Parametrizations, weights, and optimal prediction: Part 1

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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,, n and t₀,, tₙ, respectively. We propose to predict the temperature tₙ₊₁ using the data t₀,, tₙ. For each 0≤ l≤ n and each parametrization Θ⁽ˡ⁾ of the Euclidean space Rˡ⁺¹ we construct a list of weights for the data {t₀,, tₗ} based on the rows of Θ⁽ˡ⁾ which are correlated with the constant trend. Using these weights we define a list of predictors of tₗ₊₁ from the data t₀,, tₗ. 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.