• RQ1 Can k-NN-based projection methods be effectively adapted to functional time series forecasting, particularly for one-step-ahead and dynamic updating tasks?
• RQ2 How does the performance of the curve envelope method compare to k-NN and benchmark models (e.g., FPCF) in forecasting FTS with complex, nonlinear, or seasonal patterns?
• RQ3 To what extent do the proposed nonparametric methods outperform model-based approaches (e.g., FPCF, FAR models) in terms of mean squared error (MSE) across diverse FTS data structures?
• RQ4 How do the methods perform under varying levels of noise and structural complexity, such as in FTS with random shocks or periodic components?
• RQ5 Can the model-free approach maintain high accuracy even when the underlying data-generating process is not well-captured by standard parametric models?

Key findings

• The curve envelope method (EPU 0.25) achieved the lowest mean squared error (MSE) in simulations, significantly outperforming FPCF and EPF across all tested scenarios.
• For the (0.8, 0.0) parameter setting, EPF was competitive, as the future curve depends heavily on the focal curve and innovations, aligning with the method’s reliance on recent functional similarity.
• In the (0.4, 0.4) setting, the MSE difference between FPCF and EPF was approximately 0.1, representing a relative improvement of less than 7%, indicating strong performance of EPF under balanced dynamics.
• For the (0.2, 0.0) setting, where the FTS is dominated by noise, the functional average was the best predictor, and the proposed methods performed reasonably well, though not superior to the mean.
• In the stationary FTS with random shocks model, the proposed methods maintained low MSE, demonstrating robustness to irregular shock patterns and non-Gaussian noise structures.
• The methods showed superior performance on real-world datasets: daily electricity demand and NOx emissions, where they consistently outperformed benchmark models in terms of forecasting accuracy.