Vecchia approximation

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In statistics, the Vecchia approximation is a technique for approximating the joint likelihood function of several variables. Joint likelihood often has a complicated form which might lead to some commonly used mathematical transformations to require a lot of computational resources. By approximating the original function, some of these transformations can be dramatically simplified.

Vecchia approximation originated in the field of spatial statistics where variables are often assumed to have a jointly Gaussian distribution. This means that evaluating the likelihood function requires inverting their joint covariance matrix. Matrix inversion has very high computational complexity, which means that inverting large matrices takes a lot of time. By simplifying the likelihood function, this time can sometimes be dramatically reduced.

In general, if vector $\mathbf{𝐱}=(x_1,x_2,\dots ,x_n)$ follows a distribution with probability density function (PDF) $f$, then we can express its joint probability as $$f(\mathbf{𝐱})=\prod _{i=1}^{n}f(x_i|x_1,\dots ,x_{i-1}),$$ where $f(x_i|x_1,\dots ,x_{i-1})$ is the conditional distribution of $x_i$ given $x_1,\dots ,x_{i-1}$. The simplest form of the Vecchia approximation assumes that $$f(\mathbf{𝐱})=\prod _{i=1}^{n}f(x_i|x_{i-r},x_{i-r+1},\dots ,x_{i-1}),$$ or that each element of $\mathbf{𝐱}$ is independent of other elements given its $r$ preceding elements.

Vecchia approximation was originally developed in the field of spatial statistics and in the context of the joint density $f$ being Gaussian. Recently, an important generalization has been proposed which assumes that $$f(\mathbf{𝐱})=\prod _{i=1}^{n}f(x_i|x_{g(i)}),$$ where $g(i)\subset \{1,2,\dots ,i-1\}$.

In other words, the conditioning set does not have to necessarily consist of the $r$ variables immediately preceding a particular variable, but can be more general.

References

Vecchia, A. V. “Estimation and Model Identification for Continuous Spatial Processes.” Journal of the Royal Statistical Society. Series B (Methodological), vol. 50, no. 2, 1988, pp. 297–312. Katzfuss, M. and Guinness, J. (2017). A general framework for Vecchia approximations of Gaussian processes.arXiv:1708.06302.

Category:Gaussian function Category:Approximation algorithms

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