Detailed Contents"

In gremes: Estimation of Tail Dependence in Graphical Models

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Vignette "Introduction"

serves as general introduction into the scope and available methods in the package. We recommend to read it first.

Vignettes "Estimation - Note 1" and "Code - Note 1"

provide the theoretical description and the code used to compute estimates based on the method of moments. The model is a Markov tree parameterized cliquewise by Huesler-Reiss distributions.

The estimator applies to the data [ (\ln X_{i,v}- \ln X_{i,u}){v\in V\setminus u}\mid X{i,u}>n/k, \qquad i\in {1, \ldots, n}, ] for every $u\in V$.

Theoretical background on the model can be found in @asenova2021.

The method is applicable both in the case of no latent variables in the data and in the case of latent variables.

Vignettes "Estimation - Note 2" and "Code - Note 2"

provide the theoretical description and the code used to compute estimates based on the method of maximum composite likelihood function. The model is a Markov tree parameterized cliquewise by Huesler-Reiss distributions.

The estimator applies to the data [ (\ln X_{i,v}- \ln X_{i,u}){v\in V\setminus u}\mid X{i,u}>n/k, \qquad i\in {1, \ldots, n}, ] for every $u\in V$.

Theoretical background on the model can be found in @asenova2021.

There are two versions of the MLE. The two versions give similar estimates, but the first version is faster for bigger graphs. The second version may be quite slow on big problems.

The two versions are applicable both in the case of no latent variables in the data and in the case of latent variables.

Vignettes "Estimation - Note 3" and "Code - Note 3"

provide the theoretical description and the code used to compute estimates based on extremal coefficients. The model is a Markov tree parameterized cliquewise by Huesler-Reiss distributions.

Theoretical background on the model can be found in @asenova2021.

Similarly to the maximum likelihood estimator there are two versions. The first version is faster for bigger graphs. The second version may be quite slow on big problems.

Both methods are applicable in the case of no latent variables in the data and in the case of latent variables.

Vignettes "Estimation - Note 4" and "Code - Note 4"

provide the theoretical description and the code used to compute estimates based on a method introduced in @engelke2020. The model is a Markov tree parameterized cliquewise by Huesler-Reiss distributions.

The estimator applies to the data [ \Big(X_{i,v}/(n/k), v\in V\Big)\mid \max_{v \in V} X_{i,v}>n/k, \qquad i\in {1, \ldots, n}. ]

It is applicable for both cases of no latent variables in the data and in the case of latent variables.

Vignettes "Estimation - Note 5" and "Code - Note 5"

provide the theoretical description and the code used to compute estimates based on the method of moments. The model is a Markov block graph parameterized cliquewise by Huesler-Reiss distributions.

The estimator applies to the data [ (\ln X_{i,v}- \ln X_{i,u}){v\in V\setminus u}\mid X{i,u}>n/k, \qquad i\in {1, \ldots, n}, ] for every $u\in V$.

Theoretical background on the model can be found in @asejoh2.

The method is applicable both in the case of no latent variables in the data and in the case of latent variables.

Vignettes "Estimation - Note 6" and "Code - Note 6"

provide the theoretical description and the code used to compute estimates based on the method of moments and on the method of maximum likelihood for a Markov tree parameterized cliquewise by Huesler-Reiss distributions.

The estimator applies to the data [ (\ln X_{i,v}- \overline{\ln X}{i}){v\in V}\Big| \overline{\ln X}i>n/k, \qquad i\in {1, \ldots, n}. ] where $\overline{\ln X}_i=\frac{1}{d}\sum{v\in V}\ln X_{i,u}$. Note the difference with respect to the data used in MME and MLE in "Estimation - Note" 1 and 2.

The methods are applicable both in the case of no latent variables in the data and in the case of latent variables. Its theoretical motivation originates from unpublished note @segers2019mean.

Vignette "Subsets and coordinates"

contains more explanation on how to create the subsets used by many of the estimation methods and how to create the evaluation points (or coordinates) for the extremal coefficients estimator. There are several ways to construct these subsets, but there are two principles - on the neighborhood or between flow-connected points. The last notion comes from applications on river networks.

Also, the choice of subsets can impact the identifiability of the parameters in case of latent variables. We recommend to follow the principle:

The subgraph induced by the node in the subset must be such that all edge parameters that belong to the subgraph are identifiable. This means that every node within the subgraph with latent variable must be connected to at least three nodes which are also part of the subgraph.

Another principle to form subsets is

The graph induced by the nodes in the subset should form a connected graph too.

Vignette "Huesler-Reiss distributions"

presents the particular parameterization used in the package. We recommend the user to consult this to understand the relation between the magnitude of the edge parameters and the tail dependence.

Vignette "Additional functionalities"

Some functionalities intended for post-estimation analysis are: computing extremal coefficients, ail dependence coefficients, the stable tail dependence function. These can be computed parametrically or non-parametrically.

There is also a method generating random sample from a Markov tree parameterized cliquewise by Huesler-Reiss distributions.

Functionalities - summary

The main functionalities of the package are summarized here.

Regarding a model for $X$ as being Markov with respect to a tree and being parameterized cliquewise by Huesler-Reiss copulas with unit Fréchet margins:

• computes MME of edge weights based on asymptotic result about $\ln (X/X_u)\mid X_u>t$ - use method estimate.MME; applicable when there are latent variables too.

• computes MLEs based on asymptotic result about $\ln (X/X_u)\mid X_u>t$ - use methods estimate.MLE1, estimate.MLE2. They are applicable when there are latent variables.

• computes ECE based on the result that such a model is in the domain of attraction of a Huesler-Reiss distribution with some parameter matrix derived from edge weights - use method estimate.EKS, estimate.EKS_part; both are applicable when there are latent variables.

• computes cliquewise estimator by @engelke2020 based on an asymptotic result about $(X/t)\mid \max_{v\in V}X_v>t$ - use method estimate.MME; applicable when there are latent variables.

• computes MME and MLE based on an asymptotic result about $(\ln X_v- \overline{\ln X})\mid \overline{\ln X}>t$ - use method estimate.MME_ave or estimate.MLE_ave; applicable when there are latent variables.

• computes confidence intervals for the ECE - use method confInt.EKS;

• generator of random observations from $X$ - see method rHRM.HRMtree;

• for post-estimation analysis methods for computing parametric and non-parametric extremal coefficients and tail dependence coefficients - use methods taildepCoeff and extrCoeff;

Regarding a model for $X$ as being Markov with respect to a block graph and being parameterized cliquewise by Huesler-Reiss copulas with unit Fréchet margins:

• computes MME based on asymptotic result about $\ln (X/X_u)\mid X_u>t$ - use method estimate.HRMBG;

• generator of random observations from the max-stable attractor of $X$ - see method rHRM.HRMBG;

• for post-estimation analysis methods for computing parametric and non-parametric extremal coefficients extrCoeff;

References

gremes documentation built on Feb. 16, 2023, 8:06 p.m.