R: Multivariate Laplace Distribution: Cholesky Parameterization

dist.Multivariate.Laplace.Cholesky {LaplacesDemon} | R Documentation

## Multivariate Laplace Distribution: Cholesky Parameterization

### Description

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

### Usage

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

### Arguments

`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 x k upper-triangular matrix that is Cholesky factor U of covariance matrix Sigma.
`log` | Logical. If `log=TRUE`, then the logarithm of the density is returned.

### Details

- Application: Continuous Multivariate

- Density:

  p(theta) = (2 / ((2*pi)^(k/2) * |Sigma|^(1/2))) ((sqrt(pi/(2*sqrt(2*(theta-mu)^TSigma^(-1)(theta-mu)))) * exp(-sqrt(2*(theta-mu)^TSigma^(-1)(theta-mu)))) / sqrt((theta-mu)^TSigma^(-1)(theta-mu)/2)^(k/2-1))

- Inventor: Fang et al. (1990)

- Notation 1: theta ~ MVL(mu, Sigma)

- Notation 2: theta ~ L[k](mu, Sigma)

- Notation 3: p(theta) = MVL(theta | mu, Sigma)

- Notation 4: p(theta) = L[k](theta | mu, Sigma)

- Parameter 1: location vector mu

- Parameter 2: positive-definite k x 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, 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.

### Value

`dmvlc` gives the density, and `rmvlc` generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.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, 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.

### See Also

`chol`, `daml`, `dlaplace`, `dmvnc`, `dmvnpc`, `dmvpec`, `dmvtc`, `dnorm`, `dnormp`, and `dnormv`.

### Examples

    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)

[Package LaplacesDemon version 16.1.1 Index]