Gamma random field simulation by a covariance matrix transformation method

Abstract

In studies involving environmental risk assessment, Gaussian random field generators are often used to yield realizations of a Gaussian random field, and then realizations of the non-Gaussian target random field are obtained by an inverse-normal transformation. Such simulation process requires a set of observed data for estimation of the empirical cumulative distribution function (ECDF) and covariance function of the random field under investigation. However, if realizations of a non-Gaussian random field with specific probability density and covariance function are needed, such observed-data-based simulation process will not work when no observed data are available. In this paper we present details of a gamma random field simulation approach which does not require a set of observed data. A key element of the approach lies on the theoretical relationship between the covariance functions of a gamma random field and its corresponding standard normal random field. Through a set of devised simulation scenarios, the proposed technique is shown to be capable of generating realizations of the given gamma random fields.

Appendices

Appendix 1: Proof of one-to-one mapping between x and w in Eq. 16

From the Wilson–Hilferty approximation and assuming the approximation to be exact, we have

$$ x = {\frac{\alpha }{\lambda }}\left\{ {1 - {\frac{1}{9\alpha }} + w\sqrt {{\frac{1}{9\alpha }}} } \right\}^{3} . $$

(26)

Differentiating x with respect to w yields

$$ \begin{aligned} \frac{dx}{dw} & = {\frac{\alpha }{\lambda }}3\left\{ {1 - {\frac{1}{9\alpha }} + w\sqrt {{\frac{1}{9\alpha }}} } \right\}^{2} \sqrt {{\frac{1}{9\alpha }}} \\ & = {\frac{\sqrt \alpha }{\lambda }}\left\{ {1 - {\frac{1}{9\alpha }} + w\sqrt {{\frac{1}{9\alpha }}} } \right\}^{2} \ge 0 \\ \end{aligned} $$

(27)

Let $\frac{dx}{dw} = 0,$ we have

$$ w = {\frac{1 - 9\alpha }{3\sqrt \alpha }} $$

(28)

Taking the second derivative of x with respect to w, it yields

$$ {\frac{{d^{2} x}}{{dw^{2} }}} = {\frac{\sqrt \alpha }{\lambda }}2\left\{ {1 - {\frac{1}{9\alpha }} + w\sqrt {{\frac{1}{9\alpha }}} } \right\}\sqrt {{\frac{1}{9\alpha }}} $$

(29)

Substituting (28) into (29),

$$ {\frac{{d^{2} x}}{{dw^{2} }}} = 0\quad {\text{at}}\quad w = {\frac{1 - 9\alpha }{3\sqrt \alpha }} $$

(30)

From Eqs. 27 and 30, we conclude that Eq. 26 is a one-to-one mapping function with an inflection point at $w = {\frac{1 - 9\alpha }{3\sqrt \alpha }}$.

Appendix 2: Proof of Eq. 17 as a unique conversion

Assuming the approximation of Eq. 17 to be exact and taking derivative of COV(X ₁, X ₂) with respect to $\rho_{{W_{1} W_{2} }}$, it yields

$$ {\frac{d}{{d\rho_{{W_{1} W_{2} }} }}}{\text{COV}}(X_{1} ,X_{2} ) = {\frac{162}{{6561\alpha_{1}^{0.5} \alpha_{2}^{0.5} \lambda_{1} \lambda_{2} }}}\left( {\rho_{{W_{1} W_{2} }} - E} \right)^{2} \ge 0 $$

(31)

where

$$ E = - {\frac{{162\alpha_{1} \alpha_{2} - 18\alpha_{1} - 18\alpha_{2} + 2 - F^{0.5} }}{{18\alpha_{1}^{0.5} \alpha_{2}^{0.5} }}}, $$

and

$$ F = 13122\alpha_{1}^{2} \alpha_{2}^{2} - 4374\alpha_{1}^{2} \alpha_{2} - 4347\alpha_{1} \alpha_{2}^{2} + 1134\alpha_{1} \alpha_{2} + 162\alpha_{1}^{2} - 54\alpha_{1} + 162\alpha_{2}^{2} - 54\alpha_{2} + 2. $$

The covariance of two gamma random variable X ₁ and X ₂ monotonically increases with correlation coefficient of two corresponding standard normal random variables W ₁ and W ₂. Thus, Eq. 17 represents a unique conversion function.