Parametrizations, weights, and optimal prediction

Abstract

The goal of the present paper is to predict the future value $y_{n+1}$ based on previously observed time series y₀, $\dots ,$ y_n which are correlated with the constant trend, i.e. ${\sum }_{i=0}^n{y}_i\ne 0.$ We show that the construction of the weights $\mathbf{w}=(w_0,\dots ,w_n)$ of the linear predictor ${\sum }_{i=0}^n{w}_i{y}_i,$ using several stochastic models, is equivalent to predict without error a subspace of ${\mathbb{R}}^{n+2}$ of dimension n + 1. The geometry of the latter subspace depends on the model’s covariance matrix. We extract from each parametrization of the Euclidean space ${\mathbb{R}}^{n+1}$ a new list of weights which are correlated with the constant trend. Using these weights we define a new list of predictors of $y_{n+1}.$ We analyze how the parametrization affects the prediction, and provide an optimality criterion for the selection of weights and parametrization. Finally, we illustrate the proposed estimation approach by application to data set on the mean annual temperature of France and Morocco recorded for a period of 115 years (1901 to 2015).

Keywords:

• Parametrization
• basis
• weights
• cubic spline
• Mahalanobis distance
• conservative rows

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 62P12

Acknowledgments

The authors would like to thank the anonymous referee for his/her careful reading and valuable suggestions that improved the results of the paper. We are also grateful to Richard G. Krutchkoff and Syed Ejaz Ahmed for stimulating discussions on this topic. Azzouz Dermoune was supported by Laboratoire Paul-Painlevé USTL-UMR-CNRS 8524, the Labex CEMPI ANR-11-LABEX-0007-01 and the project ClinMine–ANR-13-TECS-0009.