[table]capposition=top

# Partial Tail Correlation for Extremes

Jeongjin Lee
Namur Institue for Complex Systems
University of Namur
Namur, 5000
<EMAIL_ADDRESS>
& Daniel Cooley
Department of Statistics
Colorado State University
Fort Collins, CO 80526
<EMAIL_ADDRESS>
Use footnote for providing further information about author (webpage,
alternative address)— _not_ for acknowledging funding agencies.

###### Abstract

We develop a method for investigating conditional extremal relationships
between variables at their extreme levels. We consider an inner product space
constructed from transformed-linear combinations of independent regularly
varying random variables. By developing the projection theorem for the inner
product space, we derive the concept of partial tail correlation via
projection theorem. We show that the partial tail correlation can be
understood as the inner product of the prediction errors associated with the
best transformed-linear prediction. Similar to Gaussian cases, we connect
partial tail correlation to the inverse of the inner product matrix and show
that a zero in this inverse implies a partial tail correlation of zero. We
develop a hypothesis test for the partial tail correlation of zero and
demonstrate the performance in a simulation study as well as in two
applications: high nitrogen dioxide levels in Washington DC and extreme river
discharges in the upper Danube basin.

_Keywords_ First keyword $\cdot$ Second keyword $\cdot$ More

##  1 Motivation

For Gaussian random vectors, the covariance matrix provides complete
information about dependence between variables. Even so, conditional
relationships, which are a key concept for understanding causal structures
between variables, are not directly apparent from the covariance matrix. In
Gaussian cases, conditional relationships can be completely specified since
conditional distributions are obtainable and remain Gaussian. Conditional
relationships are more readily apparent from the the precision matrix (the
inverse of the covariance matrix). The conditional relationship between
$X_{i}$ and $X_{j}$ given all other elements of a Gaussian random vector
(denoted by $\bm{X}_{\setminus{(i,j)}}$) is related to the $(i,j)$th element
of the precision matrix. Specifically, if the $(i,j)$th element of the
precision matrix is zero, that is $\Sigma_{i,j}^{-1}=0$, then $X_{i}$ and
$X_{j}$ are conditionally independent given $\bm{X}_{\setminus{(i,j)}}$.

When a distributional assumption is not made, one cannot fully characterize
conditional relationships. However, the notion of partial correlation provides
a measure of the strength of the conditional relationships between two
variables. Consider a centered $p$-dimensional random vector $\bm{X}_{p}$ with
covariance matrix $\Sigma$. Partitioning into two subvectors, let
$\bm{X}_{p}=({\bm{X}^{(1)}}^{T},{\bm{X}^{(2)}}^{T})^{T}$, where
$\bm{X}^{(1)}=(X_{i},X_{j})^{T}$ and
$\bm{X}^{(2)}=\bm{X}_{\setminus{(i,j)}}^{T}$, and partition the covariance
matrix accordingly

$\Sigma=\begin{bmatrix}\Sigma_{11}&\Sigma_{12}\\\
\Sigma_{21}&\Sigma_{22}\end{bmatrix}.$

The partial correlation can be connected to the idea of residuals. Consider
the matrix

$\displaystyle\Sigma_{1|2}$
$\displaystyle=E[(\bm{X}^{(1)}-\hat{\bm{X}})(\bm{X}^{(1)}-\hat{\bm{X}})^{T}],$

where $\hat{\bm{X}}=(\hat{X}_{i},\hat{X}_{j})^{T}$ is the vector of best
linear predictors in terms of mean squared prediction errors. The partial
correlation between $X_{i}$ and $X_{j}$ given $\bm{X}_{\setminus{(i,j)}}$ is
given by

$\rho_{ij}=\frac{[\Sigma_{1|2}]_{12}}{\sqrt{[\Sigma_{1|2}]_{11}[\Sigma_{1|2}]_{22}}}.$

Note that $\rho_{ij}=0$ if and only if $[\Sigma_{1|2}]_{12}=0$. By matrix
inversion, we can show that if the $(i,j)$th element in the precision matrix
is zero, then the partial correlation between $X_{i}$ and $X_{j}$ is also
zero.

To illustrate conditional relationships between variables, consider the simple
$4$-dimensional linear model

$\begin{bmatrix}X_{1}\\\ X_{2}\\\ X_{3}\\\
X_{4}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\\ \phi&1&0&0\\\
\phi^{2}&\phi&1&0\\\
\phi^{3}&\phi^{2}&\phi&1\end{bmatrix}\begin{bmatrix}Z_{1}\\\ Z_{2}\\\ Z_{3}\\\
Z_{4}\end{bmatrix},$

where $|\phi|<1$ and $Z_{i}$ are uncorrelated noise terms with mean $0$ and
variance 1. Another way of thinking of this model is through the equation

$X_{i}=\phi X_{i-1}+Z_{i},$

which if $X_{0}=0\;a.s.$, can generate $X_{i}$ sequentially for
$i=1,\ldots,4$. The precision matrix of $\bm{X}_{4}=(X_{1},...,X_{4})^{T}$ is

$\displaystyle Q:=\Sigma_{\bm{X}_{4}}^{-1}=\begin{bmatrix}1&-\phi&0&0\\\
-\phi&1+\phi^{2}&-\phi&0\\\ 0&-\phi&1+\phi^{2}&-\phi\\\
0&0&-\phi&1\end{bmatrix}.$

The sparsity seen in the precision matrix can lead to model simplification. In
the Gaussian setting, precision matrices have been linked to Gaussian Markov
random fields, which in turn can be linked to graphical representations for
models (Rue and Held, 2005). Following the convention in (Rue and Held, 2005)
of connecting graph nodes for non-zero entries of the precision matrix yields
the graph in Figure 1 for the illustrative model. Since we have not specified
the $X_{i}$’s to be Gaussian, the graph does not imply truly Markov behavior.
However, in terms of linear prediction, the predicted value of $X_{i}$ given
only its neighbors $X_{i-1}$ and $X_{i+1}$ is the same as if predicted on
$\bm{X}_{4}\setminus X_{i}$.

Figure 1: The graph given by the precision matrix of the illustrative model.

As it is based on covariance, partial correlation describes conditional
relationships at the center of the distribution and is not well-suited for
describing relationships in the tail. In the past few years, there has been a
concerted effort to develop simplified models for high dimensional extremes
based on graphical models and conditional relationships at extreme levels.
Gissibl and Klüppelberg (2018) develop causal structure for max linear models
via directed acyclic graphs. Directed graphs differ from Figure 1 in that the
graph edges have direction. Via max linear operations, Gissibl and Klüppelberg
(2018) connect directed acyclic graphs to max stable models, and model
sparsity is achieved from the graph structure simplifying high-dimensional
models. In other work, Engelke and Hitz (2020) develop the notion of
conditional independence for a multivariate Pareto distribution. In
particular, Engelke and Hitz (2020) focus on the Hüsler and Reiss (1989) model
which is characterized by a variogram. The graphical structure of the Hüsler
and Reiss (1989) can be described by a sparse pattern from precision matrices.
Engelke and Hitz (2020) use AIC to perform likelihood-based model selection,
and use a greedy algorithm to stepwise search of graphical models.

In this work, we develop a novel method for characterizing and investigating
extremal conditional relationships between pairs of variables. We rely on
multivariate regular variation on the positive orthant to describe extremal
dependence in the upper tail, which is assumed to be the direction of
interest. We develop the projection theorem for the inner product space
defined in Lee and Cooley (2021), and we consider subspaces spanned by a
collection of $p$ variables. Via the projection theorem, we develop the idea
of partial tail correlation. We show that partial tail correlation can be
understood as the inner product of the prediction errors associated with the
transformed linear prediction. Similar to the Gaussian case, we connect
partial tail correlation to the inverse of the inner product matrix, and show
that a zero in this inverse implies a partial tail correlation of zero. Our
approach differs from Gissibl and Klüppelberg (2018) in that our approach is
more closely linked to ideas from linear models in non-extreme statistics. Our
approach is less model-based than that of Engelke and Hitz (2020) in that we
do not specify the full model, but instead only work from summaries of
pairwise dependence.

In terms of inference, we connect the matrix of inner products to the tail
pairwise dependence matrix (TPDM) in Cooley and Thibaud (2019). We define the
observed residuals, which when considered in pairs are regularly varying in
$\mathbb{R}^{2}$ rather than on the positive orthant. Finally, we develop a
test for the hypothesis that the partial tail correlation is zero. We
demonstrate the performance of this test via a simulation study as well as in
two applications: high nitrogen dioxide levels in Washington D.C. to explore
conditional extremal relationships between stations ,and assessing flood risk
in application to extreme river discharges in the upper Danube basin which was
studied in Engelke and Hitz (2020).

##  2 Background

###  2.1 Multivariate Regular Variation

Our framework assumes multivariate regular variation, which is closely related
to classical multivariate extreme value analysis (De Haan and Ferreira, 2007,
Appendix B). Let $\bm{X}$ be a $p$-dimensional regularly varying random vector
in $\mathbb{R}_{+}^{p}=[0,\infty]^{p}$ (denoted by $RV_{+}^{p}(\alpha)$). A
formal definition is that $\bm{X}\in RV_{+}^{p}(\alpha)$ is regularly varying
if there exists a normalizing function $b(s)\rightarrow\infty$ as
$s\rightarrow\infty$ and a non-degenerate limit measure $\nu_{\bm{X}}$ for
sets in $E:=[0,\infty)^{p}\setminus\\{\bm{0}\\}$ such that

$sPr(b(s)^{-1}\bm{X}\in\cdot)\xrightarrow{v}\nu_{\bm{X}}(\cdot)$ (1)

as $s\rightarrow\infty$, where $\xrightarrow{v}$ indicates vague convergence
in the space of non-negative Radon measures on
$[0,\infty]^{p}\setminus\\{\bm{0}\\}$ (Resnick, 2007). The normalizing
function is of the form $b(s)=U(s)s^{1/\alpha}$ where $U(s)$ is slowly
varying, and the tail index $\alpha$ determines the power law of the tail.
Applying the same normalization $b(s)$ for all components of $\bm{X}$, we
assume $\bm{X}$ has common marginal distributions throughout.

Following notations in Resnick (2007), given any norm $\|\cdot\|$, let
$T:\bm{X}\mapsto(\|\bm{X}\|,\bm{X}/\|\bm{X}\|)=(R,\bm{W})$ be the polar
coordinate transformation. We can equivalently formulate the regular variation
by Resnick (2007),

$sPr((b(s)^{-1}R,\bm{W})\in\cdot)\xrightarrow{\nu}c\nu_{\alpha}\times
H_{\bm{X}},$ (2)

where $\nu_{\alpha}$ is the measure on $(0,\infty]$ and $H_{\bm{X}}$ is the
angular (or spectral measure) on the unit ball $\Theta_{p-1}^{+}=\\{\bm{w}\in
E:||\bm{w}||=1\\}$. The angular measure $H_{\bm{X}}$ fully characterizes tail
dependence in the limit; however, modeling $H_{\bm{X}}$ is challenging in high
dimensions.

The right hand side in (2) is a product measure, implying that the radial and
angular measure are independent of each other in the limit. For a set
$C(r,B)=\\{\bm{x}\in\mathbb{R}_{+}^{p}:||\bm{x}||>r,\bm{x}/||\bm{x}||\in B\\}$
defined with some high threshold $r$ and a Borel set
$B\subset\Theta_{p-1}^{+}$, the scaling property of $H_{\bm{X}}$ implies
$\nu_{\bm{X}}{(C(r,B))}=cr^{-\alpha}H_{\bm{X}}(B)$.

###  2.2 Tail Pairwise Dependence Matrix

To fully characterize the angular measure is challenging even in moderately
large dimension. Instead, we use the summary information of tail dependence in
a matrix of pairwise dependence measures, which is obtainable in high
dimensions. We choose a bivariate dependence measure which has similar
properties to covariance. Let $\bm{X}\in RV_{+}^{p}(2)$ have its angular
measure $H_{\bm{X}}.$ We consider the tail pairwise dependence matrix
$\Sigma_{\bm{X}}=\\{\sigma_{\bm{X}_{ij}}\\}_{i,j=1,\cdots,p}\in\mathbb{R}_{p\times
p}^{+}$ defined in Cooley and Thibaud (2019). The $(i,j)$th element of
$\Sigma_{\bm{X}}$ is
$\sigma_{\bm{X}_{ij}}=\int_{\Theta_{p-1}^{+}}{w_{i}w_{j}}dH_{\bm{X}}(w)$ on
$\Theta_{p-1}^{+}=\\{\bm{w}\in\mathbb{R}_{+}^{p-1}:||\bm{w}||_{2}=1\\}$ and is
essentially the extremal dependence measure of Larsson and Resnick (2012).

However, unlike Larsson and Resnick (2012), we set two main different features
in our framework. First of all, we require $\alpha=2$ and the $L_{2}$ norm to
make $\Sigma_{\bm{X}}$ have similar properties to a covariance matrix;
$\Sigma_{\bm{X}}$ is positive semi-definite (Cooley and Thibaud, 2019). In
addition, we do not require $H_{\bm{X}}$ to be a probability measure so that
the diagonal elements $\sigma_{\bm{X}_{ii}}$ imply the relative magnitudes of
the respective elements $X_{i}$ like a covariance matrix. The relation between
the scale and the magnitude of each element of $\bm{X}$ can be readily
observed by regular variation
$\lim_{s\rightarrow\infty}sPr(b(s)^{-1}X_{i}>c)=c^{-2}{\sigma_{\bm{X}}}_{ii}.$
By letting $x=cU(s)s^{1/2}$, the relation can be equivalently expressed as

$\lim_{x\rightarrow\infty}\frac{Pr(X_{i}>x)}{x^{-2}U(x)}={\sigma_{\bm{X}}}_{ii}.$
(3)

Therefore, the magnitude of $X_{i}$ tied to the diagonal element
$\sigma_{\bm{X}_{ii}}$ is based on tail probabilities. Furthermore, the sum of
diagonal elements is identical to the total mass of the angular measure since
$\sum_{i=1}^{p}\sigma_{\bm{X}_{ii}}=\int_{\Theta_{p-1}^{+}}dH_{\bm{X}}(\bm{w})$.
If the $(i,j)$th element of $\sigma_{ij}=0$ is zero, then it implies the
asymptotic independence of $(X_{i},X_{j})$ since
$H_{\bm{X}_{p}}(\\{\bm{w}\in\Theta_{p-1}^{+}:w_{i}>0,w_{j}>0\\})=0$. Unlike
covariance matrices, there is an additional property of the TPDM in that it is
completely positive, meaning there exists some $q_{*}<\infty$ and a
nonnegative $p\times q_{*}$ matrix $A_{*}$ such that
$\Sigma_{\bm{X}}=A_{*}A_{*}^{T}$. The value of $q^{*}$ is unknown, and $A_{*}$
is not unique.

###  2.3 Transformed Linear Operations

To establish a vector space in the positive orthant, Cooley and Thibaud (2019)
defined transformed linear operations. For
$\bm{x}\in\mathbb{R}_{+}^{p}=[0,\infty)^{p},$ a key idea is to consider a
monotone bijection function $t$ mapping from $\mathbb{R}$ to $\mathbb{R}_{+}$,
with $t^{-1}$ its inverse. The transform $t$ is applied to $\bm{x}$ or
$\bm{y}:=t^{-1}(\bm{x})$ componentwise. For $\bm{x}_{1}$ and
$\bm{x}_{2}\in\mathbb{R}_{+}^{p}=[0,\infty)^{p}$, consider the transformed
linear operations: vector addition
$\bm{x}_{1}\oplus\bm{x}_{2}=t\\{t^{-1}(\bm{x}_{1})+t^{-1}(\bm{x}_{2})\\}$, and
scalar multiplication $a\circ\bm{x}_{1}=t\\{at^{-1}(\bm{x}_{1})\\}$ for
$a\in\mathbb{R}$. We can easily show that $\mathbb{R}_{+}^{p}$ with these
transformed-linear operations is a vector space (Cooley and Thibaud, 2019).
Furthermore, for $x_{j}\in\mathbb{R}_{+}$ and $a_{j}\in\mathbb{R},$
$j=1,...,q$, a transformed-linear combination is defined as $a_{1}\circ
x_{1}\oplus\cdots\oplus a_{q}\circ
x_{q}=t\Big{\\{}\sum_{j=1}^{q}a_{j}t^{-1}(x_{j})\Big{\\}}.$ Let
$A=(\bm{a}_{1},...,\bm{a}_{q})$ be a $p\times q$ matrix of real numbers where
$\bm{a}_{j}\in\mathbb{R}^{p}$ is a $p$-dimensional column vector for
$j=1,\ldots,q$. Matrix multiplication is defined as
$A\circ\bm{x}=\bm{a}_{1}\circ x_{1}\oplus\cdots\oplus\bm{a}_{q}\circ
x_{q}=t\\{At^{-1}(x)\\}$ and note that $A\circ\bm{x}\in\mathbb{R}_{+}^{p}$.
For a matrix $B\in\mathbb{R}^{r\times p}$, $B\circ A\circ x=B\circ
t\\{At^{-1}(x)\\}=t\\{BAt^{-1}(x)\\}=BA\circ x$. It coincides with the
standard matrix multiplication.

More importantly, to preserve regular variation on the positive orthant,
Cooley and Thibaud (2019) consider the specific transform
$t:\mathbb{R}\rightarrow(0,\infty),$ $t(y)=\log\\{\exp(y)+1\\}$ and its
inverse $t^{-1}:(0,\infty)\rightarrow\mathbb{R},$ $t^{-1}=\log\\{\exp(x)-1\\}$
under some conditions. This transform $t$ is called the softplus function
widely used in neural networks. The important property that the transform must
meet is such that $\lim_{y\to\infty}t(y)/y=\lim_{x\to\infty}t^{-1}(x)/x=1$.
The condition implies that the transform negligibly affects large values.
Consider $\bm{X}_{i}=(X_{i1},\ldots,X_{i,p})^{T}\in RV_{+}^{p}(\alpha),$
another condition we require is a lower tail condition
$P(X_{i,j}<x)\rightarrow 0$ as $x\rightarrow 0$. This condition ensures that
$a\circ\bm{X}_{i}$ does not affect the upper tail for $a<0$. For the softplus
$t$, the lower tail condition is met since
$sPr\\{X_{i,j}\leq\exp(-kb(s))\\}\to 0$ as $s\rightarrow\infty$ for all
$k>0,j=1,\cdots,p.$ This lower tail condition ensures that none of marginals
have enough non-zero mass at $0$. The lower tail condition is met by standard
regularly varying distributions such as the Pareto and the Fréchet. Other
transforms $t:\mathbb{R}\rightarrow\mathbb{R}_{+}$ meeting both the
aforementioned limiting properties and the lower tail condition can be used to
preserve regular variation on the positive orthant.

More precisely, Cooley and Thibaud (2019) show the following.

######  Proposition 2.1.

Let
$sPr(b(s)^{-1}\bm{X}_{i}\in\cdot)\xrightarrow{\nu}\nu_{\bm{X}_{i}}(\cdot)$,
$i=1,2$ and $\bm{X}_{1},\bm{X}_{2}$ be independent, then

$sPr(b(s)^{-1}(\bm{X}_{1}\oplus\bm{X}_{2})\in\cdot)\xrightarrow{\nu}\nu_{\bm{X}_{1}}(\cdot)+\nu_{\bm{X}_{2}}(\cdot)$

######  Proposition 2.2.

Let $sPr(b(s)^{-1}\bm{X}\in\cdot)\xrightarrow{\nu}\nu_{\bm{X}}(\cdot)$, then
for $a\in\mathbb{R}$,

$\displaystyle
sPr[b(s)^{-1}(a\circ\bm{X})\in\cdot]\xrightarrow{\nu}\begin{cases}&a^{\alpha}\nu_{\bm{X}(\cdot)}\quad\text{if}\,a>0\\\
&0\quad\text{if}\,a\leq 0\end{cases}$

Furthermore, Cooley and Thibaud (2019) consider a simple and useful model
framework for $\bm{X}\in RV_{+}^{p}(\alpha)$ via transformed linear
combinations of independent regularly varying random variables. Under the
aforementioned propositions, we can construct a regularly varying random
vector $\bm{X}$ by applying an arbitrary matrix $A$ to a vector of independent
regularly varying random variables $\bm{Z}$. Let
$A=(\bm{a}_{1},\ldots,\bm{a}_{q})$ with $\max_{i=1,...p}a_{ij}>0$ for all
$j=1,...,q$, where $\bm{a}_{j}\in\mathbb{R}^{p}$ and hence
$A\in\mathbb{R}^{p\times q}$. Let

$\bm{X}=A\circ\bm{Z}=t(At^{-1}(\bm{Z})),$ (4)

where $\bm{Z}=(Z_{1},\ldots Z_{q})^{T}$ is a vector of independent and
identically distributed regularly varying random variables meeting
$sPr(b(s)^{-1}Z_{j}>z)\to z^{-\alpha}$ as $s\to\infty$ for $j=1,\ldots,q$ and
the lower tail condition. Then, $\bm{X}\in RV_{+}^{p}(\alpha)$, and when
normalized by $b(s)$, its angular measure is
$H_{\bm{X}}=\sum_{j=1}^{q}\|\bm{a}^{(0)}_{j}\|^{\alpha}\delta_{\bm{a}^{(0)}_{j}/\|\bm{a}^{(0)}_{j}\|}(\cdot),$
where $\delta$ is the Dirac mass function. The angular measure $H_{\bm{X}}$ is
consistent with point masses corresponding to the normalized columns of $A$.
The zero operation $a^{(0)}:=\max(a,0)$ is applied to vectors or matrices
componentwise throughout.

Cooley and Thibaud (2019) show that as $q\rightarrow\infty,$ the class of
angular measures constructed by this matrix multiplication is dense in the
class of possible angular measures, implying we are only required to consider
a nonnegative matrix $A$ to construct the dense class.

If $\bm{X}=A\circ\bm{Z}$ as in (4), the TPDM of the resulting vector is
$\Sigma_{\bm{A}\circ\bm{Z}}=A^{(0)}{A^{(0)}}^{T}.$ Further, if $\bm{X}\in
RV_{+}^{p}(2)$ has TPDM $\Sigma_{\bm{X}}$, the completely positive
decomposition implies that there exists a $0<q_{*}<\infty$ and a nonnegative
$p\times q_{*}$ matrix $A_{*}$ such that $\bm{X}_{*}:=A_{*}\circ\bm{Z}$ has
TPDM $\Sigma_{\bm{X}_{*}}=\Sigma_{\bm{X}}$.

##  3 Projection Theorem in Inner Product Space $\mathcal{V}^{q}$

###  3.1 Inner Product Space $\mathcal{V}^{q}$

We briefly review the vector space of regularly varying random variables
constructed from a transformed-linear combinations that was introduced by Lee
and Cooley (2021). For $\bm{a}\in\mathbb{R}^{q}$, the vector space is,

$\mathcal{V}^{q}=\big{\\{}X;X=\bm{a}^{T}\circ\bm{Z}=a_{1}\circ
Z_{1}\oplus\cdots\oplus a_{q}\circ Z_{q}\\},$ (5)

where $Z_{j}\in RV_{+}^{1}(2),$ $j=1,\ldots,q$, are independent regularly
varying random variables meeting lower tail condition
$sP(Z_{j}\leq\exp(-kb(s)))\to 0$ as $s\to\infty$ with a common normalization
$\lim_{z\to\infty}{\frac{P(Z_{j}>z)}{z^{-2}L(z)}=1}$. We note that
$\mathcal{V}^{q}$ is a stochastic vector space and a Hilbert space (Lee,
2022). For any $X_{1}$, $X_{2}$ in $\mathcal{V}^{q}$, the inner product of
$X_{1}=\bm{a}_{1}^{T}\circ\bm{Z}$ and $X_{2}=\bm{a}_{2}^{T}\circ\bm{Z}$ is
defined as

$\langle
X_{1},X_{2}\rangle:=\bm{a}_{1}^{T}\bm{a}_{2}=\sum_{i=1}^{q}a_{1i}a_{2i}.$

We define the angle between $X_{1}$ and $X_{2}$ to be

$\theta=\cos^{-1}[\langle X_{1},X_{2}\rangle/(||X_{1}||\,||X_{2}||)],$

where $\theta\in[0,\pi]$. We say $X_{1},X_{2}\in\mathcal{V}^{q}$ are
orthogonal if $\langle X_{1},X_{2}\rangle=0$. The norm of $X$ is defined as
$\|X\|_{\mathcal{V}^{q}}=\sqrt{\langle X,X\rangle}$. We use the subscript
$\mathcal{V}^{q}$ to remind that the norm is based on the coefficients which
determine the random variable and to distinguish from the usual Euclidean norm
based on a location in space. The norm induces the metric of $X_{1}$ and
$X_{2}$ as $d(X_{1},X_{2})=\|X_{1}\ominus
X_{2}\|_{\mathcal{V}^{q}}=\sqrt{\sum_{i=1}^{q}(a_{1i}-a_{2i})^{2}}.$

We recall the tail ratio of $X=\bm{a}^{T}\circ\bm{Z}\in\mathcal{V}^{q},$

$TR(X):=\lim_{z\rightarrow\infty}\frac{P(X>z)}{P(Z_{1}>z)}=\sum_{j=1}^{q}{(a_{j}^{(0)})}^{2},$

in Lee and Cooley (2021) where only the positive elements of $\bm{a}$
contribute. The square in the exponent arises because we assume $\alpha=2$.
Unlike the norm which is not estimable since the random variable’s
coefficients are not observable from data, the tail ratio is estimable.
However, the metric can be connected to the tail ratio

$TR\left(\max((X_{1}\ominus X_{2}),(X_{2}\ominus
X_{1}))\right)=\sum_{j=1}^{q}(a_{1j}-a_{2j})^{2}=d^{2}(X_{1},X_{2}),$

because $P(\max(Z_{1},Z_{2})>z)\sim P(Z_{1}>z)+P(Z_{2}>z)$ as
$z\rightarrow\infty$ for independent regularly varying random variables
$Z_{1}$ and $Z_{2}$ (cf. Jessen and Mikosch, 2006, Lemma 3.1). This equality
still holds for non-independent regularly varying random variables
$X_{1}\ominus X_{2}$ and $X_{2}\ominus X_{1}$ because of the max operation.
See proofs in Lee (2022). In general, it is required for $\alpha=2$ to connect
the inner products of $\mathcal{V}^{q}$ to quantities which are observable
from the tail behavior of the data. We will return to this discussion in
Section 5.

For a random vector whose elements are in $\mathcal{V}^{q}$:
$\bm{X}_{p}=(X_{1},\ldots,X_{p})^{T}$ where
$X_{i}=\bm{a}_{i}^{T}\circ\bm{Z}\in\mathcal{V}^{q}$ for $i=1,\ldots,p,$ it was
shown that $\bm{X}_{p}\in RV_{+}^{p}(2)$ and $\bm{X}_{p}$ is of the form
$A\circ\bm{Z}.$ We denote the matrix of inner products by

$\Gamma_{\bm{X}_{p}}=\langle X_{i},X_{j}\rangle_{i,j=1,\ldots p}=AA^{T}.$ (6)

The inner product matrix $\Gamma_{\bm{X}}$ for $X_{i}\in\mathcal{V}^{q}$ will
be connected to the TPDM $\Sigma_{\bm{X}}$ for $\bm{X}\in RV_{+}^{p}(2)$ in
section 5

###  3.2 Projection Theorem in $\mathcal{V}^{q}$

As any $X\in\mathcal{V}^{q}$ is uniquely identifiable by its vector of
coefficients $\bm{a},$ $\mathcal{V}^{q}$ is isomorphic to $\mathbb{R}^{q}$
with the same inner product. Therefore, $\mathcal{V}^{q}$ is complete and is
Hilbert space (Lee, 2022). Let
$X_{i}=\bm{a}_{i}^{T}\circ\bm{Z}\in\mathcal{V}^{q},$ $i=1,\ldots,p,$ where $p$
is assumed to be less than $q$. We consider the subspace $\mathcal{V}_{A}$
spanned by a finite set $\\{X_{1},\ldots,X_{p}\\}$, where $A$ refers to the
matrix which generates $\bm{X}_{p}=(X_{1},\ldots,X_{p})^{T}.$ Thus,

$\mathcal{V}_{A}=\\{\bm{b}^{T}\circ\bm{X}_{p};\bm{b}\in\mathbb{R}^{p}\\}.$

We develop the projection theorem in the vector space $\mathcal{V}^{q}$
constructed from transformed-linear combinations. For any
$X\in\mathcal{V}^{q}$, we define a transformed-projection mapping
$P_{\mathcal{V}_{A}}$ by

$P_{\mathcal{V}_{A}}X=\\{\bm{b}^{T}\circ\bm{X}_{p}\,\text{such
that}\,||X\ominus(\bm{b}^{T}\circ\bm{X}_{p})||_{\mathcal{V}^{q}}=\inf_{Y\in\mathcal{V_{A}}}||X\ominus
Y||_{\mathcal{V}^{q}}\\}.$

We say $P_{\mathcal{V}_{A}}$ is a transformed-linear projection mapping of
$\mathcal{V}^{q}$ onto $\mathcal{V}_{A}$. We define the orthogonal complement
of a subset $\mathcal{V}_{A}^{\perp}\subset\mathcal{V}^{q}$ as

$\mathcal{V}_{A}^{\perp}=\\{X\in\mathcal{V}^{q};\langle
X,Y\rangle=0\quad\forall\,Y\in\mathcal{V}_{A}\\};$

that is, $\mathcal{V}_{A}^{\perp}$ is the set of all elements of
$\mathcal{V}^{q}$ which are orthogonal to all elements of $\mathcal{V}_{A}$.

Lee and Cooley (2021) briefly mentioned the projection theorem as an
alternative method to find the optimal transformed linear predictor of an
unobserved $X_{p+1}$ given $\bm{X}_{p}$. Here, we present a more thorough
development of the projection theorem. The following development of the
projection theorem and its properties are similar to the the presentation in
Brockwell et al. (1991) and Cline (1983). Instead of considering transformed
linear operations of nonnegative regularly varying random variables as we do,
Cline (1983) considered standard linear combinations of symmetric regularly
varying random variables with any tail index. We defer all proofs to the
Appendix A and B.

######  Theorem 3.1.

(Projection theorem) Let $\mathcal{V}_{A}$ be the previously defined subspace
of the Hilbert space $\mathcal{V}^{q}$ and $X\in\mathcal{V}^{q}$. Let
$X_{i}=\sum_{j=1}^{q}a_{ij}\circ Z_{j}\in\mathcal{V}^{q},$ $i=1,\ldots,p,$ and
let $X=\sum_{j=1}^{q}a_{j}^{*}\circ Z_{j}\in\mathcal{V}^{q}$. Then
i) $\hat{X}:=P_{\mathcal{V}_{A}}X$ ($\hat{X}$ is the projection of $X$ onto
$\mathcal{V}_{A}$) has a unique element in $\mathcal{V}_{A}$ such that

$||X\ominus\hat{X}||_{\mathcal{V}^{q}}=\inf_{Y\in\mathcal{V}_{A}}||X\ominus
Y||_{\mathcal{V}^{q}},\mbox{ and }$

ii) $\hat{X}\in\mathcal{V}_{A}$ such that
$||X\ominus\hat{X}||_{\mathcal{V}^{q}}=\inf_{Y\in\mathcal{V}_{A}}||X\ominus
Y||_{\mathcal{V}^{q}}$ if and only if $\hat{X}\in\mathcal{V}_{A}$ and
$(X\ominus\hat{X})\in\mathcal{V}_{A}^{\perp}$.

Now, let $I$ be the identity mapping on $\mathcal{V}^{q}$. The proposition
below shows there is a unique mapping $P_{\mathcal{V}_{A}}$ of
$\mathcal{V}^{q}$ onto $\mathcal{V}_{A}$ such that $I-P_{\mathcal{V}_{A}}$
maps $\mathcal{V}^{q}$ onto $\mathcal{V}_{A}^{\perp}$ by Theorem 3.1.

######  Proposition 3.1.

(Property of Projection Mappings) Let $P_{\mathcal{V}_{A}}$ be the projection
mapping of $\mathcal{V}^{q}$ onto a subspace $\mathcal{V}_{A}$. Then,
i) $P_{\mathcal{V}_{A}}(\alpha\circ X\oplus\beta\circ Y)=\alpha\circ
P_{\mathcal{V}_{A}}X\oplus\beta\circ P_{\mathcal{V}_{A}}Y,\quad
X,Y\in\mathcal{V}^{q},\quad\alpha,\beta\in\mathbb{R}$.

[That is, the projection mapping $P_{\mathcal{V}_{A}}$ is a linear mapping.]
ii) For every $X\in\mathcal{V}^{q}$, there exist an element of
$\mathcal{V}_{A}$ and an element of $\mathcal{V}_{A}^{\perp}$ such that

$X=P_{\mathcal{V}_{A}}X\oplus(I-P_{\mathcal{V}_{A}})X$

and this decomposition is unique.

Theorem 3.1 shows that $\hat{X}\in\mathcal{V}_{A}$ is the unique element
closest to $X$ such that

$\langle X\ominus\hat{X},Y\rangle=0$ (7)

for all $Y\in\mathcal{V}_{A}$. The equation (7) is called the prediction
equation and makes $\hat{X}$ as the best predictor of $X\in\mathcal{V}^{q}$.
When we consider the problem of predicting an unobserved $X_{p+1}$ by using
the transformed-linear predictor of $(X_{1},\ldots,X_{p}),$ the goal is to
find $\hat{X}_{p+1}\in\mathcal{V}_{A}$ that minimizes $||\hat{X}_{p+1}\ominus
X_{p+1}||_{\mathcal{V}^{q}}$. The prediction equation is written as $\langle
X_{p+1}\ominus\hat{X}_{p+1},X_{i}\rangle=0$, for $i=1,\cdots,p$. This
condition can equivalently be expressed with the matrix notation by the
linearity of the inner product.

$\displaystyle\begin{bmatrix}<X_{p+1},X_{i}>\\\ \end{bmatrix}_{i=1}^{p}$
$\displaystyle=\begin{bmatrix}<X_{i},X_{j}>\end{bmatrix}_{i,j=1}^{p}\begin{bmatrix}b_{i}\end{bmatrix}_{i=1}^{p}=\begin{bmatrix}\sum_{k=1}^{q}a_{ik}a_{jk}\end{bmatrix}_{i,j=1}^{p}\begin{bmatrix}b_{i}\end{bmatrix}_{i=1}^{p}$
(8)

Solving this equation, this in turn yields the form of the best transformed
linear predictor of $\hat{X}=\bm{b}^{T}\circ\bm{X}_{p}$ in terms of minimizing
the tail metric.

###  3.3 Inner Product Matrix of Prediction Errors

Changing focus from the setting where $\bm{X}_{p}$ is observed and $X_{p+1}$
is unobserved, we continue to assume $\bm{X}_{p}=(X_{1},\ldots,X_{p})^{T}$
where $X_{i}\in\mathcal{V}^{q},$ for $i=1,\ldots,p,$, but assume we partition
the vector so that $\bm{X}_{p}=(\bm{X}^{(1)^{T}},\bm{X}^{(2)^{T}})^{T}$, where
$\bm{X}^{(1)}$ has dimension $p_{1}<p$ and $\bm{X}^{(2)}$ has dimension
$p-p_{1}$. Without loss of generality, $\bm{X}_{p}$ can be reordered so that
$\bm{X}^{(1)}$ is any subvector of elements of $\bm{X}_{p}$. Partitioning $A$
yields

$\begin{bmatrix}\bm{X}^{(1)}\\\
\bm{X}^{(2)}\end{bmatrix}=\begin{bmatrix}A^{(1)}\\\
A^{(2)}\end{bmatrix}\circ\bm{Z}_{q}.$

The matrix of inner products of $(\bm{X}^{(1)^{T}},\bm{X}^{(2)^{T}})^{T}$ is

$\Gamma_{(\bm{X}^{(1)^{T}},\bm{X}^{(2)^{T}})^{T}}=\begin{bmatrix}A^{(1)}A^{(1)^{T}}&A^{(1)}A^{(2)^{T}}\\\
A^{(2)}A^{(1)^{T}}&A^{(2)}A^{(2)^{T}}\end{bmatrix}:=\begin{bmatrix}\Gamma_{11}&\Gamma_{12}\\\
\Gamma_{21}&\Gamma_{22}\end{bmatrix}.$ (9)

We now consider the problem of finding $P_{\mathcal{V}_{A}}\bm{X}^{(1)}$ via
projection theorem. Minimizing $d(\bm{b}^{T}\circ\bm{X}^{(2)},\bm{X}^{(1)})$
is identical to minimization of
$\|\bm{b}^{T}A^{(2)}-A^{(1)}\|_{\mathcal{V}^{q}}^{2}$. Taking derivatives with
respect to $\bm{b}$ and setting equal to zero, the minimizer $\hat{\bm{b}}$
solves $(A^{(2)}A^{(2)^{T}})\hat{\bm{b}}=A^{(2)}A^{(1)^{T}}$. If
$(A^{(2)}A^{(2)^{T}})$ is invertible, then the solution $\hat{\bm{b}}$ is,

$\hat{\bm{b}}=(A^{(2)}A^{(2)^{T}})^{-1}A^{(2)}A^{(1)^{T}}=\Gamma_{22}^{-1}\Gamma_{21}.$
(10)

With the best linear predictor, we can then consider the vector of prediction
errors
$\bm{X}^{(1)}\ominus\hat{\bm{X}}=(A^{(1)}-\bm{b}^{T}A^{(2)})\circ\bm{Z}\in
RV_{p_{1}}^{+}(2)$, and whose elements are in $\mathcal{V}^{q}$. The inner
product of these prediction errors has a similar form to the conditional
covariance matrix under Gaussian assumptions.

$\begin{split}\|\bm{X}^{(1)}\ominus\hat{\bm{X}}\|_{\mathcal{V}^{q}}^{2}:&=\langle\bm{X}^{(1)}\ominus\hat{\bm{X}},\bm{X}^{(1)}\ominus\hat{\bm{X}}\rangle\\\
&=(A^{(1)}-\hat{\bm{b}}^{T}A^{(2)})(A^{(1)}-\hat{\bm{b}}^{T}A^{(2)})^{T}\\\
&=\Gamma_{11}-\Gamma_{12}\Gamma_{22}^{-1}\Gamma_{21}.\\\ \end{split}$ (11)

##  4 Partial Tail Correlation

###  4.1 Partial Tail Correlation via the Projection Theorem

We now turn attention to developing the notion of partial tail correlation
between pairs of elements of a vector $\bm{X}_{p}=(X_{1},\ldots,X_{p})^{T}$
where $X_{i}\in\mathcal{V}^{q}$ for $i=1,\ldots,p.$ Let
$\bm{X}^{(1)}=(X_{i},X_{j})^{T}$ and
$\bm{X}^{(2)}=(\bm{X}_{p}\setminus{(X_{i},X_{j})})^{T}$. From a geometric
perspective the projection theorem provides a way of defining the partial tail
correlation between $X_{i}$ and $X_{j}$ given $\bm{X}^{(2)}$ as the cosine of
the angle between the prediction errors.

Because we aim to project $\bm{X}^{(1)}$ onto the space spanned by
$\bm{X}^{(2)}$, we consider the subspace $\mathcal{V}_{A_{2}}$ spanned by a
finite set $\\{X_{1},\ldots,X_{p}\\}\setminus\\{X_{i},X_{j}\\}.$ Note that
$\mathcal{V}_{A_{2}}\subset\mathcal{V}_{A}$. We define
$P_{\mathcal{V}_{A_{2}}}$ as the projection mapping of $\mathcal{V}^{q}$ onto
$\mathcal{V}_{A_{2}}$. We denote by $P_{\mathcal{V}_{A_{2}}}\bm{X}^{(1)}$ the
projection of $\bm{X}^{(1)}$ onto the space $\mathcal{V}_{A_{2}}$. We call
$(\bm{X}^{(1)}\ominus P_{\mathcal{V}_{A_{2}}}\bm{X}^{(1)})$ prediction errors
obtained by projecting $\bm{X}^{(1)}$ onto the space $\mathcal{V}_{A_{2}}$.
The orthogonality condition says
$P_{\mathcal{V}_{A_{2}}}\bm{X}^{(1)}=\bm{b}^{T}\circ\bm{X}^{(2)}$ is such that
$\bm{X}^{(1)}\ominus P_{\mathcal{V}_{A_{2}}}\bm{X}^{(1)}$ is orthogonal to the
space $\mathcal{V}_{A_{2}}$. Proposition 3.1 says that $\bm{X}^{(1)}$ can be
uniquely expressed as the sum of $P_{\mathcal{V}_{A_{2}}}\bm{X}^{(1)}$ and
$(I-P_{\mathcal{V}_{A_{2}}})\bm{X}^{(1)}$.

######  Definition 4.1.

Let $X_{i}\in\mathcal{V}^{q}$ for $i=1,...,p$. Denote by $\mathcal{V}_{A_{2}}$
the space spanned by the set of variables
$\bm{X}^{(2)}=(\bm{X}_{p}\setminus\\{X_{i},X_{j}\\})^{T}$. Let $X_{i}\ominus
P_{\mathcal{V}_{A_{2}}}X_{i}$ and $X_{j}\ominus P_{\mathcal{V}_{A_{2}}}X_{j}$
be prediction errors obtained after projecting $X_{i}$ and $X_{j}$ onto the
space $\mathcal{V}_{A_{2}}$, respectively. Then, the partial tail correlation
between $X_{i}$ and $X_{j}$ is defined as

$\rho_{ij}^{E}=\frac{<X_{i}\ominus P_{\mathcal{V}_{A_{2}}}X_{i},X_{j}\ominus
P_{\mathcal{V}_{A_{2}}}X_{j}>}{\|X_{i}\ominus
P_{\mathcal{V}_{A_{2}}}X_{i}\|_{\mathcal{V}^{q}}\|X_{j}\ominus
P_{\mathcal{V}_{A_{2}}}X_{j}\|}_{\mathcal{V}^{q}},$ (12)

where the superscript $E$ in $\rho_{ij}^{E}$ stands for "extreme". $\langle
X_{i}\ominus P_{\mathcal{V}_{A_{2}}}X_{i},X_{j}\ominus
P_{\mathcal{V}_{A_{2}}}X_{j}\rangle=0$ iff $\rho_{ij}^{E}=0$, which we denote
by $X_{i}\perp X_{j}|\bm{X}^{(2)}$.

As before we denote the inner product matrix of prediction errors by

$\displaystyle\Gamma_{1|2}$ $\displaystyle:=<\bm{X}^{(1)}\ominus
P_{\mathcal{V}_{A_{2}}}\bm{X}^{(1)},\bm{X}^{(1)}\ominus
P_{\mathcal{V}_{A_{2}}}\bm{X}^{(1)}>$
$\displaystyle=\Gamma_{11}-\Gamma_{12}\Gamma_{22}^{-1}\Gamma_{21}.$

We define $\Gamma_{1|2}$ as the conditional inner product matrix (IPM). The
partial tail correlation can be represented by elements of the conditional
IPM,

$\rho_{i,j}^{E}=\frac{a_{ij}}{\sqrt{a_{ii}a_{jj}}},\quad i,j=1,2,$ (13)

where $\Gamma_{1|2}=[a_{ij}]_{i,j=1,2}$. Note that $\Gamma_{1|2}$ is positive
semi-definite but not completely positive.

###  4.2 Partial Tail Correlation and Transformed Linear Prediction

We return temporarily to the problem of predicting one variable
$X_{p+1}\in\mathcal{V}^{q}$ given $\bm{X}_{p}\in RV_{+}^{p}(2)$. In this
setting, the partial tail correlation corresponds to the coefficients of the
vector $\bm{b}$ for the best transformed-linear predictor. Importantly, if
$b_{i}$ denotes the $i$th element of $\bm{b}$, $b_{i}=0$ iff $X_{p+1}\perp
X_{i}|\bm{X}_{p}\setminus X_{1}.$ This implies that if $X_{p+1}$ and $X_{i}$
given $\bm{X}_{p}\setminus X_{i}$ have partial tail correlation of zero, then
$X_{i}$ adds no additional information to the transformed-linear prediction of
$X_{p+1}$. Without loss of generality, below we consider the specific case
where $i=1$.

######  Proposition 4.1.

Let $\mathcal{V}_{A}$ be the previously defined subspace of the Hilbert space
$\mathcal{V}^{q}$. Assume $X_{i}\in\mathcal{V}^{q}$, $i=1,...,p+1$. Then the
partial tail correlation between $X_{p+1}$ and $X_{1}$ is zero if and only if
the $i$th coefficient of $\bm{b}$ in the best transformed-linear predictor
$\hat{X}_{p+1}=\bm{b}^{T}\circ\bm{X}_{p}=b_{1}\circ X_{1}\oplus\cdots\oplus
b_{p}\circ X_{p}$ is zero.

###### Proof.

By projection theorem, the space $\mathcal{V}_{A}$ can be decomposed into two
orthogonal subspaces $\mathcal{V}_{A_{1}}$ spanned by $(X_{2},\cdots,X_{p})$
and $\mathcal{V}_{A_{1}^{\perp}}$ spanned by $(X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1})$, respectively. Thus, the projection of
$X_{p+1}$ onto the space $\mathcal{V}_{A}$ can also be split into two parts,

$\hat{X}_{p+1}=P_{\mathcal{V}_{A}}X_{p+1}=P_{\mathcal{V}_{A_{1}}}X_{p+1}\oplus
P_{\mathcal{V}_{A_{1}^{\perp}}}X_{p+1}=P_{\mathcal{V}_{A_{1}}}X_{p+1}\oplus
c\circ(X_{1}\ominus P_{\mathcal{V}_{A_{1}^{\perp}}}X_{1}),$ (14)

where $c=\frac{\langle X_{p+1},X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}\rangle}{||X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}||^{2}}=\frac{\langle X_{p+1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{p+1},X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}\rangle}{||X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}||^{2}}$ since
$P_{\mathcal{V}_{A_{1}}}X_{p+1}\perp X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}$. We show that $c$ is related to the partial tail
correlation between $X_{1}$ and $X_{p+1}$. To find the form of $c$, we note
that the projection of any variable in $\mathcal{V}^{q}$ onto the space
$\mathcal{V}_{A_{1}}$ is represented by the transformed-linear combination of
the remaining variables $\\{X_{2},\cdots,X_{p}\\}$. The projection of $X_{1}$
onto $\mathcal{V}_{A_{1}}$ is
$P_{\mathcal{V}_{A_{1}}}X_{1}=\bigoplus_{i=1}^{p-1}d_{i}\circ X_{i+1}$ and the
projection of $X_{p+1}$ onto $\mathcal{V}_{A_{1}}$ is
$P_{\mathcal{V}_{A_{1}}}X_{p+1}=\bigoplus_{i=1}^{p-1}e_{i}\circ X_{i+1}$.
Substituting these projections into (14), $\hat{X}_{p+1}=c\circ
X_{1}\oplus\big{(}\sum_{i=1}^{p-1}(d_{i}-ce_{i})\circ X_{i+1}\big{)}.$ By
matching the coefficient $c$ of $X_{1}$ in (14) with the $b_{1}$ of the best
transformed-linear predictor $\hat{X}_{p+1}=b_{1}\circ X_{1}\oplus\cdots
b_{p}\circ X_{p}$, the coefficient $c$ can be expressed in terms of the inner
product of residuals,

$c=b_{1}=\frac{\langle X_{p+1},X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}\rangle}{\|X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}\|_{\mathcal{V}^{q}}^{2}}=\frac{\langle
X_{p+1}\ominus P_{\mathcal{V}_{A_{1}}}X_{p+1},X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}\rangle}{\|X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}\|_{\mathcal{V}^{q}}^{2}}.$

Thus, if $b_{1}$ is zero, then the partial tail correlation between $X_{p+1}$
and $X_{1}$ is zero. ∎

Another way of understanding $P_{\mathcal{V}_{A_{1}^{\perp}}}X_{p+1}$ is
through the regression setting. We consider a simple linear regression with no
intercept $Y=X\beta+\epsilon$. The projection of $Y$ onto the space spanned by
$X$ is given by $\hat{Y}=P_{X}Y=X\hat{\beta}$ where
$\beta=(X^{T}X)^{-1}X^{T}y$. Note that $\hat{Y}$ can be expressed in terms of
the inner products, we have $\hat{Y}=\frac{\langle X,y\rangle}{\langle
X,X\rangle}X$. By replacing $Y$ and $X$ with $X_{p+1}$ and $X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}$ respectively,
$\hat{X}_{p+1}=P_{\mathcal{V}_{A_{1}}}X_{p+1}=\frac{\langle
X_{p+1},X_{1}\ominus P_{\mathcal{V}_{A_{1}}}X_{1}\rangle}{||X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1}||^{2}}(X_{1}\ominus
P_{\mathcal{V}_{A_{1}}}X_{1})$.

###  4.3 Relation between Partial Tail Correlation and the Inverse Inner
Product Matrix

In non-extreme analysis of dependence, the precision matrix (the inverse of
the covariance matrix) provides information about conditional relationships
between variables. In the non-extreme setting, the partial correlation between
$X_{i}$ and $X_{j}$ given all other elements of $\bm{X}_{\setminus{(i,j)}}$ is
related to the $(i,j)$th element of the precision matrix. Specifically,
$\Sigma_{ij}^{-1}=0\Leftrightarrow X_{i}\perp
X_{j}|\bm{X}_{\setminus\\{X_{i},X_{j}\\}}$. Analogously, we connect the idea
of partial tail correlation in (12) to the inverse of the inner product
matrix. The relation between the partial tail correlation and the inverse
inner product matrix can be shown by matrix inversion.

Let $\bm{X}_{p}\in RV_{p}^{+}(2)$ be a $p$-dimensional regularly varying
random vector where $X_{i}\in\mathcal{V}^{q},$ $i=1,\ldots,p$. As in Section
4.1, partition $\bm{X}_{p}$ into two subvectors
$\bm{X}^{(1)}:=(X_{i},X_{j})^{T}$ and
$\bm{X}^{(2)}:=\bm{X}_{\setminus{(i,j)}}^{T}$. Recall the block form of the
inner product matrix of $\bm{X}_{p}=(\bm{X}^{(1)^{T}},\bm{X}^{(2)^{T}})^{T}$

$\Gamma_{\bm{X}_{p}}:=\begin{bmatrix}\Gamma_{11}&\Gamma_{12}\\\
\Gamma_{21}&\Gamma_{22}\end{bmatrix},$ (15)

By the matrix inversion in block form,

$\displaystyle\Gamma_{\bm{X}_{p}}^{-1}$
$\displaystyle=\begin{bmatrix}\Gamma_{1|2}^{-1}&-\Gamma_{1|2}^{-1}\Gamma_{12}\Gamma_{22}^{-1}\\\
-\Gamma_{22}^{-1}\Gamma_{21}\Gamma_{1|2}^{-1}&\Gamma_{22}^{-1}+\Gamma_{22}^{-1}\Gamma_{21}\Gamma_{1|2}^{-1}\Gamma_{12}\Gamma_{22}^{-1},\end{bmatrix}$
(16)

where $\Gamma_{1|2}^{-1}\in\mathbb{R}^{2\times 2}$ is the inverse of the
conditional IPM. Note that the matrix $\Gamma_{1|2}^{-1}$ could have negative
off-diagonal elements. Since both of $\Gamma_{1|2}$ and $\Gamma_{1|2}^{-1}$
are a 2 by 2 matrix, the relation between $\Gamma_{1|2}$ and
$\Gamma_{1|2}^{-1}$ can be readily shown by the inversion formula,

$\Gamma_{1|2}^{-1}=\frac{1}{|\Gamma_{1|2}|}\begin{bmatrix}[\Gamma_{1|2}]_{22}&-[\Gamma_{1|2}]_{12}\\\
-[\Gamma_{1|2}]_{21}&[\Gamma_{1|2}]_{11}\end{bmatrix},$ (17)

where $[\Gamma_{1|2}]_{i,j=1,2}$ is the element of $\Gamma_{1|2}$ and a
determinant
$|\Gamma_{1|2}|=[\Gamma_{1|2}]_{11}[\Gamma_{1|2}]_{22}-[\Gamma_{1|2}]_{12}[\Gamma_{1|2}]_{21}$.
Thus

$\rho_{i,j}^{E}=\frac{[\Gamma_{1|2}]_{12}}{\sqrt{[\Gamma_{1|2}]_{11}[\Gamma_{1|2}]_{22}}}=\frac{-[\Gamma_{1|2}^{-1}]_{12}}{\sqrt{[\Gamma_{1|2}^{-1}]_{11}[\Gamma_{1|2}^{-1}]_{22}}}.$
(18)

Hence, the partial tail correlation between $X_{i}$ and $X_{j}$ given
$\bm{X}^{(2)}$ can be represented by the first block matrix of the inverse
IPM. Note that the direction of the partial tail correlation is of the
opposite sign of $[\Gamma_{1|2}^{-1}]_{12}$. If $[\Gamma_{1|2}^{-1}]_{12}=0$
it implies that $X_{i}$ and $X_{j}$ given $\bm{X}_{\setminus(i,j)}$ are
partially uncorrelated in terms of tail behavior.

We can also consider the case where we predict one variable
$X_{p+1}\in\mathcal{V}^{q}$ conditioned on $\bm{X}_{p}\in RV_{+}^{p}(2)$.
Similarly, let $\Gamma_{(X_{p+1},\bm{X}_{p}^{T})^{T}}$ be partitioned in block
form,

$\displaystyle\Gamma_{(X_{p+1},\bm{X}_{p}^{T})^{T}}$
$\displaystyle=\begin{bmatrix}\Gamma_{11}&\Gamma_{12}\\\
\Gamma_{21}&\Gamma_{22}\end{bmatrix},$ (19)

where $\Gamma_{11}\in\mathbb{R}$ is a scale of $X_{p+1}$ and
$\Gamma_{22}\in\mathbb{R}^{p\times p}$ is the IPM of $\bm{X}_{p}$. Suppose
$\Gamma_{22}$ is invertible, then by the inverse formula,

$\displaystyle\Gamma_{(X_{p+1},\bm{X}_{p}^{T})^{T}}^{-1}$
$\displaystyle=\begin{bmatrix}\frac{1}{k}&-\frac{1}{k}\Gamma_{12}\Gamma_{22}^{-1}\\\
-\frac{1}{k}\Gamma_{22}^{-1}\Gamma_{21}&(\Gamma_{22}-\Gamma_{21}\Gamma_{11}^{-1}\Gamma_{12})^{-1}\end{bmatrix}=\begin{bmatrix}\frac{1}{k}&-\frac{1}{k}\bm{b}^{T}\\\
-\frac{1}{k}\bm{b}&(\Gamma_{22}-\Gamma_{21}\Gamma_{11}^{-1}\Gamma_{12})^{-1}\end{bmatrix}$
(20)

where $k=\Gamma_{11}-\Gamma_{12}\Gamma_{22}^{-1}\Gamma_{21}\in\mathbb{R}$.
Thus, the inverse IPM can be expressed in terms of the vector
$\bm{b}=\Gamma_{22}^{-1}\Gamma_{21}$, and we see that if the element of
$\Gamma^{-1}$ relating $X_{p+1}$ to $X_{i}$, $\Gamma^{-1}_{1,i+1},$ equals
zero, then $b_{i}=0.$

For illustration, we now consider the transformed-linear model

$\begin{bmatrix}X_{1}\\\ X_{2}\\\ X_{3}\\\
X_{4}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\\ \phi&1&0&0\\\
\phi^{2}&\phi&1&0\\\
\phi^{3}&\phi^{2}&\phi&1\end{bmatrix}\circ\begin{bmatrix}Z_{1}\\\ Z_{2}\\\
Z_{3}\\\ Z_{4}\end{bmatrix}$ (21)

where $\\{Z_{i}\\}$ is a sequence of independent regularly varying $\alpha=2$
with unit scale. We set $\phi\in(0,1)$ to induce a positive dependence in the
$\\{X_{i}\\}.$ The sequential generating equation is

$X_{i}=\phi\circ X_{i-1}\oplus Z_{i},\mbox{ for }i=1,2,3,4,$

where $X_{0}=0\;a.s$.

By the matrix inversion in (20), the inverse IPM shows a sparse pattern.

$\displaystyle\Sigma_{(X_{p+1},\bm{X}_{p}^{T})^{T}}^{-1}$
$\displaystyle=\begin{bmatrix}1&-\phi&0&0\\\ -\phi&1+\phi^{2}&-\phi&0\\\
0&-\phi&1+\phi^{2}&-\phi\\\ 0&0&-\phi&1\end{bmatrix},$ (22)

The partial tail correlation between $X_{i}$ and $X_{i-k}$ is zero for
$|i|>1$. In terms of transformed linear prediction, consider
$\hat{X}_{4}=\bm{b}^{T}\circ\bm{X}_{3}$ where
$\bm{b}=\Gamma_{22}^{-1}\Gamma_{21}=(0,0,\phi)^{T}$. These optimized weights
imply that given $X_{3}$, knowledge of $X_{1}$ or $X_{2}$ does not provide
additional information about $X_{4}$ in terms of tail behaviors.

##  5 Positive Subset $\mathcal{V}_{+}^{q}$ as a Modeling Framework

In the vector space $\mathcal{V}^{q}$ in (5), we essentially require that the
elements of the coefficient vectors $\bm{a}$ take negative values for
$\mathcal{V}^{q}$ to be a vector space. However, we note that
$X=\bm{a}\circ\bm{Z}\in\mathcal{V}^{q}$ and $X^{+}=\bm{a}^{(0)}\circ\bm{Z}$
have the same tail ratio because negative values in $\bm{a}$ do not contribute
to tail behaviors. Thus, $\bm{X}_{p}=A\circ\bm{Z}_{q}$ and
$\bm{X}_{p}^{+}=A^{(0)}\circ\bm{Z}$ are indistinguishable in terms of their
tail behavior because they both have the same angular measure:
$H_{\bm{X}}=H_{\bm{X}_{p}^{+}}=\sum_{j=1}^{q}\|a^{(0)}_{j}\|^{\alpha}\delta_{a^{(0)}_{j}/\|a^{(0)}_{j}\|}(\cdot)$.

In terms of modeling and inference, it is reasonable to restrict attention to
the positive subset
$\mathcal{V}^{q}_{+}=\big{\\{}X;X=\bm{a}^{T}\circ\bm{Z}=a_{1}\circ
Z_{1}\oplus\cdots\oplus a_{q}\circ Z_{q}\\},$ where $a_{j}\in[0,\infty),$
$j=1,\ldots,q,$ and $\bm{Z}=(Z_{1},\ldots Z_{q})^{T}$ meeting a lower tail
condition and having a common unit scale. Critically, if
$\bm{X}_{p}=A\circ\bm{Z}$ where $A\geq 0$, then
$\Sigma_{\bm{X}}=\Gamma_{\bm{X}}=AA^{T}$. The assumption
$X_{i}\in\mathcal{V}^{q}_{+}$ for $i=1,\ldots,p$ is really essential for
modeling as the inner product matrix which forms the basis is not estimable as
the coefficients which determine $X_{1}$ are not observable; however, the TPDM
is estimable.

Perhaps most importantly, $q$ is not identifiable, nor does it need to be in
order to use the framework for modeling. In fact, we do not need to believe
that our data arise from a linear combination of $q$ regularly varying random
variables. Even if we do not believe that our data are constructed by this
linear combinations, our summary matrix or its inverse is still obtainable
from the data and provides a useful information about extreme dependence
between variables. The definition of the TPDM is not tied to
$\mathcal{V}^{q}$, these pairwise dependence summaries can be estimated for
any regularly varying random vector in $RV^{p}_{+}(2)$. If we are willing to
make the modeling assumption that
$X_{i}=\bm{a}_{i}^{T}\circ\bm{Z}_{q}\in\mathcal{V}^{q}_{+}$, for
$i=1,\ldots,p$, we then have all the tools that arise from this inner product
space. This is not such a strong assumption since angular measures arising
from $p\times q$ matrices $A$ are dense in the class of angular measures for
$p$-dimensional regularly varying random vectors as $q\rightarrow\infty$
(Cooley and Thibaud, 2019).

##  6 Hypothesis Testing for Zero Elements in the Inverse TPDM

###  6.1 Asymptotic Normality of TPDM Estimates

We aim to develop a hypothesis test for $H_{0}:\rho^{E}_{ij}=0$ versus
$H_{1}:\rho^{E}_{ij}\neq 0$. Towards that aim, we first review the asymptotic
normality of the sample TPDM $\hat{\Sigma}$ using results for the extremal
dependence measure by (Resnick (2004);Larsson and Resnick (2012)).

Let $\bm{X}_{p}\in RV_{+}^{p}(2)$ be a $p$-dimensional regularly varying
random vector such that
$nP(n^{-1/2}\bm{X}_{p}\in\cdot)\xrightarrow{\nu}\nu_{\bm{X}_{p}}(\cdot)$ and
have the angular measure $H_{\bm{X}}$. Unlike Larsson and Resnick (2012), we
do not require $H_{\bm{X}}$ to be a probability measure so that the scale of
the components of $\bm{X}_{p}$ is retained in the angular measure. We can find
the equivalent form of the extremal dependence measure given as
$\sigma_{ij}=\lim_{x\to\infty}m{E[W_{i}W_{j}|R>x]},$ where
$R=||(X_{i},X_{j})||,$ $x$ is a high threshold, and
$m=H_{\bm{X}_{p}}(\Theta_{p-1}^{+})$ is the total mass of the angular measure
by (Larsson and Resnick, 2012, Proposition4). This equivalent form provides a
natural estimator for $\sigma_{ij}$.

Let $\bm{x}_{\ell}=(x_{\ell,1},\ldots,x_{\ell,p})^{T},$ $\ell=1,\ldots,n$ be
realizations of iid copies of $\mathbb{R}_{+}^{p}$-valued regularly varying
vectors with the tail index $\alpha=2$. Letting $r_{\ell}=\|\bm{x}_{\ell}\|$
and $\bm{w}_{\ell}=r_{\ell}^{-1}\bm{x}_{\ell}$, the natural estimator for
$\sigma_{ij}$ is

$\hat{\sigma}_{ij}(n)=\frac{\hat{m}}{k}\sum_{\ell=1}^{n}w_{\ell,i}w_{\ell,j}\mathbb{I}[r_{\ell}>r_{(k)}],$
(23)

where $\hat{m}$ is an estimate of $H_{\bm{X}_{p}}(\Theta_{p-1}^{+})$,
$k:=k(n)$ is such that $\lim_{n\to\infty}{k/n}=0$ as $k\rightarrow\infty$, and
$r_{(k)}$ is the $k^{th}$ upper order statistic in the sample of size $n$. If
we preprocess the data to have a common unit scale, then $m$ is identical to
$p$. When the data are not preprocessed to have a common unit scale, an
estimator for $m$ is given as $\hat{m}=(r_{(k)}^{2}/n)k$ by Cooley and Thibaud
(2019).

Asymptotic normality is shown for the estimator of the extremal dependence
measure in the case of iid observations by Resnick (2004) and Larsson and
Resnick (2012). The asymptotic normality of $\hat{\sigma}_{ij}(n)$ is proven
under the following condition. Let $F$ be the distribution function of $R$ and
$\bar{F}$ be its tail probability.

$\lim_{n\to\infty}\sqrt{k}\Big{(}\frac{n}{k}mE[W_{i}W_{j}\mathbb{I}[R/b(n/k)\geq
t^{-\gamma}]]-E[mW_{i}W_{j}\frac{n}{k}\bar{F}(b(n/k)t^{-\gamma}]\Big{)}=0,$
(24)

holds locally uniformly for $t\in[0,\infty),$ and assume that
$\tau^{2}=Var(W_{i}W_{j})>0$. Larsson and Resnick (2012) notes that
$\tau^{2}=0$ implies asymptotic independence, meaning that the rate factor
$\sqrt{k}$ increases too slowly to obtain a non-degenerate limit. This
condition implies that the dependence between $(W_{i,l}W_{j,l})$ and $R_{l}$
must decay fast enough with $n$ as $R_{l}$ is conditioned to lie above
$b(n/k)$. The condition (24) does not require the second-order regular
variation condition and the use of the order statistic does not require to
know the normalization $b(\cdot)$. Under the condition (24), the estimator
$\hat{\sigma}_{ij}(n)$ is asymptotically normal by Larsson and Resnick (2012).

$\sqrt{k}(\hat{\sigma}_{ij}(n)-mE[W_{i}W_{j}])\sim N(0,\tau_{ij}^{2}),$

where $m$ is the total mass of $H_{\bm{X}}(\cdot)$ and
$\tau_{ij}^{2}=Var(\hat{\sigma}_{ij})$.

Following a construction method for $\bm{X}_{p}$ as in (4), that is,
$\bm{X}_{p}=A\circ\bm{Z}\in RV_{+}^{p}(2)$, we can specify an explicit form of
the variance $\tau_{ij}^{2}$ in terms of the angular measure $H_{\bm{X}}$
being consistent with point masses. For $X_{i}=\bm{a}_{i}^{T}\circ\bm{Z}$ and
$X_{j}=\bm{a}_{j}^{T}\circ\bm{Z}$ in $\mathcal{V}^{q}$, the $(i,j)$th element
of $\Sigma_{A\circ\bm{Z}}$ is given by
$\sigma_{ij}=\int_{\Theta_{p-1}^{+}}w_{i}w_{j}dH_{A\circ\bm{Z}}(\bm{w})=\sum_{l=1}^{q}a_{il}^{(0)}a_{jl}^{(0)},$
where
$H_{\bm{X}}(\cdot)=\sum_{l=1}^{q}||\bm{a}_{l}^{(0)}||^{2}\delta_{\bm{a}_{l}/||\bm{a}_{l}||}(\cdot).$
To find $\tau_{ij}^{2}=Var(\hat{\sigma}_{ij})$, we first consider

$Var(W_{i}W_{j})=\int_{\Theta_{p-1}^{+}}(w_{i}w_{j}-E[W_{i}W_{j}])^{2}dN_{\bm{X}}(\bm{w})=E[W_{i}^{2}W_{j}^{2}]-E[W_{i}W_{j}]^{2},$

where $N_{\bm{X}}(\cdot)=m^{-1}H_{\bm{X}}(\cdot)$ indicates a probability
measure, $E[W_{i}W_{j}]=\frac{1}{m}\sum_{l=1}^{q}a_{il}^{(0)}a_{jl}^{(0)}$,
and
$E[W_{i}^{2}W_{j}^{2}]=\frac{1}{m}\sum_{l=1}^{q}\frac{a_{il}^{(0)^{2}}}{||\bm{a}_{l}^{(0)}||}\frac{a_{jl}^{(0)^{2}}}{||\bm{a}_{l}^{(0)}||}.$
We thus have
$Var(\hat{\sigma}_{ij})=Var\Big{(}\frac{m}{k}\sum_{l=1}^{n}W_{il}W_{jl}\mathbb{I}[R_{l}>R_{(k)}]\Big{)}=\frac{m^{2}}{k}Var[W_{i,1}W_{j,1}]$
since $(W_{i,l},W_{j,l})$’s are i.i.d. We can obtain an estimate of
$\widehat{Var}(\hat{\sigma}_{ij})$ by estimating $Var(W_{i1}W_{j1}).$ To
obtain $\widehat{Var}(W_{i1}W_{j1})$, a natural estimate for $E[W_{i}W_{j}]$
and $E[W_{i}^{2}W_{j}^{2}]$ are
$\hat{E}[W_{i}W_{j}]=\frac{1}{k-1}\sum_{l=1}^{k}w_{i,l}w_{j,l}\mathbb{I}[R_{l}>R_{(k)}]$
and
$\hat{E}[W_{i}^{2}W_{j}^{2}]=\frac{1}{k-1}\sum_{l=1}^{k}w_{il}^{2}w_{jl}^{2}\mathbb{I}[R_{l}>R_{(k)}],$
respectively.

###  6.2 Residuals and Asymptotic Normality of the Conditional Inner Product
Matrix

The ultimate goal is to derive the asymptotic normality of the sample
conditional inner product matrix. We assume that we observe iid copies of
$\bm{X}_{p}$ whose elements are in $\mathcal{V}^{q}$, from which we obtain
$\hat{\Sigma}$, an estimate of the TPDM. A straightforward estimator of the
conditional inner product matrix is

$\hat{\Gamma}_{1|2}=[\hat{\Sigma}_{11}-\hat{\Sigma}_{12}\hat{\Sigma}_{22}^{-1}\hat{\Sigma}_{21}],$
(25)

where $\hat{\Sigma}_{ij}$ for $i,j=1,2$ are sample block matrices in (15).
However, the distribution of $\hat{\Gamma}_{1|2}$ is not straightforward to
obtain from (25).

As the partial tail correlation is tied to the inner product of prediction
errors, it is natural to consider using the observed ‘residuals’ to understand
the properties of the sample conditional inner product matrix. The prediction
errors are in $\mathcal{V}^{q}$ and
$\bm{X}^{(1)}\ominus\hat{\bm{X}}=(A^{(1)}-\bm{b}^{T}A^{(2)})\circ\bm{Z}_{q}$.
Thus

$\Gamma_{1|2}=(A^{(1)}-\bm{b}^{T}A^{(2)})(A^{(1)}-\bm{b}^{T}A^{(2)})^{T},$

and note that $\Gamma_{1|2}$ is not necessarily completely positive. Unlike
the original data where we can assume away the importance of any negative
coefficients as described in Section 5, here negative coefficients are
consequential. If we consider the TPDM of the prediction errors,

$\Sigma_{\bm{X}^{(1)}\ominus\hat{\bm{X}}}=(A^{(1)}-\bm{b}^{T}A^{(2)})^{(0)}(A^{(1)}-\bm{b}^{T}A^{(2)})^{(0)^{T}}\neq\Gamma_{1|2}.$

Furthermore, the order of the definition of the prediction errors matters as
the scale of $\bm{X}^{(1)}\ominus\hat{\bm{X}}$ is
$(A^{(1)}-\bm{b}^{T}A^{(2)})^{(0)}$, and this differs from the scale of
$\hat{\bm{X}}\ominus\bm{X}^{(1)}$ which is
$(\bm{b}^{T}A^{(2)}-A^{(1)})^{(0)}$.

As the conditional inner product matrix is not completely positive, the direct
use of transformed-residuals is not suitable for estimation. Instead, we
consider the preimages of the prediction errors in (4.1) to account for
negative coefficients. Assuming $Z_{j},$ $j=1,\ldots,q,$ is independent and
identically distributed regularly varying random variable with unit scale
meeting lower tail condition $nP(Z_{j}\leq\exp(-kn^{1/2}))\rightarrow 0$ for
any $k>0$, we define the preimages of the transformed-residuals by

$\bm{U}:=t^{-1}(\bm{X}^{(1)}\ominus\hat{\bm{X}})=(A^{(1)}-\bm{b}^{T}A^{(2)})t^{-1}(\bm{Z}_{q})$

which are not restricted to the positive orthant. Lemma A4 in the appendix in
Cooley and Thibaud (2019) implies that $\bm{U}\in RV_{2}(2).$ Let
$\bm{U}=(U_{1},U_{2})^{T}$ and continue to let $T$ denote the polar coordinate
transformation, $T(U_{1},U_{2})=(R,\bm{W}),$ where $R=||(U_{1},U_{2})||_{2}$
and $\bm{W}=(U_{1}/R,U_{2}/R).$ We can summarize the second-order behaviors of
$\bm{U}$ with respect to the angular measure $H_{\bm{U}}.$ We define

$\sigma_{\bm{U}_{ij}}=\int_{\Theta_{1}}w_{i}w_{j}dH_{\bm{U}}(\bm{w}),\quad
i,j=1,2,$

where $\Theta_{1}=\\{w\in\mathbb{R}:||\bm{w}||_{2}=1\\},$ and $H_{\bm{U}}$ is
the angular measure of $\bm{U}.$ Thus, the pairwise tail dependence matrix of
$\bm{U}$ is
$\Sigma_{\bm{U}}:=(A^{(1)}-\bm{b}^{T}A^{(2)})(A^{(1)}-\bm{b}^{T}A^{(2)})^{T}$
identical to $\Gamma_{1|2}$. In contrast to the fact that
$\sigma_{\bm{X}_{12}}=0$ implies asymptotic independence of $X_{1}$ and
$X_{2}$, $\sigma_{\bm{U}_{12}}=0$ does not necessarily mean that elements
$U_{1}$ and $U_{2}$ are asymptotically independent. Instead this implies

$\int_{\Theta_{1}:w_{1}w_{2}>0}w_{1}w_{2}dH_{\bm{U}}(\bm{w})=\int_{\Theta_{1}:w_{1}w_{2}<0}w_{1}w_{2}dH_{\bm{U}}(\bm{w}),$

meaning that angular components of $(w_{1},w_{2})$ are uncorrelated on the
$L_{2}$ unit ball and quadrants of $(w_{1},w_{2})$ plane are balanced.

The off-diagonal element $\sigma_{\bm{U}_{12}}$ in $\Gamma_{1|2}$ is of
primary interest because it is tied to the idea of the partial tail
correlation $\rho_{ij}^{E}$. Following similar steps above, let
$\bm{U}_{\ell}=(U_{\ell,1},U_{\ell,2}),$ $\ell=1,\ldots,n$, be iid copies of
$\bm{U}\in RV_{2}(2)$. We set $R_{\ell}=||\bm{U}_{\ell}||_{2}$,
$\bm{W}_{\ell}=(U_{\ell,1}/R_{\ell},U_{\ell,2}/R_{\ell})$, and
$k(n)=\sum_{\ell=1}^{n}\mathbb{I}[R_{\ell}>R_{(k)}],$ is the number of
exceedances over the $k^{th}$ upper order statistic.

Since $\bm{U}$ is a linear combination of independent $Z_{j}$’s, its angular
measure is discrete and $\sigma_{12}^{u}$ has a simple form. Let
$A^{(1)}-\bm{b}^{T}A^{(2)}:=\bm{C}=(\bm{c}_{1}^{T},\cdots,\bm{c}_{q}^{T})^{T}\in\mathbb{R}^{2\times
q}.$ The $(1,2)$ element of $\Sigma_{\bm{U}}$ is

$\sigma_{12}^{u}=\int_{\Theta_{1}}w_{1}w_{2}dH_{\bm{U}}(\bm{w})=\sum_{i=1}^{q}c_{1i}c_{2i},$

where
$H_{\bm{U}}(\cdot)=\sum_{j=1}^{q}||\bm{c}_{j}||^{2}\delta_{\bm{c}_{j}/||\bm{c}_{j}||}(\cdot)$
and $\delta(\cdot)$ is a Dirac mass function.

The natural estimator for
$\sigma_{12}^{u}=\int_{\Theta_{1}}w_{1}w_{2}dH_{\bm{U}}(\bm{w})=\tilde{m}\int_{\Theta_{1}}w_{1}w_{2}dN_{\bm{U}}(\bm{w})$
is given by

$\hat{\sigma}_{12,n}^{u}=\tilde{m}\int_{\Theta_{1}}w_{1}w_{2}d\hat{N}_{\bm{U}}(\bm{w})=\frac{\tilde{m}}{k}\sum_{l=1}^{n}w_{1l}w_{2l}\mathbb{I}[R_{l}>R_{(k)}],$
(26)

where $\tilde{m}$ is the total mass of the angular measure $H_{\bm{U}}(\cdot)$
and $N_{\bm{U}}(\cdot)=\tilde{m}^{-1}H_{\bm{U}}(\cdot)$ is a probability
measure. $k=\sum_{\ell=1}^{n}\mathbb{I}[R_{\ell}>R_{(k)}]$ is such that
$\lim_{n\to\infty}{k/n}=0$ as $k\rightarrow\infty$ and $R_{(k)}$ is the
$k^{th}$ upper order statistic in the sample of size $n$.

Under the condition (24), the estimator $\hat{\sigma}_{12,n}^{u}$ is
asymptotically normal by Larsson and Resnick (2012),

$\sqrt{k}(\hat{\sigma}_{12,n}^{u}-\tilde{m}E[W_{1,1}W_{2,1}])\sim
N(0,\tau^{u^{2}}),$

where $\tilde{m}$ is the total mass of the angular measure $H_{\bm{U}}$
identical to the sum of diagonal elements of the conditional TPDM
$\Sigma_{1|2}$. To obtain $\tau^{u^{2}}=\tilde{m}^{2}Var(W_{1}W_{2}),$ we
first consider $Var(W_{1}W_{2}),$

$Var(W_{1}W_{2})=\int_{\Theta_{1}}(w_{1}w_{2}-E[W_{1}W_{2}])^{2}dN_{\bm{U}}(\bm{w}),$

where $N_{\bm{U}}(\cdot)=\tilde{m}^{-1}H_{\bm{U}}(\cdot).$

$\displaystyle Var(\hat{\sigma}_{12,n}^{u})$
$\displaystyle=Var\Big{(}\frac{\tilde{m}}{k}\sum_{l=1}^{n}W_{1l}W_{2l}\mathbb{I}[R_{l}>R_{(k)}]\Big{)}$
$\displaystyle=\frac{\tilde{m}^{2}}{k}Var[W_{1,1}W_{2,1}]\quad\text{since}\quad(W_{1,l},W_{2,l})\text{'s
are iid}$

Our estimate $\hat{\tau}^{u^{2}}$ for $\tilde{m}^{2}Var(\sigma_{12,n}^{u})$ is
obtained in the same manner as above. Under the null hypothesis
$H_{0}:\rho^{E}_{ij}=0,$ since
$\sqrt{k}(\hat{\sigma}_{12,n}^{u}-\sigma_{12}^{u})\sim N(0,\tau^{u^{2}}),$ we
have

$\frac{\hat{\sigma}_{12,n}^{u}}{\sqrt{\hat{\tau}^{u^{2}}/k}}\sim T_{k-1},$
(27)

where $T_{k-1}$ denotes a $t$-distribution with $k-1$ degrees of freedom. With
the asymptotic result, we can construct confidence intervals and perform a
hypothesis test for zero elements in the inverse TPDM.

###  6.3 Asymptotic Normality for the Transformed-linear Extreme Illustrative
Model

We use a simulation study to illustrate asymptotic normality for the sample
conditional inner product matrix and perform a hypothesis test for zero
elements in its inverse. We again consider the four-dimensional transformed-
linear extreme model in (4.3) with generating equation

$X_{i}=\phi\circ X_{i-1}\oplus Z_{i},\quad i=1,2,3,4,$

where $\\{Z_{i}\\}$ is a sequence of independent regularly varying random
variables meeting lower tail condition, $\phi\in(0,1)$ and $X_{0}=0\;a.s.$

Our simulation study aims to estimate the partial correlation between $X_{2}$
and $X_{4}$ given $X_{1}$ and $X_{3}$, and to test whether this is
significantly different from zero. We set $\phi=0.7$ and generate $n=10,000$
four dimensional vectors $\bm{X}_{4}$. The largest $2\%$ of the samples is
used to find the estimated TPDM $\hat{\Sigma}_{\bm{X}_{4}}$.

Let $\bm{X}^{(1)}=(X_{2},X_{4})^{T}$ and $\bm{X}^{(2)}=(X_{1},X_{3})^{T}$. We
find $\hat{\bm{b}}=\hat{\Sigma}_{22}^{-1}\hat{\Sigma}_{21}$ and subsequently
find $\hat{\bm{X}}^{(1)}$. We then obtain two dimensional vectors of residuals
$\bm{U}=t^{-1}(\bm{X}^{(1)})-t^{-1}(\hat{\bm{X}}^{(1)})$. We have two methods
for estimating the conditional inner product matrix. The first is to use the
partitions of the estimated TPDM
$\hat{\Gamma}_{1|2}=\hat{\Sigma}_{11}-\hat{\Sigma}_{12}\hat{\Sigma}_{22}^{-1}\hat{\Sigma}_{21}.$
The second is to estimate ${\Gamma}_{1|2}$ from the residuals. For this
method, we use the largest $2\%$ of angular components. We focus on the off
diagonal element $[\Sigma_{1|2}]_{12}$. Figure 2 shows the comparison between
the kernel density of estimates obtained from the residuals (solid line) and
the kernel density from the partition of the TPDM (dashed line) under repeated
simulations. The figure shows little difference in these methods, and we
suggest using the estimate from the partition as this is immediately available
from the estimated TPDM.

Importantly, Figure 2 indicates that the variance of the residuals does in
fact capture the uncertainty in the estimates of the conditional inner product
matrix. Following the procedure in Section 6.1, we obtain estimated variances
for $\tau^{u^{2}}$. From the equation in (27), we construct a 95% confidence
interval for each iteration and achieve a coverage rate of 0.95.

Figure 2: The kernel density based on the residuals (solid line) versus the
kernel density from the partition of the TPDM (dashed line).

##  7 Application

###  7.1 Nitrogen Dioxide Air Pollution

We apply the idea of partial tail correlation to daily EPA $\text{NO}_{2}$
data from five stations in the Washington DC metropolitan area (see Figure 3).
We analyze 5163 daily $\text{NO}_{2}$ data between 1995 and 2016 where all
five stations have measurements. We follow the same preprocessing process by
Lee and Cooley (2021) so that we can assume each variable $X_{i}\in
RV_{+}^{1}(2)$ for $i=1,\ldots,5$. Let $X_{i}^{(orig)}$ denote the random
variable for detrended $\text{NO}_{2}$ at the $i$th location. We apply the
empirical CDF to perform marginal transformation
$X_{i}=1/\sqrt{1-\hat{F}_{i}(X_{i}^{(orig)})}-\delta$ for each location so
that $X_{i}$ follows a ’shifted’ Pareto distribution for $i=1,\ldots 5$. Each
$X_{i}\in RV_{+}^{1}(2)$ and the shift $\delta=0.9352$ is such that
$E[t^{-1}(X_{i})]=0.$ This shift makes the preimages of the transformed data
centered which helps reduce bias in the estimation of the TPDM (Lee and
Cooley, 2021). Assuming $\bm{X}=(X_{1},\ldots,X_{5})^{T}\in RV_{+}^{5}(2),$ we
let $\bm{X}_{t}=(X_{1,t},\ldots,X_{5,t})^{T}$ denote the random vector of the
daily $\text{NO}_{2}$ level on day $t$ and be iid copies of $\bm{X}.$

The goal is to test whether or not extreme $\text{NO}_{2}$ levels between each
pair of stations exhibit partial tail correlation. The first step is to
estimate the TPDM $\hat{\Sigma}_{\bm{X}}$. Let $\bm{x}_{t}$ denote the
observed daily $\text{NO}_{2}$ level on day $t$. For each $i\neq j$, let
$r_{t,ij}=\|(x_{t,i},x_{t,j})\|$ and
$(w_{t,i},w_{t,j})=(x_{t,i},x_{t,j})/r_{t,ij}$. We let
$\hat{\sigma}_{ij}=2k^{-1}\sum_{i=1}^{n}w_{t,i}w_{t,j}\mathbb{I}[r_{t,ij}>r_{i,j}^{*}],$
where $k=\sum_{t=1}^{n}\mathbb{I}[r_{t,ij}>r_{ij}^{*}]$ is the number of
exceedances. We set $r_{ij}^{*}$ as the 0.95 quantile for radial components.
Due to the pairwise estimation for the TPDM, the total mass $2$ arises from
the fact that each $X_{i}$ has the unit scale after preprocessing. We can then
calculate $\hat{\Sigma}^{-1}$, which is given in Table 1. The inverse TPDM
includes small values close to zero. Our aim can now be described as trying to
assess if each off-diagonal element is significantly different from 0.

| 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
1 | 2.10 | -0.54 | -0.19 | -0.81 | -0.23
2 | -0.54 | 2.72 | -1.14 | -0.31 | -0.58
3 | -0.19 | -1.14 | 2.28 | -0.22 | -0.38
4 | -0.81 | -0.31 | -0.22 | 2.11 | -0.47
5 | -0.23 | -0.58 | -0.38 | -0.47 | 2.01
Table 1: Inverse TPDM for all pairs of five stations

For each $i\neq j$ for $i,j=1,\ldots,5$, let
$\bm{X}_{t}^{(1)}=(X_{t,i},X_{t,j})^{T}$ and
$\bm{X}_{t}^{(2)}=(\bm{X}_{t}\setminus{(X_{t,i},X_{t,j})})^{T}.$ Given the
estimated TPDM $\hat{\Sigma}_{\bm{X}},$ we obtain
$\hat{\bm{X}}_{t}=\hat{\bm{b}}^{T}\circ\bm{X}_{t}^{(2)}\in RV_{+}^{2}(2),$
where $\hat{\bm{b}}=\hat{\Sigma}_{22}^{-1}\hat{\Sigma}_{21}.$ After computing
$\hat{\bm{X}}_{t}$ for all $t$, we take the top 5% of $\hat{\bm{X}}_{t}$ to
find residual vectors
$\bm{U}_{t}=t^{-1}(\bm{X}_{t}^{(1)})-t^{-1}(\hat{\bm{X}}_{t})=t^{-1}(\bm{X}_{t}^{(1)})-\hat{\bm{b}}^{T}t^{-1}(\bm{X}_{t}^{(2)})\in
RV^{2}(2).$ Note that we suppress the index $(i,j)$ in $\bm{U}_{t}$ for
simplicity.

For each pair of $(X_{i},X_{j})^{T}$ given all other components, we estimate
the off-diagonal element of the conditional TPDM $[\Sigma_{1|2}]_{12}$ and its
variance. Let $r_{t,12}=\|(U_{t,1},U_{t,2})\|$ and
$(w_{t,1},w_{t,2})=(u_{t,1},u_{t,2})/r_{t,12}$. We let
$\hat{\sigma}_{12}^{u}=\tilde{m}^{*}k^{-1}\sum_{i=1}^{k}w_{t,1}w_{t,2}\mathbb{I}[r_{t,12}>r_{12}^{*}],$
where $k=\sum_{t=1}^{n}\mathbb{I}[r_{t,12}>r_{12}^{*}]$ and $\tilde{m}^{*}$ is
an estimate for the total mass of $H_{\bm{U}}(\cdot).$ We choose $r_{12}^{*}$
as the 0.98 quantile for radial components.

Under the null hypothesis that $\rho_{ij}^{E}=0,$ for each $i\neq j,$ we
calculate test statistics $t=\sqrt{k}(\hat{\sigma}_{12}^{u}/\hat{\tau}^{u}),$
where $\hat{\tau}^{u}$ is an estimate for $\tilde{m}\sqrt{Var(W_{i}W_{j})}.$
We employ the Tukey’s exact procedure to adjust for multiple comparisons
because the Tukey’s exact procedure is well-suited for all pairwise
comparisons where the number of exceedances is equal across all pairwise
comparisons. We have the total number of observations $N=103\times 10=1030$
where each pairwise comparison has the equal number of threshold exccedances
of 103 and there are 10 pairwise comparisons. The degrees of freedom is
$df=N-10=1020$. Having a critical value of $t_{crit}=4.797,$ we summarize test
statistics in Table 2. If $|t|<4.797$, then we fail to reject the null
hypothesis that $\rho_{ij}^{E}=0$.

| 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
1 | - | 1.69 | 1.69 | 2.37 | 9.89
2 | 1.69 | - | 6.18 | 7.83 | 3.27
3 | 1.69 | 6.18 | – | 2.42 | 4.50
4 | 2.37 | 7.83 | 2.42 | - | 5.31
5 | 9.89 | 3.27 | 4.50 | 5.31 | -
Table 2: Test statistics for each pair of stations $i\neq j$ for
$i,j=1,\ldots,5$

We create an undirected graphical model for five stations given in Figure 3 by
assuming partial tail correlation implies conditional independence. The
extremal graph looks similar to a Markov chain. The thickness of lines is
proportional to the test statistics, and describes the strength of the
conditional relationship. The extremal graph from the partial tail correlation
has 4 edges determined to be significant from the ${{5}\choose{2}}=10$
possible edges.

Figure 3: Left: An outline of Washington, D.C. with locations of five
$\text{NO}_{2}$ monitors. Right: The extremal graph induced by partial tail
tail correlation for five stations. The thickness of edges corresponds to
absolute values of test statistics being greater than 4.797.

###  7.2 Danube River Basin

We also employ the notion of partial tail correlation to investigate
conditional relationships between the extremes of average daily river
discharges in the upper Danube basin. The Danube is Europe’s second largest
river and the upper Danube extends from its source in Germany to Bratislava in
Slovakia111https://www.icpdr.org/main/danube-basin/river-basin. To assess
flood risks caused by extreme river discharges, there are a number of gauging
stations along the river and its tributaries. The main feature in the upper
Danube basin is that there are physical flow connections among stations. This
feature allows us to compare the estimated graphical structure to the known
structure of the river network on the Danube. Figure 4 shows the river network
in the upper Danube basin where the path $10\rightarrow\cdots\rightarrow 1$ is
the main channel and the 21 other locations are on tributaries.

Figure 4: Physical flow connections in the upper Danube river basin

We analyze average daily river discharges from 31 gauging stations for
1960-2009. The data are available in the Bavarian Environmental
Agency222http://www.gkd.bayern.de. This data set has been widely used across
multiple disciplines to assess flood risk. Asadi et al. (2015) used a spatial
extremes model to fit data from these 31 stations. Engelke and Hitz (2020) fit
an extremal undirected graphical model based on the Hüsler-Reiss model to this
data.

We follow a similar preprocessing approach as Asadi et al. (2015) in order to
compare results. Engelke and Hitz (2020) and Asadi et al. (2015) only
considered June, July, and August because the main factor causing extreme
flooding is extreme precipitation in these summer months. It results in
$n=50\times 92=4600$ daily river discharges where all gauging stations have
measurements. Focusing on the summer period helps remove seasonality. The
overall trend in extreme river discharges on the Danube turns out to be
insignificant by Katz et al. (2002). Extreme discharges for each station occur
in clusters because extreme discharges at downstream may occur a few days
later from upstream stations. To remove temporal dependence, they set
nonoverlapping timewindows of length $p=9$ days and then take the largest
value within each window, resulting in a declustered time series of $n=428$
independent data from the original data. However, we decide to use the whole
sample of size $n=4600$ to get a large enough sample size by treating the data
as independent samples.

We assume each $X_{i}\in RV_{+}^{1}(2)$ in our inner product space. Let
$X_{i}^{(orig)}$ denote the random variable for average daily river discharges
at the $i$th station for $i=1,\ldots,31.$ For simplicity, we apply the
empirical CDF to perform the marginal transformation
$X_{i}=1/\sqrt{(1-\hat{F}_{i}(X_{i}^{(orig)}))}-\delta$ for each station so
that $X_{i}$ follows the shifted Pareto distribution. That is, each $X_{i}\in
RV_{+}^{1}(2)$ and the shift $\delta=0.9352$ is such that
$E[t^{-1}(X_{i})]=0.$ We assume $\bm{X}=(X_{1},\ldots,X_{31})^{T}\in
RV_{+}^{31}(2).$ We let $\bm{X}_{t}=(X_{1,t},\ldots,X_{31,t})^{T}$ denote the
random vector of the average daily river discharge on day $t$, which we treat
as iid copies of $\bm{X}.$

We first investigate a sub-network for the stations on the main channel,
$10\rightarrow\cdots\rightarrow 1$. The physical flow connections look similar
to a graphical model generated by an AR(1) model or the Markov chain. The goal
is to test whether or not extreme discharges between each pair of stations
exhibit partial tail correlation. To perform a hypothesis test for the partial
tail correlation for each pair of stations, we first estimate the TPDM
$\hat{\Sigma}_{\bm{X}}$. Let $\bm{x}_{t}$ denote the observed average daily
discharge on day $t$. For each $i\neq j$, let $r_{t,ij}=\|(x_{t,i},x_{t,j})\|$
and $(w_{t,i},w_{t,j})=(x_{t,i},x_{t,j})/r_{t,ij}$. We let
$\hat{\sigma}_{ij}=2k^{-1}\sum_{i=1}^{k}w_{t,i}w_{t,j}\mathbb{I}[r_{t,ij}>r_{i,j}^{*}],$
where $k=\sum_{t=1}^{n}\mathbb{I}[r_{t,ij}>r_{ij}^{*}]$ is the number of
exceedances. We set $r_{ij}^{*}$ as the 0.95 quantile for radial components.
We can then calculate $\hat{\Sigma}^{-1}$, which is given in Table 3. The goal
is to assess if each off-diagonal element is significantly different from 0.

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
1 | 4.33 | -5.41 | 4.62 | -2.77 | 4.74 | -4.32 | -1.72 | 2.95 | -1.41 | -0.78
2 | -5.41 | 29.24 | -32.01 | 8.24 | -4.45 | 3.00 | 2.17 | -2.57 | 0.72 | 1.72
3 | 4.62 | -32.01 | 46.69 | -19.71 | -5.91 | 7.00 | -1.53 | 4.99 | -4.34 | 0.31
4 | -2.77 | 8.24 | -19.71 | 39.78 | -28.03 | 2.08 | 1.76 | -4.24 | 1.15 | 1.57
5 | 4.74 | -4.45 | -5.91 | -28.03 | 123.04 | -101.71 | 11.22 | 6.83 | 0.78 | -7.34
6 | -4.32 | 3.00 | 7.00 | 2.08 | -101.71 | 123.03 | -29.01 | -13.46 | 10.06 | 4.36
7 | -1.72 | 2.17 | -1.53 | 1.76 | 11.22 | -29.01 | 25.30 | -3.89 | -2.96 | -1.84
8 | 2.95 | -2.57 | 4.99 | -4.24 | 6.83 | -13.46 | -3.89 | 26.37 | -22.79 | 6.03
9 | -1.41 | 0.72 | -4.34 | 1.15 | 0.78 | 10.06 | -2.96 | -22.79 | 39.47 | -20.87
10 | -0.78 | 1.72 | 0.31 | 1.57 | -7.34 | 4.36 | -1.84 | 6.03 | -20.87 | 17.66
Table 3: Inverse TPDM for the main path.

For each $i\neq j$ for $i,j=1,\ldots,10$, let
$\bm{X}_{t}^{(1)}=(X_{t,i},X_{t,j})^{T}$ and
$\bm{X}_{t}^{(2)}=(\bm{X}_{t}\setminus{(X_{t,i},X_{t,j})})^{T}.$ Given the
estimated TPDM $\hat{\Sigma}_{\bm{X}},$ we obtain
$\hat{\bm{X}}_{t}=\hat{\bm{b}}^{T}\circ\bm{X}_{t}^{(2)}\in RV_{+}^{2}(2),$
where $\hat{\bm{b}}=\hat{\Sigma}_{22}^{-1}\hat{\Sigma}_{21}$ for all
$t=1,\ldots,4600.$ We only consider those for which $\hat{\bm{X}}_{t}$ exceeds
the 0.98 quantile to find two dimensional residual vectors
$\bm{U}_{t}=t^{-1}(\bm{X}_{t}^{(1)})-t^{-1}(\hat{\bm{X}}_{t})=t^{-1}(\bm{X}_{t}^{(1)})-\hat{\bm{b}}^{T}t^{-1}(\bm{X}_{t}^{(2)})\in
RV^{2}(2).$ Note that we suppress the index $(i,j)$ in $\bm{U}_{t}$ for
simplicity.

Following the similar steps in $\text{NO}_{2}$ application, for each pair of
$(X_{i},X_{j})^{T}$ given all other components, we estimate the off-diagonal
element of the conditional TPDM $[\Sigma_{1|2}]_{12}$ and its variance. Let
$r_{t,12}=\|(U_{t,1},U_{t,2})\|$ and
$(w_{t,1},w_{t,2})=(u_{t,1},u_{t,2})/r_{t,12}$. We let
$\hat{\sigma}_{12}^{u}=\tilde{m}^{*}k^{-1}\sum_{i=1}^{k}w_{t,1}w_{t,2}\mathbb{I}[r_{t,12}>r_{12}^{*}],$
where $k=\sum_{t=1}^{n}\mathbb{I}[r_{t,12}>r_{12}^{*}]$ and $\tilde{m}^{*}$ is
an estimate for the total mass of $H_{\bm{U}}(\cdot).$ We choose $r_{12}^{*}$
as the 0.98 quantile for radial components. For each pair of
$(X_{i},X_{j})^{T}$ given all other components, we estimate the off-diagonal
element of the conditional TPDM $[\Sigma_{1|2}]_{12}$ and its variance.

Under the null hypothesis that $\rho_{ij}^{E}=0,$ for each $i\neq j,$ we
calculate test statistics $t=\sqrt{k}(\hat{\sigma}_{12}^{u}/\hat{\tau}^{u}),$
where $\hat{\tau}^{u}$ is an estimate for $\tilde{m}\sqrt{Var(W_{i}W_{j})}.$
We employ the Tukey’s exact procedure to adjust for multiple comparisons. We
consider 10 nodes and 45 possible comparisons. The total number of
observations is $N=92\times 45=4140$ where each pairwise comparison has the
equal number of threshold exccedances of 92 and there are 45 pairwise
comparisons. Hence, the degrees of freedom is $df=N-45=4095$. Finding a
critical value of $t_{crit}=5.847$, we summarize test statistics in a matrix
(4). If $|t|<5.847$, then we fail to reject the null hypothesis that
$\rho_{ij}^{E}=0$.

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
1 | - | 7.62 | -3.46 | 3.12 | -0.84 | -0.35 | 4.77 | -1.65 | -0.18 | 1.29
2 | 7.62 | - | 21.40 | -3.52 | -1.02 | 0.43 | -0.77 | -1.24 | -0.01 | -0.52
3 | -3.46 | 21.40 | - | 6.94 | 0.46 | -0.53 | -1.91 | 0.80 | -0.06 | 0.81
4 | 3.12 | -3.52 | 6.94 | - | 8.76 | -1.66 | 0.85 | 3.51 | -1.03 | -1.73
5 | -0.84 | -1.02 | 0.46 | 8.76 | - | 16.93 | -1.55 | -0.88 | 0.78 | -0.20
6 | -0.35 | 0.43 | -0.53 | -1.66 | 16.93 | - | 8.92 | -0.26 | -0.80 | -0.45
7 | 4.77 | -0.77 | -1.91 | 0.85 | -1.55 | 8.92 | - | 2.82 | 2.00 | 0.06
8 | -1.65 | -1.24 | 0.80 | 3.51 | -0.88 | -0.26 | 2.82 | - | 9.62 | -3.81
9 | -0.18 | -0.01 | -0.06 | -1.03 | 0.78 | -0.80 | 2.00 | 9.62 | - | 15.21
10 | 1.29 | -0.52 | 0.81 | -1.73 | -0.20 | -0.45 | 0.06 | -3.81 | 15.21 | -
Table 4: Test statistics for each pair of stations $i\neq j$ for
$i,j=1,\ldots,10$ in the main path.

Let $\mathcal{G}=(V,E)$ be an undirected graphic with a node set
$V=\\{1,\ldots,10\\}$ and its edge set $E.$ Based on the test statistics in
the table above, we create an undirected graphical model for the main path by
assuming partial tail correlation implies conditional relationships in Figure
5. Each circle indicates a node. Extreme discharges at nearby stations tend to
be partially correlated in terms of tail behaviors. The thickness of lines is
proportional to the test statistics, and describes the strength of the
conditional relationship. Focusing on the main stream line, our extremal graph
shows a resemblance to the physical flow connection. The graph from the
partial tail correlation has 8 edges determined to be significant. The
disconnection between a station 8 and 7 is not a big surprise because the
actual distance between them is far away from each other.

Figure 5: The known physical flow connections (above) versus the extremal
graph induced by partial tail correlation (below) for the main path
$10\rightarrow\cdots\rightarrow 1$ in the upper Danube river basin. The
thickness of edges corresponds to absolute values of test statistics being
greater than 5.847.

We also investigate the whole river network in the upper Danube basin.
Following the similar steps above, we standardize the off-diagonal element of
the conditional TPDM for each pair of stations. We obtain the critical value
of 7.189 via the Tukey’s exact procedure. We then create an undirected graph
on the whole river network. The extremal graph constructed from the partial
tail correlation has 192 significant edges out of the 465 possible edges. The
graph has a noteworthy reduction from all possible edges but is quite
complicated to interpret.

The graph found by Engelke and Hitz (2020) was much more simple and closely
resembled the physical flow network. However, their approach was much more
model-based and used knowledge of the physical flow graph to perform stepwise
model selection based on AIC. In recent work, Röttger et al. (2021) use a
somewhat less model-based approach to fit a graphical model on this same
Danube data and find a more connected network than their earlier estimate, but
one which is still more simple than the one in Figure 5. We believe that
including too many variables may introduce noise when estimating the TPDM.

##  8 Summary and Discussion

A vector space constructed from transformed linear combinations of independent
regularly varying random variables provides a framework for linear prediction
for extremes. We develop projection theorem as a natural way of defining
partial tail correlation in the context of extremes. Similar to Gaussian
cases, sparsity in the inverted TPDM can also be tied to the idea of the
partial tail correlation. Without a distributional assumption, the notion of
partial tail correlation can provide much less model-based approach for
describing conditional relationships between variables at extreme levels.

Using the asymptotic normality result for the estimator of the conditional
TPDM, we develop a hypothesis test for zero elements in the inverse extremal
matrix. We observed the performance of conditional extremal relationships
between variables in simulation study as well as in both applications: high
$\text{NO}_{2}$ levels in Washington D.C. and extreme river discharges in the
upper Danube river basin.

Especially, in the upper Daunbe river basin application, focusing on the main
channel $10\rightarrow\cdots\rightarrow 1,$ one can explore further the
oscillating pattern in the table of test statistics for each pair of stations.
The study of this pattern may be tied to the idea of directed graphs.
Furthermore, taking the spatial dependence into account can be another
important factor when describing conditional relationships between variables
at extreme levels.

## Acknowledgements

We are aware that there has been independent and parallel work by Gong, Huser,
and Opitz which also investigates partial tail correlation for extremes. Our
understanding is that inference in their work is done from a perspective of
model selection rather than hypothesis testing. This paper was completed
during a Ph.D. at Colorado State University.

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##  Appendix A Projection Theorem

######  Theorem A.1.

(Projection theorem) Let $\mathcal{V}_{A}$ be the previously defined subspace
of the Hilbert space $\mathcal{V}^{q}$ and $X\in\mathcal{V}^{q}$. Let
$X_{i}=\sum_{j=1}^{q}a_{ij}\circ Z_{j}\in\mathcal{V}^{q},$ $i=1,\ldots,p,$ and
let $X=\sum_{j=1}^{q}a_{j}^{*}\circ Z_{j}\in\mathcal{V}^{q}$. Then
i) $\hat{X}:=P_{\mathcal{V}_{A}}X$ ($\hat{X}$ is the projection of $X$ onto
$\mathcal{V}_{A}$) has a unique element in $\mathcal{V}_{A}$ such that

$||X\ominus\hat{X}||_{\mathcal{V}^{q}}=\inf_{Y\in\mathcal{V}_{A}}||X\ominus
Y||_{\mathcal{V}^{q}},\mbox{ and }$

ii) $\hat{X}\in\mathcal{V}_{A}$ such that
$||X\ominus\hat{X}||_{\mathcal{V}^{q}}=\inf_{Y\in\mathcal{V}_{A}}||X\ominus
Y||_{\mathcal{V}^{q}}$ if and only if $\hat{X}\in\mathcal{V}_{A}$ and
$(X\ominus\hat{X})\in\mathcal{V}_{A}^{\perp}$.

###### Proof.

i) Consider $X_{i}=\sum_{j=1}^{q}a_{ij}\circ Z_{j},$ $i=1,\ldots,p,$ and
$X=\sum_{j=1}^{q}a_{j}^{*}\circ Z_{j}$ in $\mathcal{V}^{q}$. For
$\bm{X}_{p}=(X_{1},\ldots,X_{p})^{T},$ consider
$\bm{b}^{T}\circ\bm{X}_{p}\in\mathcal{V}_{A}$.
$||X\ominus(\bm{b}^{T}\circ\bm{X}_{p})||_{\mathcal{V}^{q}}^{2}=\sum_{j=1}^{q}(a_{j}^{*}-\bm{b}^{T}\bm{a}_{\cdot
j})^{2}$ where $\bm{a}_{\cdot j}$ is the $j^{th}$ column vector of $\bm{A}$.
We assume Rank$(\bm{A})=p$. Let $S_{j}=\\{\bm{b}\in\mathbb{R}^{p}\,\text{such
that}\,\bm{b}^{T}\bm{a}_{\cdot j}=a^{*}\\}$ and
$f_{j}(\bm{b})=(a_{j}^{*}-\bm{b}^{T}\bm{a}_{\cdot j})^{2}$. For $\bm{b}\notin
S_{j}$, $\partialderivative{f_{j}(\bm{b})}{\bm{b}}=2\bm{a}_{\cdot
j}[\bm{b}^{T}\bm{a}_{\cdot j}-a_{j}^{*}]$ and
$\partialderivative{f_{j}(\bm{b})}{\bm{b}}{\bm{b}^{T}}=2\bm{a}_{\cdot
j}\bm{a}_{\cdot j}^{T}(\bm{b}\bm{a}_{\cdot j}-a_{j}^{*})$. As $\bm{a}_{\cdot
j}\bm{a}_{\cdot j}^{T}$ is nonnegative definite, $f_{j}$ is convex off of
$S_{j}$. Since $f_{j}$ is minimized on $S_{j}$, $f_{j}$ is convex everywhere.
Thus for $\bm{b}_{1}$ and $\bm{b}_{2}$ and any $w\in(0,1)$,

$wf_{j}(\bm{b}_{1})+(1-w)f_{j}(\bm{b}_{2})\geq
f_{j}(w\bm{b}_{1}+(1-w)\bm{b}_{2}),$

where equality above implies $\bm{b}_{1}^{T}\bm{a}_{\cdot
j}=\bm{b}_{2}^{T}\bm{a}_{\cdot j}$. Equality does not hold for every $j$.
$||X\ominus(\bm{b}^{T}\circ\bm{X}_{p})||_{\mathcal{V}^{q}}^{2}=\sum_{j=1}^{q}f_{j}$
is strictly convex since $\bm{A}$ is full rank.
$||X\ominus(\bm{b}^{T}\circ\bm{X}_{p})||_{\mathcal{V}^{q}}\rightarrow\infty$
as $\max_{1\leq j\leq p}|a_{j}^{*}|\rightarrow\infty$. Thus,
$||X\ominus(\bm{b}^{T}\circ\bm{X}_{p})||_{\mathcal{V}^{q}}$ must have a unique
minimum.
ii) Suppose $\hat{X}\in\mathcal{V}_{A}$ and
$(X\ominus\hat{X})\in\mathcal{V}_{A}^{\perp}.$ For any $Y\in\mathcal{V}_{A}$,

$\displaystyle||X\ominus Y||_{\mathcal{V}^{q}}^{2}$
$\displaystyle=\langle(X\ominus\hat{X})\oplus(\hat{X}\ominus
Y),(X\ominus\hat{X})\oplus(\hat{X}\ominus Y)\rangle$
$\displaystyle=||X\ominus\hat{X}||_{\mathcal{V}^{q}}^{2}+||\hat{X}\ominus
Y||_{\mathcal{V}^{q}}^{2}$
$\displaystyle\geq||X\ominus\hat{X}||_{\mathcal{V}^{q}}^{2},$

with equality iff $Y=\hat{X}.$ Thus, $\hat{X}$ is such that
$||X\ominus\hat{X}||_{\mathcal{V}^{q}}=\inf_{Y\in\mathcal{V}_{A}}||X\ominus
Y||_{\mathcal{V}^{q}}.$

Conversely if $\hat{X}\in\mathcal{V}_{A}$ and
$(X\ominus\hat{X})\notin\mathcal{V}_{A}$, then $\hat{X}$ is not the element of
$\mathcal{V}_{A}$ closest to $X$ since there exists

$\tilde{X}=\hat{X}\oplus a\circ Y/||Y||_{\mathcal{V}^{q}}^{2}$

closer to $X$ where $Y$ is any element of $\mathcal{V}^{q}$ such that $\langle
X\ominus\hat{X},Y\rangle\neq 0$ and $a=\langle X\ominus\hat{X},Y\rangle$.

$\displaystyle||X\ominus\tilde{X}||_{\mathcal{V}^{q}}^{2}$
$\displaystyle=\langle
X\ominus\hat{X}\oplus\hat{X}\ominus\tilde{X},X\ominus\hat{X}\oplus\hat{X}\ominus\tilde{X}\rangle$
$\displaystyle=||X\ominus\hat{X}||_{\mathcal{V}^{q}}^{2}+a^{2}\circ\frac{1}{||Y||_{\mathcal{V}^{q}}^{2}}+2\langle
X\ominus\hat{X},\hat{X}\ominus\tilde{X}\rangle$
$\displaystyle=||X\ominus\hat{X}||_{\mathcal{V}^{q}}^{2}-a^{2}\circ\frac{1}{||Y||_{\mathcal{V}^{q}}^{2}}$
$\displaystyle<||X\ominus\hat{X}||_{\mathcal{V}^{q}}^{2}.$

∎

##  Appendix B Property of Projection Mappings

######  Proposition B.1.

(Property of Projection Mappings) Let $P_{\mathcal{V}_{A}}$ be the projection
mapping of $\mathcal{V}^{q}$ onto a subspace $\mathcal{V}_{A}$. Then,
i) $P_{\mathcal{V}_{A}}(\alpha\circ X\oplus\beta\circ Y)=\alpha\circ
P_{\mathcal{V}_{A}}X\oplus\beta\circ P_{\mathcal{V}_{A}}Y,\quad
X,Y\in\mathcal{V}^{q},\quad\alpha,\beta\in\mathbb{R}$.

[That is, the projection mapping $P_{\mathcal{V}_{A}}$ is a linear mapping.]
ii) For every $X\in\mathcal{V}^{q}$, there exist an element of
$\mathcal{V}_{A}$ and an element of $\mathcal{V}_{A}^{\perp}$ such that

$X=P_{\mathcal{V}_{A}}X\oplus(I-P_{\mathcal{V}_{A}})X$

and this decomposition is unique.

###### Proof.

i) $(\alpha\circ P_{\mathcal{V}_{A}}X)\oplus(\beta\circ
P_{\mathcal{V}_{A}}Y)\in\mathcal{V}_{A}$ since $\mathcal{V}_{A}$ is a linear
subspace of $\mathcal{V}^{q}$. In addition,

$\alpha\circ X\oplus\beta\circ Y\ominus(\alpha\circ
P_{\mathcal{V}_{A}}X\oplus\beta\circ
P_{\mathcal{V}_{A}}Y)=\alpha\circ(X\ominus
P_{\mathcal{V}_{A}}X)\oplus\beta\circ(Y\ominus
P_{\mathcal{V}_{A}}Y)\in\mathcal{V}_{A}^{\perp}$

since $\mathcal{V}_{A}^{\perp}$ is a linear subspace of $\mathcal{V}^{q}$.
Thus, these two properties indicate $\alpha\circ
P_{\mathcal{V}_{A}}X\oplus\beta\circ P_{\mathcal{V}_{A}}Y$ is the projection
of $P_{\mathcal{V}_{A}}(\alpha\circ X\oplus\beta\circ Y).$ We note that this
linear mapping is not necessarily true when $\alpha\neq 2.$
ii). To show uniqueness of decomposition, let $X=Y\oplus Z,$
$Y\in\mathcal{V}_{A}$, $Z\in\mathcal{V}_{A}^{\perp}$ be another decomposition,
then

$Y\ominus P_{\mathcal{V}_{A}}X\oplus Z\ominus(I-P_{\mathcal{V}_{A}})X=0.$

By taking inner products of each side with $Y\ominus P_{\mathcal{V}_{A}}$,
$||Y\ominus P_{\mathcal{V}_{A}}X||_{\mathcal{V}^{q}}^{2}=0$ since
$Z\ominus(I-P_{\mathcal{V}_{A}})X\in\mathcal{V}_{A}^{\perp}$. Hence
$Y=P_{\mathcal{V}_{A}}X$ and $Z=(I-P_{\mathcal{V}_{A}})X$. ∎