Matrix- and vector-valued generalizations of Gamma processes



\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\corr}{\operatorname{Corr}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\gvn}{\mid} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

Processes that generalise Gamma processes but to take vector or matrix values.

We start by considering trivial processes which have an empty index set, i.e. multivariate gamma distributions. So here is the simplest multivariate case:

Vector Gamma process

How can we turn the a multivariate gamma distribution into a vector valued gamma process?

An associated Lévy process is easy. Are there any Ornstein-Uhlenbeck-type processes?

Ornstein-Uhlenbeck Dirichlet process

TBD. Is that what Griffin (2011) achieves.

Wishart processes

Wishart distributions are commonly claimed to generalise Gamma distributions, although AFAICT they are not so similar. “Wishart processes” are indeed a thing (Pfaffel 2012; Wilson and Ghahramani 2011); although the Wishart distribution is not a special case of these it seems (?). It generalises the square Bessel process, which is marginally \(\chi^2\) distributed.

Inverse Wishart

Does the Inverse Wishart Process relate? (Shah, Wilson, and Ghahramani 2014; Tracey and Wolpert 2018) TODO

HDP Matrix Gamma Process

Matrix-valued Lévy-Gamma process analogue. See (Meier, Kirch, and Meyer 2020, sec. 2), which uses the multivariate construction of Pérez-Abreu and Stelzer (2014) to construct a family of matrix-variate Gamma processes That construction is an extremely general, and somewhat abstract, and is easy to handle usually only through its Lévy measure.

AΓ Process

Meier, Kirch, and Meyer (2020) mentions a construction less general than the HDP Matrix Gamma which is nonetheless broad and quite useful. We could this of it as the tractable HDP.:

A special case of the \(\operatorname{Gamma}_{d \times d}(\alpha, \lambda)\) distribution is the so-called \(A \Gamma\) distribution, that has been considered in Pérez-Abreu and Stelzer (2014) and generalized to the Hpd setting in (Meier 2018, sec. 2.4). To elaborate, the \(A \Gamma(\eta, \omega, \Sigma)\) distribution is defined with the parameters \(\eta>d-1, \omega>0\) and \(\Sigma \in\) \(\mathcal{S}_{d}^{+}\) as the \(\operatorname{Gamma}_{d \times d}\left(\alpha_{\eta, \Sigma}, \lambda_{\Sigma}\right)\) distribution, with \[ \alpha_{\eta, \boldsymbol{\Sigma}}(d \boldsymbol{U})=|\boldsymbol{\Sigma}|^{-\eta} \operatorname{tr}\left(\boldsymbol{\Sigma}^{-1} \boldsymbol{U}\right)^{-d \eta} \Gamma(d \eta) \tilde{\Gamma}_{d}(\eta)^{-1}|\boldsymbol{U}|^{\eta-d} d \boldsymbol{U}, \] where \(\Gamma\) denotes the Gamma function and \(\tilde{\Gamma}_{d}\) the complex multivariate Gamma function (see Mathai and Provost 2005), and \(\lambda_{\boldsymbol{\Sigma}}(\boldsymbol{U})=\operatorname{tr}\left(\boldsymbol{\Sigma}^{-1} \boldsymbol{U}\right)\). It has the advantage that for \(\boldsymbol{X} \sim A \Gamma(\eta, \omega, \Sigma)\), the formulas for mean and covariance structure are explicitly known: \[ \mathrm{E} \boldsymbol{X}=\frac{\omega}{d} \boldsymbol{\Sigma}, \quad \operatorname{Cov} \boldsymbol{X}=\frac{\omega}{d(\eta d+1)}\left(\eta \boldsymbol{I}_{d^{2}}+\boldsymbol{H}\right)(\boldsymbol{\Sigma} \otimes \boldsymbol{\Sigma}), \] where \(\boldsymbol{H}=\sum_{i, j=1}^{d} \boldsymbol{H}_{i, j} \otimes H_{j, i}\) and \(\boldsymbol{H}_{i, j}\) being the matrix having a one at \((i, j)\) and zeros elsewhere, see (Meier 2018 Lemma 2.8). Thus the \(A\Gamma\)-distribution is particularly well suited for Bayesian prior modeling if the prior knowledge is given in terms of mean and covariance structure.

References

Barndorff-Nielsen, Ole E., Makoto Maejima, and Ken-Iti Sato. 2006. Some Classes of Multivariate Infinitely Divisible Distributions Admitting Stochastic Integral Representations.” Bernoulli 12 (1): 1–33.
Barndorff-Nielsen, Ole E., Jan Pedersen, and Ken-Iti Sato. 2001. Multivariate Subordination, Self-Decomposability and Stability.” Advances in Applied Probability 33 (1): 160–87.
Bladt, Mogens, and Bo Friis Nielsen. 2010. Multivariate Matrix-Exponential Distributions.” Stochastic Models 26 (1): 1–26.
Buchmann, Boris, Benjamin Kaehler, Ross Maller, and Alexander Szimayer. 2015. Multivariate Subordination Using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing.” arXiv:1502.03901 [Math, q-Fin], February.
Das, Sourish, and Dipak K. Dey. 2010. On Bayesian Inference for Generalized Multivariate Gamma Distribution.” Statistics & Probability Letters 80 (19-20): 1492–99.
Foti, Nicholas, Joseph Futoma, Daniel Rockmore, and Sinead Williamson. 2013. A Unifying Representation for a Class of Dependent Random Measures.” In Artificial Intelligence and Statistics, 20–28.
Griffin, J.E. 2011. The Ornstein–Uhlenbeck Dirichlet Process and Other Time-Varying Processes for Bayesian Nonparametric Inference.” Journal of Statistical Planning and Inference 141 (11): 3648–64.
Grigelionis, Bronius. 2013. Student’s t-Distribution and Related Stochastic Processes. SpringerBriefs in Statistics. Berlin, Heidelberg: Springer Berlin Heidelberg.
Grunwald, G K, R J Hyndman, and L M Tedesco. n.d. “A Unified View of Linear AR(1) Models,” 26.
Kirch, Claudia, Matthew C. Edwards, Alexander Meier, and Renate Meyer. 2019. Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis.” Bayesian Analysis 14 (4): 1037–73.
Laverny, Oskar, Esterina Masiello, Véronique Maume-Deschamps, and Didier Rullière. 2021. Estimation of Multivariate Generalized Gamma Convolutions Through Laguerre Expansions.” arXiv:2103.03200 [Math, Stat], July.
Lawrence, Neil D., and Raquel Urtasun. 2009. Non-Linear Matrix Factorization with Gaussian Processes.” In Proceedings of the 26th Annual International Conference on Machine Learning, 601–8. ICML ’09. New York, NY, USA: ACM.
Liou, Jun-Jih, Yuan-Fong Su, Jie-Lun Chiang, and Ke-Sheng Cheng. 2011. Gamma Random Field Simulation by a Covariance Matrix Transformation Method.” Stochastic Environmental Research and Risk Assessment 25 (2): 235–51.
Mathai, A. M., and P. G. Moschopoulos. 1991. On a Multivariate Gamma.” Journal of Multivariate Analysis 39 (1): 135–53.
Mathai, A. M., and Serge B. Provost. 2005. Some Complex Matrix-Variate Statistical Distributions on Rectangular Matrices.” Linear Algebra and Its Applications, Tenth Special Issue (Part 2) on Linear Algebra and Statistics, 410 (November): 198–216.
Mathal, A. M., and P. G. Moschopoulos. 1992. A Form of Multivariate Gamma Distribution.” Annals of the Institute of Statistical Mathematics 44 (1): 97–106.
Meier, Alexander. 2018. A matrix Gamma process and applications to Bayesian analysis of multivariate time series.”
Meier, Alexander, Claudia Kirch, Matthew C. Edwards, and Renate Meyer. 2019. beyondWhittle: Bayesian Spectral Inference for Stationary Time Series (version 1.1.1).
Meier, Alexander, Claudia Kirch, and Renate Meyer. 2020. Bayesian Nonparametric Analysis of Multivariate Time Series: A Matrix Gamma Process Approach.” Journal of Multivariate Analysis 175 (January): 104560.
Pérez-Abreu, Victor, and Robert Stelzer. 2014. Infinitely Divisible Multivariate and Matrix Gamma Distributions.” Journal of Multivariate Analysis 130 (September): 155–75.
Pfaffel, Oliver. 2012. Wishart Processes.” arXiv:1201.3256 [Math], January.
Ranganath, Rajesh, and David M. Blei. 2018. Correlated Random Measures.” Journal of the American Statistical Association 113 (521): 417–30.
Rao, Vinayak, and Yee Whye Teh. 2009. “Spatial Normalized Gamma Processes.” In Proceedings of the 22nd International Conference on Neural Information Processing Systems, 1554–62. NIPS’09. Red Hook, NY, USA: Curran Associates Inc.
Sato, Ken-iti. 1999. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Semeraro, Patrizia. 2008. A Multivariate Variance Gamma Model for Financial Applications.” International Journal of Theoretical and Applied Finance 11 (01): 1–18.
Shah, Amar, Andrew Wilson, and Zoubin Ghahramani. 2014. Student-t Processes as Alternatives to Gaussian Processes.” In Artificial Intelligence and Statistics, 877–85. PMLR.
Sim, C.H. 1993. Generation of Poisson and Gamma Random Vectors with Given Marginals and Covariance Matrix.” Journal of Statistical Computation and Simulation 47 (1-2): 1–10.
Singpurwalla, Nozer D., and Mark A. Youngren. 1993. Multivariate Distributions Induced by Dynamic Environments.” Scandinavian Journal of Statistics 20 (3): 251–61.
Thibaux, Romain, and Michael I. Jordan. 2007. Hierarchical Beta Processes and the Indian Buffet Process.” In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, 564–71. PMLR.
Tracey, Brendan D., and David H. Wolpert. 2018. Upgrading from Gaussian Processes to Student’s-T Processes.” 2018 AIAA Non-Deterministic Approaches Conference, January.
Walker, Stephen G. 2021. On Infinitely Divisible Multivariate Gamma Distributions.” Communications in Statistics - Theory and Methods 0 (0): 1–7.
Warren, D. 1992. A Multivariate Gamma Distribution Arising from a Markov Model.” Stochastic Hydrology and Hydraulics 6 (3): 183–90.
Warren, David. 1986. Outflow Skewness in Non-Seasonal Linear Reservoirs with Gamma-Distributed Markovian Inflows.” Journal of Hydrology 85 (1): 127–37.
Wilson, Andrew Gordon, and Zoubin Ghahramani. 2011. Generalised Wishart Processes.” In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence, 736–44. UAI’11. Arlington, Virginia, United States: AUAI Press.
Wolpert, R., and Katja Ickstadt. 1998. Poisson/Gamma Random Field Models for Spatial Statistics.” Biometrika 85 (2): 251–67.
Wolpert, Robert L. 2021. Lecture Notes on Stationary Gamma Processes.” arXiv:2106.00087 [Math], May.
Xuan, Junyu, Jie Lu, Guangquan Zhang, Richard Yi Da Xu, and Xiangfeng Luo. 2015. Nonparametric Relational Topic Models Through Dependent Gamma Processes.” arXiv:1503.08542 [Cs, Stat], March.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.