Mathematical Statistics and Stochastic Processes

Chapter 15

Prediction

15.1. Generalities

Let be a real, square-integrableprocess with basis space , be the sub-σ-algebra generated by Xn−j, j ≥ 0, and be the closed linear subspace of generated by the same variables and the constant 1.

We wish to predict Xn+h from the observed variables X1,…,Xn. The strictly positive integer h is called the horizon of the prediction.

With respect to the quadratic error, the best predictor given Xn−j, j ≥ 0, is the conditional expectation

This is the orthogonal projection in of Xn+h onto .

The best linear predictor of Xn+h is its orthogonal projection onto . If (Xt) is Gaussian, it coincides with .

A statistical predictor is a known function of the data: ...