Stochastic simulation of bivariate gamma distribution: a frequency-factor based approach

Abstract

A frequency-factor based approach for stochastic simulation of bivariate gamma distribution is proposed. The approach involves generation of bivariate normal samples with a correlation coefficient consistent with the correlation coefficient of the corresponding bivariate gamma samples. Then the bivariate normal samples are transformed to bivariate gamma samples using the well-known general equation of hydrological frequency analysis. We demonstrate that the proposed bivariate gamma simulation approach is capable of generating random sample pairs which not only have the desired marginal densities of component random variables but also their correlation coefficient. Scatter plots of simulated bivariate sample pairs also exhibit appropriate linear patterns (dependence structure) that are commonly observed in environmental and hydrological applications. Caution should also be exercised when specifying combinations of coefficients of skewness and the correlation coefficient for bivariate gamma simulation.

Gamma random field simulation by a covariance matrix transformation method

“…Similar to the conversional relationship between correlation coefficients of the bivariate gamma and bivariate normal densities derived by Cheng et al (2010), we present in this section another form of the conversional relationship.…”

“…Based on the bivariate gamma simulation approach proposed by Cheng et al (2010), simulation of a gamma random field can be achieved in a similar, yet more complicated, manner. Through a theoretical conversion between q UV and q XY , random samples of a pair of bivariate gamma random variates (X, Y) with desired marginal densities [which are fully characterized by the means (l X , l Y ) and coefficients of skewness (c X , c Y )] and correlation coefficient q XY can be obtained by firstly generating random samples of a corresponding pair of bivariate variates (U, V) with standard normal density and correlation coefficient q UV , and a subsequent Gaussian-to-gamma transformation.…”

“…the covariance function) of the Gaussian random field, given the spatial variation structure of the non-Gaussian random field. Cheng et al (2010) proposed a frequency-factor-based bivariate gamma simulation approach. The approach basically generates a random sample {fðu i ; v i Þ; i ¼ 1; .…”

Multisite Spatiotemporal Streamflow Simulation - With an Application to Irrigation Water Shortage Risk Assessment

“…(8) can be transformed to a corresponding standard normal random field. Cheng et al (2011) showed that such transformation requires the following conversion between the correlation coefficient of a bivariate PT3 distribution and the correlation coefficient of a bivariate standard normal distribution: =`j . For convenience, {X(s 1 , t 1 ), X(s 2 , t 2 ), X(s 3 , t 3 )} is replaced by X = {X 1 , X 2 , X 3 } from here.…”

“…While it may be difficult to preserve all statistical properties of complex phenomenon, the minimum goal is to preserve up to the secondmoment properties, i.e., the expectation and variance/covariance. Many environmental and natural variables, such as precipitation amount, storm duration, and streamflow exhibit significant degrees of non-Gaussian randomness (Parada and Liang 2010;Cheng et al 2011). Univariate simulation of such non-Gaussian random variables can be achieved by using the probability integral transformation (Cheng et al 2007), and many statistical software packages provide builtin commands for such univariate simulations.…”