dist.Multivariate.Laplace {LaplacesDemon}  R Documentation 
Multivariate Laplace Distribution
Description
These functions provide the density and random number generation for the multivariate Laplace distribution.
Usage
dmvl(x, mu, Sigma, log=FALSE)
rmvl(n, mu, Sigma)
Arguments
x 
This is data or parameters in the form of a vector of length

n 
This is the number of random draws. 
mu 
This is mean vector 
Sigma 
This is the 
log 
Logical. If 
Details
Application: Continuous Multivariate
Density:
p(\theta) = \frac{2}{(2\pi)^{k/2} \Sigma^{1/2}} \frac{(\pi/(2\sqrt{2(\theta  \mu)^T \Sigma^{1} (\theta  \mu)}))^{1/2} \exp(\sqrt{2(\theta  \mu)^T \Sigma^{1} (\theta  \mu)})}{\sqrt{((\theta  \mu)^T \Sigma^{1} (\theta  \mu) / 2)}^{k/21}}
Inventor: Fang et al. (1990)
Notation 1:
\theta \sim \mathcal{MVL}(\mu, \Sigma)
Notation 2:
\theta \sim \mathcal{L}_k(\mu, \Sigma)
Notation 3:
p(\theta) = \mathcal{MVL}(\theta  \mu, \Sigma)
Notation 4:
p(\theta) = \mathcal{L}_k(\theta  \mu, \Sigma)
Parameter 1: location vector
\mu
Parameter 2: positivedefinite
k \times k
covariance matrix\Sigma
Mean:
E(\theta) = \mu
Variance:
var(\theta) = \Sigma
Mode:
mode(\theta) = \mu
The multivariate Laplace distribution is a multidimensional extension of the onedimensional or univariate symmetric Laplace distribution. There are multiple forms of the multivariate Laplace distribution.
The bivariate case was introduced by Ulrich and Chen (1987), and the first form in larger dimensions may have been Fang et al. (1990), which requires a Bessel function. Alternatively, multivariate Laplace was soon introduced as a special case of a multivariate Linnik distribution (Anderson, 1992), and later as a special case of the multivariate power exponential distribution (Fernandez et al., 1995; Ernst, 1998). Bayesian considerations appear in HaroLopez and Smith (1999). Wainwright and Simoncelli (2000) presented multivariate Laplace as a Gaussian scale mixture. Kotz et al. (2001) present the distribution formally. Here, the density is calculated with the asymptotic formula for the Bessel function as presented in Wang et al. (2008).
The multivariate Laplace distribution is an attractive alternative to the multivariate normal distribution due to its wider tails, and remains a twoparameter distribution (though alternative threeparameter forms have been introduced as well), unlike the threeparameter multivariate t distribution, which is often used as a robust alternative to the multivariate normal distribution.
Value
dmvl
gives the density, and
rmvl
generates random deviates.
Author(s)
Statisticat, LLC. software@bayesianinference.com
References
Anderson, D.N. (1992). "A Multivariate Linnik Distribution". Statistical Probability Letters, 14, p. 333–336.
Eltoft, T., Kim, T., and Lee, T. (2006). "On the Multivariate Laplace Distribution". IEEE Signal Processing Letters, 13(5), p. 300–303.
Ernst, M. D. (1998). "A Multivariate Generalized Laplace Distribution". Computational Statistics, 13, p. 227–232.
Fang, K.T., Kotz, S., and Ng, K.W. (1990). "Symmetric Multivariate and Related Distributions". Monographs on Statistics and Probability, 36, ChapmanHall, London.
Fernandez, C., Osiewalski, J. and Steel, M.F.J. (1995). "Modeling and Inference with vspherical Distributions". Journal of the American Statistical Association, 90, p. 1331–1340.
Gomez, E., GomezVillegas, M.A., and Marin, J.M. (1998). "A Multivariate Generalization of the Power Exponential Family of Distributions". Communications in StatisticsTheory and Methods, 27(3), p. 589–600.
HaroLopez, R.A. and Smith, A.F.M. (1999). "On Robust Bayesian Analysis for Location and Scale Parameters". Journal of Multivariate Analysis, 70, p. 30–56.
Kotz., S., Kozubowski, T.J., and Podgorski, K. (2001). "The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance". Birkhauser: Boston, MA.
Ulrich, G. and Chen, C.C. (1987). "A Bivariate Double Exponential Distribution and its Generalization". ASA Proceedings on Statistical Computing, p. 127–129.
Wang, D., Zhang, C., and Zhao, X. (2008). "Multivariate Laplace Filter: A HeavyTailed Model for Target Tracking". Proceedings of the 19th International Conference on Pattern Recognition: FL.
Wainwright, M.J. and Simoncelli, E.P. (2000). "Scale Mixtures of Gaussians and the Statistics of Natural Images". Advances in Neural Information Processing Systems, 12, p. 855–861.
See Also
daml
,
dlaplace
,
dmvn
,
dmvnp
,
dmvpe
,
dmvt
,
dnorm
,
dnormp
, and
dnormv
.
Examples
library(LaplacesDemon)
x < dmvl(c(1,2,3), c(0,1,2), diag(3))
X < rmvl(1000, c(0,1,2), diag(3))
joint.density.plot(X[,1], X[,2], color=TRUE)