These functions provide the density and random number generation for the multivariate Laplace distribution, given the Cholesky parameterization.

```
dmvlc(x, mu, U, log=FALSE)
rmvlc(n, mu, U)
```

x

This is data or parameters in the form of a vector of length \(k\) or a matrix with \(k\) columns.

n

This is the number of random draws.

mu

This is mean vector \(\mu\) with length \(k\) or matrix with \(k\) columns.

U

This is the \(k \times k\) upper-triangular matrix that is Cholesky factor \(\textbf{U}\) of covariance matrix \(\Sigma\).

log

Logical. If `log=TRUE`

, then the logarithm of the
density is returned.

`dmvlc`

gives the density, and
`rmvlc`

generates random deviates.

Application: Continuous Multivariate

Density: $$p(\theta) = \frac{2}{(2\pi)^{k/2} |\Sigma|^{1/2}} \frac{(\pi/(2\sqrt{2(\theta - \mu)^T \Sigma^{-1} (\theta - \mu)}))^{1/2} \exp(-\sqrt{2(\theta - \mu)^T \Sigma^{-1} (\theta - \mu)})}{\sqrt{((\theta - \mu)^T \Sigma^{-1} (\theta - \mu) / 2)}^{k/2-1}}$$

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: positive-definite \(k \times k\) 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 one-dimensional 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 Haro-Lopez 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 two-parameter distribution (though alternative three-parameter forms have been introduced as well), unlike the three-parameter multivariate t distribution, which is often used as a robust alternative to the multivariate normal distribution.

In practice, \(\textbf{U}\) is fully unconstrained for proposals
when its diagonal is log-transformed. The diagonal is exponentiated
after a proposal and before other calculations. Overall, the Cholesky
parameterization is faster than the traditional parameterization.
Compared with `dmvl`

, `dmvlc`

must additionally
matrix-multiply the Cholesky back to the covariance matrix, but it
does not have to check for or correct the covariance matrix to
positive-definiteness, which overall is slower. Compared with
`rmvl`

, `rmvlc`

is faster because the Cholesky decomposition
has already been performed.

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, Chapman-Hall, London.

Fernandez, C., Osiewalski, J. and Steel, M.F.J. (1995). "Modeling and
Inference with v-spherical Distributions". *Journal of the
American Statistical Association*, 90, p. 1331--1340.

Gomez, E., Gomez-Villegas, M.A., and Marin, J.M. (1998). "A
Multivariate Generalization of the Power Exponential Family of
Distributions". *Communications in Statistics-Theory and
Methods*, 27(3), p. 589--600.

Haro-Lopez, 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 Heavy-Tailed 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.

`chol`

,
`daml`

,
`dlaplace`

,
`dmvnc`

,
`dmvnpc`

,
`dmvpec`

,
`dmvtc`

,
`dnorm`

,
`dnormp`

, and
`dnormv`

.

```
# NOT RUN {
library(LaplacesDemon)
Sigma <- diag(3)
U <- chol(Sigma)
x <- dmvlc(c(1,2,3), c(0,1,2), U)
X <- rmvlc(1000, c(0,1,2), U)
joint.density.plot(X[,1], X[,2], color=TRUE)
# }
```

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