Multivariate Laplace distribution

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]

Symmetric multivariate Laplace distribution

Multivariate Laplace (symmetric)

Parameters	μ ∈ Rk — location Σ ∈ Rk×k — covariance (positive-definite matrix)
Support	x ∈ μ + span(Σ) ⊆ Rk
PDF	If ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$, ${\displaystyle {\frac {2}{(2\pi )^{k/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}}\left({\frac {\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }{2}}\right)^{v/2}K_{v}\left({\sqrt {2\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }}\right),}$ where ${\displaystyle v=(2-k)/2}$ and ${\displaystyle K_{v}}$ is the modified Bessel function of the second kind.
Mean	μ
Mode	μ
Variance	Σ
Skewness	0
CF	${\displaystyle {\frac {\exp(i{\boldsymbol {\mu }}'\mathbf {t} )}{1+{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} }}}$

A typical characterization of the symmetric multivariate Laplace distribution has the characteristic function:

${\displaystyle \varphi (t;{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {\exp(i{\boldsymbol {\mu }}'\mathbf {t} )}{1+{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} }},}$

where ${\displaystyle {\boldsymbol {\mu }}}$ is the vector of means for each variable and ${\displaystyle {\boldsymbol {\Sigma }}}$ is the covariance matrix. ^[2]

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]

Probability density function

If ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$, the probability density function (pdf) for a k-dimensional multivariate Laplace distribution becomes:

${\displaystyle f_{\mathbf {x} }(x_{1},\ldots ,x_{k})={\frac {2}{(2\pi )^{k/2}|{\boldsymbol {\Sigma }}|^{0.5}}}\left({\frac {\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }{2}}\right)^{v/2}K_{v}\left({\sqrt {2\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }}\right),}$

where:

${\displaystyle v=(2-k)/2}$ and ${\displaystyle K_{v}}$ is the modified Bessel function of the second kind. ^[1]

In the correlated bivariate case, i.e., k = 2, with ${\displaystyle \mu _{1}=\mu _{2}=0}$ the pdf reduces to:

${\displaystyle f_{\mathbf {x} }(x_{1},x_{2})={\frac {1}{\pi \sigma _{1}\sigma _{2}{\sqrt {1-\rho ^{2}}}}}K_{0}\left({\sqrt {\frac {2\left({\frac {x_{1}^{2}}{\sigma _{1}^{2}}}-{\frac {2\rho x_{1}x_{2}}{\sigma _{1}\sigma _{2}}}+{\frac {x_{2}^{2}}{\sigma _{2}^{2}}}\right)}{1-\rho ^{2}}}}\right),}$

where:

${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{2}}$ are the standard deviations of ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$, respectively, and ${\displaystyle \rho }$ is the correlation coefficient of ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$. ^[1]

For the uncorrelated bivariate Laplace case, that is k = 2, ${\displaystyle \mu _{1}=\mu _{2}=\rho =0}$ and ${\displaystyle \sigma _{1}=\sigma _{2}=1}$, the pdf becomes:

${\displaystyle f_{\mathbf {x} }(x_{1},x_{2})={\frac {1}{\pi }}K_{0}\left({\sqrt {2(x_{1}^{2}+x_{2}^{2})}}\right).}$ ^[1]

Asymmetric multivariate Laplace distribution

Multivariate Laplace (asymmetric)

Parameters	μ ∈ Rk — location Σ ∈ Rk×k — covariance (positive-definite matrix)
Support	x ∈ μ + span(Σ) ⊆ Rk
PDF	${\displaystyle {\frac {2e^{\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{(2\pi )^{\frac {k}{2}}|{\boldsymbol {\Sigma }}|^{0.5}}}{\Big (}{\frac {\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }{2+{\boldsymbol {\mu }}'{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{\Big )}^{\frac {v}{2}}K_{v}{\Big (}{\sqrt {(2+{\boldsymbol {\mu }}'{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }})(\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} )}}{\Big )}}$ where ${\displaystyle v=(2-k)/2}$ and ${\displaystyle K_{v}}$ is the modified Bessel function of the second kind.
Mean	μ
Variance	Σ + μ ' μ
Skewness	non-zero unless μ=0
CF	${\displaystyle {\frac {1}{1+{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} -i{\boldsymbol {\mu }}\mathbf {t} }}}$

A typical characterization of the asymmetric multivariate Laplace distribution has the characteristic function:

${\displaystyle \varphi (t;{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {1}{1+{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} -i{\boldsymbol {\mu }}\mathbf {t} }}.}$ ^[1]

As with the symmetric multivariate Laplace distribution, the asymmetric multivariate Laplace distribution has mean ${\displaystyle {\boldsymbol {\mu }}}$, but the covariance becomes ${\displaystyle {\boldsymbol {\Sigma }}+{\boldsymbol {\mu }}'{\boldsymbol {\mu }}}$. ^[3] The asymmetric multivariate Laplace distribution is not elliptical unless ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$, in which case the distribution reduces to the symmetric multivariate Laplace distribution with ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$. ^[1]

The probability density function (pdf) for a k-dimensional asymmetric multivariate Laplace distribution is:

${\displaystyle f_{\mathbf {x} }(x_{1},\ldots ,x_{k})={\frac {2e^{\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{(2\pi )^{k/2}|{\boldsymbol {\Sigma }}|^{0.5}}}{\Big (}{\frac {\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }{2+{\boldsymbol {\mu }}'{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{\Big )}^{v/2}K_{v}{\Big (}{\sqrt {(2+{\boldsymbol {\mu }}'{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }})(\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} )}}{\Big )},}$

where:

${\displaystyle v=(2-k)/2}$ and ${\displaystyle K_{v}}$ is the modified Bessel function of the second kind. ^[1]

The asymmetric Laplace distribution, including the special case of ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$, is an example of a geometric stable distribution. ^[3] It represents the limiting distribution for a sum of independent, identically distributed random variables with finite variance and covariance where the number of elements to be summed is itself an independent random variable distributed according to a geometric distribution. ^[1] Such geometric sums can arise in practical applications within biology, economics and insurance. ^[1] The distribution may also be applicable in broader situations to model multivariate data with heavier tails than a normal distribution but finite moments. ^[1]

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 ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$). Simulate a bivariate normal random variable vector ${\displaystyle \mathbf {Y} }$ from a distribution with ${\displaystyle \mu _{1}=\mu _{2}=0}$ and covariance matrix ${\displaystyle {\boldsymbol {\Sigma }}}$. Independently simulate an exponential random variable ${\displaystyle \mathbf {W} }$ from an Exp(1) distribution. ${\displaystyle \mathbf {X} ={\sqrt {W}}\mathbf {Y} +W{\boldsymbol {\mu }}}$ will be distributed (asymmetric) bivariate Laplace with mean ${\displaystyle {\boldsymbol {\mu }}}$ and covariance matrix ${\displaystyle {\boldsymbol {\Sigma }}}$. ^[1]

References

1. Kotz. Samuel; Kozubowski, Tomasz J.; Podgorski, Krzysztof (2001). The Laplace Distribution and Generalizations. Birkhauser. pp. 229–245. ISBN   0817641661.
2. Fragiadakis, Konstantinos & Meintanis, Simos G. (March 2011). "Goodness-of-fit tests for multivariate Laplace distributions". Mathematical and Computer Modelling. 53 (5–6): 769–779. doi: 10.1016/j.mcm.2010.10.014 .
3. Kozubowski, Tomasz J.; Podgorski, Krzysztof; Rychlik, Igor (2010). "Multivariate Generalized Laplace Distributions and Related Random Fields". Journal of Multivariate Analysis. 113. University of Gothenburg: 59–72. doi: 10.1016/j.jmva.2012.02.010 . S2CID   206252976.