Inference and Sampling for Archimax Copulas
Anonymous Author(s)
AfﬁliationAddress email
Abstract
Understanding multivariate dependencies in both the bulk and the tails of a distribution is an important problem for many applications, such as ensuring algorithms arerobust to observations that are infrequent but have devastating effects.
Archimaxcopulas are a family of distributions endowed with a precise representation thatallows simultaneous modeling of the bulk and the tails of a distribution.
Ratherthan separating the two as is typically done in practice, incorporating additionalinformation from the bulk may improve inference of the tails, where observationsare limited.
Building on the stochastic representation of Archimax copulas, wedevelop a non-parametric inference method and sampling algorithm.
Our proposedmethods, to the best of our knowledge, are the ﬁrst that allow for highly ﬂexibleand scalable inference and sampling algorithms, enabling the increased use of
Archimax copulas in practical settings.
We experimentally compare to state-of-theart density modeling techniques, and the results suggest that the proposed methodeffectively extrapolates to tails while scaling to higher dimensional data.
Our ﬁndings suggest that the proposed algorithms can be used in a variety of applicationswhere understanding the interplay between the bulk and the tails of a distribution isnecessary, such as healthcare and safety.
Introduction
Modeling dependencies between random variables is a central task in statistics and understandingthe dependence between covariates throughout the distribution is important in characterizing adistribution outside of its areas of highest density.
For example, in machine learning contexts, a majortopic of interest lies in enforcing dependencies between covariates, such as spatial dependence inconvolutional neural networks or temporal dependence in recurrent neural networks.
Copulas arefunctions that model the joint dependence of random variables, and they have been successfullyemployed in a variety of practical modeling settings due to their ease of use and intuitive properties.
Moreover, since copulas are used to represent cumulative distribution functions (CDFs), they havebeen particularly useful in situations where tail events are important – events that have high impactbut low probability.
For example, in computer vision applications, local dependence modeled byconvolutions may be sufﬁcient for the bulk of the data, but for data in the tails of the distributionnon-local dependencies may be present and need to be modeled.
Current successful applications ofcopulas have largely been limited to a few basic parametric families and low dimensional settings,preventing their widespread adoption in settings where the dependencies are complicated or thedimension is large.
Archimax copulas are a class of copulas that merges a tractable form with sufﬁcient expressiveness.
Archimax copulas effectively balance the representation of data within the bulk of the distributionwhile extrapolating to the tails by combining the tractability of Archimedean copulas with the tailproperties of extreme value copulas .
Notably, they remove the simpliﬁed symmetry assumption among covariates that is present in
Archimedean copulas and the max-stable property, a result of which is lower tail independence, ofextreme value copulas.
These are major assumptions often not held in practice.
The use of Archimaxcopulas has resulted in better ﬁt to data in applications such as healthcare [25] and hydrology [3, 11].
However, existing computational methods do not allow feasible inference and sampling for Archimaxcopulas, preventing their widespread use.
Therein lies the motivation behind this work.
We constructefﬁcient and ﬂexible inference and sampling methods using deep learning techniques and discusshow they compare to traditional means of density estimation that use existing copula methods anddeep generative models.
In addition, we provide numerical studies where the proposed Archimaxtechniques extrapolate to the tails better than existing methods.
1.1 Related work
Modeling distributions is a major task in machine learning, with techniques such as generativeadversarial networks (GANs) [12], normalizing ﬂows (NFs) [30], and variational autoencoders(VAEs)
[21] being the major developments for representing complex distributions.
However, these techniques are largely used for modeling the bulk of a distribution and may notextrapolate well to out-of-distribution samples or samples within the tails, as discussed in [23].
Somemethods have been proposed to represent only the tails, for example in [1, 6].
However, thesemethods are used only for sampling, and do not provide a way of quantifying the dependence of theobservation.
Moreover, they disregard the information in the bulk of the distribution and only focuson the tails.
Copulas are an important technique for representing distribution functions since they allow easyseparation of the marginals and the joint dependence structure of a distribution.
Copulas have beenapplied in machine learning wherein techniques from machine learning have been used in conjunctionwith traditional copula theory to model more general classes of densities with examples of such workfound in [24, 28, 35], please see Appendix A for more background on copulas and application ofcopulas in machine learning.
However, these have generally focused on simpliﬁed assumptions suchas a symmetric dependence between covariates, or a hierarchy of bivariate dependencies.
Moreover,these do not extrapolate to tail distributions and do not readily appear to generalize to high dimensions.
With regards to Archimax copulas, several theoretical works have been proposed analyzing thedistributions [9, 10, 27].
In Chatelain et al.
[11], the authors proposed a method for inferring a stabletail dependence function (stdf) when given an Archimedean generator.
However the method assumesknowledge of the Archimedean generator or infers a one-parameter Archimedean generator frompairwise Kendall’s taus.
Past applications of Archimax copulas were of low dimensions, such asdimensions 2, 3 and 3 in the studies of river
ﬂow rates [3], rainfall [11] and nutrient intake [25].
Our contributions We propose methods for ﬁlling the gaps in the existing literature by
Developing methods for inferring both the Archimedean generator and stdf;2.
Developing methods for sampling from Archimax copulas;
Providing ﬂexible representations for both the radial and spectral components.
Background
Copulas are given by separating the marginal distributions from the joint dependence of a randomvariable.
Speciﬁcally, consider
[Equation]
where F is a d -variate cumulative distribution function (CDF) of the random variable X =( X 1 , · · · , X d ) ∈ R d , F j is the j th univariate margin, and C is the copula describing the dependencebetween the uniform random variables
U = ( U 1 , · · · , U d ) = ( F 1 ( X 1 ) , · · · , F d ( X d )) ∈ [0 , 1] d .
Moreover, if the marginals F j are continuous, then the copula C is unique [34].
As discussed in the introduction, Archimax copulas describe a generalization of Archimedean andextreme value copulas.
They are deﬁned as:
Deﬁnition 1 (Archimax copula)
An Archimax copula is given by
[Equation]
) where ϕ : [0 , ∞ ) → [0 , 1] is an Archimedean generator and (cid:96) : [0 , ∞ )
d → [0 , ∞ ) is a stable taildependence function (stdf) [8, 10, 27].
From the deﬁnition, the Archimax copula is completely characterized by these two functions andthe objective during inference lies in estimating both the stdf and the Archimedean generator .
Bothfunctions have speciﬁc properties that must be fulﬁlled in order to obtain a valid Archimax copula.
The stdf is deﬁned as:
Deﬁnition 2 (stable tail dependence function (stdf))
A d -variate stdf (cid:96) : [0 , ∞ )
d → [0 , ∞ ) isgiven by:
[Equation]
with a spectral random variable W ∈ ∆ d − 1 satisfying moment constraints E W [ W j ] = 1 /d forj ∈ { 1 , · · · , d } [13, 20, 33].
Intuitively, the stdf dictates the asymptotic dependence between covariates.
Notably, the deﬁnition completely relies on the distribution of W , and the stdf is homogeneous of order one, i.e.(cid:96) ( cx 1 , · · · ,
[Equation] d ) = c [Equation]
( x 1 , · · · , x d ) for c > 0 .
On the other hand, the Archimedean generator is deﬁned by:
Deﬁnition 3 (Archimedean generator)
An Archimedean generator ϕ : [0 , ∞ ) → [0 , 1] is the
Williamson d -transform of a random variable R > 0 :
[Equation]
where ( y ) + := max(0 , y ) for y ∈ R [26, 37].
The Archimedean generator has a one-to-one correspondence with the distribution of R [26] anddictates the shape of the radial envelope applied across all covariates.
With these deﬁnitions in mind, we leverage Charpentier et al.
[10, Theorem 3.3] for inference andsampling, which established that any random variable
[Equation]
follows an Archimax copula, where R and S are independent and known as the radial and simplexcomponents, with supports R > 0 , S ∈ [0 , 1] d , (cid:96) ( S ) ≤ 1 .
Moreover, every Archimax copula givenin (2) has a decomposition given by (5).
The radial component R is the same R in Deﬁnition 3, andthe simplex component S has a one-to-one correspondence with the spectral component W from Deﬁnition 2.
Therefore, representing the R and W in Deﬁnitions 3 and 2 respectively providescomplete understanding of the distribution. [Figure]
Method
As established in Section 2, the stdf and Archimedean generator are expectations of functions ofthe spectral component W and radial component R respectively.
We ﬁrst model these expectationswhere W and R are discrete random variables with ﬁnite support [15, 17].
We then let them beoutputs of generative networks in the limit of inﬁnite support.
We begin by describing the inferencealgorithm for the stdf followed by the inference algorithm for the Archimedean generator.
Throughsampling the stdf we deﬁne the relationship between S and W which we make use of in the inferencefor the Archimedean generator.
We ﬁnally show how the representations we use can be easilyadapted for sampling from Archimax copulas.
A ﬂow chart describing the relationship betweenstdf and Archimedean generator parameter estimation is in Figure 2.
The source code with detaileddocumentation will be made available online.
3.1 Stable tail dependence function inference and sampling
For this section, we suppose that the Archimedean generator ϕ ( · ) is known and the goal is to infer orsample from the stdf.
3.1.1 Inference for stable tail dependence function
We consider a stdf following Deﬁnition 2, where (cid:96) is speciﬁed by the spectral decomposition withspectral component W [13, 20, 33].
To summarize the inference for the stdf: We ﬁrst establish arepresentation of the stdf through the empirical expectation of samples w from a generative network
G W .
We then deﬁne the likelihood of transformed data observations and optimize the parameters of
G W to maximize this likelihood.
Following Chatelain et al.
[11], when given the Archimedean generator ϕ we can deﬁne the transformation ξ ( u , x ) for an observation u and a pseudo-observation x = ( x 1 , · · · , x d ) ∈ ∆ d − 1 :
[Equation]
a transformation used for estimating extreme value copulas [8, 31].
In the case of Archimax copulas, and by using the homogeneity property of the stdf (3), the CDF ofthe random variable ξ
( U , x ) is expressed as:
[Equation]
The full derivation is given in Appendix A.
Following Hasan et al.
[18], we differentiate the CDF of ξ ( U , x ) with respect to x and obtain thelog-likelihood of the transformed observation as:
[Equation]
Recalling Deﬁnition 2, let (cid:96) θ be parameterized by a generative network G W such that:
[Equation]
where w k = ( w k 1 , · · · , w kd ) for k = { 1 , · · · , l } are l samples from G W with dimension d andoutput activation given by the Softmax ( · ) function to respect the support W ∈ ∆ d − 1 .
The momentconstraints E W [ W j ] = 1 /d for j = { 1 , · · · , d } are scale conveniences and not necessities [15].
Toapproximate the moment constraints, we penalize the residual
We may then train the generative network G W such that (cid:96) θ approximates (cid:96) by minimizing the negativelog-likelihood of transformed observations ξ ( u , x ) , with x uniformly sampled on the unit simplex∆ d − 1 .
The full technique is presented in Algorithm 1 in Appendix A.
The inverse ϕ − 1 in the computation of ξ ( u , x ) in (6) can be computed numerically by NewtonRaphson.
The derivative ϕ (cid:48) in the computation of log L ( x ; ϕ, (cid:96) ) in (8) can be calculated explicitlyfrom Deﬁnition 3 as:
[Equation]
Putting these computations together results in the likelihood of the stdf given the generator.
3.1.2 Sampling the simplex component
We now consider sampling the simplex component S .
We noted in the introduction that there is aone-to-one relationship between S and W which we explicitly note here.
Using this relationship andthe result of Charpentier et al. [10], the stdf may also be written in terms of the simplex componentthrough the following deﬁnition:
Deﬁnition 4 (survival distribution of simplex component)
The survival distribution function(SDF) of the simplex component S is:
[Equation]
11) which relates to the so-called generalized Pareto copulas deﬁned as:
Deﬁnition 5 (generalized Pareto copula)
A generalized Pareto copula is deﬁned as the copula of ageneralized Pareto distribution with support on ( −∞ , 0] d and CDF speciﬁed by (cid:96) as:
[Equation]
Moreover, it has stochastic representation as:
[Equation]
where U is uniformly distribution on [0 , 1] , independent of spectral component W =
( W 1 , · · · , W d )from the spectral decomposition of (cid:96) in Deﬁnition 2 [7, 14].
Therefore, we can frame the sampling of the simplex component through the sampling from ageneralized Pareto copula, such that, given samples ( x 11 , · · · , x 1 d ) , · · · , ( x ( d − 1)1 , · · · , x ( d − 1) d ) ofsize d − 1 from a generalized Pareto distribution with CDF in (12), we may obtain a sample of s ,where coordinates s j for j = { 1 , · · · , d } are computed as [10]:
[Equation]
The coordinate-wise maxima is taken over d − 1 samples to get the d − 1 exponent in the SDF of S ,corresponding to the d − 1 exponent in the expression of the Archimedean generator ϕ .
Due to our convenient representation of (cid:96) as an expectation of a function of W from generativenetwork G W , we are able to sample a generalized Pareto distribution from its stochastic representation.
The full technique is presented in Algorithm 2 in Appendix A.
3.2 Archimedean generator inference and sampling
We now assume that the stdf is known and the goal is to infer the Archimedean generator.
Weconsider a general representation of the Archimedean generator following Deﬁnition 3, where ϕ is ad − monotone function speciﬁed by the Williamson d − transform of the radial component R [10, 26].
To provide an outline for the overall approach, we ﬁrst consider the so-called Kendall distribution,which describes an integral transform of the copula.
We then use the fact that the empirical Kendalldistribution converges to the true Kendall distribution as the empirical copula converges to the truecopula, providing a means of estimation.
The Kendall distribution is formally deﬁned as:
Deﬁnition 6 (Kendall distribution) Let C be a copula, and let the CDF of the random variable
U ∈ [0 , 1]
d be the copula C .
Deﬁne the random variable W :=
C ( U ) .
The Kendall distribution of
C is the multivariate probability integral transform given by the CDF of W :
[Equation]
In the case of Archimax copulas, using the stochastic representation (5) and the homogeneity propertyof the stdf (3), the Kendall distribution is expressed as:
[Equation]
[Equation]
The full derivation is provided in Appendix A.
The empirical Kendall distribution K n for n givenobservations ( u 11 , · · · , u 1 d ) , · · · , ( u n 1 , · · · , u nd ) is deﬁned as:
[Equation]
wherew i =
[Equation]
[Equation] In the case of symmetric dependence in the non-parametric inference of Archimedean copulas [17],
S is uniformly distributed on the simplex ∆ d − 1 and (cid:96) ( S 1 , · · · , S d ) =
S 1 + · · · + S d ≡ 1 .
Then, Ris discrete with the cardinality of the support the same as W and K n is the Kendall distribution of aunique Archimedean copula [17].
In the non-parametric inference of Archimax copulas, we deﬁne the random variables:
[Equation]
(19) where T is a discrete random variable with support the same size as W and K n is the Kendalldistribution of an Archimax copula.
Note that R and Z are independent as R and S are independent.
Using K n , we reconstruct the support of R , given (cid:96) and the support of S , which in turn provides thesupport of Z .
With the ﬁnal objective of letting R and S be independent and identically distributed(iid) outputs of generative networks, we ﬁrst (linearly) interpolate K n to be equispaced, such thateach w i has the same probability k i
= 1 / ( n r n z ) , where n r and n z are the chosen sizes of supportsfor R and Z .
We describe the reconstruction procedure for the general case with non-iid randomvariables including cases where r j z l = r j (cid:48)
z l (cid:48) in Appendix A.
The sizes of supports n r and n z arechosen empirically with examples given in Appendix B.
Suppose the supports of the distributions of R, Z, W, T are ﬁnite and respectively denoted by:
[Equation]
R , z l ∈ Z} = { t 1 , t 2 , · · · , t n r n z } , where the n r n z elements of W are sorted in decreasing (nonincreasing) order, and the n r n z elements of T are sorted in increasing (non-decreasing) order.
Thisreverse ordering is due to ϕ being a decreasing function.
We minimize the mean sum of square residuals motivated by the uniform convergence of the empiricalprocess √ n ( K n − K ) , n → ∞ established by Barbe et al.
[Equation]
where, following the deﬁnition of ϕ in (4),
[Equation]
The ﬁnite support assumption of R and Z is not necessary, and we consider a modiﬁcation where thesupports R , Z are speciﬁed by samples from generative networks.
The main objective is to learn theparameters of generative network G R given the empirical Kendall distribution K n and samples of Z .
Since scaling the support R by a constant c > 0 does not change the copula, we add a regularizationterm for E R
[ R ] = 1 .
The algorithm can be understood as an alternating minimization algorithm,where the map between R and Z to W via T , and the support R are updated in an alternating fashion,please see Algorithm 3 in Appendix A for details.
The learned G R also provides a source of samples for the radial component R , which we will use togenerate full samples from the Archimax copula.
3.3 Inference and sampling for Archimax copulas
We ﬁnally summarize the combination of inference and sampling for both the Archimedean andsimplex component to obtain the full algorithm for inference and sampling for Archimax copulas.
This culminates into an iterative technique that successively updates each component.
3.3.1 Inference for Archimax copulas
We initialize with ϕ ( x ) = exp( − x ) and infer (cid:96) to learn G W , following Section 3.1.
This specialcombination of ϕ and (cid:96) corresponds to extreme-value dependence with the max-stable property.
Toaid inference for this initialization step, we pre-process our data to have extreme-value dependence.
We do so by computing the block maxima , a technique from extreme value copulas where we groupobservations and take the coordinate-wise maximas within each group.
Speciﬁcally, given observations ( x 11 , · · · , x 1 d ) , · · · , ( x n 1 , · · · , x nd ) , the block-maximas ( m 11 , · · · , m 1 d ) , · · · , ( m k 1 , · · · ,m kd ) for k blocks of size n/k are computed as:
[Equation]
(22) for i = { 1 , · · · , k } and j = { 1 , · · · , d } .
To determine the block size, with larger block sizesn/k better approximating extreme-value dependence and more blocks k for more observations,we test the block maximas for extreme-value dependence via the max-stable property using thetest by Kojadinovic et al.
[22], and select the ﬁrst k where the null hypothesis of extreme-valuedependence is not rejected, starting from k = n .
Given an estimate of (cid:96) and a learned G W , we are able to generate many samples of S , therebyproviding a way to compute Z = (cid:96) ( S ) .
We then infer ϕ , learning G R , following Section 3.2.
Atthis point, we repeat the estimation for G W with the updated ϕ to improve our estimate for (cid:96) .
Thealgorithms may be iterated as needed with suggested convergence criteria such as Cramér–von Mises(CvM) distance between successively estimated copulas.
3.3.2 Sampling for Archimax copulas
Given samples of the radial and simplex components, the stochastic representation of Archimaxcopulas (5) gives a straightforward method for sampling Archimax copulas [10].
Speciﬁcally, wesimply sample S and R and multiply to obtain a sample from the Archimax copula.
Experiments
To understand the modeling and sampling capabilities of the proposed algorithms, we conduct anumber of empirical studies.
We compare the proposed method to a number of existing copula-basedand deep learning-based methods for density estimation.
Our experiments relate to the main focusof the proposed method where we are interested in understanding the dependencies between thevariables in both the bulk and the tail.
In that sense, we conduct a number of experiments where wewish to extrapolate to the tail.
Further experimental details may be found in Appendix B.
The metric we use to compare the methods is based on the Cramér-von Mises (CvM) statistic, which computes the L 2 distance between theempirical copula and the estimated copula.
This statistic is commonlyused to determine differences between distributions in goodness-of-ﬁttests [32].
We use a version where we compare the empirical copula oftrue samples to the empirical copula of generated samples.
Speciﬁcally,it can be written as: CvM = (cid:82) ( C ∗ ,n ( u ) − C θ,n ( u )) 2 d u , where C ∗ ,nis the empirical copula of true samples and C θ,n is the empirical copulaof generated samples [32].
Inference for Archimedean generator Our ﬁrst set of experimentsinvolve inference for the Archimedean generator following the proposedmethod in Section 3.2.
We consider data with dimension d = 10 andsample size of n = 1000 given the true stdf (cid:96) .
To the best of our knowledge, our proposed methodis the ﬁrst method for non-parametric inference of ﬂexible Archimedean generators in Archimaxcopulas.
As such, there are no baselines for comparison.
Instead, we compute the results in terms of the map λ ( w ) =
ϕ − 1 ( w ) / ( ϕ − 1 ( w )) (cid:48) , w ∈ (0 , 1) due to its scale invariant property and knownasymptotic variance, which was described as a useful metric for how well ϕ is ﬁt in [16, 17].
Weevaluated our proposed method on the Clayton, Frank, Joe and Gumbel generators, representingdifferent radial envelopes, for Kendall’s tau of τ = { 0 . 2 , 0 . 5 } , representing different associations.
The stdf comes from the family of negative scaled extremal Dirichlet (NSD), a ﬂexible class ofstdf [5].
All estimates were within the asymptotic variance of λ ( w ) , w ∈ (0 , 1) .
The next best methodis comparing to a Clayton generator estimated by pairwise Kendall’s tau [11].
We give the resultsin terms of MSE to λ in Table 1, and plot estimates of λ ( w ) in Figure 3.
Additional experimentresults, including small sample performance n = 200 , and choices of support sizes n r and n z , are in Appendix B.
Inference for stable tail dependence function and sampling for simplex component We nowconsider the opposite case where we wish to estimate the stdf (cid:96) given the true Archimedean generatorϕ .
We compute the integrated relative absolute error (IRAE) between the estimated (cid:96) θ and the true(cid:96) .
The IRAE is given by IRAE ( (cid:96), (cid:96) θ ) =
[Table]
|
[Table]
| (cid:82) [Table] | (cid:96) ( x ) − (cid:96) θ ( x ) | /(cid:96) ( x ) d x [11].
For the samesettings as above, results given true ϕ are in Table 1. [Table] − 3 [11] [Table]
[Table] MSE × [Table] [Table]
URS )
[Table]
IRAE
[Table]
[Table]
[Table] IRAE [Table] Modeling nutrient intake The USDA studied the nutrient intake of women [36].
One particular task is understanding the dependenciesbetween the intake of different nutrients.
Wecan model this using an Archimax copula tounderstand the dependencies between the nutrients in the bulk and the tail.
To assess this,we ﬁt and compare models from the literatureon representing distributions and compute the CvM goodness-of-ﬁt statistic for each model.
Speciﬁcally, we break our comparison into twodifferent types of models: copula based modelsand deep network based models.
For the copulamodels, we use the same marginal distributionsfor all methods including the proposed methodbut consider various forms for the copula parameterization.
For methods marked with *, we usethe ϕ described in Section 3.2 and for methodsmarked with † we use the sampling method described in Section 3.1.2.
For the deep generativemodels, we use standard methods based on the Wasserstein GAN [2], masked autoregressiveﬂow (MAF) [29], and variational autoencoders(VAE)
The results are presented in Table 2.
The proposed method (Gen-AX) with the Archimaxhas the lowest CvM distance among all the competing methods, suggesting that the proposed methodis recovering the true dependency structure.
The state of the art Clayton Archimax (C-AX) did notperform well possibly due to the difﬁculty in scaling the single parameter Clayton generator to higherdimension.
The Archimedean copula (AC *) possibly beneﬁted from the use of our proposed ϕ .
Weadditionally provide examples of samples versus the ground truth in Appendix B for the differentmethods as well as an explanation of the different abbreviations.
Extrapolating to extreme rainfall We also consider another realistic dataset which models therainfall in French Britanny as studied by Chatelain et al. In this experiment, we are interested in
[Table] testing how well the proposed method can extrapolate to the extremes from non-extreme data.
Thisdataset consists of 3 locations (Belle-Ile, Groix, and Lorient) where the rainfall was recorded over thetime period.
For this experiment, we ﬁrst train the models on the full dataset.
We then generate manysamples from the trained model, compute the block maxima, then estimate the extremal dependencefrom the block maxima using the CFG estimator [8].
The results are presented in Table 3 and Figure 5where we compare the proposed method to deep generative models.
We did not compare to copulabased models since classical copulas are generally Gumbel or independent in the extremes and notsuitable for this application.
[Table]
High dimensional modeling We ﬁnally consider an experiment where we sample data from a100-dimensional Clayton-NSD Archimax copula.
Scaling to high dimensions is an important propertyof the proposed method since many existing copula methods fail to scale beyond lower dimensions.
As such, we only compare with deep generative models since the existing copula models resulted innumerical errors during optimization.
We report the CvM statistic in the second line of Table 2.
Weadditionally provide examples of the samples in Appendix B.
Conclusion
We developed highly ﬂexible and scalable inference and sampling algorithms, facilitating the useof Archimax copulas in practical settings.
We experimentally compare to state-of-the-art densitymodeling techniques, and the results suggest that the proposed method effectively extrapolates to tailswhile scaling to higher dimensional data.
The methods are especially useful in scenarios requiringextrapolations to the tails while also incorporating data from the bulk.
Limitations and future work
A single Archimedean generator ϕ to describe the radial envelopeacross all coordinates may not be sufﬁciently expressive for certain datasets.
For these cases,hierarchical Archimax copulas may be more appropriate and a direction of future work [19].
Potential negative societal impacts Model misspeciﬁcation may lead to misspeciﬁcation of risks,leading to potentially catastrophic outcomes in areas such as healthcare, safety and ﬁnance.
Risksmay be mitigated by conﬁrming a reasonable ﬁt between the observations generated by the modeland the data.
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[Table]
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available
[-We infer and sample a 100-dimensional Clayton-NSD Archimax copula, with parameters θ = 2 , α =( α 0 , · · · , α 0 ) , α 
 = (1 , 1 , 1 , 1 , 2 , 2 , 2 , 3 , 3 , 4) , ρ = 0 .-][Table]
[Table]