The Autocovariance Prediction of the Earth Rotation Parameters

Abstract

Let X _t = X ₁,X ₂,...,X _N be an equidistant stationary stochastic process of N observations. Our main interest is in estimating the prediction ${{\hat{X}}_{{N + 1}}}$ in a time of N + 1. In the autocovariance prediction method the first prediction point satisfies the following condition:

$$ P = \sum\limits_{k = 0}^{N - 1} {(\hat c'_k - \hat c_k )^2 = \min } $$

(1)

where

$$ \hat c_k = h_k /\left( {N - k} \right)\sum\limits_{t = 1}^{N - k} {X_t } X_{t + k} ,{\text{ }}for{\text{ }}k = 0,1,...,N - 1 $$

(2)

is the autocovariance estimation of N number of data, $\hat c'_k$ is the autocovariance estimation of N + 1 number of data including the first prediction point ${{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}_{{N + 1}}}$ and h _k is a lag window.