README

# install.packages("mvp")  # uncomment this to install the package
library("cmvnorm")

[ f\left({\mathbf x};\Sigma\right) = \frac{ \exp\left(-\frac{1}{2}{\mathbf x}^T\Sigma{-1}{\mathbf x}\right) }{ \sqrt{\left|2\pi\Sigma\right|} } \qquad{\mathbf x}\in{\mathbb R}^n](https://latex.codecogs.com/png.latex?%0Af%5Cleft%28%7B%5Cmathbf%20x%7D%3B%5CSigma%5Cright%29%20%3D%0A%5Cfrac%7B%0A%20%20%5Cexp%5Cleft%28-%5Cfrac%7B1%7D%7B2%7D%7B%5Cmathbf%20x%7D%5ET%5CSigma%5E%7B-1%7D%7B%5Cmathbf%20x%7D%5Cright%29%0A%7D%7B%0A%20%20%5Csqrt%7B%5Cleft%7C2%5Cpi%5CSigma%5Cright%7C%7D%0A%7D%0A%5Cqquad%7B%5Cmathbf%20x%7D%5Cin%7B%5Cmathbb%20R%7D%5En%0A " f({x};) = ^n ")

where

is an

positive-definite variance matrix. Now compare the complex version with

Hermitian positive-definite:

[ f\left({\mathbf z};\Gamma\right) = \frac{ \exp\left( -{\mathbf z}^\dag\Gamma{-1}{\mathbf z}\right) }{ \left|\pi\Gamma\right| } \qquad{\mathbf z}\in\mathbb{C}^n](https://latex.codecogs.com/png.latex?%0Af%5Cleft%28%7B%5Cmathbf%20z%7D%3B%5CGamma%5Cright%29%20%3D%0A%5Cfrac%7B%0A%20%20%5Cexp%5Cleft%28%20-%7B%5Cmathbf%20z%7D%5E%5Cdag%5CGamma%5E%7B-1%7D%7B%5Cmathbf%20z%7D%5Cright%29%0A%7D%7B%0A%20%5Cleft%7C%5Cpi%5CGamma%5Cright%7C%0A%7D%0A%5Cqquad%7B%5Cmathbf%20z%7D%5Cin%5Cmathbb%7BC%7D%5En%0A " f({z};) = ^n ")

See how much nicer the complex version is! No awkward, unsightly factors of two and no inconvenient square roots. This is essentially due to Gauss’s integral operating more cleanly over the complex plane than the real line:

[ { \int_\mathbb{C}e^{-z\dag z}\,dz= \iint_{(x,y)\in\mathbb{R}^{2}\!\!\!\!\!\!\!\!\!\!\!\!\!\! e}{-(x^2+y2)}\,dx\,dy= \int_{\theta=0}^{{2\pi}\int_{r=0}}\infty e^{-r2}r\,dr\,d\theta= 2\pi\int_{r=0}^\infty e^{-r2}r\,dr=\pi. }](https://latex.codecogs.com/png.latex?%0A%7B%0A%5Cint_%5Cmathbb%7BC%7De%5E%7B-z%5E%5Cdag%20z%7D%5C%2Cdz%3D%0A%5Ciint_%7B%28x%2Cy%29%5Cin%5Cmathbb%7BR%7D%5E2%7D%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%0Ae%5E%7B-%28x%5E2%2By%5E2%29%7D%5C%2Cdx%5C%2Cdy%3D%0A%5Cint_%7B%5Ctheta%3D0%7D%5E%7B2%5Cpi%7D%5Cint_%7Br%3D0%7D%5E%5Cinfty%20e%5E%7B-r%5E2%7Dr%5C%2Cdr%5C%2Cd%5Ctheta%3D%0A2%5Cpi%5Cint_%7Br%3D0%7D%5E%5Cinfty%20e%5E%7B-r%5E2%7Dr%5C%2Cdr%3D%5Cpi.%0A%7D%0A " { e^{-zz},dz= {(x,y)^{2}!!!!!!!!!!!!!! e}{-(x^2+y2)},dx,dy= _{=0}^{{2}{r=0}^e{-r^2}r,dr,d= 2{r=0}}e^{-r2}r,dr=. } ")

It can be shown that

, so

really is the variance of the distribution. We can also introduce a nonzero mean,

in the natural way.

The cmvnorm package furnishes some R functionality for dealing with the complex multivariate Gaussian distribution.

The package in use

The simplest case would be the univariate standard normal distribution, that is is a complex random variable

with PDF

. Random samples are given by rcnorm():

rcnorm(10)
#>  [1]  0.6181509+0.0783307i -0.2617821-1.1716151i  0.4188516+0.9129822i
#>  [4] -0.0499387-0.6009572i  0.4166873-0.2954984i  0.1986088-0.5429393i
#>  [7]  0.5185832+0.6300364i  0.7295693-0.3090116i -1.4870623+0.7090736i
#> [10] -0.0452294-0.1559835i

Observations are circularly symmetric in the sense that

has the same distribution as

for any

, as we may verify visually:

par(pty="s")
plot(rcnorm(10000),asp=1,xlim=c(-3,3),ylim=c(-3,3),pch=16,cex=0.2)

We may sample from this distribution and verify that it has zero mean and unit variance:

z <- rcnorm(1e6)
mean(z)   # zero, subject to sample error
#> [1] -9.71348e-05+6.269418e-04i
var(z)    # one, subject to sample error
#> [1] 1.000525

z <- rcnorm(1e6)

var(Re(z))
#> [1] 0.4991952
var(Im(z))
#> [1] 0.4999403

tm <- c(1,1i)  # true mean
tS <- matrix(c(3,1i,-1i,2),2,2)  # true variance
rcmvnorm(10,mean=tm, sigma=tS)
#>                        [,1]                  [,2]
#>  [1,]  3.2217513+1.6826868i -1.1991170+2.5806750i
#>  [2,]  2.2866000-1.0382517i  0.1287602-0.3371253i
#>  [3,]  2.9225545+1.0450105i -2.0117682+1.3707301i
#>  [4,]  2.8709187+1.1591237i  1.3770551+0.1685449i
#>  [5,]  2.6686119+2.0988344i -1.5598805+1.7444876i
#>  [6,] -0.3600905+0.1005174i  0.9452877+0.5510341i
#>  [7,]  1.8070602+1.0582978i  0.2338880+0.2351671i
#>  [8,]  0.5616892-0.4861250i  0.3593575-0.6405177i
#>  [9,]  1.5572701+0.3994906i -1.6689283-0.0957045i
#> [10,] -0.7120009+1.2346053i -0.6296520+1.6596427i

n <- 1e6  # sample size
z <- rcmvnorm(n,mean=tm, sigma=tS)
colMeans(z)   # should be close to tm=[1,i]
#> [1]  1.000704-0.000501i -0.000205+1.001046i
z <- scale(z,scale=FALSE) # sweep out the mean
cprod(z)/n  # should be close to tS
#>                     [,1]                [,2]
#> [1,]  3.001797+0.000000i -0.002471+1.000779i
#> [2,] -0.002471-1.000779i  1.999561+0.000000i

The complex multivariate Gaussian distribution in R

Overview

The package in use

Further information