Stochastic generators of multi-site daily temperature: comparison of performances in various applications

Abstract

We present a multi-site stochastic model for the generation of average daily temperature, which includes a flexible parametric distribution and a multivariate autoregressive process. Different versions of this model are applied to a set of 26 stations located in Switzerland. The importance of specific statistical characteristics of the model (seasonality, marginal distributions of standardized temperature, spatial and temporal dependence) is discussed. In particular, the proposed marginal distribution is shown to improve the reproduction of extreme temperatures (minima and maxima). We also demonstrate that the frequency and duration of cold spells and heat waves are dramatically underestimated when the autocorrelation of temperature is not taken into account in the model. An adequate representation of these characteristics can be crucial depending on the field of application, and we discuss potential implications in different contexts (agriculture, forestry, hydrology, human health).

Appendix A: Skew exponential distribution

The exponential power (EP) distribution, also known as the generalized error distribution, is a generalization of the normal distribution. A shape parameter, ν > 0, leads to tails that are either heavier than normal if ν < 2 or lighter if ν > 2, and ν = 2 corresponds to the normal case. For a standardized variate z (with a mean of 0 and a standard deviation of 1), its probability density is defined as

$$f_{EP}(z|\nu) = \frac{\nu}{\lambda_{\nu} \times 2^{1 + 1/\nu} \times {\Gamma}(1/\nu)} \times \exp \left( - \frac{1}{2} \times \left|\frac{z}{\lambda_{\nu}}\right|^{\nu} \right) $$

with

$$\lambda_{\nu} = \sqrt{2^{-2/\nu} \times \frac{{\Gamma}(1/\nu)}{{\Gamma}(3/\nu)}}. $$

The skew exponential power (SEP) distribution is a generalization of the EP distribution. An additional shape parameter, ξ, can lead to different skewness when it differs from 1. For a normalized variate z and parameters ν > 0 and ξ > 0, the probability density of the SEP distribution is defined as

$$ f_{SEP}(z|\nu,\xi) = \frac{2}{\xi + 1/\xi} \, f_{EP}(\tilde{z}/\xi^{\text{sign}(\tilde{z})}|\nu) $$

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with

$$\begin{array}{@{}rcl@{}} \tilde{z} &=& \mu_{z} + \sigma_{z} \times z \\ \mu_{z} &=& \gamma_{\nu} \times (\xi - 1/\xi) \\ \sigma_{z} &=& \sqrt{(1-\gamma_{\nu}^{2}) (\xi^{2}+ 1/\xi^{2}) + 2\gamma_{\nu}^{2} - 1} \\ \gamma_{\nu} &=& 2^{1/\nu} \times \lambda_{\nu} \times \frac{{\Gamma}(2/\nu)}{{\Gamma}(1/\nu)} \end{array} $$