Predictive Inference for Locally Stationary Time Series With an Application to Climate Data

Abstract

The model-free prediction principle of Politis has been successfully applied to general regression problems, as well as problems involving stationary time series. However, with long time series, for example, annual temperature measurements spanning over 100 years or daily financial returns spanning several years, it may be unrealistic to assume stationarity throughout the span of the dataset. In this article, we show how model-free prediction can be applied to handle time series that are only locally stationary, that is, they can be assumed to be stationary only over short time-windows. Surprisingly, there is little literature on point prediction for general locally stationary time series even in model-based setups, and there is no literature whatsoever on the construction of prediction intervals of locally stationary time series. We attempt to fill this gap here as well. Both one-step-ahead point predictors and prediction intervals are constructed, and the performance of model-free is compared to model-based prediction using models that incorporate a trend and/or heteroscedasticity. Both aspects of the article, model-free and model-based, are novel in the context of time-series that are locally (but not globally) stationary. We also demonstrate the application of our model-based and model-free prediction methods to speleothem climate data which exhibits local stationarity and show that our best model-free point prediction results outperform that obtained with the RAMPFIT algorithm previously used for analysis of this type of data. Supplementary materials for this article are available online.

KEYWORDS:

• Kernel smoothing
• Linear predictor
• Nonstationary series
• Prediction intervals

Acknowledgments

The authors would like to thank Richard Davis, Stathis Paparoditis, Yiren Wang, and Yunyi Zhang for their helpful questions and suggestions. The constructive comments of two anonymous reviewers and the associate editor are also acknowledged. Finally, the authors would like to thank Dr. Manfred Mudelsee for kindly sharing the FORTRAN code for the RAMPFIT algorithm used for analyzing the speleothem dataset used in this article.

Supplementary Materials

Supplementary Materials contain the following: model-based and model-free algorithms for the construction of prediction intervals are described in detail in Appendix A. The RAMPFIT algorithm used to generate point prediction results for comparison with our model-free and model-based methods is described in Appendix B. Finally, Appendix C describes some diagnostics useful for model-free predictive inference.

Notes

1 If f_t(w) can be assumed monotone in w, then it does not need to be continuous in w; see Corollary 3.1 in what follows.

Additional information

Funding

This research was partially supported by NSF grants DMS 12-23137 and DMS 16-13026. The authors would like to acknowledge the Pacific Research Platform, NSF Project ACI-1541349 and Larry Smarr (PI, Calit2 at UCSD) for providing the computing infrastructure used in this project.