Exponential Smoothing for Time Series of Random Objects

2026-09-17

AI Summary

本文提出了一种针对随机对象时间序列的测地指数平滑方法，通过在Hadamard空间中沿测地线移动来更新预测值，适用于不依赖标准算术运算的时间序列预测。

Abstract

Time series of random objects, such as covariance matrices, probability distributions, and functional data, call for forecasting methods that do not rely on standard arithmetic operations. We introduce geodesic exponential smoothing, a generalization of exponential smoothing to time series in Hadamard spaces: the forecast level moves a fixed fraction of the way along the geodesic toward each new observation. The smoothing parameter is estimated by minimizing the average squared distance between observations and their forecasts. We further introduce an innovations mechanism under which each observation has conditional Fréchet mean equal to the current level, providing the metric-space analog of the innovations state-space model. In contrast to autoregressive models for object-valued time series, the framework involves a single scalar parameter, assumes no stationarity, and updates online in constant time per observation. Under this mechanism, we establish sample-path properties of the generative process via the quasilinearization available in Hadamard spaces, and prove almost-sure consistency of the smoothing-parameter estimator. Three real-data applications, spanning covariance-matrix, distributional, and functional time series, assess the forecasting performance of the method against structurally heavier alternatives.

Problem

• time series
• random objects
• forecasting methods
• Hadamard spaces
• exponential smoothing

Innovation

• geodesic exponential smoothing
• Hadamard spaces
• Fréchet mean
• innovations state-space model