# Exact unconditional sampling from max-stable random vectors

#### Léo Belzile, HEC Montréal

#### 2024-07-08

The `mev` package was originally introduced to implement the exact unconditional sampling algorithms in Dombry, Engelke, and Oesting (2016). The two algorithms therein allow one to simulate simple max-stable random vectors. The implementation will work efficiently for moderate dimensions.

# Functions and use

There are two main functions, `rmev` and `rmevspec`. `rmev` samples from simple max-stable processes, meaning it will return an \(n \times d\) matrix of samples, where each of the column has a sample from a unit Frechet distribution. In constrast, `rmevspec` returns sample on the unit simplex from the spectral (or angular) measure. One could use this to test estimation based on spectral densities, or to construct samples from Pareto processes.

The syntax is

    library(mev)
    #Sample of size 1000 from a 5-dimensional logistic model
    x <- rmev(n=1000, d=5, param=0.5, model="log")
    #Marginal parameters are all standard Frechet, meaning GEV(1,1,1)
    apply(x, 2, function(col){ismev::gev.fit(col, show=FALSE)$mle})

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 1.0150748 1.0191299 1.0022764 0.9660028 0.9820029
    ## [2,] 1.0086591 1.0052804 0.9963394 0.9599275 0.9841253
    ## [3,] 0.9568289 0.9390485 0.9384795 0.9969916 0.9893118

    #Sample from the corresponding spectral density
    w <- rmevspec(n=1000, d=5, param=0.5, model="log")
    #All rows sum to 1 by construction
    head(rowSums(w))

    ## [1] 1 1 1 1 1 1

    #The marginal mean is 1/d
    round(colMeans(w),2)

    ## [1] 0.20 0.20 0.20 0.19 0.21

# Description of the models implemented

The different models implemented are described in Dombry, Engelke, and Oesting (2016), but some other models can be found and are described here. Throughout, we consider \(d\)-variate models and let \(\mathbb{B}_d\) be the collection of all nonempty subsets of \(\{1, \ldots, d\}\).

## Logistic

The logistic model (`log`) of Gumbel (1960) has distribution function \[\begin{align*} \Pr(\boldsymbol{X} \leq \boldsymbol{x})= \exp \left[ - \left(\sum_{i=1}^{n} {x_i}^{-\alpha}\right)^{\frac{1}{\alpha}}\right] \end{align*}\] for \(\alpha>1\). The spectral measure density is \[\begin{align*} h_{\boldsymbol{W}}(\boldsymbol{w})=\frac{1}{d}\frac{\Gamma(d-\alpha)}{\Gamma(1-\alpha)}\alpha^{d-1}\left( \prod_{j=1}^d w_j\right)^{-(\alpha+1)}\left(\sum_{j=1}^d w_j^{-\alpha}\right)^{1/\alpha-d}, \qquad \boldsymbol{w} \in \mathbb{S}_d \end{align*}\]

## Asymmetric logistic distribution

The `alog` model was proposed by Tawn (1990). It shares the same parametrization as the `evd` package, merely replacing the algorithm for the generation of logistic variates. The distribution function of the \(d\)-variate asymmetric logistic distribution is \[\begin{align*} \Pr(\boldsymbol{X} \le \boldsymbol{x}) = \exp \left[ -\sum_{b \in \mathbb{B}_d}\left(\sum_{i \in b} \left(\frac{\theta_{i, b}}{x_i}\right)^{\alpha_b}\right)^{\frac{1}{\alpha_b}}\right], \end{align*}\]

The parameters \(\theta_{i, b}\) must be provided in a list and represent the asymmetry parameter. The sampling algorithm, from Stephenson (2003) gives some insight on the construction mechanism as a max-mixture of logistic distributions. Consider sampling \(\boldsymbol{Z}_b\) from a logistic distribution of dimension \(|b|\) (or Fréchet variates if \(|b|=1)\) with parameter \(\alpha_b\) (possibly recycled). Each marginal value corresponds to the maximum of the weighted corresponding entry. That is, \(X_{i}=\max_{b \in \mathbb{B}_d}\theta_{i, b}Z_{i,b}\) for all \(i=1, \ldots, d\). The max-mixture is valid provided that \(\sum_{b \in \mathbb{B}_d} \theta_{i,b}=1\) for \(i=1, \ldots, d.\) As such, empirical estimates of the spectral measure will almost surely place mass on the inside of the simplex rather than on subfaces.

## Negative logistic distribution

The `neglog` distribution function due to Galambos (1975) is \[\begin{align*} \Pr(\boldsymbol{X} \le \boldsymbol{x}) = \exp \left[ -\sum_{b \in \mathbb{B}_d} (-1)^{|b|}\left(\sum_{i \in b} {x_i}^{\alpha}\right)^{-\frac{1}{\alpha}}\right] \end{align*}\] for \(\alpha \geq 0\) (Dombry, Engelke, and Oesting 2016). The associated spectral density is \[\begin{align*} h_{\boldsymbol{W}}(\boldsymbol{w}) = \frac{1}{d} \frac{\Gamma(1/\alpha+1)}{\Gamma(1/\alpha + d-1)} \alpha^d\left(\prod_{i=1}^d w_j\right)^{\alpha-1}\left(\sum_{i=1}^d w_i^{\alpha}\right)^{-1/\alpha-d} \end{align*}\]

## Asymmetric negative logistic distribution

The asymmetric negative logistic (`aneglog`) model is alluded to in Joe (1990) as a generalization of the Galambos model. It is constructed in the same way as the asymmetric logistic distribution; see Theorem~1 in Stephenson (2003). Let \(\alpha_b \leq 0\) for all \(b \in \mathbb{B}_d\) and \(\theta_{i, b} \geq 0\) with \(\sum_{b \in \mathbb{B}_d} \theta_{i, b} =1\) for \(i=1, \ldots, d\); the distribution function is \[\begin{align*} \Pr(\boldsymbol{X} \le \boldsymbol{x}) = \exp \left[ -\sum_{b \in \mathbb{B}_d}(-1)^{|b|} \left\{\sum_{i \in b} \left(\frac{\theta_{i, b}}{x_i}\right)^{\alpha_b}\right\}^{\frac{1}{\alpha_b}}\right]. \end{align*}\] In particular, it does not correspond to the ``negative logistic distribution’’ given in e.g., Section 4.2 of Coles and Tawn (1991) or Section3.5.3 of Kotz and Nadarajah (2000). The latter is not a valid distribution function in dimension \(d \geq 3\) as the constraints therein on the parameters \(\theta_{i, b}\) are necessary, but not sufficient.

Joe (1990) mentions generalizations of the distribution as given above but the constraints were not enforced elsewhere in the literature. The proof that the distribution is valid follows from Theorem~1 of Stephenson (2003) as it is a max-mixture. Note that the parametrization of the asymmetric negative logistic distribution does not match the bivariate implementation of `rbvevd`.

## Multilogistic distribution

This multivariate extension of the logistic, termed multilogistic (`bilog`) proposed by M.-O. Boldi (2009), places mass on the interior of the simplex. Let \(\boldsymbol{W} \in \mathbb{S}_d\) be the solution of \[\begin{align*} \frac{W_j}{W_d}=\frac{C_jU_j^{-\alpha_j}}{C_dU_d^{-\alpha_d}}, \quad j=1, \ldots, d \end{align*}\] where \(C_j=\Gamma(d-\alpha_j)/\Gamma(1-\alpha_j)\) for \(j=1, \ldots, d\) and \(\boldsymbol{U} \in \mathbb{S}_d\) follows a \(d\)-mixture of Dirichlet with the \(j\)th component being \(\mathcal{D}(\boldsymbol{1}-\delta_{j}\alpha_j)\), so that the mixture has density function \[\begin{align*} h_{\boldsymbol{U}}(\boldsymbol{u})=\frac{1}{d} \sum_{j=1}^d \frac{\Gamma(d-\alpha_j)}{\Gamma(1-\alpha_j)} u_j^{-\alpha_j} \end{align*}\] for \(0<\alpha_j <1, j=1, \ldots, d\). The spectral density of the multilogistic distribution is thus \[\begin{align*} h_{\boldsymbol{W}}(\boldsymbol{w}) = \frac{1}{d} \left(\sum_{j=1}^d \alpha_ju_j\right)^{-1} \left(\prod_{j=1}^d \alpha_ju_d \right)\left(\sum_{j=1}^d \frac{\Gamma(d-\alpha_j)}{\Gamma(1-\alpha_j)}u_j^{-\alpha_j}\right)\prod_{j=1}^d w_j^{-1} \end{align*}\] for \(\alpha_j \in (0,1)\) \((j=1, \ldots, d)\).

## Coles and Tawn Dirichlet distribution

The Dirichlet (`ct`) model of Coles and Tawn (1991) \[\begin{align*} h_{\boldsymbol{W}}(\boldsymbol{w}) = \frac{1}{d} \frac{\Gamma \left(1+\sum_{j=1}^d \alpha_j\right)}{\prod_{j=1}^d \alpha_jw_j} \left(\sum_{j=1}^d \alpha_jw_j\right)^{-(d+1)}\prod_{j=1}^d \alpha_j \prod_{j=1}^d \left(\frac{\alpha_jw_j}{\sum_{k=1}^d \alpha_kw_k}\right)^{\alpha_j-1} \end{align*}\] for \(\alpha_j>0.\)

## Scaled extremal Dirichlet

The angular density of the scaled extremal Dirichlet (`sdir`) model with parameters \(\rho > -\min(\boldsymbol{\alpha})\) and \(\boldsymbol{\alpha} \in \mathbb{R}^{d}_{+}\) is given, for all \(\boldsymbol{w} \in \mathbb{S}_d\), by \[\begin{align*} h_{\boldsymbol{W}}(\boldsymbol{w})=\frac{\Gamma(\bar{\alpha}+\rho)}{d\rho^{d-1}\prod_{i=1}^d\Gamma(\alpha_i)} \bigl\langle\{\boldsymbol{c}(\boldsymbol{\alpha},\rho)\}^{1/\rho},\boldsymbol{w}^{1/\rho}\bigr\rangle^{-\rho-\bar{\alpha}}\prod_{i=1}^{d} \{c(\alpha_i,\rho)\}^{\alpha_i/\rho}w_i^{\alpha_i/\rho-1}. \end{align*}\] where \(\boldsymbol{c}(\boldsymbol{\alpha},\rho)\) is the \(d\)-vector with entries \(\Gamma(\alpha_i+\rho)/\Gamma(\alpha_i)\) for \(i=1, \ldots, d\) and \(\langle \cdot, \cdot \rangle\) denotes the inner product between two vectors.

## Huesler–Reiss

The Huesler–Reiss model (`hr`), due to Hüsler and Reiss (1989), is a special case of the Brown–Resnick process. While Engelke et al. (2015) state that H"usler–Reiss variates can be sampled following the same scheme, the spatial analog is conditioned on a particular site (\(\boldsymbol{s}_0\)), which complicates the comparisons with the other methods.

Let \(I_{-j}=\{1, \ldots, d\} \setminus \{j\}\) and \(\lambda_{ij}^2 \geq 0\) be entries of a strictly conditionally negative definite matrix \(\boldsymbol{\Lambda}\), for which \(\lambda_{ij}^2=\lambda_{ji}^2\). Then, following Nikoloulopoulos, Joe, and Li (2009) (Remark~2.5) and Huser and Davison (2013), we can write the distribution function as \[\begin{align*} \Pr(\boldsymbol{X} \le \boldsymbol{x}) = \exp \left[ -\sum_{j=1}^d \frac{1}{x_j} \Phi_{d-1, \boldsymbol{\Sigma}_{-j}} \left( \lambda_{ij}- \frac{1}{2\lambda_{ij}} \log\left(\frac{x_j}{x_i}\right), i \in I_{-j}\right)\right]. \end{align*}\] where the partial correlation matrix \(\boldsymbol{\Sigma}_{-j}\) has elements \[\begin{align*} \varrho_{i,k; j}= \frac{\lambda_{ij}^2+\lambda_{kj}^2-\lambda_{ik}^2}{2\lambda_{ij}\lambda_{kj}} \end{align*}\] and \(\lambda_{ii}=0\) for all \(i \in I_{-j}\) so that the diagonal entries \(\varrho_{i,i; j}=1\). Engelke et al. (2015) uses the covariance matrix with entries are \(\varsigma=2(\lambda_{ij}^2+\lambda_{kj}^2-\lambda_{ik}^2)\), so the resulting expression is evaluated at \(2\boldsymbol{\lambda}_{.j}^2-\log({x_j}/{\boldsymbol{x}_{-j}})\) instead. We recover the same expression by standardizing, since this amounts to division by the standard deviations \(2\boldsymbol{\lambda}_{.j}\)

The package implementation has a bivariate implementation of the H"usler–Reiss distribution with dependence parameter \(r\), with \(r_{ik}=1/\lambda_{ik}\) or \(2/r=\sqrt{2\gamma(\boldsymbol{h})}\) for \(\boldsymbol{h}=\|\boldsymbol{s}_i-\boldsymbol{s}_i\|\) for the Brown–Resnick model. In this setting, it is particularly easy since the only requirement is non-negativity of the parameter. For inference in dimension \(d>2\), one needs to impose the constraint \(\boldsymbol{\Lambda}=\{\lambda_{ij}^2\}_{i, j=1}^d \in \mathcal{D}\) (cf. Engelke et al. (2015), p.3), where \[\begin{multline*} \mathcal{D}=\Biggl\{\mathbf{A}\in [0, \infty)^{d\times d}: \boldsymbol{x}^\top\!\!\mathbf{A}\boldsymbol{x} <0, \ \forall \ \boldsymbol{x} \in \mathbb{R}^{d} \setminus\{\boldsymbol{0}\} \\ \qquad \text{ with } \sum_{i=1}^d x_i=0, a_{ij}=a_{ji}, a_{ii}=0 \ \forall \ i, j \in \{1,\ldots, d\}\Biggr\} \end{multline*}\] denotes the set of symmetric conditionally negative definite matrices with zero diagonal entries. An avenue to automatically satisfy these requirements is to optimize over a symmetric positive definite matrix parameter \(\boldsymbol{\varSigma}=\mathbf{L}^\top\mathbf{L}\), where \(\mathbf{L}\) is an upper triangular matrix whose diagonal element are on the log-scale to ensure uniqueness of the Cholesky factorization; see Pinheiro and Bates (1996). By taking \[\begin{align*} \boldsymbol{\Lambda}(\boldsymbol{\varSigma})= \begin{pmatrix} 0 & \mathrm{diag} (\boldsymbol{\varSigma})^\top \\ \mathrm{diag}(\boldsymbol{\varSigma}) & \boldsymbol{1}\mathrm{diag}(\boldsymbol{\varSigma})^\top + \mathrm{diag}(\boldsymbol{\varSigma})\boldsymbol{1}^\top - 2 \boldsymbol{\varSigma} \end{pmatrix} \end{align*}\] one can perform unconstrained optimization for the non-zero elements of \(\mathbf{L}\) which are in one-to-one correspondence with those of \(\boldsymbol{\Lambda}\).

It easily follows that generating \(\boldsymbol{Z}\) from a \(d-1\) dimensional log-Gaussian distribution with covariance \(\mathsf{Co}(Z_i, Z_k)=2(\lambda_{ij}^2+\lambda_{kj}^2-\lambda_{ik}^2)\) for \(i, k \in I_{-j}\) with mean vector \(-2\lambda_{\bullet j}^2\) gives the finite dimensional analog of the Brown–Resnick process in the mixture representation of Dombry, Engelke, and Oesting (2016).

The function checks conditional negative definiteness of the matrix. The easiest way to do so negative definiteness of \(\boldsymbol{\Lambda}\) with real entries is to form \(\tilde{\boldsymbol{\Lambda}}=\mathbf{P}\boldsymbol{\Lambda}\mathbf{P}^\top\), where \(\mathbf{P}\) is an \(d \times d\) matrix with ones on the diagonal, \(-1\) on the \((i, i+1)\) entries for \(i=1, \ldots d-1\) and zeros elsewhere. If the matrix \(\boldsymbol{\Lambda} \in \mathcal{D}\), then the eigenvalues of the leading \((d-1) \times (d-1)\) submatrix of \(\tilde{\boldsymbol{\Lambda}}\) will all be negative.

For a set of \(d\) locations, one can supply the variogram matrix as valid input to the method.

## Brown–Resnick process

The Brown–Resnick process (`br`) is the functional extension of the H"usler–Reiss distribution, and is a max-stable process associated with the log-Gaussian distribution. It is often in the spatial setting conditioned on a location (typically the origin). Users can provide a variogram function that takes distance as argument and is vectorized. If `vario` is provided, the model will simulate from an intrinsically stationary Gaussian process. The user can alternatively provide a covariance matrix `sigma` obtained by conditioning on a site, in which case simulations are from a stationary Gaussian process. See Engelke et al. (2015) or Dombry, Engelke, and Oesting (2016) for more information.

## Extremal Student

The extremal Student (`extstud`) model of Nikoloulopoulos, Joe, and Li (2009), eq. 2.8, with unit Fréchet margins is \[\begin{align*} \Pr(\boldsymbol{X} \le \boldsymbol{x}) = \exp \left[-\sum_{j=1}^d \frac{1}{x_j} T_{d-1, \nu+1, \mathbf{R}_{-j}}\left( \sqrt{\frac{\nu+1}{1-\rho_{ij}^2}} \left[\left(\frac{x_i}{x_j}\right)^{1/\nu}\!\!\!-\rho_{ij}\right], i \in I_{-j} \right)\right], \end{align*}\] where \(T_{d-1}\) is the distribution function of the $d-1 $ dimensional Student-\(t\) distribution and the partial correlation matrix \(\mathbf{R}_{-j}\) has diagonal entry \[r_{i,i;j}=1, \qquad r_{i,k;j}=\frac{\rho_{ik}-\rho_{ij}\rho_{kj}}{\sqrt{1-\rho_{ij}^2}\sqrt{1-\rho_{kj}^2}}\] for \(i\neq k, i, k \in I_{-j}\).

The user must provide a valid correlation matrix (the function checks for diagonal elements), which can be obtained from a variogram.

## Dirichlet mixture

The Dirichlet mixture (`dirmix`) proposed by M.-O. Boldi and Davison (2007), see Dombry, Engelke, and Oesting (2016) for details on the mixture. The spectral density of the model is \[\begin{align*} h_{\boldsymbol{W}}(\boldsymbol{w}) = \sum_{k=1}^m \pi_k \frac{\Gamma(\alpha_{1k}+ \cdots + \alpha_{dk})}{\prod_{i=1}^d \Gamma(\alpha_{ik})} \left(1-\sum_{i=1}^{d-1} w_i\right)^{\alpha_{dk}-1}\prod_{i=1}^{d-1} w_{i}^{\alpha_{ik}-1} \end{align*}\] The argument `param` is thus a \(d \times m\) matrix of coefficients, while the argument for the \(m\)-vector `weights` gives the relative contribution of each Dirichlet mixture component.

## Smith model

The Smith model (`smith`) is from the unpublished report of Smith (1990). It corresponds to a moving maximum process on a domain \(\mathbb{X}\). The de Haan representation of the process is \[\begin{align*} Z(x)=\max_{i \in \mathbb{N}} \zeta_i h(x-\eta_i), \qquad \eta_i \in \mathbb{X} \end{align*}\] where \(\{\zeta_i, \eta_i\}_{i \in \mathbb{N}}\) is a Poisson point process on \(\mathbb{R}_{+} \times \mathbb{X}\) with intensity measure \(\zeta^{-2}\mathrm{d} \zeta \mathrm{d} \eta\) and \(h\) is the density of the multivariate Gaussian distribution. Other \(h\) could be used in principle, but are not implemented.

# References

Boldi, Marc-Olivier. 2009. “A Note on the Representation of Parametric Models for Multivariate Extremes.” Extremes 12 (3): 211–18. https://doi.org/10.1007/s10687-008-0076-0.
Boldi, M.-O., and A. C. Davison. 2007. “A Mixture Model for Multivariate Extremes.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69 (2): 217–29. https://doi.org/10.1111/j.1467-9868.2007.00585.x.
Coles, Stuart G., and Jonathan A. Tawn. 1991. “Modelling Extreme Multivariate Events.” Journal of the Royal Statistical Society. Series B (Methodological) 53 (2): 377–92. http://www.jstor.org/stable/2345748.
Dombry, Clément, Sebastian Engelke, and Marco Oesting. 2016. “Exact Simulation of Max-Stable Processes.” Biometrika 103 (2): 303–17. https://doi.org/10.1093/biomet/asw008.
Engelke, Sebastian, Alexander Malinowski, Zakhar Kabluchko, and Martin Schlather. 2015. “Estimation of Hüsler–Reiss Distributions and Brown–Resnick Processes.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 (1): 239–65. https://doi.org/10.1111/rssb.12074.
Galambos, János. 1975. “Order Statistics of Samples from Multivariate Distributions.” J. Amer. Statist. Assoc. 70 (351, part 1): 674–80.
Gumbel, Émile J. 1960. “Distributions Des Valeurs Extrêmes En Plusieurs Dimensions.” Publ. Inst. Statist. Univ. Paris 9: 171–73.
Huser, R., and A. C. Davison. 2013. “Composite Likelihood Estimation for the Brown–Resnick Process.” Biometrika 100 (2): 511–18. https://doi.org/10.1093/biomet/ass089.
Hüsler, Jürg, and Rolf-Dieter Reiss. 1989. “Maxima of Normal Random Vectors: Between Independence and Complete Dependence.” Statist. Probab. Lett. 7 (4): 283–86. https://doi.org/10.1016/0167-7152(89)90106-5.
Joe, Harry. 1990. “Families of Min-Stable Multivariate Exponential and Multivariate Extreme Value Distributions.” Statistics & Probability Letters 9 (1): 75–81. https://doi.org/10.1016/0167-7152(90)90098-R.
Kotz, Samuel, and Saralees Nadarajah. 2000. Extreme Value Distributions. London: Imperial College Press. https://doi.org/10.1142/9781860944024.
Nikoloulopoulos, Aristidis K., Harry Joe, and Haijun Li. 2009. “Extreme Value Properties of Multivariate \(t\) Copulas.” Extremes 12 (2): 129–48.
Pinheiro, José C., and Douglas M. Bates. 1996. “Unconstrained Parametrizations for Variance-Covariance Matrices.” Statistics and Computing 6 (3): 289–96. https://doi.org/10.1007/BF00140873.
Smith, Richard L. 1990. “Max-Stable Processes and Spatial Extremes.” https://rls.sites.oasis.unc.edu/postscript/rs/spatex.pdf.
Stephenson, Alec. 2003. “Simulating Multivariate Extreme Value Distributions of Logistic Type.” Extremes 6 (1): 49–59. https://doi.org/10.1023/A:1026277229992.
Tawn, Jonathan A. 1990. “Modelling Multivariate Extreme Value Distributions.” Biometrika 77 (2): 245–53. https://doi.org/10.1093/biomet/77.2.245.