$\chi^2$ Random Fields in Space and Time

Spherically Invariant Vector Random Fields in Space and Time

“…An important feature of (2) is that it allows for any correlation structure, whenever a mixing variable has a second-order moment. Another important feature is that it does not have a restriction or a tight connection between its mean and covariance functions, unlike log-Gaussian [42] or cases [39], so that the spherically invariant random field may be relatively more flexible for applications. The latter feature of being analytically easy to manipulate would make second-order spherically invariant random fields work effectively for studying various correlation effects in science and engineering.…”

“…, [11], [12], the non-Gaussianity cannot be neglected [7], and there are often specific reasons for assuming particular non-Gaussian finite-dimensional distributions [2], [4], [9], [17], [20]- [22], [27], [30], [32], [34], [35], [38], [39], [44]- [46], [48]- [50], [52], [55], [56]. For example, "One of the main problems in statistical analysis of multi-component images and multidimensional signals is the choice of relevant statistical parametric laws" [7], and the electromagnetic environment encountered by receiver systems is often non-Gaussian in nature [62].…”

On modeling positive continuous data with spatiotemporal dependence

“…The stationary process X m is a scaled version of a 2 random process (Adler, 1981;Ma, 2010) with marginal distribution Gamma(m/2,m/2), where the pairs m/2,m/2 are the shape and rate parameters. By definition, E(X m (s)) = 1 and Var(X m (s)) = 2∕m for all s.…”

“…In this article, we shall look at processes that are derived by Gaussian processes but differently from the trans-Gaussian random processes and the copula models we do not consider just one copy of the Gaussian process. We suggest to model positive continuous data by transforming 2 processes (Adler, 1981;Ma, 2010), that is, a sum of squared of independent copies of a standard Gaussian process. Even though probabilistic properties of a sum of squared Gaussian processes have been studied several years ago, less attention has been paid to use this for statistical modeling of dependent positive data.…”

Multivariate transformed Gaussian processes

“…However, Genton and Zhang (2012) pointed out identifiability problems related to the above construction. Ma (2010) built the χ 2 random process by summation of the squares of m independent Gaussian processes and extended it to the multivariate case (Ma, 2011). A series of papers with the same author combined the above two ideas and constructed many special cases for non-Gaussian vector random fields, such as the hyperbolic vector random fields (Du et al, 2012) and the K-distributed process (Ma, 2013).…”