Distribution and Moments of a Normalized Dissimilarity Ratio for two Correlated Gamma Variables

TL;DR

This work derives a complete statistical characterization of the Fujii index $D(X,Y)=\frac{|X-Y|}{X+Y}$ for two correlated Gamma variables with identical shape $k$ and scale, grounded in the representation $X,Y$ as sums of $k$ correlated exponentials from circular complex Gaussian fields. The authors obtain a closed-form PDF for $D$ via a sequence of transformations: from the joint PDF of $(X,Y)$ to the ratio $Z=X/Y$, and finally to $D$, accompanied by three equivalent expressions for the $m$-th moment in hypergeometric form. Key contributions include explicit joint PDFs $f_{\text{Gamma}}(x_1,x_2)$, the ratio PDF $f_Z(z)$, and the $D$-PDF $f_D(r)$, together with flexible moment formulas that cover the special cases $\rho=0$ and $k=1$ and are validated through numerical simulations. The results offer a rigorous statistical framework for dynamic speckle imaging analyses and can be extended to other contexts involving correlated Gamma-distributed variables and their contrast measures.

Abstract

We consider two random variables $X$ and $Y$ following correlated Gamma distributions, characterized by identical scale and shape parameters and a linear correlation coefficient $ρ$. Our focus is on the parameter: \[ D(X,Y) = \frac{|X - Y|}{X + Y}, \] which appears in applied contexts such as dynamic speckle imaging, where it is known as the \textit{Fujii index}. In this work, we derive a closed-form expression for the probability density function of $D(X,Y)$ as well as analytical formulas for its moments of order $k$. Our derivation starts by representing $X$ and $Y$ as two correlated exponential random variables, obtained from the squared magnitudes of circular complex Gaussian variables. By considering the sum of $k$ independent exponential variables, we then derive the joint density of $(X,Y)$ when $X$ and $Y$ are two correlated Gamma variables. Through appropriate varable transformations, we obtain the theoretical distribution of $D(X,Y)$ and evaluate its moments analytically. These theoretical findings are validated through numerical simulations, with particular attention to two specific cases: zero correlation and unit shape parameter.

Figures (7)

• Figure 1: Empirical correlation between the Gamma-distributed intensities $X = |Z_x|^2$ and $Y = |Z_y|^2$ as a function of the input correlation $\rho$ of the complex fields. The empirical values superimpose on the theoretical curve $\rho^2$, confirming the expected correlation structure.
• Figure 2: Comparison between empirical and theoretical joint probability density of $(X,Y)$ in the exponential case ($k=1$) for $\rho_z=0.8$ and $\sigma_z=0.7$.
• Figure 3: Comparison between empirical and theoretical joint probability density of $(X,Y)$ in the Gamma case for $\rho_z=0.8, \rho=0.64$, $\sigma_z=0.7,\sigma=2.88$ and $k=12$
• Figure 4: Visualization of the theoretical joint probability density of $(X,Y)$ for the Exponential case (left) and the Gamma case (right). The joint PDFs are represented in a horizontal plane, while the third dimension is used to display the corresponding marginal distributions along the $X$- and $Y$-axes.
• Figure 5: Comparison between the empirical histogram and the theoretical density of the Normalized Dissimilarity Ratio $D(X,Y)$ for fixed values of $\rho$, $\sigma$, and $k$.