In the mathematical theory of probability, multivariate Laplace distributions are extensions of the Laplace distribution and the asymmetric Laplace distribution to multiple variables. The marginal distributions of symmetric multivariate Laplace distribution variables are Laplace distributions. The marginal distributions of asymmetric multivariate Laplace distribution variables are asymmetric Laplace distributions.[1]
A typical characterization of the symmetric multivariate Laplace distribution has the characteristic function:
\varphi(t;\boldsymbol\mu,\boldsymbol\Sigma)=
\exp(i\boldsymbol\mu't) | |
1+\tfrac{1 |
{2}t'\boldsymbol\Sigmat
where
\boldsymbol\mu
\boldsymbol\Sigma
Unlike the multivariate normal distribution, even if the covariance matrix has zero covariance and correlation the variables are not independent.[1] The symmetric multivariate Laplace distribution is elliptical.[1]
If
\boldsymbol\mu=0
fx(x1,\ldots,xk)=
2 | |
(2\pi)k/2|\boldsymbol\Sigma|0.5 |
\left(
x'\boldsymbol\Sigma-1x | |
2 |
\right)v/2Kv\left(\sqrt{2x'\boldsymbol\Sigma-1x
where:
v=(2-k)/2
Kv
In the correlated bivariate case, i.e., k = 2, with
\mu1=\mu2=0
fx(x1,x2)=
1 | |
\pi\sigma1\sigma2\sqrt{1-\rho2 |
where:
\sigma1
\sigma2
x1
x2
\rho
x1
x2
For the uncorrelated bivariate Laplace case, that is k = 2,
\mu1=\mu2=\rho=0
\sigma1=\sigma2=1
fx(x1,x2)=
1 | |
\pi |
K0\left(\sqrt{
2 | |
2(x | |
1 |
+
2) | |
x | |
2 |
}\right).
A typical characterization of the asymmetric multivariate Laplace distribution has the characteristic function:
\varphi(t;\boldsymbol\mu,\boldsymbol\Sigma)=
1 | |
1+\tfrac{1 |
{2}t'\boldsymbol\Sigmat-i\boldsymbol\mut}.
As with the symmetric multivariate Laplace distribution, the asymmetric multivariate Laplace distribution has mean
\boldsymbol\mu
\boldsymbol\Sigma+\boldsymbol\mu'\boldsymbol\mu
\boldsymbol\mu=0
\boldsymbol\mu=0
The probability density function (pdf) for a k-dimensional asymmetric multivariate Laplace distribution is:
fx(x1,\ldots,xk)=
| |||||||
(2\pi)k/2|\boldsymbol\Sigma|0.5 |
(
x'\boldsymbol\Sigma-1x | |
2+\boldsymbol\mu'\boldsymbol\Sigma-1\boldsymbol\mu |
)v/2Kv(\sqrt{(2+\boldsymbol\mu'\boldsymbol\Sigma-1\boldsymbol\mu)(x'\boldsymbol\Sigma-1x)}),
where:
v=(2-k)/2
Kv
The asymmetric Laplace distribution, including the special case of
\boldsymbol\mu=0
The relationship between the exponential distribution and the Laplace distribution allows for a simple method for simulating bivariate asymmetric Laplace variables (including for the case of
\boldsymbol\mu=0
Y
\mu1=\mu2=0
\boldsymbol\Sigma
X=\sqrt{W}Y+W\boldsymbol\mu
\boldsymbol\mu
\boldsymbol\Sigma