en>JamesMilnerWhite: /* Integrality */

2014-10-15T14:59:50Z

Integrality

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2012-05-27T15:01:07Z

Integrality: updated references (hopefully)

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In [[probability theory]], the family of '''complex normal distributions''' characterizes [[complex random variable]]s whose real and imaginary parts are jointly [[Multivariate normal distribution|normal]],<ref>{{harvtxt|Goodman|1963}}</ref> i.e., normally distributed and independent. The complex normal family has three parameters: ''location'' parameter ''μ'', ''covariance'' matrix Γ, and the ''relation'' matrix ''C''. The '''standard complex normal''' is the univariate distribution with ''μ'' = 0, Γ = 1, and ''C'' = 0.

An important subclass of complex normal family is called the '''circularly-symmetric complex normal''' and corresponds to the case of zero relation matrix and zero mean: <math> \mu = 0 \ \text{and} \ C=0 </math>.<ref>[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf ''bookchapter, Gallager.R''], pg9.</ref> Circular symmetric complex normal random variables are used extensively in [[signal processing]], and are sometimes referred to as just '''complex normal''' in signal processing literature.

==Definition==
Suppose ''X'' and ''Y'' are random vectors in '''R'''''k'' such that vec[''X Y''] is a 2''k''-dimensional [[normal random vector]]. Then we say that the complex random vector
: <math>
Z = X + iY \,
</math>
has the '''complex normal distribution'''. This distribution can be described with 3 parameters:<ref name="picinbono">{{harvtxt|Picinbono|1996}}</ref>
: <math>
\mu = \operatorname{E}[Z], \quad
\Gamma = \operatorname{E}[(Z-\mu)(\overline{Z}-\overline\mu)'], \quad
C = \operatorname{E}[(Z-\mu)(Z-\mu)'],
</math>
where ''Z'' ′ denotes [[matrix transpose]], and Z denotes [[complex conjugate]]. Here the ''location'' parameter ''μ'' can be an arbitrary k-dimensional complex vector; the ''covariance'' matrix Γ must be [[Hermitian matrix|Hermitian]] and [[non-negative definite]]; the ''relation'' matrix ''C'' should be [[symmetric matrix|symmetric]]. Moreover, matrices Γ and ''C'' are such that the matrix
: <math>
P = \overline\Gamma - \overline{C}'\Gamma^{-1}C
</math>
is also non-negative definite.<ref name="picinbono"/>

Matrices Γ and ''C'' can be related to the covariance matrices of ''X'' and ''Y'' via expressions
: <math>\begin{align}
& V_{xx} \equiv \operatorname{E}[(X-\mu_x)(X-\mu_x)'] = \tfrac{1}{2}\operatorname{Re}[\Gamma + C], \quad
V_{xy} \equiv \operatorname{E}[(X-\mu_x)(Y-\mu_y)'] = \tfrac{1}{2}\operatorname{Im}[-\Gamma + C], \\
& V_{yx} \equiv \operatorname{E}[(Y-\mu_y)(X-\mu_x)'] = \tfrac{1}{2}\operatorname{Im}[\Gamma + C], \quad\,
V_{yy} \equiv \operatorname{E}[(Y-\mu_y)(Y-\mu_y)'] = \tfrac{1}{2}\operatorname{Re}[\Gamma - C],
\end{align}</math>
and conversely
: <math>\begin{align}
& \Gamma = V_{xx} + V_{yy} + i(V_{yx} - V_{xy}), \\
& C = V_{xx} - V_{yy} + i(V_{yx} + V_{xy}).
\end{align}</math>

==Density function==
The probability density function for complex normal distribution can be computed as

: <math>\begin{align}
f(z) &= \frac{1}{\pi^k\sqrt{\det(\Gamma)\det(P)}}\,
\exp\!\left\{-\frac12 \begin{pmatrix}(\overline{z}-\overline\mu)' & (z-\mu)'\end{pmatrix}
\begin{pmatrix}\Gamma&C\\\overline{C}'&\overline\Gamma\end{pmatrix}^{\!\!-1}\!
\begin{pmatrix}z-\mu \\ \overline{z}-\overline{\mu}\end{pmatrix}
\right\} \\[8pt]
&= \tfrac{\sqrt{\det\left(\overline{P^{-1}}-\overline{R}'P^{-1}R\right)\det(P^{-1})}}{\pi^k}\,
e^{ -(\overline{z}-\overline\mu)'\overline{P^{-1}}(z-\mu) +
\operatorname{Re}\left((z-\mu)'R'\overline{P^{-1}}(z-\mu)\right)},
\end{align}</math>

where ''R'' = C′ Γ −1 and ''P'' = Γ − ''RC''.

==Characteristic function==
The [[characteristic function (probability theory)|characteristic function]] of complex normal distribution is given by <ref name="picinbono"/>
: <math>
\varphi(w) = \exp\!\big\{i\operatorname{Re}(\overline{w}'\mu) - \tfrac{1}{4}\big(\overline{w}'\Gamma w + \operatorname{Re}(\overline{w}'C\overline{w})\big)\big\},
</math>
where the argument <math>w</math> is a ''k''-dimensional complex vector.

==Properties==

* If ''Z'' is a complex normal ''k''-vector, ''A'' an ''ℓ×k'' matrix, and ''b'' a constant '' ℓ''-vector, then the linear transform {{nowrap|''AZ + b''}} will be distributed also complex-normally:
: <math>
Z\ \sim\ \mathcal{CN}(\mu,\, \Gamma,\, C) \quad\Rightarrow\quad AZ+b\ \sim\ \mathcal{CN}(A\mu+b,\, A\Gamma\overline{A}',\, ACA')
</math>

* If ''Z'' is a complex normal ''k''-vector, then
: <math>
2\Big[ (\overline{Z}-\overline\mu)'\overline{P^{-1}}(Z-\mu) -
\operatorname{Re}\big((Z-\mu)'R'\overline{P^{-1}}(Z-\mu)\big)
\Big]\ \sim\ \chi^2(2k)
</math>

* '''Central limit theorem'''. If ''z''1, …, ''z''''T'' are independent and identically distributed complex random variables, then
: <math>
\sqrt{T}\Big( \tfrac{1}{T}\textstyle\sum_{t=1}^Tz_t - \operatorname{E}[z_t]\Big) \ \xrightarrow{d}\
\mathcal{CN}(0,\,\Gamma,\,C),
</math>
where Γ = E[ ''z''z′ ] and ''C'' = E[ ''zz′'' ].

==Circularly-symmetric complex normal distribution==
The circularly-symmetric complex normal distribution <ref>[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf ''bookchapter, Gallager.R'']</ref> corresponds to the case of zero mean and zero relation matrix, ''μ=0'', ''C=0''. If {{nowrap|1=''Z'' = ''X'' + ''iY''}} is circularly-symmetric complex normal, then the vector vec[''X Y''] is multivariate normal with covariance structure
: <math>
\begin{pmatrix}X \\ Y\end{pmatrix} \ \sim\
\mathcal{N}\Big( \begin{bmatrix}
\operatorname{Re}\,\mu \\
\operatorname{Im}\,\mu
\end{bmatrix},\
\tfrac{1}{2}\begin{bmatrix}
\operatorname{Re}\,\Gamma & \operatorname{Im}\,\Gamma \\
\operatorname{Im}\,\Gamma & \operatorname{Re}\,\Gamma
\end{bmatrix}\Big)
</math>
where {{nowrap|1=''μ'' = E[ ''Z'' ] = 0}} and {{nowrap|1=Γ = E[ ''ZZ′'' ]}}. This is usually denoted
:<math>Z \sim \mathcal{CN}(0,\,\Gamma)</math>
and its distribution can also be simplified as
: <math>
f(z) = \tfrac{1}{\pi^k\det(\Gamma)}\, e^{ -\overline{z}'\; \Gamma^{-1}\; z }.
</math>
The '''standard complex normal''' corresponds to the distribution of a scalar random variable with ''μ'' = 0, ''C'' = 0 and Γ = 1. Thus, the standard complex normal distribution has density

: <math>
f(z) = \tfrac{1}{\pi} e^{-\overline{z}z} = \tfrac{1}{\pi} e^{-|z|^2}.
</math>

This expression demonstrates why the case ''C'' = 0, ''μ'' = 0 is called “circularly-symmetric”. The density function depends only on the magnitude of ''z'' but not on its [[Arg (mathematics)|argument]]. As such, the magnitude ''|z|'' of standard complex normal random variable will have the [[Rayleigh distribution]] and the squared magnitude ''|z|''2 will have the [[Exponential distribution]], whereas the argument will be distributed [[Uniform distribution (continuous)|uniformly]] on [−''π'', ''π''].

If {''z''1, …, ''z''''n''} are independent and identically distributed ''k''-dimensional circular complex normal random variables with ''μ'' = 0, then random squared norm
: <math>
Q = \sum_{j=1}^n \overline{z_j'} z_j = \sum_{j=1}^n \| z_j \|^2
</math>
has the [[Generalized chi-squared distribution]] and the random matrix
: <math>
W = \sum_{j=1}^n z_j\overline{z_j'}
</math>
has the [[complex Wishart distribution]] with ''n'' degrees of freedom. This distribution can be described by density function
: <math>
f(w) = \frac{\det(\Gamma^{-1})^n\det(w)^{n-k}}{\pi^{k(k-1)/2}\prod_{j=1}^p(n-j)!}\
e^{-\operatorname{tr}(\Gamma^{-1}w)}
</math>
where ''n ≥ k'', and ''w'' is a ''k×k'' nonnegative-definite matrix.

==See also==
* [[Directional statistics#Distribution of the mean]]
* [[Normal distribution]]
* [[Multivariate normal distribution]]
* [[Generalized chi-squared distribution]]
* [[Wishart distribution]]

==References==
{{reflist}}

{{refbegin}}
* {{cite journal
| first = N.R.
| last = Goodman
| year = 1963
| title = Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)
| journal = The Annals of Mathematical Statistics
| volume = 34
| issue = 1
| pages = 152–177
| jstor = 2991290
}}
* {{cite journal
| last = Picinbono
| first = Bernard
| year = 1996
| title = Second-order complex random vectors and normal distributions
| journal = IEEE Transactions on Signal Processing
| volume = 44
| issue = 10
| pages = 2637–2640
}}
{{refend}}
{{ProbDistributions|continuous-infinite}}
[[Category:Continuous distributions]]
[[Category:Multivariate continuous distributions]]
[[Category:Complex numbers]]
[[Category:Probability distributions]]

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en>JamesMilnerWhite: /* Integrality */

en>Joel B. Lewis: /* Integrality */ updated references (hopefully)