Parametrizations, weights, and optimal prediction

Azzouz Dermoune, Khalifa Es-Sebaiy, Mohammed Es.Sebaiy and Jabrane Moustaaid

Communications in Statistics - Theory and Methods, 2021, vol. 50, issue 4, 815-836

Abstract: The goal of the present paper is to predict the future value yn+1 based on previously observed time series y0, …, yn which are correlated with the constant trend, i.e. ∑i=0nyi≠0. We show that the construction of the weights w=(w0,…,wn) of the linear predictor ∑i=0nwiyi, using several stochastic models, is equivalent to predict without error a subspace of Rn+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 Rn+1 a new list of weights which are correlated with the constant trend. Using these weights we define a new list of predictors of yn+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).

Date: 2021
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DOI: 10.1080/03610926.2019.1642489

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