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Abstract
Here I introduce cmvnorm, a complex generalization of the mvtnorm package. A complex generalization of the Gaussian process is suggested and numerical results presented using the package. An application in the context of approximating the Weierstrass sigma function using a complex Gaussian process is given.
Highlights
Complex-valued random variables find applications in many areas of science such as signal processing (Kay, 1989), radio engineering (Ozarow, 1994), and atmospheric physics (Mandic et al, 2009)
The real multivariate Gaussian distribution is well supported in R by package mvtnorm (Genz et al, 2014), having density function f (x; m, Σ)
In the context of the emulator, a Gaussian process is usually defined as a random function η : Rp −→ R which, for any set of points {x1, . . . , xn} in its domain D the random vector {η (x1), . . ., η} is multivariate Gaussian
Summary
Complex-valued random variables find applications in many areas of science such as signal processing (Kay, 1989), radio engineering (Ozarow, 1994), and atmospheric physics (Mandic et al, 2009) In this short paper I introduce cmvnorm (Hankin, 2015), a package for investigating one commonly encountered complex-valued probability distribution, the complex Gaussian. One natural generalization would be to consider Z ∼ N Cn (m, Γ), the complex multivariate Gaussian, with density function f (z; m, Γ). The complex generalization is to write z = Zβ + , Z ∈ Cn×p, β ∈ Cp, ∼ N Cn (0, Γ) which gives β = Z∗Γ−1Z −1 Z∗Γ−1z Such considerations suggest a natural complex generalization of the Gaussian process. The package in use Random complex vectors are generated using the rcmvnorm() function, analogous to rmvnorm():. This shows reasonable agreement with the true value of the mean and the analytic value of the MLE,
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