# install.packages("mvp") # uncomment this to install the package
library("cmvnorm")Consider the (zero mean) multivariate Gaussian distribution
[ 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{-(x2+y^2)}\,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{-(x2+y^2)},dx,dy= _{}{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 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] -0.0452294-0.1559835iObservations 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] 1.000525Note that the real and imaginary components of have variance :
z <- rcnorm(1e6)
var(Re(z))
#> [1] 0.4991952
var(Im(z))
#> [1] 0.4999403We may sample from the multivariate case similarly. Suppose and :
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+1.6596427iWe may perform elementary inference. For the mean, we would calculate the
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+0.000000iFor further information, see the package vignette: type
vignette("cmvnorm")
at the R command line.