text:
\section{Introduction}

Extreme value statistics deal with the characterization of extreme events, such as stock market crashes, that occur with low frequency but with a potentially severe impact. A typical extreme value question is to estimate the (low) probability of such an extreme event. In the context of multivariate extremes, we are interested in estimating the probability that a $d$-dimensional random vector $\vc{X} = (X_1,\ldots,X_d)$ falls into some ``extreme region'' $C$ (also called failure region). For example, in the example of portfolio risk, the components of $\vc{X}$ represent negative log-returns of $d$ stocks in a portfolio. Another example is in catastrophic climate events, where $\vc{X}$ may represent maximum daily rainfall amounts at $d$ monitoring stations. To estimate such a probability $\mathbb{P}[\vc{X} \in C]$, it is important to take the tail dependence across the margins of $\vc{X}$ into account, i.e., the tendency of extremes to occur in several components simultaneously.

In practice, the extreme region of interest might contain few or even no observed data points, especially when the dimension $d$ is large.
One potential solution is to employ appropriate parametric models for multivariate extremes and to estimate the model parameters using intermediate level observations. However, as the dimension of the vector $\vc{X}$ increases, such a parametric approach becomes increasingly complex, which leads to the following dilemma. On the one hand, models with a small number of parameters tend to oversimplify the tail dependence structure. On the other hand, models with many parameters are difficult to estimate and may even lead to overfitting.

This paper provides a parametric model for the case where the underlying random vector is high dimensional. The model can capture the so-called tail dependence coefficients for all pairs of components of the $d-$dimensional random vector. Even though it has a large number of parameters (up to $d(d+1)/2$), they are easily obtained based on a fast algorithm, allowing one to approximate the failure probability using the estimated model.
Other estimators of (multivariate) failure probabilities are proposed in \citet{drees2015} (for the bivariate case) and \citet{valk2016}, among others.

In the existing literature, tail dependence of $\vc{X}$ is characterized in various ways, such as by the exponent measure, the angular measure, and the stable tail dependence function, among others \citep{beirlant2004, dehaanferreira2006}.  While such objects fully characterize the extremal behavior of a random vector, their estimation in high dimensions suffers from the ``curse of dimensionality''. Instead,
one may consider a partial summary of tail dependence by focusing on  the tail dependence coefficients $\chi$ or the extremal coefficients $\theta$ for all pairs, see e.g. \citet{coles1999}. This is similar to summarizing the dependence structure of a random vector by the covariance or correlation matrix.

In this paper, we focus on a generalization of the so-called \emph{tail pairwise dependence matrix} (TPDM), which collects all pairwise tail dependence coefficients. This dependence matrix was originally proposed by \citet{larsson2012} and has gained in popularity since the work of \citet{cooley2019}; for example,  \citet{kluppelberg2019} use it to estimate the parameters of an extreme Bayesian network, \citet{fomichov2020} propose a clustering algorithm inspired by spherical $k$-means, where the means are replaced by the Perron–Frobenius eigenvectors of the TPDM, and \citet{kim2022} study the extremal dependence between vectors of scores of functional data.

The TPDM is particularly appropriate when $\vc{X}$ is multivariate regularly varying with tail index $\alpha = 2$. By contrast, our approach allows for multivariate regularly variation with a general tail index $\alpha$, without standardizing the data to $\alpha=2$. Since marginal standardization does not affect dependence modeling, it is common in practice to separate marginal and dependence modeling by first estimating the marginal distributions (parametrically or non-parametrically), standardizing each marginal to a common distribution, and then modeling the dependence structure only. However, such an approach can be suboptimal in some applications. For example, in the portfolio risk example, if one intends to estimate the Value-at-Risk or the Expected Shortfall of a portfolio loss $L = \sum_{j=1}^d v_j X_j$, where $v_1,\ldots,v_d$ represent weights of the stocks \citep[see, e.g.,][]{mainik2013}, the failure region is of the form $\{ \sum_j v_j X_j > x \}$ with $x$ large.  In this case, we need to directly model the original vector $\vc{X}$: a nonparametric transformation for the marginals will lead to untractable failure regions. Our approach without standardizing a priori is therefore necessary. Such failure regions are also of interest in climate applications, where $\sum_{j=1}^d v_j X_j$ represents the spatial average over a given region of some environmental variable \citep[see, e.g.][]{kiriliouk2020nr2}.

The tail pairwise dependence matrix is closely related to the max-linear model, a simple but rather flexible multivariate extreme-value model. In such a model, each component of a $d$-dimensional vector can be interpreted as the maximum shock among a set of $q$ independent heavy-tailed factors. The max-linear model has $d \times q$ parameters (or $d \times(q-1)$ when margins are standardized), usually stocked in a parameter matrix $A \in \mathbb{R}^{d \times q}$.
Max-linear models are, among other applications, used to model extremes on directed acyclic graphs \citep{gissibl2015} or latent factor structures for financial returns \citep{cui2018}. If the number of parameters is limited, one can use the minimum distance estimation  \citep{einmahl2018} or spherical $k$-means \citep{janssen2020} to estimate a max-linear model. \citet{kiriliouk2020} presents hypothesis tests aiding in deciding the size of $q$.

Theoretically, \citet{fougeres2013} showed that the max-linear model is dense in the class of $d$-dimensional multivariate extreme-value distributions. Moreover, \citet{cooley2019} showed  that there exists a finite $q \in \mathbb{N}$ such that the tail pairwise dependence matrix $\Sigma_{\vc{X}}$ of any multivariate regularly varying random vector $\vc{X}$ with $\alpha = 2$ is equal to that of a max-linear model with $q$ factors via the relation $\Sigma_{\vc{X}}=AA^T$. In practice, the parameter matrix $A$ of this max-linear model can be estimated through a completely positive decomposition of $\Sigma_{\vc{X}}$. A matrix $S$ is called \emph{completely positive} if there exists a non-negative matrix $A$ such that $S = A A^T$; finding this matrix $A$ amounts to finding a \emph{completely positive decomposition} of $S$. Positive decompositions are well studied in the linear algebra literature \citep{groetzner2020,bot2021}, where it is shown that the maximum number of columns of $A$ is $d(d+1)/2 - 4$. The positive decomposition can be obtained via a decomposition algorithm, nevertheless, the algorithm is computationally intensive.

In this paper, we propose an algorithm for an \emph{approximate} completely positive decomposition $\Sigma_{\vc{X}} \approx A A^T$ with $A  \in \mathbb{R}^{d \times d}$.
The decomposition is easy to compute and applicable to moderate to high dimensions. In addition, it is fast enough to obtain various decompositions corresponding to a single TPDM and hence to characterize the model uncertainty.
We show that the decomposition is exact when $\vc{X}$ follows a max-linear model with a triangular coefficient matrix $A$.
Moreover, in all other cases, the decomposition is exact for all off-diagonal entries of $\Sigma_{\vc{X}}$, but may overestimate its diagonal entries.
We apply the proposed decomposition algorithm to industry portfolio returns and maximal wind speeds to illustrate its applicability. A third application, concerning rainfall in Switzerland, has been deferred to Appendix~\ref{appB}.

\section{Background}

\subsection{Multivariate regular variation and the tail pairwise dependence matrix}
Let $\vc{X} \in [0,\infty)^d$ be a random vector with joint cumulative distribution function $F$ and continuous marginal distributions $F_1, \ldots, F_d$.
Write $\mathbb{E}_0 = [0,\infty]^{d} \setminus \{\vc{0}\}$.
We assume that $\vc{X}$ is \emph{multivariate regularly varying}, i.e., there exists a sequence $b_n \rightarrow \infty$ and a limit measure $\nu_{\vc{X}}$ such that
\begin{equation}\label{eq:regvar}
n \, \mathbb{P} \left[ b_n^{-1} \vc{X}  \in \,\cdot \, \right] \overset{v}{\rightarrow} \nu_{\vc{X}} (\, \cdot \,), \qquad \text{as } n \rightarrow \infty,
\end{equation}
where $\overset{v}{\rightarrow}$ denotes vague convergence in the space of non-negative Radon measures on $\mathbb{E}_0$ and $\nu_{\vc{X}}$ is called the \emph{exponent measure} \citep[Chapter 6]{resnick2007}.  Then there exists an $\alpha > 0$, called the \emph{tail index} of $\vc{X}$, such that the
limit measure $\nu_{\vc{X}}$ satisfies the homogeneity property of order $\alpha$, i.e., $\nu_{\vc{X}}  (tC) = t^{-\alpha} \nu_{\vc{X}} (C)$ for any constant $t > 0$ and any ($\nu_{\vc{X}}$-measurable) Borel set $C \subset \mathbb{E}_0$. The sequence $b_n$ can be shown to satisfy $b_n \sim L(n) n^{1/\alpha}$, where $L$ denotes a slowly varying function.

The homogeneity property suggests that it is possible to express $\vc{X}$ and correspondingly the limit relation in \eqref{eq:regvar} in polar coordinates.
Let $\norm{\, \cdot \,}$ denote any norm on $\mathbb{R}^d$ and consider the transformation $T: 	\mathbb{E}_0 \mapsto (0, \infty) \times \mathbb{S}_{d-1}$, where $\mathbb{S}_{d-1} = \{\vc{w} \in \mathbb{E}_0: \norm{ \vc{w}} = 1 \}$, and
\[
T(\vc{X}) = \left( \norm{ \vc{X} }, \frac{\vc{X}}{ \norm{ \vc{X}}} \right) =: (R, \vc{W}).
\]
Then \eqref{eq:regvar} is equivalent to the existence of a finite measure $H_{\vc{X}}$ (called the \emph{angular measure}) on $\mathbb{S}_{d-1}$ and a sequence $b_{n} \rightarrow \infty$ such that
\begin{equation}\label{eq:polar}
n \, \mathbb{P} \left[ \left( b_{n}^{-1} R, \vc{W} \right) \in \,  \cdot \, \right]  \overset{v}{\rightarrow} \mu_{\alpha} \times H_{\vc{X}}  (\, \cdot \,) , \qquad
\text{as } n \rightarrow \infty,
\end{equation}
in the space of non-negative Radon measures on $(0,\infty] \times \mathbb{S}_{d-1} $, where $\mu_{\alpha}$ is a measure on $[0, \infty)$ given by
$\mu_{\alpha} \left( (x, \infty] \right) = x^{-\alpha}$ for $x > 0$ \citep[Theorem 6.1]{resnick2007}.
Let $B \subset \mathbb{S}_{d-1}$ be $H_{\vc{X}}$-measurable and consider the set
\[
C_{s, B} := \{\vc{x} \in \mathbb{E}_0: \norm{\vc{x}}  > s,  \vc{x}/\norm{\vc{x}} \in B \}.
\]
Equation \eqref{eq:polar} implies $\nu_{\vc{X}} (C_{s,B}) = s^{-\alpha} H_{\vc{X}}(B)$ and $\nu_{\vc{X}} (\, \textnormal{d} r \times \, \textnormal{d} \vc{w}) = \alpha r^{-\alpha - 1}\, \textnormal{d} r \, \textnormal{d} H_{\vc{X}} (\vc{w})$.
The angular measure $H_{\vc{X}}$ is related to the probability measure $S_{\vc{X}}$ in \citet[Chapter 6]{resnick2007} through $H_{\vc{X}} = m S_{\vc{X}}$ for
$m := \nu (C_{1,\mathbb{S}_{d-1}})= H_{\vc{X}} (\mathbb{S}_{d-1}) > 0$. The sequence $b_n$ in \eqref{eq:polar} is determined by
\[
n \, \mathbb{P}[R > b_n x] \rightarrow m x^{-\alpha }, \qquad \text{ as } n \rightarrow \infty,
\]
and the marginals of $\vc{X}$ satisfy
\begin{equation}\label{eq:scale}
\lim_{n \rightarrow \infty }n \, \mathbb{P} \left[  X_j > b_n x \right] =    x^{- \alpha} \int_{\mathbb{S}_{d-1}} w_j^{\alpha} \, \textnormal{d} H_{\vc{X}} (\vc{w}).
\end{equation}

To summarize the dependence of a multivariate regularly-varying vector $\vc{X}$, we define the \emph{tail pairwise dependence matrix (TPDM)} $\Sigma_{\vc{X}}$ as
\begin{equation}\label{eq:tpdm}
\Sigma_{\vc{X}} = (\sigma_{\vc{X}_{jk}})_{j,k=1,\ldots,d}, \qquad \text{with   } \,\,\, \sigma_{\vc{X}_{jk}} = \int_{\mathbb{S}_{d-1} } w_j^{\alpha/2} w_k^{\alpha/2} \, \textnormal{d} H_{\vc{X}} (\vc{w}).
\end{equation}
Note that this constitutes a generalization of the matrix introduced in \citet{larsson2012} whose integrand does not depend on $\alpha$; see also \citet{samorodnitsky1994}.
We will write $\vc{X} \sim \Sigma_{\vc{X}}$ if $\Sigma_{\vc{X}}$ is the TPDM of $\vc{X}$.
When no confusion can arise, we will write the elements of $\Sigma_{\vc{X}}$ as $\sigma_{jk}$. Two variables $X_j,X_k$ are \emph{tail dependent} if and only if $\sigma_{jk} > 0$. From \eqref{eq:scale} we observe that
\[
\lim_{n \rightarrow \infty }n \, \mathbb{P} \left[  X_j > b_n x \right] = x^{-\alpha} \sigma_{{jj}}.
\]
Hence, the (limiting) scale of $X_j$ is $\left( \sigma_{{jj}} \right)^{1/\alpha}$, and we say that the margins of $\vc{X}$ are \emph{standardized} with index $\alpha$ whenever $\sigma_{jj} = 1$ for all $j=1,\ldots,d$.

Finally, let $\norm{\, \cdot \,}_{\alpha}$ denote the $L_{\alpha}$ norm, i.e., for $\vc{x} \in [0, \infty)^d$, $\norm{\vc{x}}_{\alpha} = \left( \sum_{j=1}^d \lvert x_j \rvert^{\alpha} \right)^{1/\alpha}$ for $\alpha \geq 1$. In the following, we will always use the $L_{\alpha}$ norm in the definition of the angular measure $H_{\vc{X}}$.  The total mass of the angular measure then equals
\begin{equation*}
m = H_{\vc{X}}(\mathbb{S}_{d-1}) = \int_{\mathbb{S}_{d-1}} \norm{\vc{w}}_{\alpha}^{\alpha} \, \textnormal{d} H_{\vc{X}} (\vc{w})  = \sum_{j=1}^{d}  \sigma_{{jj}}.
\end{equation*}
\begin{prop}[Generalization of \citet{cooley2019}]\label{prop:semidefinite}
The tail pairwise dependence matrix $\Sigma_{\vc{X}}$, as defined in \eqref{eq:tpdm}, is positive semi-definite.
\end{prop}

\subsection{Max-linear model}\label{sec:ML}
We are interested in calculating the probability that $\vc{X}$ falls into some extreme failure region  $C \subset \mathbb{E}_0$. In practice, to estimate such probabilites $\mathbb{P}[\vc{X} \in C]$, we need to assume a parametric model for the exponent measure $\nu$, or, equivalently, for the angular measure $H$. A simple yet effective parametric model for the goal of estimating failure probabilities is the \emph{max-linear model}.
Let $A$ denote a $d \times q$ matrix with non-negative entries, with columns $\vc{a}_{1},\ldots,\vc{a}_{q}$, and suppose that $\max(\vc{a}_l) = \max(a_{1l}, \ldots, a_{dl}) > 0$ for all $l \in \{1,\ldots,q\}$. Let $Z_1,\ldots,Z_q$ be independent random variables whose distribution is Fr\'echet with scale 1 and shape $\alpha$, $\mathbb{P}[Z_l \leq z] = \exp \left\{ -z^{-\alpha}\right\}$, and define
\begin{align*
 A \times_{\max} \vc{Z}  := \left( \max_{l=1,\ldots,q} a_{1l} Z_l, \ldots,  \max_{l=1,\ldots,q} a_{dl} Z_l \right)^T,
\end{align*}
for $\vc{Z} = (Z_1,\ldots,Z_q)$. Denoting $\vc{Y} =  A \times_{\max} \vc{Z} $  and choosing $b_n = n^{1/\alpha}$, the angular measure of $\vc{Y}$ is
\begin{equation}\label{eq:maxlinH}
 H_{\vc{Y}} (\, \cdot \,) = \sum_{l=1}^{q} \norm{\vc{a}_{l}}_{\alpha}^{\alpha} \, \delta_{\vc{a}_{l}/\norm{\vc{a}_{l}}_{\alpha}} ( \, \cdot \,).
\end{equation}
Note that the order of the columns of $A$ does not matter.
The marginal distribution of $Y_j$ is Fr\'echet with scale $\left( \sum_{l=1}^q a_{jl}^{\alpha} \right)^{1/\alpha}$ and shape $\alpha$ for $j = 1,\ldots,d$.  The TPDM of $\vc{Y}$ has elements
\[
\sigma_{\vc{Y}_{jk}} = \sum_{l=1}^q \norm{\vc{a}_l}_{\alpha}^{\alpha} \left( \frac{a_{jl}}{\norm{\vc{a}_l}_{\alpha }}\right)^{\alpha/2}  \left( \frac{a_{kl}}{\norm{\vc{a}_l}_{\alpha }}\right)^{\alpha/2}  = \sum_{l=1}^q a_{jl}^{\alpha/2} a_{kl}^{\alpha/2},
\]
and hence
\begin{align*
\Sigma_{\vc{Y}} =  A_* A^T_*, \qquad \text{where } \, A_* := \left( a_{jk}^{\alpha/2} \right)_{j,k=1,\ldots,d}
\end{align*}

\citet{cooley2019} show that the class of max-linear angular measures, as defined in \eqref{eq:maxlinH} is dense in the class of possible angular measures (for any norm),  i.e., any multivariate regularly varying vector $\vc{X}$ can be approximated by a max-linear model with a $(d \times q)$ parameter matrix as $q \rightarrow \infty$.
Moreover, if attention is restricted to the TPDM, a max-linear model with \emph{finite} $q$ is sufficient to exactly match $\Sigma_{\vc{X}}$; this result has been proven in \citet{cooley2019} for $\alpha = 2$, but can be easily generalized to any $\alpha>0$.
\begin{prop}[\citet{cooley2019}, Propositions 4 \& 5]
\label{prop:maxlin}
\text{}
\begin{enumerate}
\item Let $Z_1,\ldots,Z_q$ be Fr\'echet$(1, \alpha)$ random variables. Given any angular measure $H$, there exists a sequence of non-negative $(d \times q)$ matrices $\{ A_q \}$, for $q = 1, 2, \ldots$ such that $H_{A_q \times_{\max} \vc{Z}} \rightarrow H$ weakly as $q \rightarrow \infty$.
\item If $\vc{X} \in \mathbb{R}^d$ has TPDM $\Sigma_{\vc{X}}$, there exists a non-negative $(d \times \widetilde{q})$ matrix $A_*$ with $\widetilde{q} < \infty$, such that $\Sigma_{\vc{X}} = A_* A_*^T = \Sigma_{A \times_{\max} \vc{Z}}$.
\end{enumerate}
\end{prop}

Whenever a matrix $\Sigma$ can be decomposed as $\Sigma=AA^T$, and the matrix $A$ has non-negative entries, we say that $\Sigma$ is \emph{completely positive}. The second part of Proposition \ref{prop:maxlin} not only shows that the TPDM is completely positive, but also that for any vector $\vc{X}$, there exists a max-linear vector $\vc{Y}$ whose TPDM is equal to that of $\vc{X}$. This result is key for the estimation of failure probabilites.

\subsection{The probability of a failure region under the max-linear model}\label{sec:failure}
We intend to estimate the failure probability $\mathbb{P}[\vc{Y} \in C]$ for a max-linear vector $\vc{Y}$. Indeed, whenever the set $C \subset \mathbb{E}_0$ is ``extreme enough'', we have
\begin{equation}\label{eq:failure}
\mathbb{P} \left[ \vc{Y} \in C \right] = \frac{1}{n} \left\{ n \,  \mathbb{P} \left[ n^{-1/\alpha} \vc{Y} \in n^{-1/\alpha} C \right]  \right\} \approx \frac{1}{n} \nu_{\vc{Y}} \left( n^{-1/\alpha} C \right).
\end{equation}

The choice of the region $C$ in \eqref{eq:failure} depends on the application of
interest. Nevertheless, for some practically relevant failure regions, the probability can be calcuated directly from the parameters of the max-linear model. For environmental data, e.g., if $\vc{Y}$ represents a vector of maximum daily wind speeds, rainfall amounts, or temperatures in $d$ locations, the region
\begin{align*}
C_{(\max)} (\vc{x}) & = \{ \vc{y} \in \mathbb{E}_0: y_1 > x_1 \text{ or } \ldots \text{ or } y_d > x_d \},
\end{align*}
is a common choice if an extreme event at at least one location can lead to a climatological catastrophe.
The exponent measure for this region equals
\begin{align*}
\nu( C_{(\max)} (\vc{x}) ) = \sum_{l=1}^q \max_{j=1,\ldots,d} \left( \frac{a_{jl}}{x_j} \right)^{\alpha}.
\end{align*}
On the other hand, if $\vc{Y}$ represents for example the negative log-returns of $d$ stocks in a portfolio, one might be interested in estimating the probability of all stocks crashing together,
\begin{align*}
C_{(\min)} (\vc{x}) & = \{ \vc{y} \in \mathbb{E}_0: y_1 > x_1 \text{ and } \ldots \text{ and } y_d > x_d \}.
\end{align*}
Similarly to $C_{(\max)}$, we can calculate
\begin{align*}
\nu( C_{(\min)} (\vc{x}) ) = \sum_{l=1}^q \min_{j=1,\ldots,d} \left( \frac{a_{jl}}{x_j} \right)^{\alpha}.
\end{align*}

Many simple regions (including the ones above when $x_1 = \ldots = x_d$) can be written as follows. Let $C_f (x) = \{ \vc{y} \in \mathbb{E}_0: f(\vc{y}) > x\}$ denote a failure region where $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is some function. For fixed $w$, let $r_* = r_*(\vc{w})$ be such that $f(r_* \vc{w}) = x$. Then
\begin{equation*
\nu (C_f (x)) = \sum_{l=1}^q \norm{\vc{a}_l}^{\alpha}_{\alpha} \left\{ r_*( \vc{a}_l / \norm{\vc{a}_l}_{\alpha}) \right\}^{-\alpha}.
\end{equation*}
Hence, we can obtain an explicit expression for the exponent measure of any failure region that can be written in the form of $C_f$.  For example, if $f(\vc{y})= \vc{v}^T \vc{y}$ for $\vc{v} \in \mathbb{E}_0$ such that $v_1 + \ldots + v_d = 1$, we get
\begin{align*}
 C_{(\textnormal{sum})} (\vc{v},x) & = \{ \vc{y} \in \mathbb{E}_0: v_1y_1 + \ldots + v_d y_d > x \}.
\end{align*}
Such a region is both of interest for climate events (e.g. in attribution studies such as \citet{kiriliouk2020nr2}, among others) and in the financial context (e.g. to calculate the Value-at-Risk for a portfolio), and we calculate
\begin{align*}
\nu \left( C_{(\textnormal{sum})} (\vc{v}, x)\right) =  x^{- \alpha} \sum_{l=1}^q \left( \vc{v}^T \vc{a}_l  \right)^{\alpha}.
\end{align*}
Moreover, we find
\[
\nu \left( C_{(\textnormal{sum})} (\vc{v}, x)\right) =
\begin{dcases}
  x^{-1}  \sum_{j=1}^d v_j \sigma_{jj} & \text{if } \alpha = 1, \\
  x^{-2} \left( \vc{v}^T \Sigma_{\vc{X}} \vc{v} \right) & \text{if } \alpha = 2.
 \end{dcases}
\]
In these two special cases, $\nu \left( C_{(\textnormal{sum})} (\vc{v}, x) \right)$ can be calculated based on the TPDM only. In other words, the random variable $\vc{v}^T \vc{Y}$ approximately follows a Pareto distribution whose scale is a function of the TPDM. By contrast, when $\alpha \notin \{1,2\}$, the parameter matrix $A$ needs to be recovered first.

\section{Decomposing the pairwise tail dependence matrix}\label{sec:alg}

Proposition \ref{prop:maxlin} shows that, for any random vector $\vc{X} $, a max-linear vector $\vc{Y}$ exists such that $\Sigma_{\vc{X}} = \Sigma_{\vc{Y}}$. Obtaining such a model requires a \emph{completely positive decomposition} of $\Sigma_{\vc{X}}$, i.e., finding a $(d \times q)$ matrix $A_*$ with non-negative entries such that $\Sigma_{\vc{X}}   = A_* A_*^T$ (see also Section \ref{sec:ML}). \citet{barioli2003} show that the number of columns of such a matrix $A_*$ is at most $d (d+1)/2 - 4$ for $d \geq 5$. Current algorithms performing completely positive decompositions are computationally heavy \citep{groetzner2020,bot2021}, and hence they cannot be applied repeatedly to obtain multiple factorizations when the dimension $d$ is high. Notice that the factorization of a completely positive matrix is not unique. Obtaining multiple different decompositions could be useful to quantify the model uncertainty regarding estimations of the probability of falling into a failure region.

In this section, we propose a simple algorithm to construct a $(d \times d)$ matrix $A_*$, which leads to an \emph{approximate} completely positive decomposition in the following sense. Firstly, we define the notation ``$\preceq$'': for two TPDMs, we write $\Sigma_{\vc{X}} \preceq \Sigma_{\vc{Y}}$ if
\[
\sigma_{\vc{X}_{jk}} = \sigma_{\vc{Y}_{jk}} \,\,\,\,\, \text{for all } j,k \in \{1, \ldots, d \}, \, j \neq k, \quad \text{and } \, \sigma_{\vc{X}_{jj}} \leq \sigma_{\vc{Y}_{jj}} \,\,\,\,\,   \text{for all } j \in \{1, \ldots, d \}.
\]
Then, for any random vector $\vc{X}$ with TPDM $\Sigma_{\vc{X}}$, we aim at constructing a max-linear vector $\vc{Y}$ with a $(d \times d)$ coefficient matrix $A_*$, such that $\Sigma_{\vc{X}} \preceq \Sigma_{\vc{Y}}$. The max-linear model can be viewed as an approximate model for the original random vector $\vc{X}$, ``matching'' the pairwise dependence structure of $\vc{X}$.

The details of the proposed algorithm are as follows. Define, for any TPDM $\Sigma_{\vc{X}} = (\sigma_{{jk}})_{j,k=1,\ldots,d}$ and any $i \in \{1,\ldots,d\}$,
\begin{equation*}
D_i (\Sigma_{\vc{X}}) :=   \max \left\{j,k \in \{1,\ldots,d\} \setminus \{i\}: \, \frac{\sigma_{ji} \sigma_{ki}}{\sigma_{jk} \sigma_{ii}} \right\}.
\end{equation*}
We write $D_i = D_i (\Sigma_{\vc{X}})$ when no confusion can arise. Let $\vc{\tau}_i = (\tau_{1,i},\ldots,\tau_{d,i})^T$ with
\begin{equation} \label{eq:tau}
\tau_{j,i}  =
\begin{dcases}
\sigma_{ji} \left(\sigma_{ii} \max(D_i,1)\right)^{-1/2} & \text{ if } j \neq i,\\
\left(\sigma_{ii} \max(D_i,1)\right)^{1/2} & \text{ if } j = i.
\end{dcases}
\end{equation}
Let $\vc{\tau}_{-i} := (\tau_{1,i},\ldots,\tau_{i-1,i},\tau_{i+1,i},\ldots,\tau_{d,i})^T  \in \mathbb{R}^{d-1}$ and define
\begin{equation}\label{eq:sigmai}
\Sigma^{(i)}_{\vc{X}} := \Sigma^{(-i,-i)}_{\vc{X}} - \vc{\tau}_{-i} \vc{\tau}_{-i}^T \quad \in \mathbb{R}^{(d-1) \times (d - 1)},
\end{equation}
where $\Sigma^{(-i,-i)}_{\vc{X}} \in \mathbb{R}^{(d-1) \times (d - 1)}$ is the matrix obtained by removing the $i$-th row and column from $\Sigma_{\vc{X}}$. The following lemma shows that $\Sigma^{(i)}_{\vc{X}}$ is still a valid TPDM.
\begin{lem}\label{lem1}
For all $i \in \{1,\ldots,d\}$ with $D_i \neq 1$, the matrix $\Sigma^{(i)}_{\vc{X}}$ in \eqref{eq:sigmai} is a TPDM since
\begin{enumerate}
\item $\Sigma^{(i)}_{\vc{X}} > 0$ componentwise;
\item $\Sigma^{(i)}_{\vc{X}}$ is positive semi-definite.
\end{enumerate}
\end{lem}
The next proposition shows that if we can construct a $(d-1)$-dimensional random vector with TPDM $\Sigma^{(i)}_{\vc{X}}$, we can then use this vector to construct $\vc{Y}$ such that $\Sigma_{\vc{X}}\preceq \Sigma_{\vc{Y}}$. In addition, the proposition illustrates the circumstances under which $\Sigma_{\vc{X}}=\Sigma_{\vc{Y}}$ is achieved. The proofs of Lemma \ref{lem1} and of Proposition \ref{prop:peel} are postponed to the appendix.
\begin{prop}\label{prop:peel}
Let $\Sigma_{\vc{X}} = (\sigma_{jk})_{j,k=1,\ldots,d} \in \mathbb{R}^{d \times d}$ be a TPDM. Let $i \in \{1,\ldots,d\}$ and let $Z_i$ be a Fr\'echet random variable with scale 1 and shape $\alpha$. Suppose there exists a random vector $\vc{\tilde Y}_{-i} = (\tilde Y_1,\ldots,\tilde Y_{i-1},\tilde Y_{i+1},\ldots,\tilde Y_{d})^T$ with TPDM $\Sigma^{(i)}_{\vc{X}}$.
By defining
\begin{equation*}
Y_j  := \begin{dcases}
\max \left( \tau_{j,i}^{2/\alpha} Z_i , \tilde Y_j \right) & \text{ if } j \in \{1,\ldots,d\} \setminus \{i\}, \\
\tau_{i,i}^{2/\alpha} Z_i & \text{ if } j = i,
\end{dcases}
\end{equation*}
we have that $\Sigma_{\vc{X}}\preceq \Sigma_{\vc{Y}}$. In particular, if $D_i (\Sigma_{\vc{X}}) < 1$, then $\Sigma_{\vc{Y}} = \Sigma_{\vc{X}}$.
\end{prop}
Hence, we can repeatedly apply the procedure in expression \eqref{eq:sigmai} to $\Sigma^{(i)}_{\vc{X}}$ to further decrease the dimension. This leads to an iterative algorithm, throughout which we obtain the matrix $A_*$. First, let $i_1 \mapsto i_2 \mapsto \ldots \mapsto i_d$ denote a \emph{path}, where $(i_1,\ldots,i_d)$ is a permutation of $(1,\ldots,d)$. The order determines which column of $\Sigma_{\vc{X}}$ will be treated first using Proposition \ref{prop:peel}. For a given path, the iterative algorithm is as follows.
\begin{itemize}
\item In the first step, we obtain the vector $\vc{\tau}_{i_1}$ from \eqref{eq:tau} by taking $i=i_1$. We fill in the first column of the matrix $A_*$ with the obtained vector $\vc{\tau}_{i_1}$ (as the order of the columns of $A_*$  does not matter, we do not necessarily need to fill the $i_1$-th column). The targeted TPDM is then reduced to a $(d-1)\times (d-1)$ matrix $\Sigma_{\vc{X}}^{(-i_1)}$ as defined in \eqref{eq:sigmai}.
\item In the second step, we apply expressions \eqref{eq:tau} and \eqref{eq:sigmai} to $\Sigma_{\vc{X}}^{(-i_1)}$, with the focal dimension now indexed by $i_2$. This step results in a $(d-1)$-dimensional vector. For the second column of the matrix $A_*$, we set the $i_1$-th row to zero and fill in the rest with the obtained $(d-1)$-dimensional vector. After the second step, the targeted TPDM is reduced to a $(d-2)\times(d-2)$ matrix.
\item The previous step is iterated as follows. In each step $j$, we aim at filling in the $j$-th column of the matrix $A_*$. First, we set the elements in the $i_1,i_2,\cdots,i_{j-1}$-th row to zero. Then, we fill in the other $(d-j+1)$ elements by the obtained $(d-j+1)$-dimensional vector of this iteration. After each step, the dimension of the targeted TPDM is reduced by one.
\item At step $d$, the targeted TPDM is reduced to $1\times 1$, i.e. a single positive number. We simply fill in the square root of that value to the $d$-th column and $i_d$-th row of $A_*$, while setting all other elements in the $d$-th column to zero.
\end{itemize}
Proposition \ref{prop:peel} guarantees that the matrix $A_*$ obtained with this algorithm has only non-negative elements and satisfies $\Sigma_{\vc{X}}\preceq A_*A_*^T$. In addition, define $A=A_*^{2/\alpha}$ (i.e. each element of the matrix $A_*$ is raised to the power $2/\alpha$) and let $\vc{Z}=(Z_1,\cdots,Z_d)^T$ where $Z_1,\ldots,Z_d$ are
i.i.d. standard Fr\'{e}chet distributed random variables with scale 1 and shape $\alpha$. The max-linear vector $\vc{Y}=A \times_{\max} \vc{Z}$ has a TPDM that satisfies $\Sigma_{\vc{X}}\preceq \Sigma_{\vc{Y}}$. In other words, we can construct a max-linear vector $\vc{Y}$ based on only $d$ independent factors to approximate the TPDM of $\vc{X}$.

Below is an example for this procedure when $d=3$ and the chosen path is $1 \mapsto 2 \mapsto 3$.
\begin{exa}[$d = 3, \alpha = 2$, path $1 \mapsto 2 \mapsto 3$]\label{example:3d}
Let $\Sigma_{\vc{X}} = (\sigma_{jk})_{j,k = 1,2,3}$ be a known (or estimated) TPDM. We want to find a coefficient matrix $A$ (since $\alpha=2$, we have $A=A_*$) such that the corresponding max-linear model $\vc{Y} =  A \times_{\max} (Z_1,Z_2,Z_3)^T$ has TPDM $\Sigma_{\vc{Y}}$ satisfying $\Sigma_{\vc{X}} \preceq \Sigma_{\vc{Y}}$. Starting with $i  =1 $, we obtain from \eqref{eq:tau} that
\[
\begin{pmatrix} \tau_{1,1} \\ \tau_{2,1} \\ \tau_{3,1} \end{pmatrix} = \frac{1}{\sqrt{\sigma_{11} \max(D_1,1)}} \begin{pmatrix} \sigma_{11} \max(D_1,1) \\ \sigma_{12} \\ \sigma_{13} \end{pmatrix}
\]
and
\begin{equation*}
\Sigma_{\vc{X}}^{(-1)} := \begin{pmatrix}
s_{22} & s_{23} \\
s_{23} & s_{33} \\
\end{pmatrix}
:=
\begin{pmatrix}
\sigma_{22} - \frac{\sigma_{12}^2}{\sigma_{11} \max(D_1,1)} & \sigma_{23} - \frac{\sigma_{12} \sigma_{13}}{\sigma_{11} \max(D_1,1)} \\
\sigma_{23} - \frac{\sigma_{12} \sigma_{13}}{\sigma_{11} \max(D_1,1)} & \sigma_{33} - \frac{\sigma_{13}^2}{\sigma_{11} \max(D_1,1)} \\
\end{pmatrix}.
\end{equation*}
Hence, the first column of $A$ is given by $(\tau_{1,1}, \tau_{2,1}, \tau_{3,1})^T$.

To find the second column of $A$, we iterate the same algorithm to the newly obtained TPDM $\Sigma_{\vc{X}}^{(-1)}$ to get that
\[
\begin{pmatrix} \tau_{2,2} \\ \tau_{3,2} \end{pmatrix} = \frac{1}{\sqrt{s_{22} }} \begin{pmatrix} s_{22}  \\ s_{23} \end{pmatrix}.
\]
Note that since $\Sigma_{\vc{X}}^{(-1)}$ is positive semidefinite, in this case we have $D_2(\Sigma_{\vc{X}}^{(-1)})=\frac{s_{23}^2}{s_{33}s_{22}}\leq 1$. After this iteration, the targeted TPDM is reduced to a single value $s_{33} - s_{23}^2/ s_{22}\geq 0$. Hence, we can define $\tau_{3,3}:= \sqrt{s_{33} - s_{23}^2/ s_{22}}$.

Eventually, we can construct the max-linear model as $\vc{Y} = A \times_{\max} (Z_1,Z_2,Z_3)$ where
\begin{equation*}
A = \begin{pmatrix}
\tau_{1,1} & 0 & 0 \\
\tau_{2,1} & \tau_{2,2} & 0 \\
\tau_{3,1} & \tau_{3,2} & \tau_{3,3} \\
\end{pmatrix}, \qquad \text{and} \qquad \Sigma_{\vc{Y}} = A A^T =
 \begin{pmatrix}
\sigma_{11} \max(D_1,1) & \sigma_{12} & \sigma_{13} \\
\sigma_{12} & \sigma_{22} & \sigma_{23} \\
\sigma_{13} & \sigma_{23} & \sigma_{33}
\end{pmatrix}.
\end{equation*}
We have that $\Sigma_{\vc{Y}}= \Sigma_{\vc{X}} $ if $D_1 < 1$ and $\Sigma_{\vc{X}} \preceq \Sigma_{\vc{Y}} $ otherwise.
\end{exa}

If the matrix $\Sigma_{\vc{X}}$ has a zero eigenvalue, it means that the components of $\vc{X}$ are linearly related when the radius $R$ is high. In practice, an eigenvalue of $\Sigma_{\vc{X}}$ can indeed be close to zero when the variables are strongly dependent given a high radius. This is comparable with the concept of multicolinearity in multivariate statistics. In that case, at least one column of $A_*$ will be zero: the column corresponding to the first variable encountered that is close to a linear combination of previous variables, given that the radius is high. We can remove that column in $A_*$ and correspondingly $A$. Eventually, the number of non-zero columns of $A_*$ corresponds to the rank of $\Sigma_{\vc{X}}$.

In Example \ref{example:3d}, we applied the iterative algorithm implied by Proposition \ref{prop:peel} with the path $1 \mapsto 2 \mapsto 3$. The resulting matrix $A_*$ (and eventually $A$) is then lower triangular. By considering others paths we will obtain different matrices $A_*$. Hence, in general, for different choices of paths, a completely positive decomposition (approximate or not) is not unique. It is also possible that one path leads to an exact decomposition while another path leads to an approximation in the sense of $\preceq$. Proposition \ref{prop:unique} shows that if the vector $\vc{X}$ indeed follows a max-linear model constructed from a lower-triangular parameter matrix $A$, by choosing the path \emph{appropriately}, it is possible to recover the exact max-linear model. Given that the order of columns of $A$ does not matter, the proposition can of course be generalized to parameter matrices $A$ in which each column $j$ has at least one more zero than column $(j-1)$, for $j=2,\ldots,d$.
\begin{prop}\label{prop:unique}
Suppose that $\vc{X} = A \times_{\max} \vc{Z}$ follows a max-linear model with lower-triangular parameter matrix $A \in \mathbb{R}^{d \times d}$ and $\vc{Z} = (Z_1,\ldots,Z_q)$, where $Z_1,\ldots,Z_q$ are iid Fréchet with scale 1 and shape $\alpha$. If we choose the canonical path, i.e., $1 \mapsto \ldots \mapsto d$ in the decomposition of $\Sigma_{\vc{X}}$ to obtain the matrix $A_*$, then $A_*A_*^T=\Sigma_{\vc{X}}$.
\end{prop}
For a high-dimensional TPDM $\Sigma_{\vc{X}}$, it is not feasible to exhaust all $d!$ possible paths in search for an exact decomposition.
In practice, we need an efficient searching algorithm to find a path that leads to a max-linear model $\vc{Y}$ such that $\Sigma_{\vc{Y}} = \Sigma_{\vc{X}}$, if possible. We consider three potential algorithms to achieve the goal.
\begin{enumerate}
\item A simple approach: In step $j$ we pick the next dimension $i_j$ corresponding to the lowest value of $D_i$ among all remaining dimensions.
\item An exhaustive approach: In step $j$, we denote $T_j$ as the \emph{set} of remaining dimensions corresponding to a value of $D_i$ below 1, i.e. $T_j := \{ i : D_i < 1, i\neq i_1,i_2,\cdots,i_{j-1}\}$. For each $i\in T_j$, we proceed with the iterative process for $\Sigma^{(-i)}_{\vc{X}}$. This will result in a tree of possibilities after each step. For some ``branches'' of the tree, the algorithm may end at a ``leaf'' where no $D_i$ lies below 1, i.e. $T_j=\varnothing$. If some ``branch'' can iterate $d$ steps, it leads to a path that corresponds to an exact decomposition. We retain all paths (and hence all possible matrices $A_*$) that lead to an exact decomposition.
\item A pragmatic approach: In step $j$, we denote $T_j$ as the \emph{set} of remaining dimensions corresponding to a value of $D_i$ below 1, i.e. $T_j := \{ i : D_i < 1, i\neq i_1,i_2,\cdots,i_{j-1}\}$. We randomly choose a dimension from this set as $i_j$ and proceed with the iteration. If the procedure stops after less than $d$ iterations and ends with a scenario where no $D_i$ lies below 1, we restart the entire procedure from the first iteration. We keep ``restarting'' the algorithm until finding an exact decomposition.
\end{enumerate}
Note that the simple approach may not lead to an exact decomposition because there is no guarantee that the minimum of the $D_i$'s is below $1$ in all steps. The exhaustive approach will find all exact decompositions if there exists at least one. However, it is limited to moderate dimensions $d$ because of its computational intensity. The pragmatic approach does not guarantee finding an exact decomposition either. Nevertheless, for most examples in Section \ref{sec:appl}, an exact decomposition is quickly found after a few ``restarts'', and hence it is the approach that we will most use in practice.

\newpage

\section{Estimation}\label{sec:estim}

\subsection{Estimation of the TPDM}\label{sec:estim1}
Let $\vc{X}_i = (X_{i1},\ldots,X_{id})$ for $i = 1,\ldots,n$ be iid copies of $\vc{X}$. First, we estimate the tail indices $\alpha_j$, or, equivalently, the extreme value indices $\gamma_j = 1/\alpha_j$, for each $j= 1,\ldots,d$ using the sample $\left\{X_{1j},\ldots,X_{nj}\right\}$. Note that the assumption of regular variation \eqref{eq:regvar} implies that all margins have equal tail indices, $\alpha_1 = \ldots = \alpha_d := \alpha$, which is a common assumption, since different tail indices will lead to the largest one dominating the others  and masking tail dependence.

Let $X_{(1),j} \leq \ldots \leq X_{(n),j}$ denote the order statistics of $X_{1j},\ldots,X_{nj}$ and let $k = k_n$ be an intermediate sequence such that $k \rightarrow \infty$ and $k/n \rightarrow 0$ as $n \rightarrow \infty$. The \emph{Hill estimator},
\[
\widehat{\gamma}_j = \frac{1}{k} \sum_{i=1}^k \log \left( \frac{X_{(n-i+1),j}}{X_{(n-k),j}} \right)
\]
is the best-known estimator of the extreme-value index \citep{hill1975}, and will be used to obtain the estimated tail indices $\widehat{\alpha}_j = 1/\widehat{\gamma}_j$.
If the confidence intervals of $\widehat{\alpha}_1,\ldots,\widehat{\alpha}_d$ are overlapping, a single tail index estimate $\widehat{\alpha}$ might be justified. In that case, the entire sample $X_{11},\ldots,X_{n1}, X_{12},\ldots,X_{n2},\ldots,X_{1d},\ldots,X_{nd}$ of length $nd$ can be used to obtain the estimate $\widehat{\alpha}$ \citep{einmahl2022}.

Next, for $i  = 1,\ldots,n$, set $R_i =  \norm{\vc{X}_i}_{\widehat{\alpha}}$ and $\vc{W}_i = \vc{X}_i/ R_i$. Let $m = H_{\vc{X}}(\mathbb{S}_{d-1})$ denote the total mass of the angular measure. If the data are standardized to ensure equal scales for all $j$ (see subsection \ref{sec:estim2}), then $m = d$.  Otherwise, we observe that
\[
m =  \nu_{\vc{X}} (C_{1, \mathbb{S}_{d-1}}) = \lim_{n \rightarrow \infty} n \, \mathbb{P} [b_n^{-1} \vc{X}_i \in C_{1,\mathbb{S}_{d-1}}] = \lim_{n \rightarrow \infty} n \, \mathbb{P}[R _i> b_n]
\]
Following \cite{cooley2019}, we estimate $b_n$ by a high quantile of the empirical distribution of $R_1,\ldots,R_n$, denoted as $r_0$. By setting $n \approx r_0^{\widehat{\alpha}}$, $m$ can be estimated by  $\widehat{m} = r_0^{\widehat{\alpha}}  (n_{\textnormal{exc}}/n)$ for $n_{\textnormal{exc}} = \sum_{i=1}^n \mathbbm{1} \left\{ R_i > r_0 \right\}$.  A value for $r_0$ could be chosen by plotting different quantiles against the estimates of $m$ and searching for a region where estimates are stable.
Finally, since $\sigma_{jk} = m \int_{\mathbb{S}_{d-1}} w_j^{\alpha/2} w_k^{\alpha/2} \, \textnormal{d} \widetilde{H}_{\vc{X}} (\vc{w})$ where $\widetilde{H}_{\vc{X}} ( \, \cdot \,) := m^{-1} H_{\vc{X}} (\, \cdot \,)$ is a probability measure, the elements of the TPDM can be estimated by
\[
\widehat{\sigma}_{jk} = \frac{\widehat{m}}{n_{\textnormal{exc}}} \sum_{i=1}^n W_{ij}^{\widehat{\alpha}/2} W_{ik}^{\widehat{\alpha}/2} \mathbbm{1}\left\{ R_i > r_0 \right\} = \frac{r_0^{\widehat{\alpha}}}{n} \sum_{i: R_i > r_0} \left( W_{ij} W_{ik} \right)^{\widehat{\alpha}/2},
\]
giving $\widehat{\Sigma}_{\vc{X}} =  (\widehat{\sigma}_{jk})_{j,k=1,\ldots,d}$. \cite{larsson2012} show the asymptotic normality of $\widehat{\sigma}_{jk}$ when $\alpha = 2$ is known.

\subsection{Estimation of failure probabilities and standardization}\label{sec:estim2}
Once the TPDM is estimated, we calculate the $d \times d$ matrix $\widehat{A}_*$ obtained by applying the decomposition algorithm outlined in Section \ref{sec:alg} to $\widehat{\Sigma}_{\vc{X}}$, and the empirical $(d \times n_{\textnormal{exc}})$ estimate
\[
\widetilde{A}_*= \left(\frac{\widehat{m}}{n_{\textnormal{exc}}} \right)^{1/2} \widehat{W}^{\widehat{\alpha}/2},
\]
where $\widehat{W}^{\widehat{\alpha}/2}$ is the matrix whose columns are the vectors $\vc{W}_i^{\widehat{\alpha}/2}$ for which $R_i > r_0$. The matrix $\widetilde{A}_* $ is ``inefficient'' in the sense that it has much more columns than $\widehat{A}_*$, especially when $n_{\textnormal{exc}}$ is large.  However, it is easy to calculate and satisfies $\widehat{\Sigma}_{\vc{X}} = \widetilde{A}_*  \widetilde{A}_* ^T$; we will use it as a benchmark.  We obtain two estimators of $A$ as
\[
\widetilde{A} = \left( \widetilde{A}_* \right)^{2/\widehat{\alpha}} \quad \textnormal{and} \quad \widehat{A}= \left(\widehat{A}_* \right)^{2/\widehat{\alpha}}.
\]
Now, $\widetilde{A}$ and $\widehat{A}$ can be used to estimate the probability of falling into one of the failure regions presented in Section~\ref{sec:failure}.
We will denote the estimators of these probabilities by $\widehat{p}$ for $\widehat{A}$ and
$\widetilde{p}$ for $\widetilde{A} $.
In the applications,  we will usually take the pragmatic approach described in Section \ref{sec:alg} to generate $N$ different matrices $\widehat{A}_{*1}, \ldots, \widehat{A}_{*N}$ that correspond to
\emph{exact} decompositions of $\widehat{\Sigma}_{\vc{X}}$. By an exact decomposition, we mean that $\lVert \widehat{\Sigma}_{\vc{X}}  -  \widehat{A}_{*l} \widehat{A}^T_{*l} \rVert_F \leq 10^{-12}$ for some $l = 1,\ldots,N$, where $\lVert \, \cdot \, \rVert_F$ denotes  the Frobenius norm.

Finally, note that for certain failure regions we could standardize the margins of $\vc{X}_1,\ldots,\vc{X}_n$ without losing tractability of the shape of the failure set.
We set
\begin{align*}
X_{ij}^{\#} & = h_j (X_{ij}) = \left( - \log \widehat{F}_j^{P} (X_{ij}) \right)^{-1/2}, \qquad j = 1,\ldots,d, i = 1,\ldots,n ,
\end{align*}
where $\widehat{F}_j^{P}$ denotes the semi-parametric estimate of $F_j$,
\begin{equation*}
\widehat{F}_j^P (y) = \begin{dcases}
\widehat{F}_j (y) & \text{ if } y \leq u_j ; \\
1 - \left(1 - \widehat{F}_j (u_j) \right) \overline{H}_{\widehat{\sigma}_j, \widehat{\gamma}_j} (y - u_j) & \text{ if } y > u_j; \\
\end{dcases}
\end{equation*}
where $\widehat{F}_j$ denotes the empirical distribution function of $X_{1j}, \ldots,X_{nj}$ and $\overline{H}_{\widehat{\sigma}_j,{\widehat{\gamma}}_j}$ the survival function of a generalized Pareto distribution (GPD) with estimated scale $\widehat{\sigma}_j$ and shape $\widehat{\gamma}_j$, obtained by fitting $(X_{ij} - u_j \mid X_{ij} > u_j)_{i=1,\ldots,n}$ to the GPD for some high threshold $u_j$.
The distribution of $X^{\#}_{1j}, \ldots, X^{\#}_{nj}$ is Fr\'echet with scale $1$ and shape $\alpha = 2$. Hence,	 we have $\sigma_{jj} = 1$ for $j \in \{1,\ldots,d\}$ and $m = H_{\vc{X}} (\mathbb{S}_{d-1}) = d$ does not need to be estimated.  Subsequently, we can decompose the TPDM $\widehat{\Sigma}_{\vc{X}^{\#}}$ and construct the associated max-linear model.

\section{Applications}\label{sec:appl}

\subsection{Failure probabilities computed from repeated decompositions}\label{sec:expmeas}
The TPDM forms an incomplete characterization of tail dependence; hence,  we would like to assess by how much the exponent measure of a given failure region varies if it is computed from repeated decompositions.  We proceed as follows, inspired by Section 2 of the supplementary material of \cite{cooley2019}.  Let
\begin{align*}
A_1 & = \begin{pmatrix}
 1.00 & 0.50 & 1.75 & 0.00 & 0.50  &0.75 & 1.00 & 0.25 \\
 2.00 & 0.00 & 1.00 & 1.50 & 0.25 & 1.00 & 1.00 & 1.00 \\
 1.75 & 1.25 & 0.50 & 0.75 & 2.00  & 1.75 & 0.25 & 0.25 \\
 1.25 & 0.25 & 1.25 & 2.00 & 2.00 & 0.50&  0.25 & 0.25 \\
 1.75 & 0.50&  0.25 & 0.75&  0.50&  1.75 & 0.00&  1.25 \\
\end{pmatrix}, \\
A_2 & = \begin{pmatrix}
 1.00 & 0.00 &  0.50 &  0.00 & 0.50 \\
 2.00 & 1.50 & 0.00 & 1.00 & 0.00 \\
 1.75 & 0.75 & 1.25 & 0.25 & 0.75 \\
 1.25 & 2.00 & 0.25 &  0.25 &  0.25 \\
 1.75 & 0.75 & 0.50 & 1.25 & 0.50 \\
\end{pmatrix}, \qquad \textnormal{and }
A_3 =\begin{pmatrix}
1.00 & 0.00  & 0.00 \\
 2.00 & 1.50&   0.00 \\
 1.75 & 0.75 & 0.25 \\
 1.25 & 2.00 & 1.00 \\
 1.75 & 0.75 & 0.50 \\
\end{pmatrix}
\end{align*}
Matrix $A_1$ is as in \cite{cooley2019} (randomly generated elements).  Matrices $A_2$ and $A_3$ are similar but contain less columns and more zeroes; hence, we expect to obtain more exact decompositions for $A_2$ and $A_3$ than for $A_1$.  Let $\nu_i$ denote the exponent measure of a max-linear random vector with coefficient matrix $A_i$, $i=1,2,3$ and $\alpha = 4$. Let $\Sigma_i = A_i A_i^T$; we obtain the $d! = 5! = 120$ decompositions of  $\Sigma_i$ and consider the 120 max-linear vectors with corresponding coefficient matrices for each $i=1,2,3$.  From the 120 possible paths,  we use only 94 paths for $A_1$,  86 for $A_2$,  and 88 for $A_3$, since the others lead to paths with $D_i = 1$; see Section \ref{sec:alg}.

We assess the limiting exponent measure of $C_f  (x) = \{ \vc{y} \in \mathbb{E}_0: f(\vc{y}) > x\}$ where $f$ is some function; see Section \ref{sec:failure}. We use the functions $f_1(\vc{y}) = \frac{1}{d} \sum_{j=1}^d y_j$, $f_2 (\vc{y}) = \prod_{j=1}^d y_j$, $f_3 (\vc{y}) = \min(y_1,\ldots,y_d)$ and $f_4 (\vc{y}) = \max(y_1,\ldots,y_d)$. We take $\alpha = 4$ and $x$ is always chosen such that $\nu_j(C_f) = 0.1$.  Figure \ref{fig:expmeasures} shows boxplots for the measures $\nu_i$, $i=1,2,3$, of the sets $C_{f_1},C_{f_2}$, $C_{f_3}$ and $C_{f_4}$, obtained through the different paths. We show both the results when taking only exact decompositions into account (12 for $A_1$, 16 for $A_2$, and 24 for $A_3$) and when taking all decompositions such that $\norm{\Sigma_{\vc{X}} - A A^T}_F \leq 5$ (58 for $A_1$,  72 for $A_2$,  and 76 for $A_3$).  For $A_3$, 24 decompositions contain only 3 columns and are exactly equal to $A_3$, while 40 other decompositions contain 4 columns. For the product-region ($f_2$) and the min-region ($f_3$), only the first column of the obtained coefficient matrices is used in the calculation of $\nu_i$ (since all subsequent columns contain at least one zero; see also Section \ref{sec:failure}). Overall, the TPDM seems a reasonable summary of tail dependence. We see that the tail dependence properties of the failure regions based on $f_1$ and $f_4$ are better summarized through the TPDM than those of the failure regions based on $f_2$ and $f_3$. This can be explained by the fact that the regions $C_{f_1}$ and $C_{f_4}$ ``cut'' the axes and hence concern more scenarios that the regions $C_{f_2}$ and $C_{f_3}$ that only concern scenarios involving simultaneous extremes.  In general, variability increases when using approximate decompositions rather than exact decompositions.

\begin{figure}[ht]
\centering
\includegraphics[width=0.49\textwidth]{expomeas1.pdf}
\includegraphics[width=0.49\textwidth]{expomeas2.pdf} \\
\includegraphics[width=0.49\textwidth]{expomeas3.pdf}
\includegraphics[width=0.49\textwidth]{expomeas4.pdf}
\caption{Boxplots of limiting exponent measures of exact and approximate decompositions of $\Sigma_1$, $\Sigma_2$, $\Sigma_3$  and $\Sigma_4$ for the four functions $f_1$, $f_2$, $f_3$, and $f_4$, and $\alpha = 4$. The numbers in parentheses represent the number of decompositions on which the boxplot is based.}
\label{fig:expmeasures}
\end{figure}

\subsection{Wind gusts in the Netherlands}
Extreme windstorms are one of the important natural hazards affecting Europe. Impacts are not primarily caused by wind speed itself but by the gusts, i.e., the maxima of the wind speed during a few seconds.
In February 2022, storm Eunice caused massive damages in Western, Central, and Northern Europe. In the Netherlands, the maximum wind gust measured during storm Eunice was 144 km/h at station Cabauw, which set the record for harshest inland wind to have ever been measured in the Netherlands.

We consider daily maximal speeds of wind gusts, measured in km/h, observed at $d = 35$ weather stations in the Netherlands during extended winter (October--March), from October 2001 up to and including March 2022.  The data set is freely available from the Royal Netherlands Meteorological Institute (KNMI), \url{https://climexp.knmi.nl/}.  Please note that up to 50 stations are available in the Netherlands, but 15 were removed because of a large amount of missing data. The remaining 35 stations have between $0$ and $15$ missing data points out of $n = 3827$; these do not occur around clusters of extreme wind gusts and hence they do not impact the following analysis.
During the extended winter season, the KNMI issues a code red alarm (for any Dutch province) when there is at least 90 \% certitude that wind gusts will exceed 120 km/h .
The $35$ weather stations are initially divided into coastal stations (17) and inland stations (18) based on the same criteria used in \citet{lenderink2009}, because most environmental variables are known to behave differently over land or close to the sea; see Figure \ref{fig:wind2} (left) for a map of the stations used in this study.

We first estimate the marginal tail indices $\alpha_j = 1/\gamma_j$ using the standard Hill estimator with automated threshold selection (the so-called ``automated eye-ball method'' in \citet{danielsson2016}, where we consider $k \geq 15$). However, the Hill estimator is very erratic because of the many ties present in the data (due to rounding). To overcome this issue, we calculate the smoothed Hill estimator \citep{resnick1997} before applying the automated eye-ball method to choose a threshold.  Figure \ref{fig:wind1} (right) shows the estimates, with upper and lower bootstrap confidence limits for a $95 \%$ confidence level, obtained through a block bootstrap using the \textsf{R} package \textsf{boot} \citep{boot}. The first 17 bars correspond to coastal stations, the following 18 bars to inland stations. We see no apparent differences between the tail heaviness of maximum wind gusts at the coast and inland. The horizontal line shows the smoothed Hill estimate $\widehat{\alpha} = 9.289$ (where $k$ is again chosen following the automated eye-ball method), based on pooling all data from the 35 weather stations.

The next step concerns the choice of $r_0$. Figure \ref{fig:wind1} (right) show estimates of $m$, the total mass of the angular measure, as a function of $r_0$. We continue with $r_0$ corresponding to the $95 \%$ quantile and hence $\widehat{m} = 0.175$, corresponding to using the 192 largest values.

\begin{figure}[ht]
\centering
\includegraphics[width=0.49\textwidth]{WINDalphas.pdf}
\includegraphics[width=0.49\textwidth]{WINDr0.pdf}
\caption{Left: estimates $\widehat{\alpha}_j$ with upper and lower $95 \%$ confidence limits; the horizontal line represents the estimate $\widehat{\alpha}$. Right: estimated mass $\widehat{m}$ of the angular measure as a function of $r_0$, a high quantile of $R_1,\ldots,R_n$; the horizontal line represents the estimate for $r_0 = 0.95$.}
\label{fig:wind1}
\end{figure}

Next, we estimate pairwise tail dependence coefficients non-parametrically to examine the difference in pairwise dependence structure between the coastal and the inland stations. Figure \ref{fig:wind2} shows both the pairwise tail dependence coefficients $\chi_{jk}$ (middle) and the tail pairwise dependence matrix entries $\sigma_{jk}$ (right), for $j,k =1,\ldots,d$. The plot suggests that there is no significant difference between pairs of inland or pairs of coastal stations, and that the wind gusts are asymptotically dependent, even for large distances.

\begin{figure}[ht]
\centering
\includegraphics[width=0.33\textwidth]{WINDmap.pdf}
\hspace{-0.5cm}
\includegraphics[width=0.33\textwidth]{WINDchi.pdf}
\includegraphics[width=0.33\textwidth]{WINDtpdm.pdf}
\caption{Left: map of the Netherlands with 17 coastal stations in brown and 18 inland stations in green. Middle: empirical coefficients of tail dependence $\chi_{jk}$ for all pairs (black), pairs of coastal stations (blue), and pairs of inland stations (red), with loess curves. Right: same but for the TPDM entries $\sigma_{jk}$.}
\label{fig:wind2}
\end{figure}

Finally, we estimate $A$ as described in Section \ref{sec:estim2}; in total, we obtain $N = 200$ exact decompositions of $\widehat{\Sigma}_{\vc{X}}$. Based on $\widetilde{A}$ and on $\widehat{A}_1, \ldots, \widehat{A}_{200}$, we estimate the probability $p_{(\max)} = \mathbb{P}[\vc{X} \in C_{\max} (\vc{x})]$ for $\vc{x} = x \vc{1}_d$ and $x \in \{1, 1.2, 1.44\}$; the first two values are the thresholds for a code orange and a code red alarm respectively (issued by the KNMI during winter season), while the last value corresponds to the record wind gusts measured during storm Eunice.

Figure \ref{fig:wind3} shows boxplots of the estimates based on $\widehat{A}_1, \ldots, \widehat{A}_{200}$, together with estimates based on $\widetilde{A}$ and empirical estimates, for the three thresholds (from left to right), on the original scale and the transformed scale. Upper panels represent the analysis using coastal stations only, middle panels are for inland stations only, while lower panels use the full dataset ($d = 35$). While in the univariate and pairwise analysis, no apparent difference was present between coastal and inland stations, when estimating failure probabilities based on the $17$- and $18$-dimensional datasets, we observe that values for coastal stations are almost five times as high than for inland stations when performing the analysis on the original scale. Smaller differences are found when using the standardized data $\vc{X}_1^{\#}, \ldots,\vc{X}_n^{\#}$. The estimates based on the full $35-$dimensional dataset are closer to the coastal estimates than to the inland estimates. Finally, we see larger discrepancy with the empirical estimates for the 144 km/h threshold, which is due to the estimation uncertainty associated to such a high threshold.

\begin{figure}[ht]
\centering
\includegraphics[width=0.33\textwidth]{WINDmaxseaq1.pdf} \hspace{-0.2cm}
\includegraphics[width=0.33\textwidth]{WINDmaxseaq2.pdf}  \hspace{-0.2cm}
\includegraphics[width=0.33\textwidth]{WINDmaxseaq3.pdf}  \\
\includegraphics[width=0.33\textwidth]{WINDmaxlandq1.pdf} \hspace{-0.2cm}
\includegraphics[width=0.33\textwidth]{WINDmaxlandq2.pdf}  \hspace{-0.2cm}
\includegraphics[width=0.33\textwidth]{WINDmaxlandq3.pdf}  \\
\includegraphics[width=0.33\textwidth]{WINDmaxq1.pdf} \hspace{-0.2cm}
\includegraphics[width=0.33\textwidth]{WINDmaxq2.pdf}  \hspace{-0.2cm}
\includegraphics[width=0.33\textwidth]{WINDmaxq3.pdf}
\caption{Boxplots of estimates of failure probabilities for the region $C_{\textnormal{max}}$ for the 17 coastal stations (upper panels), the 18 inland stations (middle panels), and all 35 stations combined (lower panels).}
\label{fig:wind3}
\end{figure}

\subsection{Financial industry portfolios}\label{sec:fip}
We focus on the daily negative value-averaged returns of $d = 30$ industry portfolios,  downloaded from the Kenneth French Data Library.  Data is available for the period January 1970 until December 2019,  leading to a sample size $n = 12 \, 613$.

We start by estimating the marginal tail indices $\alpha_j = 1/ \gamma_j$ using the standard Hill estimator with automated threshold selection \citep{danielsson2016}. Figure \ref{fig:FIP} (left) show the estimates, with upper and lower bootstrap confidence limits for a $95 \%$ confidence level. Overlapping confidence intervals suggest that the hypothesis of a single index of regular variation $\alpha$ is not rejected. The marginal estimates are rather similar, and the straight line corresponding to the estimate $\widehat{\alpha}$, based on all data combined, is within all confidence intervals. Hence, in the following, we use $\widehat{\alpha}  = 3.479$.

\begin{figure}[ht]
\centering
\includegraphics[width=0.49\textwidth]{FIPalphas.pdf}
\includegraphics[width=0.49\textwidth]{FIPr0.pdf}
\caption{Left: estimates $\widehat{\alpha}_j$ with upper and lower $95 \%$ confidence limits; the horizontal line represents the estimate $\widehat{\alpha}$. Right: estimated mass $\widehat{m}$ of the angular measure as a function of $r_0$, a high quantile of $R_1,\ldots,R_n$; the horizontal line represents the estimate for $r_0 = 0.975$.}
\label{fig:FIP}
\end{figure}

The next step concerns the choice of $r_0$. Figure \ref{fig:FIP} (right) show estimates of $m$, the total mass of the angular measure, as a function of $r_0$. We continue with $r_0$ corresponding to the $97.5 \%$ quantile and hence $\widehat{m} = 29.99$, corresponding to using the 316 largest values.

Finally, we estimate $A$ as described in Section \ref{sec:estim2}; in total, we obtain $N = 200$ exact decompositions of $\widehat{\Sigma}_{\vc{X}}$. Based on $\widetilde{A}$ and on $\widehat{A}_1, \ldots, \widehat{A}_{200}$, we estimate the following quantities:
\begin{enumerate}
\item the probability $p_{\textnormal{sum}} = \mathbb{P}[\vc{X} \in C_{(\textnormal{sum})} (\vc{v}, x)] = \mathbb{P}[\vc{v}^T \vc{X} > x]$, where $x$ corresponds to the empirical $0.995$ quantile of $(\vc{v} ^T \vc{X}_i)_{i=1,\ldots,n}$; in other words, we validate the empirical $99.5\%$ value-at-risk (VaR) associated with a weighted sum of industry portfolio returns.
\item the probability $p_{\textnormal{minsum}} = \mathbb{P}[\vc{X} \in C_{f_1} (x)]$  associated with $C_{f_1} (x) = \{ \vc{y} \in \mathbb{E}_0: f_1(\vc{y}) > x \}$ and
\[
f_1(\vc{y}) = \min \left( \sum_{j=1}^{10} v_j y_j, \sum_{j=11}^{20} v_j y_j, \sum_{j=21}^{30} v_j y_j \right),
\]
where $x$ is the empirical $0.995$ quantile of $f_1(\vc{X}_1), \ldots, f_1(\vc{X}_n)$; i.e., we validate the empirical $99.5\%$ VaR associated with the minimum of three weighted sums of industry portfolio returns.
\item the probability $p_{\textnormal{maxsum}} = \mathbb{P}[\vc{X} \in C_{f_2} (x)]$
associated with $C_{f_2} (x) = \{ \vc{y} \in \mathbb{E}_0: f_2(\vc{y}) > x \}$ and
\[
f_2(\vc{y}) = \max \left( \sum_{j=1}^{10} v_j y_j, \sum_{j=11}^{20} v_j y_j, \sum_{j=21}^{30} v_j y_j \right),
\]
where $x$ is the empirical $0.995$ quantile of $f_2(\vc{X}_1), \ldots, f_2(\vc{X}_n)$.
\end{enumerate}
For all three failure regions, we compare equal weights, $\vc{v} = (1/d,\ldots,1/d)$, and unequal weights, $\vc{v} = (0.3, 1.7, 1, 0.3, 1.7, 1, \ldots, 0.3, 1.7, 1)$. All probabilities can be calculated analytically based on the estimates $\widetilde{A}$,  $\widehat{A}_1, \ldots, \widehat{A}_{200}$; see Section~\ref{sec:failure}.

Figure \ref{fig2} shows boxplots of the estimates based on $\widehat{A}_1, \ldots, \widehat{A}_{200}$ and $\widetilde{A}$. Some deviation is visible with respect to the empirical VaR, which could also be due to the empirical VaR suffering from estimation uncertainty. We note that variability is lowest for the sum-region, and highest for the minsum-region.

\begin{figure}[ht]
\centering
\includegraphics[width=0.33\textwidth]{res1sum.pdf} \hspace{-0.2cm}
\includegraphics[width=0.33\textwidth]{res1minsum.pdf}  \hspace{-0.2cm}
\includegraphics[width=0.33\textwidth]{res1maxsum.pdf}
\caption{Boxplots of estimates of failure probabilities for the regions $C_{(\textnormal{sum})}$, $C_{f_1}$, and $C_{f_2}$ for the financial industry portfolios.}
\label{fig2}
\end{figure}

\section{Discussion}
We proposed a general method for the estimation of failure probabilities based on pairwise measures of tail dependence. Our approach is fast and convenient for moderate to high dimensions. When $d \geq 40$, it becomes increasingly hard to find exact decompositions; hence, the (asymptotic) scales of the variables are overestimated, and the approach is appropriate for certain classes of failure regions only.

When summarizing the tail dependence structure through bivariate summary measures, it is natural that some information is lost. However, the ability to obtain repeated decompositions helps in quantifying such uncertainty.
Another source of error stems from the estimation of the tail index $\alpha$ and the tail pairwise dependence matrix $\Sigma$. Although a multitude of estimators exist for $\alpha$,  estimation of $\Sigma$ has yet to be investigated further and a better estimator could potentially improve our failure probability estimates.

\begin{appendix}
\section{Proofs}\label{appA}

\begin{proof}[Proof of Proposition \ref{prop:semidefinite}]
For a vector $\vc{X} \in \mathbb{R}^d$, let $m = H_{\vc{X}}  (\mathbb{S}_{d-1})$ denote the total mass of the angular measure. Then $\widetilde{H}_{\vc{X}} ( \, \cdot \,) := m^{-1} H_{\vc{X}} (\, \cdot \,)$ is a probability measure. Let $\vc{W}$ be a random vector that satisfies $\mathbb{P}[\vc{W} \in B] = \widetilde{H}_{\vc{X}} (B)$ for $B \subset \mathbb{S}_{d-1}$. Then $m  \, \mathbb{E} \left[ \vc{W}^{\alpha/2} \left( \vc{W}^{\alpha/2} \right)^T \right] = \Sigma_{\vc{X}}$. Finally, for $\vc{y} \in \mathbb{R}^d \setminus \{\vc{0} \}$,
\[
\vc{y}^T \Sigma_{\vc{X}} \vc{y} = m \, \mathbb{E} \left[  \vc{y}^T \vc{W}^{\alpha/2} \vc{W}^{\alpha/2} \vc{y} \right] = m \, \mathbb{E} \left[ \left( \vc{y}^T \vc{W}^{\alpha/2} \right)^2 \right] \geq 0.
\]
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem1}]
We have $\Sigma^{(i)} > 0$ if and only if
\begin{equation*}
\sigma_{jk} - \frac{\sigma_{ji} \sigma_{ki}}{\sigma_{ii} \max(D_i,1)} > 0 \iff \max(D_i,1) > \frac{\sigma_{ji} \sigma_{ki}}{\sigma_{ii} \sigma_{jk}}, \qquad \text{for all } j,k \in \{1,\ldots,d\} \setminus \{i\},
\end{equation*}
which is true by definition of $D_i$. Next, for $\vc{x}_{-i} = (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{d})^T \in \mathbb{R}^{d-1}$ we want to show that $\vc{x}^T_{-i} \Sigma^{(i)} \vc{x}_{-i} \geq 0$.
Let $\vc{y} = (y_1,\ldots,y_{d})^T$ with
\begin{equation*}
y_j = \begin{dcases}
x_j & \text{ if } j \in \{1,\ldots,d\} \setminus \{i\}; \\
 \frac{ -\vc{x}^T_{-i} \vc{\sigma}_{-i} }{\sigma_{ii} \max(D_i,1)} & \text{ if } j = i;
\end{dcases}
\end{equation*}
where $\vc{\sigma}_{-i} = (\sigma_{1,i},\ldots,\sigma_{i-1,i},\sigma_{i+1,i},\ldots,\sigma_{d,i})^T$, the $i$-th column of $\Sigma$ with the $i$-th element removed.
Because $\Sigma$ is positive semi-definite, we have $\vc{y}^T \Sigma \vc{y} \geq 0$. Observing that
\begin{align*}
0 \leq \vc{y}^T \Sigma \vc{y} & =\vc{x}^T_{-i} \Sigma_{(-i,-i)} \vc{x}_{-i} + 2 y_i \left( \vc{x}^T_{-i} \vc{\sigma}_{-i} \right) + y_i^2 \sigma_{ii} \\
& =\vc{x}^T_{-i} \Sigma_{(-i,-i)} \vc{x}_{-i} -  \frac{2 \left( \vc{x}^T_{-i} \vc{\sigma}_{-i} \right)^2}{\sigma_{ii} \max(D_i, 1)} +  \frac{\left( \vc{x}^T_{-i} \vc{\sigma}_{-i} \right)^2}{\sigma_{ii} \max(D_i, 1)^2}  \\
& = \vc{x}^T_{-i} \Sigma_{(-i,-i)} \vc{x}_{-i} + \left( \vc{x}^T_{-i} \tau_{-i} \right)^2 \left( \frac{1}{\max(D_i,1)} - 2 \right) \\
& \leq \vc{x}^T_{-i} \Sigma_{(-i,-i)} \vc{x}_{-i} - \left( \vc{x}^T_{-i} \tau_{-i
} \right)^2   = \vc{x}^T_{-i} \Sigma^{(i)} \vc{x}_{-i},
\end{align*}
where the third equality is obtained by observing that $\tfrac{\left( \vc{x}_{-i}^T \vc{\sigma}_{-i} \right)^2}{\sigma_{ii} \max(D_i, 1)} = \left( \vc{x}_{-i}^T \vc{\tau}_{-i} \right)^2$. Hence, we have shown that $\Sigma^{(i)}$ is positive semi-definite.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:peel}]
Suppose without loss of generality that $i = 1$. Since $\vc{Y}_{-1}$ is a max-linear random vector with TPDM $\Sigma^{(1)}$, for some $q \in \{2, 3, \ldots \}$, there exists a coefficient matrix $A_{(-1,-1)}  = (a_{jk})_{j = 2, \ldots,d, k = 2, \ldots,q} \in \mathbb{R}^{(d-1) \times (q-1)}$ and Fr\'echet$(1, \alpha)$ random variables $Z_2,\ldots,Z_q$ such that
\[
\vc{Y}_{-1} = A_{(-1,-1)} \times_{\max} (Z_2,\ldots,Z_q)^T, \qquad \text{and } \, \Sigma^{(1)} = A^{\alpha/2}_{(-1,-1)} \left( A^{\alpha/2}_{(-1,-1)} \right)^T
\]
Then
\[
\vc{X} = \begin{pmatrix}
\tau_{1,1}^{2/\alpha} Z_1 \\ \max(\tau_{2,1}^{2/\alpha} Z_1, Y_2) \\ \vdots \\ \max(\tau_{d,1}^{2/\alpha} Z_1, Y_d)  \end{pmatrix} = A \times_{\max} \vc{Z}, \qquad \text{ for } A :=
\left(\begin{array}{c@{}|@{}c}
\tau_{1,1}^{2/\alpha} &  \begin{matrix}
   & 0 & \cdots & 0
  \end{matrix} \\
  \hline
  \begin{matrix}
  \tau_{2,1}^{2/\alpha} \\ \vdots \\ \tau_{d,1}^{2/\alpha}
  \end{matrix}
  & \, \, \,  A_{(-1,-1)}
\end{array}\right),
\]
and $\vc{Z} := (Z_1,\ldots,Z_q)^T$. We have
\begin{align*}
\Sigma_{\vc{X}} = A^{\alpha/2} \left( A^{\alpha/2} \right)^T =
\left(\begin{array}{c@{}|@{}c}
\tau_{1,1}^{2} &  \begin{matrix}
   & \tau_{1,1} \tau_{2,1} & \cdots & \tau_{1,1} \tau_{d,1}
  \end{matrix} \\
    \hline
  \begin{matrix}
\tau_{1,1}  \tau_{2,1} \\ \vdots \\ \tau_{1,1 }\tau_{d,1}
  \end{matrix}
  & \, \,  \vc{\tau}_{-1} \vc{\tau}_{-1}^T + \Sigma^{(1)}
\end{array}\right) = \begin{pmatrix}
\sigma_{11} { \textstyle \max(D_1,1)} & \sigma_{12} & \ldots & \sigma_{1d} \\
\sigma_{12} & \sigma_{22} & \ldots & \sigma_{2d} \\
\vdots & \vdots & \ddots & \vdots \\
\sigma_{1d} & \sigma_{2d} & \ldots& \sigma_{dd} \end{pmatrix},
\end{align*}
by the definitions of $\Sigma^{(1)}$, $\vc{\tau}_1$ and $\vc{\tau}_{-1}$. Hence, $\Sigma = \Sigma_{\vc{X}}$ if $D_1 < 1$ and $\sigma_{11} < \sigma_{\vc{X}_{11}}$ if $D_1 > 1$.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:unique}]
  We prove the Proposition by mathematical induction with respect to $d$. First, we prove for $d=2$. Since $A$ is lower triangular, so is the corresponding matrix $A^{\alpha/2}$.

  For $d=2$, assume that
  \begin{align*}
    A^{\alpha/2}=\begin{pmatrix}
      a_{11} & 0 \\ a_{21} &a_{22}
        \end{pmatrix}.
  \end{align*}
  Then
  \begin{align*}
    \Sigma_{\vc{X}}=\begin{pmatrix}
      a^{2}_{11} & a_{11}a_{21} \\ a_{11}a_{21}  &a_{21}^{2}+a^{2}_{22}
        \end{pmatrix}.
  \end{align*}
  By taking the path $1 \mapsto 2$, we calculate
  \begin{align*}
    D_1(\Sigma_{\vc{X}})=\frac{(a_{11}a_{21})^2}{a^{2}_{11}(a_{21}^{2}+a^{2}_{22}) } = \frac{a_{21}^2}{a_{21}^{2}+a^{2}_{22}}.
  \end{align*}
  Since $D_1(\Sigma_{\vc{X}})<1$, we find $\vc{\tau}_1=(a_{11},a_{21})^T$. Correspondingly, we calculate that
  $$\Sigma_{\vc{X}}^{(-1)}=a_{21}^{2}+a^{2}_{22}-a_{21}^2=a_{22}^{2}.$$
  Therefore, the iterative algorithm leads to the matrix
  \begin{align*}
    A_*=\left(\vc{\tau}_1, \begin{matrix}0 \\ a_{22}\end{matrix}\right)=A^{\alpha/2}.
  \end{align*}
  The proposition is thus proven for $d=2$.

  Next, assume that the Proposition is valid for $d-1$ with $d\geq 3$, we show that it is also valid for $d$. Denote the lower triangular matrix $A^{\alpha/2}$ as
  \begin{align*}
    A^{\alpha/2}=\begin{pmatrix}
      a_{11} & 0 &\cdots &0 \\
      a_{21} &a_{22} & \cdots &0\\
      \vdots & \vdots & \ddots & \vdots \\
      a_{n1} &a_{n2} &\cdots &a_{nn}
        \end{pmatrix}=:\begin{pmatrix}
          a_{11} & \vc{0}^T \\
          \vc{a}_{-1} &\tilde{\vc{A}}^{(-1,-1)}
            \end{pmatrix},
  \end{align*}
  where $\vc{0}$ is a $(d-1)-$ dimensional vector with all elements zero, $\vc{a}_{-1}=(a_{21},\cdots, a_{n1})^T$ and $\tilde{{A}}^{(-1,-1)}$ is a $(d-1)\times(d-1)$ lower-triangular matrix. Then
  \begin{align*}
    \Sigma_{\vc{X}}=\begin{pmatrix}
      a^{2}_{11} & a_{1,1}\vc{a}_{-1}^T  \\ a_{11}\vc{a}_{-1}  &\vc{a}_{-1}\vc{a}_{-1}^T+\tilde{\vc{A}}^{(-1,-1)}(\tilde{A}^{(-1,-1)})^T
        \end{pmatrix}.
  \end{align*}
 Because $\tilde{A}^{(-1,-1)}$ contains only non-negative elements, we get $\sigma_{jk} > a_{j1}a_{k1}$  for any $j,k\neq 1$. In addition, $\sigma_{j1}=a_{11}a_{j1}$.  Since we consider the path $1 \mapsto 2\mapsto\cdots\mapsto d$, we first calculate $D_1$; for any $j,k\neq 1$,
  $$\frac{\sigma_{j1}\sigma_{k1}}{\sigma_{jk}\sigma_{11}} < \frac{a_{11}a_{j1}a_{11}a_{k1}}{a_{j1}a_{k1}a_{11}^2}=1,$$
  which implies that $D_1 < 1$ and thus $\vc{\tau}_1=(a_{11},a_{21},\cdots,a_{d1})^T$ following the algorithm specified in \eqref{eq:tau}. Correspondingly, from \eqref{eq:sigmai} we get
  \begin{align*}
  \Sigma_{\vc{X}}^{(1)}  =\Sigma_{\vc{X}}^{(-1,-1)}-\vc{\tau}_{-1}\vc{\tau}_{-1}^T& =\vc{a}_{-1}\vc{a}_{-1}^T+\tilde{{A}}^{(-1,-1)}(\tilde{{A}}^{(-1,-1)})^T-\vc{a}_{-1}\vc{a}_{-1}^T \\
   & =\tilde{{A}}^{(-1,-1)}(\tilde{{A}}^{(-1,-1)})^T.
  \end{align*}
  In other words, the $(d-1)\times(d-1)$ matrix $\Sigma_{\vc{X}}^{(1)}$ is of the form $\tilde{{A}}^{(-1,-1)}(\tilde{{A}}^{(-1,-1)})^T$ where $\tilde{{A}}^{(-1,-1)}$ is a lower-triangular matrix with non-negative elements only. By invoking the assumption that the proposition holds for $d-1$, we complete the proof of the induction step and conclude that the proposition hold for any positive integer $d$.
  \end{proof}

\section{Extreme precipitation in Switzerland}\label{appB}
We focus on daily precipitation amounts in mm for June, July and August for the years 1962--2012 at $d = 44$ weather stations near Z\"urich, Switzerland, leading to a sample size $n = 4691$. This data was analyzed in \citet{thibaud2015} and \citet{cooley2019} and found to be stationary, independent in time, 	and asymptotically dependent in space. Estimates of marginal tail indices tend to vary so that a common index of regular variation $\alpha$ is not justified; we standardize the data to $\vc{X}_1^\#, \ldots, \vc{X}_n^\#$.
We take $r_0$ corresponding to the $95 \%$ quantile, yielding $235$ ``extreme'' observations. Finally, we search for estimates of $A$ corresponding to exact decompositions of $\widehat{\Sigma}_{\vc{X}}$; a very large number of exact decompositions exists (but with identical paths for the first 14 iterations); we keep $N = 2000$ of them. Based on $\widetilde{A}$ and $\widehat{A}_1,\ldots,\widehat{A}_N$,  we estimate
\begin{enumerate}
\item the probability  $p_{(\min)} = \mathbb{P}[\vc{X} \in C_{\min} (\vc{x})]$, where $\vc{x}$ is chosen as the 30 mm threshold for precipitation on the original scale;
\item the probability  $p_{(\max)} = \mathbb{P}[\vc{X} \in C_{\max} (\vc{x})]$, where $\vc{x}$ is chosen as the 60 and 80 mm threshold for precipitation on the original scale.
\end{enumerate}
The first probability has also been estimated in \citet{cooley2019}, who found a value of $4.8 \times 10^{-4}$. For this region, the $N$ decompositions give the same estimate of $\widehat{p}_{(\min)} = 3.3 \times 10^{-3}$ since only the first column of $\widehat{A}_1,\ldots,\widehat{A}_N$ matters. We can compare these to the empirical estimate of $4.3 \times 10^{-4}$ and the estimate based on $\tilde{A}$ which gives $1.7 \times 10^{-3}$.
Figure \ref{fig:rain} shows boxplots of the estimates based on $\widehat{A}_1, \ldots, \widehat{A}_{N}$, together with estimates based on $\widetilde{A}$ and empirical estimates for $p_{\max}$. We see that the estimates based on $\widetilde{A}$  underestimate the probability of a heavy rainfall event.

\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{rainmax1.pdf}
\includegraphics[width=0.4\textwidth]{rainmax2.pdf}
\caption{Boxplots of estimates of failure probabilities $p_{\max}$ for the the Swiss precipitation data, for a 60 mm threshold (left) and a 80 mm threshold (right) on the original scale.}
\label{fig:rain}
\end{figure}

\end{appendix}

\begin{acks}[Acknowledgments]
The authors would like to thank Emeric Thibaud for providing the Swiss rainfall data used in Appendix \ref{appB}.
\end{acks}

\begin{funding}
Anna Kiriliouk was supported in part by the Fonds de la Recherche Scientifique - FNRS (grant number J.0124.20).
\end{funding}

\renewcommand\refname{REFERENCES}
\bibliographystyle{imsart-nameyear}

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text:
Home>Physical Science>Astronomy
Who Discovered the Crab Nebula
By: Mike Mueller - 512349
Mike Mueller - 512349's image for:
"Who Discovered the Crab Nebula"
The crab nebula, also known as M1, is the gaseous remnant of a supernova that was observed and documented by the Chinese on July 4, 1054 AD and was reportedly visible in the full light of day for 23 days and remained visible at night to the unaided eye for 653 days, according to their records. The subsequent gaseous nebula was first observed by the British amateur astronomer John Bevis in 1731 and added to his sky atlas Uranographia Britannica. It was later observed, independently by the French astronomer Charles Messier in 1758 while searching for Halley's Comet. He named it M1, the first in his catalog of M-objects that he compiled for the use of comet observers in order that they not confuse these as yet undefined fuzzy objects with a comet. In Messier's times the science of astronomy was still primarily positional astronomy and the field of astrophysics had not yet taken off so the objects in Messier's Connoissance des Temps were invariably refered to as nebula and star clusters which future workers would come to understand as a wide variety of deep space objects.
The name "Crab Nebula" for M1 emerged after a drawing made in 1844 by Lord Rosse. The drawing reveals the filamentary structure of the nebula and Roose along with other early astronomers believed they were looking at a faint open cluster of stars that they could not resolve without bigger telescopes. Brighter open star clusters such as the Pleiades (or "Seven Sisters") served as their example of what the Crab Nebula should look like if only they had bigger telescopes.
By the late 19th century, as the field of spectroscopy emerged, it was realized that the crab nebula was a gaseous entity but M1 was not confirmed to be the actual gaseous remnants of the 1054 supernova until observations made by Walter Baade in 1942 on the 100 inch Hooker Telescope at the Mount Wilson Observatory when he back extrapolated the expansion of the gases to the year 1180 using doppler analysis. That year of origin was later refined to 1140 by later observations The relatively small 86 year disparity between the years 1054 and 1140 are widely interpreted to imply that the gas underwent an early rapid expansion and then slowed down. The proximity of the Crab Nebula in both time and space to the event recorded by the Chinese as the "guest star" are key pieces of evidence cited in supernova theory by astrophysicists and M1 remains an object of great interest in the scientific community.
Further observations of the Crab Nebula by radio astronomers at the Arecibo Observatory 300-meter radio telescope in Puerto Rico reveal a pulsating radio source, a variable star with a period of 33.085 milliseconds. Astrophysicists believe this object to be a neutron star, rotating over 30 times per second. It's diameter is only 30 kilometers and yet it is as bright as our own sun in the visible spectrum, meaning it is actually 100,000 times more energetic than our sun. This object has a density greater than an atomic nucleus and is interpreted to be the solid remnant of Super Nova 1054.
Main Source: http://www.maa.clell.de/Messier/E/m001.html
More about this author: Mike Mueller - 512349
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http://www.maa.clell.de/Messier/E/m001.html

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text:
Mystery solved: Kapurthala college student had died of drug overdose
The 21-year-old college student, whose body was recovered under mysterious circumstances from the Jalandhar-Kapurthala main road near Khojewal village on Wednesday morning, had reportedly died due to overdose of drugs.
punjab Updated: Sep 19, 2014 20:38 IST
Parampreet Singh Narula
Deceased Ranjit Rai, a resident of Kapurthala, was studying in Lord Krishna Polytechnic College in Kapurthala and was living with his grandparents. He left his home on Tuesday at around 2pm by telling his family that he was going to meet his friends in Jalandhar.
Police suspected the case to be a murder when they found traces of vomit on his face and his valuables, including mobile phone, helmet, bike and t-shirt, were also missing.
After tracing his mobile phone details and enquiring about Ranjit from his friends, police got to know that he had gone to his friend, Piyush Ranjan Singh's, paying guest accommodation near Ladowali Road in Jalandhar for a party after giving his exam. Ranjit had two exams on Tuesday but skipped the second one.
Accompanied by another friend Varun Bhardwaj, the boys allegedly took heroin at the guest house. After some time, Varun left the PG and Ranjit started vomiting due to overdose of drugs. Piyush immediately called Varun and took Ranjit to a nearby hospital for treatment, but he died on the way.
After that, they both planned to deliver Ranjit's body at his home and Piyush hired an auto on the pretext of an epilepsy attack. However, the auto driver suspected something was amiss and left Piyush half way.
On the other hand, Varun texted Ranjit's family from his mobile phone, saying he was on the way and later sent another message saying he had met with an accident and requested his family to take him back home.
Piyush dropped the body half way and came back to his PG. Later, he booked a room at Devi Talab Temple in Jalandhar to hide from the police.
Jagjit Singh Saroya, superintendent of police (detective), said they had nabbed both the accused from Jalandhar and a case will be registered. He said the accused had been identified as Piyush Ranjan Singh, a migrant from Bihar, and Varun Bhardwaj, a resident of Kapurthala. Case under Sections 304, 34, 201 of the Indian Penal Code has been registered at the Sadar police station.
Saroya said Varun was also facing a case registered against him under sections of the NDPS Act wherein he was arrested with six gm heroin.


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John Arthur Malcolm Aldridge (1905 – 1983)
THE MOORS 1946 John Arthur Malcolm Aldridge (1905 – 1983) © The Artist's Estate
John Aldridge was educated at Uppingham School and at Corpus Christi College, University of Oxford. From 1928 to 1933 he lived in London, making frequent visits to Holland, France, Italy, Germany and Spain. In 1933 he moved to Essex, settling in Great Barfield. At the time he was one of a group of artists including Eric Ravilious and Edward Bawden who found inspiration in the Essex countryside. They later collected together an exhibition of their own pictures which toured the villages of Essex. His first one-man show was at the Leicester Galleries, London in 1933, he later exhibited with the Seven and Five Society alongside artists such as Ben Nicholson, Ivon Hitchens, David Jones and John Piper. He was elected a Royal Academician in 1963. Although he never went to art school, Aldridge became a remarkable and valued part-time teacher at the Slade School of Fine Art. Besides painting in oils, Aldridge designed textiles and wallpapers, and illustrated books. His pictures are built up out of the commonplace ingredients that any observant person could have found in the villages and fields and back gardens of Essex. Aldridge proved again, what so many artists have proved before him, that subject matter is no more than a starting point for adventure.
Work of art made with paint on a surface. Often the surface, also called a support, is a tightly stretched piece of canvas, paper or a wooden panel. Painting involves a wide range of techniques and materials, along with the artist's intellectual concerns effecting the content of a work.
FROM DESTRUCTION TO ABSTRACTION BRITISH ART IN THE 1940S AND 1950S
Syria, Damascus, University Of Damascus
CONTEMPORARY ARTISTS' LITHOGRAPHS
Kenya, Nairobi, Sorsbie Gallery
India, Mumbai, Jehangir Nicholson Gallery
Cyprus, Nicosia, British Council Office - Nicosia
Cyprus, Not known
Pakistan, Lahore, Alhamra Art Gallery
Turkey, Turkey
Spain, Gijon, Salon De Exposiciones Del Real Instituto Jovellanos
Spain, Oviedo, Galeria De Exposiciones De La Obra Social Y Sultural
Spain, Madrid, Ateneo De Madrid
Spain, Salon De Exposiciones De La Sociedad De Amigos Del Arte
Portugal, Lisbon, Palacio Foz
Portugal, Porto, Escola Superior De Belas Artes Do Porto
Ceylon, Colombo, Ceylon Art Gallery
Switzerland, Fribourg, Musee d'art et d'histoire
Germany, Berlin, Kunst Templehof
Canada, Québec, Montreal Museum Of Fine Arts
Austria, Alpbach, Summer School
CONTEMPORARY BRITISH PAINTING
South Africa, Port Elizabeth, King George VI Art Gallery
Rhodesia, Ndola
Uganda, Kampala, British Council Office - Kampala
Kenya, Mombassa, Mombassa
Kenya, Nairobi, National Gallery And Museum
Blantyre-Limbe
Tanzania, Moshi


The Obusier de 520 modèle 1916 should become the biggest French railway gun in the First World War, but the first prototype exploded, the second one was not ready in time to be used.
In the course of the continuing war of positions on the western front and the increasingly fortified positions and bunkers of the German army, it became apparent that even the heavy railway guns of the French army which gradually arrived did not have sufficient penetrating power. Thus the French High Command demanded ever larger guns with ever more penetrating power from the economy.
The company Schneider began thereupon in the middle of 1916 with the development of the Obusier de 520 modèle 1916. This railway gun was to receive a caliber of 520 mm and represented thereby the largest gun built up to then. The gun should have fired grenades weighing between 1.370 and 1.654 kilograms to destroy even the strongest German bunkers.
Such a large railway gun, however, required a significantly modified construction of the wagons. Until now only two wagons were connected with a steel scaffold. Now a total of 4 4-axle wagons had to be taken and connected in order to carry and move the weight of 263 tons.
In order to be able to absorb the enormous recoil, the designers used several recoil systems. Thus, a combination of cradle recoil and sliding recoil was used in conjunction with anchoring the wagons to the tracks. After one shot, the gun's recoil system served to absorb most of the recoil, while at the same time the entire gun slid backwards about one meter over the tracks. To insert a new grenade into the breech, the barrel had to be lowered completely and a new grenade could be introduced via a trolley system mounted behind the gun. After loading, the gun had to be re-aligned.
Due to the many innovations and technical adaptations, the construction of the first prototype was delayed more and more until it was finally completed at the end of 1917.
It was not until July 1917 that testing and shooting exercises were carried out until an accident occurred on the test site in Quiberon and the gun was destroyed. A fired grenade exploded early when it was still in the barrel of the weapon.
Although the second prototype was completed during the war, the tests were completed after the German Empire had already capitulated. Subsequently, the railway gun was stored by the French army in one of its depots.
After the outbreak of the Second World War, the railway gun was part of the French mobilization plans. However, it was too late to make the railway gun operational again, so that the German Wehrmacht could capture it after the invasion of France in the factory of the Schneider company before it was even fully operational.
Under the designation 52 cm long howitzer (E) 871 (f) it was put into the service of the Wehrmacht and after the beginning of the war against the Soviet Union it was also brought to the Eastern Front. There it was used from 21 November 1941 for the bombardment of Leningrad after its encirclement. As already with the first prototype also on 5 January 1942 with the second gun a grenade exploded still while this was in the barrel and destroyed thereby likewise the entire weapon. The remains were found by the Soviet army in 1943, but they were no longer used for the destroyed railway gun.


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National College is located in Salem, Virginia and is a private college. National College is a four year college and offers Associate's Degrees, Bachelor's Degrees, Master's Degrees, Doctoral Degrees, and a number of different programs and courses.
National College is in a relatively urban area (in or near a city), which may be something you prefer if you like a city lifestyle as a student.
National College does not have a rolling admission policy, and you will want to make sure that you get your application in before September 4.
National College is relatively smaller in size with an enrollment of only 735 students.
National College accepts about 98% of its applicants on average, and 80% of the students receive some sort of financial aid for college at National College.
If you are looking for more information on financial aid at National College, you can may want to contact Pamela Cotton, who is the Director of Financial Aid and Compliance Officer at National College. You may also qualify for scholarships for college in Virginia to attend National College or grants for college in Virginia to attend National College.
National College offers military credit for military courses that have been completed successfully by students.
If you have taken some advanced placement courses with an applicable test, or obtained credit from an other college, you may be eligible to transfer that credit to National College.
Do a lot of students come from out of state to attend National College? Well, about 7% of the student body at National College comes from outside the state of Virginia.
QUICK FACTS ABOUT NATIONAL COLLEGE
National College Address:
Salem, Virginia 24153
Web Site: http://www.ncbt.edu
National College admission closing date:
Does National College offer Associate's degrees?
Does National College offer Bachelor's degrees?
Does National College offer Master's degrees?
Does National College offer Doctoral degrees?
National College tuition cost (estimate):
National College room & board cost (estimate):
Is National College a private college?
Is National College a coed college?
National College school calendar:
Is National College a 2 year or 4 year college?
National College enrollment:
Percentage of applicants accepted to National College
Percentage of students at National College receiving financial aid:


text:
package gov.nasa.arc.spife.ui.timeline;

import gov.nasa.arc.spife.timeline.TimelineFactory;
import gov.nasa.arc.spife.timeline.model.Section;
import gov.nasa.arc.spife.timeline.model.TimelineBuilderContent;

import org.eclipse.emf.common.notify.impl.AdapterImpl;
import org.eclipse.jface.resource.ImageDescriptor;

public abstract class TimelineBuilder<T> extends AdapterImpl {

	public static final ImageDescriptor IMAGE_TIMELINE_DESCRIPTOR = ImageDescriptor.createFromURL(Activator.getDefault().getBundle().getEntry("icons/timeline_part.gif"));
	public static final ImageDescriptor IMAGE_TIMELINE_DESCRIPTOR_NEW = ImageDescriptor.createFromURL(Activator.getDefault().getBundle().getEntry("icons/new_timeline_part.gif"));

	public abstract String getName();

	public ImageDescriptor getActionImageDescriptor() {
		return IMAGE_TIMELINE_DESCRIPTOR_NEW;
	}

	public Section createTimelineSection() {
		TimelineBuilderContent section = TimelineFactory.eINSTANCE.createTimelineBuilderContent();
		section.setIdentifier(getName());
		return section;
	}

	public abstract TimelineViewer build(Timeline timeline, T model);

}

redpajama_set_name: RedPajamaGithub


Clean pool and grotto area open 24/7 for your enjoyment!
Large amounts of natural light comes through the beautiful bay windows in our town homes two bedroom apartments.
These floor plans allow for a large master bedroom including large walk in closets, attached full bath, ceiling fans, and large windows.
Our large living rooms include a wood burning fireplace, sliding patio doors to balconies, and some floor plans have vaulted ceilings.
Beautifully appointed courtyard adjacent to our pool area. Its a hummingbirds haven!
All appliances are either new or 1 year old, tiled flooring, pantry, and large cooking area.


text:
You are currently using the site but have requested a page in the site. Would you like to change to the site?
Travel: General
Travel: Europe / Great Britain & Ireland
Scotland For Dummies, 6th Edition
Chapter (PDF) Index (PDF) Table of Contents (PDF)
Selected type: Paperback
Barry Shelby
ISBN: 978-0-470-88870-4 April 2011 456 Pages
Starting at just $22.99
Enjoy sightseeing and shopping in bustling Edinburgh and Glasgow or explore unspoiled scenery and welcoming towns in the Hebridean Islands, Southern Scotland, Tayside, and the Northeast. Go from the Highlands to the Lowlands. Hike, canoe, or just relax at Loch Lomand. This friendly guide gives you the scoop on:
Edinburgh Old Town, with its intriguing winding alleyways
Accommodations that range from sumptuous 17th century hotel furnished with Gothic antiques to a secluded seaside escape, and from a 17th century laird's house to a sleek, modern and minimalist hotel
Enjoying a pint of lager in a rustic pub where the barmen wear kilts and you don't tip or touring distinctive distilleries
Cathedrals, castles and historic sites like the Calanais Standing Stones (the "Scottish Stonehenge"), Edinburgh Castle that holds the historic Stone of Destiny and Scotland's crown jewels, Doune Castle, made famous by the film Monty Python and the Holy Grail, and Glasgow Cathedral
Storied golf courses such as Muirfield, Royal Troon, and St. Andrews in the country credited with developing the sport
Touring Sir Walter Scott's mansion, Abbotsford, with it's incredible library, relics, and mementos, or paying homage to poet Robert Burns at numerous sites
Shopping for everything from fine wool knits to Caithness glass paper weights to Edinburgh Crystal to tartans and kilts to Highland Stoneware
Like every For Dummies travel guide, Scotland For Dummies includes:
Down-to-earth trip-planning advice
What you shouldn't miss - and what you can skip
The best hotels and restaurants for every budget
Whether you're looking for fun nightlife or the legendary Loch Ness monster...whether you want to explore art galleries and museums or walk craggy seacoasts, this guide gives you the flavor of Scotland so enchantingly you can almost hear the bagpipes.
Barry Shelby was born 1960 in Berkeley, California, where he later attended the University of California. He received a master's degree in journalism in 1984 from Northwestern University in Illinois. From 1984 to June 1997, he was an editor at World Press Review magazine in Manhattan. He moved to Scotland in 1997,?where he has worked as a caretaker at small and privately owned castle on the Clyde Coast, as a 'temp' with the privatised national railway company as well as a food and drink writer and editor for newspapers and magazines, including the Guardian, Glasgow Herald, and The List. He is married to a Scot and lives in Glasgow's East End with her, two dogs, and a cat.
Request permission to reuse content from this site
Part I: Introducing Scotland.
Chapter 1: Discovering the Best of Scotland.
Chapter 2: Digging Deeper into Scotland.
Chapter 3: Deciding Where and When to Go.
Chapter 4: Following an Itinerary: Five Fine Options.
Part II: Planning Your Trip to Scotland.
Chapter 5: Managing Your Money.
Chapter 6: Getting to Scotland.
Chapter 7: Getting Around Scotland.
Chapter 8: Booking Your Accommodations.
Chapter 9: Catering to Special Travel Needs or Interests.
Chapter 10: Taking Care of the Remaining Details.
Part III: Edinburgh and Glasgow.
Chapter 11: Edinburgh.
Chapter 12: Glasgow.
Chapter 13: Going Beyond Edinburgh and Glasgow: Day Trips.
Part IV: The Major Regions.
Chapter 14: Southern Scotland.
Chapter 15: Ayrshire and Argyll.
Chapter 16: The Trossachs to Fife.
Chapter 17: Tayside and the Northeast.
Chapter 18: The Highlands.
Chapter 19: Hebridean Islands.
Chapter 20: Orkney and Shetland Islands.
Part V: The Part of Tens.
Chapter 21: Ten Outstanding Golf Courses.
Chapter 22: Ten Can't-Miss Castles and Historic Sites.
Chapter 23: Ten Distinctive Distilleries.
Chapter 24: Ten Stunning Natural Attractions.
Appendix: Quick Concierge.
Dummies Travel


Myriam Schottenstein, founder of the sheitel review website ShayTell, and her baby.
First there was "Tinder for bras." Now there's a Yelp for sheitels, the wigs worn by many married Orthodox Jewish women.
ShayTell may be the first-ever on-line customer review site for wigs. It's certainly the most cleverly named.
And if you've never considered the demand for such a site, consider this: When was the last time you bought, say, a computer without knowing a thing about its quality, components, and warranty?
"Within a few months I had split ends everywhere and the wig just feels dry and straw-like," she writes.


Entertainment Weekly today posted an extensive article of 7 things they've learned about the Netflix series of Marvel's Daredevil.
The article talks about the tone of the series in addition to talking about some things we already knew. You can read the whole thing here. But one thing, in particular, did stand out: Bullseye will not be in the first season of Daredevil, but that doesn't rule him out for the future.
As for Fisk, Jeph Loeb said that "this season is about both the rise of the hero and the rise of the villain," and DeKnight expanded on that notion.
Marvel's Daredevil premieres in 2015, with most rumors pointing to a May premiere.


Buddy the Elf Flies High!
Buddy has it all wrapped up!
Mini Police Success at the POP Awards!
Fly me to the North Pole!
He believes he can fly!
I'm an elf; get me out of here!
Now what has he been up to?
Now look what our Elf has done!
Santa Sends a Mischievous Elf!


The cycle appears in all our lives. Each of us strikes a different balance.
To whom do you listen? Where do you find stability? Where do you seek it? When has radical change surprised you? When have you yearned for it?
My own answers come easily. I listen to my family. That's where I find stability. In the rest of my life, however, I'm driven by change. Change where I live. Change how I work. Change what I learn.
On Thanksgiving, I felt the weight of change. My extended family seems too small. Many have died, few have been born. Our little nuclear family of four came to Canada looking for change, but we're all alone here. I'm so grateful for my spouse and co-parent, and for our two young adult children.
Of course I worry about them, sometimes. I ask myself: Are they interested in their studies and passing their classes? Do they show up at work to fulfill their responsibilities? Do they have any loyal friendships? Do they do some volunteer work in the community? Are they kind to animals? Do they stay in touch with one another?
I wish I had such clarity when evaluating my own life! Wait, I think I did just list criteria for evaluating my own life. And I did it by judging someone else!
With these questions, I enter Yom Kippur. And I pray for the ability to listen for answers.
Many thanks to Anglican Bishop Melissa Skelton for teaching about the three Benedictine principles at the Vancouver School of Theology, and to Rabbi Howard Seigel for asking me about parenting at Congregation Beth Tikvah in Richmond, BC.


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Home » Tickets » Off-Broadway » 'Twas the Night Before... by Cirque du Soleil Tickets
'Twas the Night Before... by Cirque du Soleil
Hulu Theater at Madison Square Garden
Tickets are on sale now for 'Twas the Night Before... at the Hulu Theater at Madison Square Garden.
Cirque du Soleil debuts an all-new, family-friendly, world premiere theatrical production and the first Cirque show themed to the Holiday season.
(Photos by Errisson Lawrence)
1hr 20mins (no intermission)
'Twas the Night Before... by Cirque du Soleil Image and Video Gallery
About 'Twas the Night Before... by Cirque du Soleil:
'Twas the Night Before… is an exhilarating new spin on the beloved Christmas classic as only Cirque du Soleil can imagine. This vibrant acrobatic spectacle about the joy of sharing and friendship promises to spark lasting memories in the hearts of children and adults alike. Cirque du Soleil's first-ever Christmas show is a flurry of love, Christmas cheer, rip-roaring fun and hugely lovable characters that will become a new tradition for families to cherish. Join Isabella, a curious girl who's become jaded by the hoopla surrounding Christmas, as she is whisked away to an upside-down, inside-out world where she meets characters inspired by the original timeless poem. Discover what happened before Santa dropped in!
'Twas the Night Before... by Cirque du Soleil CAST AND CREATIVE
James Hadley
Cirque du Soleil (Marie-Josee Adam, Yasmine Khalil & Jayna Neagle)
Nicolas Brion
Ben Scheff
Genevieve Lizotte
Choreography:
Vinh Nguyen Kinjaz
Tuedon Ariri, Katharine Arnold, Rosie Axon, Adam Jukes, Louis Chen, Peter Lin, Tim Wang, Alexandar Yu, Michele Clark, Nicole Faubert, Guillaume Paquin, Quentin Greco, Jacob Gregorie, Chauncey Kroner, Timothé Vincent, Jinge Wang, Evan Tomlinson Weintraub, José Ignacio Flores Lopez, Alexis Vigneault, Jenna Beltran, Lyndsay Marie Farias, Kalila Hermant, Samuel Moore, Paul Ross, Diana Schoenfield, Paul Ross and Bianca Vallar
Creative Direction by Manuel Bissonnette, Musical Composition by Jean-Phi Goncalves, and Acrobatic Choreography & Performance Design by Edesia Moreno
'Twas the Night Before... by Cirque du Soleil Performance Dates & Times
Monday 2pm 6pm
Tuesday - -
Wednesday - -
Thursday - -
Friday - -
Saturday - -
Sunday - -
Performance schedules for all shows are subject to change.
4 Pennsylvania Plaza, New York, NY 10001
34th Street - Penn Station
Our 'Twas the Night Before... by Cirque du Soleil Review
Review of 'Twas the Night Before...by Cirque du Soleil at The Hulu Theater at Madison Square Garden
We all know it. At least parts of it. 'Twas the Night Before Christmas originally titled A Visit from St. Nicholas by Clement Clarke Moore. The poem first published in 1823 became an instant classic, an American tradition. It is the basis for all that we have come to think of as Santa - the reindeers, the chimney, dressed in red, all of it. It is read throughout the country in homes and... Read more


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MONTIEL, STRUB TO PLAY FOR TEAM AMERICA IN AUSTRALIA
TOM PERRY, Special to the Sun-SentinelSUN-SENTINEL
Dave Montiel and Brian Strub, both Cardinal Gibbons graduates say they may be experiencing a chance of a lifetime.
The twosome will travel to Australia, July 3-18, to play baseball for Team America.
"We are going over to help Australia's baseball program out," said Strub, who will be attending Brevard Community College in the fall on a full baseball scholarship. "This will be a great experience."
"I think it is a real honor to be selected," Montiel said. "But it is real good the way other nations like Russia and now Australia are trying to improve their baseball programs."
They were selected by Sports Exchange USA, the sponsor of Team America, to play for the 16-man team. But Montiel and Strub will have to raise $2,200 to make the trip.
"Brian and I have a raffle that we have put together and we plan to raise more money through sponsorships," Montiel said.
"We selected Dave and Brian to play for our team because they were noted as being some of the top players in the country," said David Eby of the Sports Exchange USA.


The Rockpanel Brilliant boards are produced from compressed natural basalt, a sustainable and readily available volcanic rock and bonded with an organic binder from which all Rockpanel products derive their unique properties. The products combine the advantages of stone and workability wood.
We advise you to order the required amount of Rockpanel Brilliant for a certain project as one batch to avoid any slight variation between batches.
Rockpanel boards are treated with a high quality water-based coating that keeps their appearance, colour and finish for years to come. The ProtectPlus layer gives extra protection against UV and graffiti.