From ee7b98e3f45fae7edecbf197efa27f19bbb7f289 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?L=C3=A9o=20Belzile?= Date: Mon, 8 Jul 2024 22:53:39 -0400 Subject: [PATCH] Transform vignette to Rmarkdown --- .Rbuildignore | 3 + .gitignore | 3 + DESCRIPTION | 1 + vignettes/mev-vignette.R | 15 + vignettes/mev-vignette.Rmd | 239 +++++++++++ vignettes/mev-vignette.Rnw | 281 ------------- vignettes/mev-vignette.html | 792 ++++++++++++++++++++++++++++++++++++ vignettes/mev-vignette.pdf | Bin 313685 -> 0 bytes vignettes/mev-vignette.tex | 339 --------------- vignettes/mevvignette.bib | 2 +- 10 files changed, 1054 insertions(+), 621 deletions(-) create mode 100644 vignettes/mev-vignette.R create mode 100644 vignettes/mev-vignette.Rmd delete mode 100644 vignettes/mev-vignette.Rnw create mode 100644 vignettes/mev-vignette.html delete mode 100644 vignettes/mev-vignette.pdf delete mode 100644 vignettes/mev-vignette.tex diff --git a/.Rbuildignore b/.Rbuildignore index d68fb73..ac14054 100644 --- a/.Rbuildignore +++ b/.Rbuildignore @@ -35,8 +35,11 @@ vignettes/01-univariate_cache/* vignettes/01-univariate_files/* vignettes/04-simulation_cache/* vignettes/04-simulation_files/* +vignettes/mev-vignette-back.pdf cran-comments.md R/mgp.R R/taildep2.R ^revdep$ ^CRAN-SUBMISSION$ +^doc$ +^Meta$ diff --git a/.gitignore b/.gitignore index 2cc13bd..df15cf4 100644 --- a/.gitignore +++ b/.gitignore @@ -16,3 +16,6 @@ inst/rawdata cran-comments.md revdep/* CRAN-SUBMISSION +mev-vignette-back.pdf +/doc/ +/Meta/ diff --git a/DESCRIPTION b/DESCRIPTION index 277ee1c..791e0de 100644 --- a/DESCRIPTION +++ b/DESCRIPTION @@ -25,6 +25,7 @@ Suggests: mvtnorm, gmm, revdbayes, + rmarkdown, ismev, tinytest, TruncatedNormal (>= 1.1) diff --git a/vignettes/mev-vignette.R b/vignettes/mev-vignette.R new file mode 100644 index 0000000..19bd4f3 --- /dev/null +++ b/vignettes/mev-vignette.R @@ -0,0 +1,15 @@ +## ----------------------------------------------------------------------------- +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}) + + +#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)) +#The marginal mean is 1/d +round(colMeans(w),2) + diff --git a/vignettes/mev-vignette.Rmd b/vignettes/mev-vignette.Rmd new file mode 100644 index 0000000..d292df3 --- /dev/null +++ b/vignettes/mev-vignette.Rmd @@ -0,0 +1,239 @@ +--- +title: "Exact unconditional sampling from max-stable random vectors" +author: "Léo Belzile, HEC Montréal" +date: "`r Sys.Date()`" +output: + rmarkdown::html_vignette: + toc: false +vignette: > + %\VignetteIndexEntry{Exact unconditional sampling from max-stable random vectors} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +bibliography: mevvignette.bib +--- + +The `mev` package was originally introduced to implement the exact unconditional sampling algorithms in @Dombry: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 +```{r} +#| eval: true +#| echo: true +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}) + + +#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)) +#The marginal mean is 1/d +round(colMeans(w),2) +``` + +# Description of the models implemented + +The different models implemented are described in @Dombry: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: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:1991 or Section3.5.3 of @Kotz: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 @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: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 @Husler:1989, is a special case of the Brown--Resnick process. +While @Engelke: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:2009 +(Remark~2.5) and @Huser: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: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 \texttt{evd} 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: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: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:2016. + +The \texttt{rmev} 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:2015 or @Dombry:2016 for +more information. + +## Extremal Student + +The extremal Student (`extstud`) model of @Nikoloulopoulos: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 @Boldi:2007, see @Dombry: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 {-} diff --git a/vignettes/mev-vignette.Rnw b/vignettes/mev-vignette.Rnw deleted file mode 100644 index 2c29d29..0000000 --- a/vignettes/mev-vignette.Rnw +++ /dev/null @@ -1,281 +0,0 @@ -\documentclass{article} -\usepackage[T1]{fontenc} -\usepackage[utf8]{inputenc} -\usepackage{amsmath, amssymb, amsfonts} -\usepackage{times} -\usepackage{geometry} -\usepackage{natbib} -\usepackage{hyperref} -%\geometry{left=2cm, right=2cm, top=2.5cm, bottom=2.25cm} - -\newcommand{\all}{ \; \forall \;} -\newcommand{\bs}[1]{\boldsymbol {#1}} -\newcommand{\R}{\mathbb{R}} -\renewcommand{\P}[2][]{{\mathsf P}_{#1}\left(#2\right)} -\newcommand{\E}[2][]{{\mathsf E}_{#1}\left(#2\right)} -\newcommand{\Va}[2][]{{\mathsf{Var}_{#1}}\left(#2\right)} -\newcommand{\Co}[2][]{{\mathsf{Cov}_{#1}}\left(#2\right)} -\newcommand{\code}[1]{\texttt{#1}} -\newcommand{\sumi}{\sum_{i=1}^n} -\newcommand{\pfrac}[2]{\left(\frac{#1}{#2}\right)} -\DeclareMathOperator{\diag}{diag} - -\begin{document} -<>= -library(knitr) -## set global chunk options -opts_chunk$set(fig.path='figure/manual-', cache.path='cache/manual-', fig.align='center', fig.show='hold', par=TRUE) -## I use = but I can replace it with <-; set code/output width to be 68 -options(formatR.arrow=TRUE, width=68, digits=4) -## tune details of base graphics (http://yihui.name/knitr/hooks) -knit_hooks$set(par=function(before, options, envir){ -if (before && options$fig.show!='none') par(mar=c(4,4,.1,.1),cex.lab=.95,cex.axis=.9,mgp=c(2,.7,0),tcl=-.3) -}) -@ -%\VignetteIndexEntry{Exact unconditional sampling from max-stable random vectors} -%\VignetteEngine{knitr::knitr_notangle} - -\title{Exact unconditional sampling from max-stable random vectors} -\author{Léo Belzile} -\date{} -\maketitle -\begin{center} -{ \small -Department of Decision Sciences \\ HEC Montréal \\\href{leo.belzile@hec.ca}{\texttt{leo.belzile@hec.ca}} -} -\end{center} - -The `\code{mev}' package was originally introduced to implement the exact unconditional sampling algorithms in \cite{Dombry:2016}. The two algorithms therein allow one to simulate simple max-stable random vectors. The implementation will work efficiently for moderate dimensions. - -\section{Functions and use} - -There are two main functions, \code{rmev} and \code{rmevspec}. \code{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, \code{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}) - - -#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)) -#The marginal mean is 1/d -round(colMeans(w),2) -@ - -\section{Description of the models implemented} - -The different models implemented are described in \cite{Dombry: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 $\{1, \ldots, d\}$. -\begin{enumerate} - -\item \textbf{logistic distribution} (\code{log}) -The logistic model of \cite{Gumbel:1960} (the Gumbel Archimedean copula) -\begin{align*} - \P{\bs{X} \leq \bs{x}}= \exp \left[ - \left(\sumi \pfrac{1}{x_i}^{\alpha}\right)^{\frac{1}{\alpha}}\right] -\end{align*} -for $\alpha>1.$ By default, \code{rmev} will transform an argument in $(0,1)$ without warning, to conform with the -implementation. The spectral measure density is -\begin{align*} - h_{\bs{W}}(\bs{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 \bs{w} \in \mathbb{S}_d -\end{align*} - -\item \textbf{asymmetric logistic distribution} (\code{alog}) -This model was proposed by \cite{Tawn:1990}. It shares the same parametrization as the \code{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*} - \P{\bs{X} \leq \bs{x}} = \exp \left[ -\sum_{b \in \mathbb{B}_d}\left(\sum_{i \in b} \pfrac{\theta_{i, -b}}{x_i}^{\alpha_b}\right)^{\frac{1}{\alpha_b}}\right], -\end{align*} -%while the spectral density is -%\begin{align*} -% h_{\bs{W}}(\bs{w})=\frac{1}{d}\frac{\prod_{j=1}^{d-1} (j\alpha_b-1)}{\prod_{i\in b} w_i} \left(\prod_{i %\in -%b}\frac{\theta_{i,b}}{w_i}\right)^{\alpha_b} \left(\sum_{i \in b} \pfrac{\theta_{i, b}}{w_i}^{\alpha_b}\right)^{1/\alpha_b-d} -%\end{align*} - -The parameters $\theta_{i, b}$ must be provided in a list and represent the asymmetry parameter. -The sampling algorithm, from \cite{Stephenson:2003} gives some insight on the construction mechanism as a max-mixture of logistic distributions. Consider sampling $\bs{Z}_b$ 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. - -\item \textbf{negative logistic distribution} (\code{neglog}) -The distribution function of the min-stable distribution due to \cite{Galambos:1975} is -\begin{align*} - \P{\bs{X} \leq \bs{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 $\alpha \geq 0$ \citep{Dombry:2016}. The associated spectral density is -\begin{align*} - h_{\bs{W}}(\bs{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*} - -\item \textbf{asymmetric negative logistic distribution} (\code{aneglog}) -The asymmetric negative logistic model is alluded to in \cite{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 \cite{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*} - \P{\bs{X} \leq \bs{x}} = \exp \left[ -\sum_{b \in \mathbb{B}_d}(-1)^{|b|} -\left(\sum_{i \in b} -\pfrac{\theta_{i, b}}{x_i}^{\alpha_b}\right)^{\frac{1}{\alpha_b}}\right]. -\end{align*} -%The density is -%\begin{align*} -% h_{\bs{W}}(\bs{w}) = \frac{1}{d} \sum_{\substack{b \in \mathbb{B}_d \\c \subset b}} (-1)^{|b|} -%\left(\prod_{j=1}^{d-1}(1-j\alpha)\right)\left(\prod_{i \in c}w_j\right)^{-(\alpha+1)}\left( \prod_{j \in %b} \theta_{j, b}\right)^{\alpha_b}\left(\sum_{i \in c} -%\pfrac{\theta_{i, b}}{w_i}^{\alpha_b}\right)^{1/\alpha_b-d} -%\end{align*} -In particular, it does not correspond to the ``negative logistic distribution'' given in e.g. \S 4.2 of \cite{Coles:1991} or \S 3.5.3 of \cite{Kotz: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. - -\cite{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 \cite{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 \code{rbvevd}. -\item \textbf{multilogistic distribution} (\code{bilog}) -This multivariate extension of the logistic, proposed by \cite{Boldi:2009}, places mass on the interior of the simplex. Let $\bs{W} \in \mathbb{S}_d$ 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 $\bs{U} \in \mathbb{S}_d$ follows a $d$-mixture of Dirichlet with the $j$th component being $\mathcal{D}(\bs{1}-\delta_{j}\alpha_j)$, so that the mixture has density function -\begin{align*} - h_{\bs{U}}(\bs{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_{\bs{W}}(\bs{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 $\alpha_j \in (0,1)$ $(j=1, \ldots, d)$. -\item \textbf{Coles and Tawn Dirichlet distribution} (\code{ct}) -The Dirichlet model of \cite{Coles:1991} -\begin{align*} - h_{\bs{W}}(\bs{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 $\alpha_j>0.$ -\item \textbf{scaled extremal Dirichlet} (\code{sdir}) -The angular density of the scaled extremal Dirichlet model with parameters $\rho > -\min(\bs{\alpha})$ and $\bs{\alpha} \in \R^{d}_{+}$ is given, for all $\bs{w} \in \mathbb{S}_d$, by -\begin{align*} - h_{\bs{W}}(\bs{w})=\frac{\Gamma(\bar{\alpha}+\rho)}{d\rho^{d-1}\prod_{i=1}^d\Gamma(\alpha_i)} -\bigl\langle\{\bs{c}(\bs{\alpha},\rho)\}^{1/\rho},\bs{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 $\bs{c}(\bs{\alpha},\rho)$ 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. - -\item \textbf{H\"usler--Reiss} (\code{hr}), due to \cite{Husler:1989}. It is a special case of the Brown--Resnick process. -While \cite{Engelke:2015} state that H\"usler--Reiss variates can be sampled following the same scheme, the spatial analog is -conditioned on a particular site ($\bs{s}_0$), 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 $\bs{\Lambda}$, for which $\lambda_{ij}^2=\lambda_{ji}^2$. Then, following \cite{Nikoloulopoulos:2009} -(Remark~2.5) and \cite{Huser:2013}, we can write the distribution function as - \begin{align*} - \P{\bs{X} \leq \bs{x}} = \exp \left[ -\sum_{j=1}^d \frac{1}{x_j} \Phi_{d-1, \bs{\Sigma}_{-j}} \left( \lambda_{ij}- -\frac{1}{2\lambda_{ij}} \log\pfrac{x_j}{x_i}, i \in I_{-j}\right)\right]. - \end{align*} - where the partial correlation matrix $\bs{\Sigma}_{-j}$ 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 $\varrho_{i,i; j}=1$.\footnote{\cite{Engelke: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 $2\bs{\lambda}_{.j}^2-\log\pfrac{x_j}{\bs{x}_{-j}}$ instead. We recover the same expression by -standardizing, since this amounts to division by the standard deviations $2\bs{\lambda}_{.j}$} - - - -The \texttt{evd} package implementation has a bivariate implementation -of the H\"usler--Reiss distribution with dependence parameter $r$, with $r_{ik}=1/\lambda_{ik}$ or -$2/r=\sqrt{2\gamma(\bs{h})}$ for $\bs{h}=\|\bs{s}_i-\bs{s}_i\|$ 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 $\bs{\Lambda}=\{\lambda_{ij}^2\}_{i, j=1}^d \in -\mathcal{D}$ (cf. \cite{Engelke:2015}, p.3), where -\begin{multline*} - \mathcal{D}=\Biggl\{\mathbf{A}\in [0, \infty)^{d\times d}: \bs{x}^\top\!\!\mathbf{A}\bs{x} <0, \all \bs{x} \in \R^{d} -\setminus\{\bs{0}\} \\ \qquad -\text{ with } \sum_{i=1}^d x_i=0, a_{ij}=a_{ji}, a_{ii}=0 \all 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 -$\bs{\varSigma}=\mathbf{L}^\top\mathbf{L}$, where $\mathbf{L}$ is an upper triangular matrix whose diagonal element are on the -log-scale to ensure uniqueness of the Cholesky factorization; see \cite{Pinheiro:1996}. By taking -\begin{align*} - \bs{\Lambda}(\bs{\varSigma})= \begin{pmatrix} 0 & \diag (\bs{\varSigma})^\top \\ \diag(\bs{\varSigma}) & -\bs{1}\diag(\bs{\varSigma})^\top -+ \diag(\bs{\varSigma})\bs{1}^\top - 2 \bs{\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 $\bs{\Lambda}$. - -It easily follows that generating $\bs{Z}$ from a $d-1$ dimensional log-Gaussian distribution with covariance $\Co{Z_i, -Z_k}=2(\lambda_{ij}^2+\lambda_{kj}^2-\lambda_{ik}^2)$ 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 \cite{Dombry:2016}. - -The \texttt{rmev} function checks conditional negative definiteness of the matrix. The easiest way to do so -negative definiteness of $\bs{\Lambda}$ with real entries is to form $\tilde{\bs{\Lambda}}=\mathbf{P}\bs{\Lambda}\mathbf{P}^\top$, 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 $\bs{\Lambda} \in \mathcal{D}$, then the eigenvalues of the leading $(d-1) \times (d-1)$ submatrix of $\tilde{\bs{\Lambda}}$ -will all be negative. - -For a set of $d$ locations, one can supply the variogram matrix as valid input to the method. - -% using a variogram or if specifying a -% correlation matrix $\bs{\Sigma}$ with entries $\varrho_{ij}$, by taking $2/r_{ij}=\sqrt(2-2\varrho_{ij}).$ -\item \textbf{Brown--Resnick} (\code{br}) -The Brown--Resnick process is the extension of the H\"usler--Reiss distribution, and is a max-stable process associated with the -log-Gaussian distribution. -% One of its spectral representation is -% \begin{align*} -% \max_{i \geq 1} \zeta_i \psi_i, \qquad \psi_i(x)= \exp(\varepsilon(x)-\gamma(x)) -% \end{align*} -% where $\eps(x)$ is an intrinsically stationary Gaussian process with semivariogram $\gamma(x)$ constrained so that $\eps(o)=0$ -% almost surely. - -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 \code{vario} is provided, the model will simulate from an intrinsically stationary Gaussian process. The user can alternatively provide a covariance matrix \code{sigma} obtained by conditioning on a site, in which case simulations are from a stationary Gaussian process. See \cite{Engelke:2015} or \cite{Dombry:2016} for -more information. -\item \textbf{Extremal Student} (\code{extstud}) of \cite{Nikoloulopoulos:2009}, eq. 2.8, with unit Fréchet margins is -\begin{align*} - \P{\bs{X} \leq \bs{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[\pfrac{x_i}{x_j}^{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. - - -\item \textbf{Dirichlet mixture} (\code{dirmix}) proposed by \cite{Boldi:2007}, see \cite{Dombry:2016} for details on the -mixture. -The spectral density of the model is -\begin{align*} -h_{\bs{W}}(\bs{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 \code{param} is thus a $d \times m$ matrix of coefficients, while the argument for the $m$-vector \code{weights} gives the relative contribution of each Dirichlet mixture component. - -\item \textbf{Smith model} (\code{smith}), from the unpublished report of \cite{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 $\R_{+} \times \mathbb{X}$ 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. - -\end{enumerate} - -\clearpage -\bibliographystyle{apalike} -\bibliography{mevvignette} - - -\end{document} diff --git a/vignettes/mev-vignette.html b/vignettes/mev-vignette.html new file mode 100644 index 0000000..42457a5 --- /dev/null +++ b/vignettes/mev-vignette.html @@ -0,0 +1,792 @@ + + + + + + + + + + + + + + + + +Exact unconditional sampling from max-stable random vectors + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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,] 1.0521463 1.017444 1.012105 1.008567 1.040459
+## [2,] 1.0495947 1.036283 1.050616 1.011403 1.056270
+## [3,] 0.9749297 1.048115 1.016379 1.024842 1.000469
+
#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] 0.2 0.2 0.2 0.2 0.2
+
+
+

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. +
+
+
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-\newcommand{\hlkwd}[1]{\textcolor[rgb]{0.737,0.353,0.396}{\textbf{#1}}}% -\let\hlipl\hlkwb - -\usepackage{framed} -\makeatletter -\newenvironment{kframe}{% - \def\at@end@of@kframe{}% - \ifinner\ifhmode% - \def\at@end@of@kframe{\end{minipage}}% - \begin{minipage}{\columnwidth}% - \fi\fi% - \def\FrameCommand##1{\hskip\@totalleftmargin \hskip-\fboxsep - \colorbox{shadecolor}{##1}\hskip-\fboxsep - % There is no \\@totalrightmargin, so: - \hskip-\linewidth \hskip-\@totalleftmargin \hskip\columnwidth}% - \MakeFramed {\advance\hsize-\width - \@totalleftmargin\z@ \linewidth\hsize - \@setminipage}}% - {\par\unskip\endMakeFramed% - \at@end@of@kframe} -\makeatother - -\definecolor{shadecolor}{rgb}{.97, .97, .97} -\definecolor{messagecolor}{rgb}{0, 0, 0} -\definecolor{warningcolor}{rgb}{1, 0, 1} -\definecolor{errorcolor}{rgb}{1, 0, 0} -\newenvironment{knitrout}{}{} % an empty environment to be redefined in TeX - -\usepackage{alltt} -\usepackage[T1]{fontenc} -\usepackage[utf8]{inputenc} -\usepackage{amsmath, amssymb, amsfonts} -\usepackage{times} -\usepackage{geometry} -\usepackage{natbib} -\usepackage{hyperref} -%\geometry{left=2cm, right=2cm, top=2.5cm, bottom=2.25cm} - -\newcommand{\all}{ \; \forall \;} -\newcommand{\bs}[1]{\boldsymbol {#1}} -\newcommand{\R}{\mathbb{R}} -\renewcommand{\P}[2][]{{\mathsf P}_{#1}\left(#2\right)} -\newcommand{\E}[2][]{{\mathsf E}_{#1}\left(#2\right)} -\newcommand{\Va}[2][]{{\mathsf{Var}_{#1}}\left(#2\right)} -\newcommand{\Co}[2][]{{\mathsf{Cov}_{#1}}\left(#2\right)} -\newcommand{\code}[1]{\texttt{#1}} -\newcommand{\sumi}{\sum_{i=1}^n} -\newcommand{\pfrac}[2]{\left(\frac{#1}{#2}\right)} -\DeclareMathOperator{\diag}{diag} -\IfFileExists{upquote.sty}{\usepackage{upquote}}{} -\begin{document} - -%\VignetteIndexEntry{Exact unconditional sampling from max-stable random vectors} -%\VignetteEngine{knitr::knitr_notangle} - -\title{Exact unconditional sampling from max-stable random vectors} -\author{Léo Belzile} -\date{} -\maketitle -\begin{center} -{ \small -Department of Decision Sciences \\ HEC Montréal \\\href{leo.belzile@hec.ca}{\texttt{leo.belzile@hec.ca}} -} -\end{center} - -The `\code{mev}' package was originally introduced to implement the exact unconditional sampling algorithms in \cite{Dombry:2016}. The two algorithms therein allow one to simulate simple max-stable random vectors. The implementation will work efficiently for moderate dimensions. - -\section{Functions and use} - -There are two main functions, \code{rmev} and \code{rmevspec}. \code{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, \code{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 -\begin{knitrout} -\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} -\begin{alltt} -\hlkwd{library}\hlstd{(mev)} -\hlcom{#Sample of size 1000 from a 5-dimensional logistic model} -\hlstd{x} \hlkwb{<-} \hlkwd{rmev}\hlstd{(}\hlkwc{n}\hlstd{=}\hlnum{1000}\hlstd{,} \hlkwc{d}\hlstd{=}\hlnum{5}\hlstd{,} \hlkwc{param}\hlstd{=}\hlnum{0.5}\hlstd{,} \hlkwc{model}\hlstd{=}\hlstr{"log"}\hlstd{)} -\hlcom{#Marginal parameters are all standard Frechet, meaning GEV(1,1,1)} -\hlkwd{apply}\hlstd{(x,} \hlnum{2}\hlstd{,} \hlkwa{function}\hlstd{(}\hlkwc{col}\hlstd{)\{ismev}\hlopt{::}\hlkwd{gev.fit}\hlstd{(col,} \hlkwc{show}\hlstd{=}\hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{mle\})} -\end{alltt} -\begin{verbatim} -## [,1] [,2] [,3] [,4] [,5] -## [1,] 1.0156 0.9726 0.9413 0.9762 0.9844 -## [2,] 0.9794 0.9376 0.9100 0.9759 0.9712 -## [3,] 0.9073 0.9737 0.9529 0.9968 0.9567 -\end{verbatim} -\begin{alltt} -\hlcom{#Sample from the corresponding spectral density} -\hlstd{w} \hlkwb{<-} \hlkwd{rmevspec}\hlstd{(}\hlkwc{n}\hlstd{=}\hlnum{1000}\hlstd{,} \hlkwc{d}\hlstd{=}\hlnum{5}\hlstd{,} \hlkwc{param}\hlstd{=}\hlnum{0.5}\hlstd{,} \hlkwc{model}\hlstd{=}\hlstr{"log"}\hlstd{)} -\hlcom{#All rows sum to 1 by construction} -\hlkwd{head}\hlstd{(}\hlkwd{rowSums}\hlstd{(w))} -\end{alltt} -\begin{verbatim} -## [1] 1 1 1 1 1 1 -\end{verbatim} -\begin{alltt} -\hlcom{#The marginal mean is 1/d} -\hlkwd{round}\hlstd{(}\hlkwd{colMeans}\hlstd{(w),}\hlnum{2}\hlstd{)} -\end{alltt} -\begin{verbatim} -## [1] 0.20 0.21 0.20 0.20 0.20 -\end{verbatim} -\end{kframe} -\end{knitrout} - -\section{Description of the models implemented} - -The different models implemented are described in \cite{Dombry: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 $\{1, \ldots, d\}$. -\begin{enumerate} - -\item \textbf{logistic distribution} (\code{log}) -The logistic model of \cite{Gumbel:1960} (the Gumbel Archimedean copula) -\begin{align*} - \P{\bs{X} \leq \bs{x}}= \exp \left[ - \left(\sumi \pfrac{1}{x_i}^{\alpha}\right)^{\frac{1}{\alpha}}\right] -\end{align*} -for $\alpha>1.$ By default, \code{rmev} will transform an argument in $(0,1)$ without warning, to conform with the -implementation. The spectral measure density is -\begin{align*} - h_{\bs{W}}(\bs{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 \bs{w} \in \mathbb{S}_d -\end{align*} - -\item \textbf{asymmetric logistic distribution} (\code{alog}) -This model was proposed by \cite{Tawn:1990}. It shares the same parametrization as the \code{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*} - \P{\bs{X} \leq \bs{x}} = \exp \left[ -\sum_{b \in \mathbb{B}_d}\left(\sum_{i \in b} \pfrac{\theta_{i, -b}}{x_i}^{\alpha_b}\right)^{\frac{1}{\alpha_b}}\right], -\end{align*} -%while the spectral density is -%\begin{align*} -% h_{\bs{W}}(\bs{w})=\frac{1}{d}\frac{\prod_{j=1}^{d-1} (j\alpha_b-1)}{\prod_{i\in b} w_i} \left(\prod_{i %\in -%b}\frac{\theta_{i,b}}{w_i}\right)^{\alpha_b} \left(\sum_{i \in b} \pfrac{\theta_{i, b}}{w_i}^{\alpha_b}\right)^{1/\alpha_b-d} -%\end{align*} - -The parameters $\theta_{i, b}$ must be provided in a list and represent the asymmetry parameter. -The sampling algorithm, from \cite{Stephenson:2003} gives some insight on the construction mechanism as a max-mixture of logistic distributions. Consider sampling $\bs{Z}_b$ 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. - -\item \textbf{negative logistic distribution} (\code{neglog}) -The distribution function of the min-stable distribution due to \cite{Galambos:1975} is -\begin{align*} - \P{\bs{X} \leq \bs{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 $\alpha \geq 0$ \citep{Dombry:2016}. The associated spectral density is -\begin{align*} - h_{\bs{W}}(\bs{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*} - -\item \textbf{asymmetric negative logistic distribution} (\code{aneglog}) -The asymmetric negative logistic model is alluded to in \cite{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 \cite{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*} - \P{\bs{X} \leq \bs{x}} = \exp \left[ -\sum_{b \in \mathbb{B}_d}(-1)^{|b|} -\left(\sum_{i \in b} -\pfrac{\theta_{i, b}}{x_i}^{\alpha_b}\right)^{\frac{1}{\alpha_b}}\right]. -\end{align*} -%The density is -%\begin{align*} -% h_{\bs{W}}(\bs{w}) = \frac{1}{d} \sum_{\substack{b \in \mathbb{B}_d \\c \subset b}} (-1)^{|b|} -%\left(\prod_{j=1}^{d-1}(1-j\alpha)\right)\left(\prod_{i \in c}w_j\right)^{-(\alpha+1)}\left( \prod_{j \in %b} \theta_{j, b}\right)^{\alpha_b}\left(\sum_{i \in c} -%\pfrac{\theta_{i, b}}{w_i}^{\alpha_b}\right)^{1/\alpha_b-d} -%\end{align*} -In particular, it does not correspond to the ``negative logistic distribution'' given in e.g. \S 4.2 of \cite{Coles:1991} or \S 3.5.3 of \cite{Kotz: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. - -\cite{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 \cite{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 \code{rbvevd}. -\item \textbf{multilogistic distribution} (\code{bilog}) -This multivariate extension of the logistic, proposed by \cite{Boldi:2009}, places mass on the interior of the simplex. Let $\bs{W} \in \mathbb{S}_d$ 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 $\bs{U} \in \mathbb{S}_d$ follows a $d$-mixture of Dirichlet with the $j$th component being $\mathcal{D}(\bs{1}-\delta_{j}\alpha_j)$, so that the mixture has density function -\begin{align*} - h_{\bs{U}}(\bs{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_{\bs{W}}(\bs{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 $\alpha_j \in (0,1)$ $(j=1, \ldots, d)$. -\item \textbf{Coles and Tawn Dirichlet distribution} (\code{ct}) -The Dirichlet model of \cite{Coles:1991} -\begin{align*} - h_{\bs{W}}(\bs{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 $\alpha_j>0.$ -\item \textbf{scaled extremal Dirichlet} (\code{sdir}) -The angular density of the scaled extremal Dirichlet model with parameters $\rho > -\min(\bs{\alpha})$ and $\bs{\alpha} \in \R^{d}_{+}$ is given, for all $\bs{w} \in \mathbb{S}_d$, by -\begin{align*} - h_{\bs{W}}(\bs{w})=\frac{\Gamma(\bar{\alpha}+\rho)}{d\rho^{d-1}\prod_{i=1}^d\Gamma(\alpha_i)} -\bigl\langle\{\bs{c}(\bs{\alpha},\rho)\}^{1/\rho},\bs{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 $\bs{c}(\bs{\alpha},\rho)$ 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. - -\item \textbf{H\"usler--Reiss} (\code{hr}), due to \cite{Husler:1989}. It is a special case of the Brown--Resnick process. -While \cite{Engelke:2015} state that H\"usler--Reiss variates can be sampled following the same scheme, the spatial analog is -conditioned on a particular site ($\bs{s}_0$), 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 $\bs{\Lambda}$, for which $\lambda_{ij}^2=\lambda_{ji}^2$. Then, following \cite{Nikoloulopoulos:2009} -(Remark~2.5) and \cite{Huser:2013}, we can write the distribution function as - \begin{align*} - \P{\bs{X} \leq \bs{x}} = \exp \left[ -\sum_{j=1}^d \frac{1}{x_j} \Phi_{d-1, \bs{\Sigma}_{-j}} \left( \lambda_{ij}- -\frac{1}{2\lambda_{ij}} \log\pfrac{x_j}{x_i}, i \in I_{-j}\right)\right]. - \end{align*} - where the partial correlation matrix $\bs{\Sigma}_{-j}$ 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 $\varrho_{i,i; j}=1$.\footnote{\cite{Engelke: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 $2\bs{\lambda}_{.j}^2-\log\pfrac{x_j}{\bs{x}_{-j}}$ instead. We recover the same expression by -standardizing, since this amounts to division by the standard deviations $2\bs{\lambda}_{.j}$} - - - -The \texttt{evd} package implementation has a bivariate implementation -of the H\"usler--Reiss distribution with dependence parameter $r$, with $r_{ik}=1/\lambda_{ik}$ or -$2/r=\sqrt{2\gamma(\bs{h})}$ for $\bs{h}=\|\bs{s}_i-\bs{s}_i\|$ 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 $\bs{\Lambda}=\{\lambda_{ij}^2\}_{i, j=1}^d \in -\mathcal{D}$ (cf. \cite{Engelke:2015}, p.3), where -\begin{multline*} - \mathcal{D}=\Biggl\{\mathbf{A}\in [0, \infty)^{d\times d}: \bs{x}^\top\!\!\mathbf{A}\bs{x} <0, \all \bs{x} \in \R^{d} -\setminus\{\bs{0}\} \\ \qquad -\text{ with } \sum_{i=1}^d x_i=0, a_{ij}=a_{ji}, a_{ii}=0 \all 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 -$\bs{\varSigma}=\mathbf{L}^\top\mathbf{L}$, where $\mathbf{L}$ is an upper triangular matrix whose diagonal element are on the -log-scale to ensure uniqueness of the Cholesky factorization; see \cite{Pinheiro:1996}. By taking -\begin{align*} - \bs{\Lambda}(\bs{\varSigma})= \begin{pmatrix} 0 & \diag (\bs{\varSigma})^\top \\ \diag(\bs{\varSigma}) & -\bs{1}\diag(\bs{\varSigma})^\top -+ \diag(\bs{\varSigma})\bs{1}^\top - 2 \bs{\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 $\bs{\Lambda}$. - -It easily follows that generating $\bs{Z}$ from a $d-1$ dimensional log-Gaussian distribution with covariance $\Co{Z_i, -Z_k}=2(\lambda_{ij}^2+\lambda_{kj}^2-\lambda_{ik}^2)$ 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 \cite{Dombry:2016}. - -The \texttt{rmev} function checks conditional negative definiteness of the matrix. The easiest way to do so -negative definiteness of $\bs{\Lambda}$ with real entries is to form $\tilde{\bs{\Lambda}}=\mathbf{P}\bs{\Lambda}\mathbf{P}^\top$, 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 $\bs{\Lambda} \in \mathcal{D}$, then the eigenvalues of the leading $(d-1) \times (d-1)$ submatrix of $\tilde{\bs{\Lambda}}$ -will all be negative. - -For a set of $d$ locations, one can supply the variogram matrix as valid input to the method. - -% using a variogram or if specifying a -% correlation matrix $\bs{\Sigma}$ with entries $\varrho_{ij}$, by taking $2/r_{ij}=\sqrt(2-2\varrho_{ij}).$ -\item \textbf{Brown--Resnick} (\code{br}) -The Brown--Resnick process is the extension of the H\"usler--Reiss distribution, and is a max-stable process associated with the -log-Gaussian distribution. -% One of its spectral representation is -% \begin{align*} -% \max_{i \geq 1} \zeta_i \psi_i, \qquad \psi_i(x)= \exp(\varepsilon(x)-\gamma(x)) -% \end{align*} -% where $\eps(x)$ is an intrinsically stationary Gaussian process with semivariogram $\gamma(x)$ constrained so that $\eps(o)=0$ -% almost surely. - -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 \code{vario} is provided, the model will simulate from an intrinsically stationary Gaussian process. The user can alternatively provide a covariance matrix \code{sigma} obtained by conditioning on a site, in which case simulations are from a stationary Gaussian process. See \cite{Engelke:2015} or \cite{Dombry:2016} for -more information. -\item \textbf{Extremal Student} (\code{extstud}) of \cite{Nikoloulopoulos:2009}, eq. 2.8, with unit Fréchet margins is -\begin{align*} - \P{\bs{X} \leq \bs{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[\pfrac{x_i}{x_j}^{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. - - -\item \textbf{Dirichlet mixture} (\code{dirmix}) proposed by \cite{Boldi:2007}, see \cite{Dombry:2016} for details on the -mixture. -The spectral density of the model is -\begin{align*} -h_{\bs{W}}(\bs{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 \code{param} is thus a $d \times m$ matrix of coefficients, while the argument for the $m$-vector \code{weights} gives the relative contribution of each Dirichlet mixture component. - -\item \textbf{Smith model} (\code{smith}), from the unpublished report of \cite{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 $\R_{+} \times \mathbb{X}$ 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. - -\end{enumerate} - -\clearpage -\bibliographystyle{apalike} -\bibliography{mevvignette} - - -\end{document} diff --git a/vignettes/mevvignette.bib b/vignettes/mevvignette.bib index 6797c1b..2c3be69 100644 --- a/vignettes/mevvignette.bib +++ b/vignettes/mevvignette.bib @@ -490,7 +490,7 @@ @unpublished{Smith:1990 author = {Smith, Richard L.}, year = 1990, month = {October}, - url = {http://www.unc.edu/depts/statistics/postscript/rs/spatex.pdf}, + url = {https://rls.sites.oasis.unc.edu/postscript/rs/spatex.pdf}, note = {Unpublished technical report.}, } @article{Stephenson:2003,