Exponential Smoothing for Time Series of Random Objects
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.
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BibTeX
@misc{matsubara2026exponential,
title = {{Exponential Smoothing for Time Series of Random Objects}},
author = {Takuo Matsubara and Peiwen Jiang and Wilson Ye Chen and Minh-Ngoc Tran},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20274},
eprint = {2609.20274},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20274},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exponential Smoothing for Time Series of Random Objects
AU - Takuo Matsubara
AU - Peiwen Jiang
AU - Wilson Ye Chen
AU - Minh-Ngoc Tran
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20274
UR - https://arxiv.org/abs/2609.20274
AB - 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.
ER -