# Multivariate Gamma Distributions

Multivariate Gamma Distributions are extensions of the univariate Gamma Distributions. In general, a random vector has the multivariate gamma distribution when it has gamma marginals .

A gamma marginal refers to the marginal distribution of each individual component (variable) of the random vector, which follows a univariate gamma distribution. More specifically, a random vector is said to have a multivariate gamma distribution if all its marginal distributions (i.e., the distributions of each individual variable when considered separately) are gamma distributions. In other words, when examining each variable independently, the probability distribution for that variable follows a gamma distribution.

The generalized multivariate gamma distribution probability density function is defined by :

Where Γpα is the multivariate gamma function — an extension of the gamma function for multiple variables. It is often defined as

where

• ? is the space of m x m real, positive definite (and hence matrices). A positive definite matrix is a type of symmetric matrix which has all its eigenvalues as positive numbers. Eigenvalues, which are special scalars, are obtained when a matrix is multiplied by a vector resulting in the same vector and a new scalar.
• dS = product Lebesgue measure of ½p(p + 1) distinct elements in S.

## Types of Multivariate Gamma Distributions

• Dussauchoy and Berland’s  multivariate gamma distribution is defined by a characteristic function. In the context of multivariate distributions, a characteristic function is a mathematical tool used to describe and analyze the distribution. The characteristic function provides a compact and convenient way to represent the distribution and allows for easy computation of various properties, such as moments and correlations between variables. In the case of Dussauchoy and Berland’s multivariate distribution, the characteristic function is employed to define the distribution and to derive its properties, making it easier to work with and analyze the distribution.
• Gaver’s multivariate gamma distribution  generates a multivariate distribution with a mixture of gamma variables with negative binomial weights. In the context of a multivariate distribution defined with a mixture of gamma variables, negative binomial weights refer to the probability weights assigned to each component (gamma variable) in the mixture based on a negative binomial distribution.
• Krishnamoorthy and Parthasarathy’s  p-variate gamma distribution — also called the Wishart-chi-square distribution — exists for all positive integer degrees of freedom v and at least for all real values v > p – 2, p ≥ 2.

## Real life applications of multivariate gamma distributions

The multivariate gamma distribution has various real-life applications across different fields, including finance, engineering, and the natural sciences. Some examples include:

1. Finance and risk management: In portfolio management and risk assessment, the multivariate gamma distribution can be used to model the joint behavior of multiple financial assets or the correlated risks associated with them. This helps in understanding the relationships between assets, optimizing portfolio allocation, and estimating the probability of extreme events, such as market crashes.
2. Engineering and reliability theory: In systems with multiple components, the multivariate gamma distribution can be applied to model the lifetimes or failure times of these components when they are dependent on each other. This information is crucial for designing efficient maintenance strategies, improving system reliability, and minimizing downtime.
3. Environmental science and meteorology: The multivariate gamma distribution can be used to model the joint distribution of multiple correlated variables, such as temperature, humidity, and precipitation, in climate studies or weather forecasting. This enables a better understanding of the interactions between these variables and helps in predicting extreme weather events and their potential impacts.
4. Biology and medicine: In the study of gene expression levels or protein concentrations, the multivariate gamma distribution can be employed to model the joint distribution of multiple correlated biomolecules. This information can aid in understanding the complex biological processes, identifying biomarkers, and developing targeted therapies.
5. Image processing and computer vision: The multivariate gamma distribution can be applied to model the joint distribution of pixel intensities in multi-spectral or multi-channel images, such as RGB or hyperspectral images. This allows for effective noise reduction, image segmentation, and feature extraction, which are essential tasks in various computer vision applications.

## References

 Viraswami, K. (1991). On Multivariate Gamma Distributions. Retrieved January 1, 2022 from: https://escholarship.mcgill.ca/downloads/76537233r?locale=en

 Das, S. & Dey, D. On Bayesian Inference for Generalized Multivariate Gamma Distribution.

 Dussauchoy, A. & Berland, R. (1974). A multivariate gamma type distribution whose marginal laws are gamma, and which has a property similar to a characteristic property of the normal case. Statistical Distributions in Scientific Work, Vol. 1. Patil, G. et al (eds.). D. Reidel, Boston. 319-328.

 Gaver, D. P. (1970). Multivariate gamma distributions generated by mixture. Sankyhya Series A, 32, 123-126.

 Krishnamoorthy, A. S. and Parthasarathy, M., A multivariate gamma type distribution, Ann. Math. Stat. 22
(1951), 549-557.

Scroll to Top