Parametrizations, weights, and optimal prediction: Part 1

We consider the problem of the annual mean temperature prediction. The years taken into account and the corresponding annual mean temperatures are denoted by $0,\ldots, n$ and $t_0$, $\ldots$, $t_n$, respectively. We propose to predict the temperature $t_{n+1}$ using the data $t_0$, $\ldots$, $t_n$. For each $0\leq l\leq n$ and each parametrization $\Theta^{(l)}$ of the Euclidean space $\mathbb{R}^{l+1}$ we construct a list of weights for the data $\{t_0,\ldots, t_l\}$ based on the rows of $\Theta^{(l)}$ which are correlated with the constant trend. Using these weights we define a list of predictors of $t_{l+1}$ from the data $t_0$, $\ldots$, $t_l$. We analyse how the parametrization affects the prediction, and provide three optimality criteria for the selection of weights and parametrization. We illustrate our results for the annual mean temperature of France and Morocco.