# VineCopula class

The class of vine copula models (yet C-Vines and D-Vines only)

## Properties

### dimension

The dimension (=d) of the vine copula.

### type

The type of the vine copula, i.e., 'C-Vine', 'CVine' or 0 and 'D-Vine' ,'DVine' or 1.

### simplified

A logical or vector of logicals, which provides the information, whether the simplifying assumption is fulfilled or not. Either the whole vine is set to simplified, i.e., all conditional copulas are chosen to be unconditional bivariate copulas (pair- copulas) or each conditional copula being part of the vine copula model is set to 1 for a pair-copula, i.e., an unconditional bivariate copula or 0 for a conditional copula.

Possible values for simplified are 1 (for unconditional bivariate copulas; pair-copulas) and 0 for conditional copulas. If simplified is set to 1 then every pair-copula being part of the PCC is set to be a partial copula. In contrast, if the simplified property is chosen to be 0 then every conditional copula is set to be a conditional copula with functional parameter (note that this can also be an unconditional copula whenever the functional parameter is a constant). Alternatively, the simplified property can also be given as a 1 x ((d-1)*(d-2)/2) matrix, where each entry is either 0 or 1 and therefore every copula is set to be a pair-copula or a conditional copula with functional parameter, separately.

### structure

A vector containing the information about the structure of the vine copula model.

Possible values for the structure in the case of a C-Vine copula are all permutations of the (1,...,d) sequence. The first entry of the structure is the root node of tree 1 (i.e., the unique node of degree d-1), the second entry is the root node of degree d-2 in tree 2, etc. For a D-Vine possible values for the structure vector are also all permutations of the (1,...,d) sequence. The choice [1 2 3 4 5] corresponds to the D-Vine consisting of the pair-copulas C_12, C_23, C_34, C_45, C_13|2, C_24|3, C_35|4, C_14|23, C_25|34 and C_15|234.

### families

A 2 x (d*(d-1)/2) cell array or matrix, where in the first row the copula family for each pair-copula is stored and in the second row, one can optionally store the degree of rotation for the pair-copula.

Possible values for families is either a 1 x (d(d-1)/2) cell array (if there are no rotated pair-copulas within the PCC) or a 2 x (d(d-1)/2) cell array, where the first row consists of the pair-copula families and the second one of the degrees of rotation of the different pair-copulas. If rotation is empty for a pair-copula, then it is the standard (unrotated) copula of the specified family. The copula families and rotation degrees can also be given in the numerical coding (e.g. 7 for the Clayton copula and 90 for 'r90'). The order of the entries into the columns is tree by tree and within the tree it is ordered according to the structure. For example if the structure is [2,1,3,4] then the entries in the family cell array correspond to the pair-copulas in the following ordering (for the C-Vine): C_21, C_23, C_24, C_13|2, C_14|2 and C_34_21. Here it is important to note that the root in each tree is always the first variable of each pair- copula, which is important for the specification of pair-copulas with asymetric dependence. For a five-dimensional D-Vine copula with structure [1 2 3 4 5], the ordering of the families has to correspond to the pair-copulas C_12, C_23, C_34, C_45, C_13|2, C_24|3, C_35|4, C_14|23, C_25|34 and C_15|234.

### parameters

A 1 x (d*(d-1)/2) cell array containing all the parameters of the unconditional pair-copulas. Alternatively, the same information (the parameters) can also be given simply in vectorized format, where the parameters are contained in the vector in the same ordering as for example the pair-copula families.

Possible values for parameters are either a 1 x (d*(d-1)/2) cell array, where each entry is a vector of the corresponding pair-copula parameter(s). Alternatively, one can also specify the parameters by vectorizing the information contained in the cell array of parameters.

### condparameterfunctionals

A 1 x ((d-1)*(d-2)/2) cell array containing parameter functionals for the conditional copulas being part of the vine. The parameter functionals can also be provided in vectorized form.

Possible values for the condparameterfunctionals are either a 1 x ((d-1)*(d-2)/2) cell array, where each entry is a vector of the corresponding pair-copula parameter functional(s). Alternatively one can also specify the parameter functional(s) by vectorizing the information contained in the cell array of parameter functional(s).

### MaxLLs

The values of the maximized log- likelihoods of the PCC evaluated at the sequentially and jointly estimated parameter vectors.

The first entry always corresponds to the maximized value of the vine copula log-likelihood evaluated at the sequentially estimated parameters, while the second entry corresponds to the jointly estimated parameters.

### SeqEstParameters

The sequentially estimated parameters for the vine copulas.

The estimates from estimating the vine copula by applying the sequential estimation approach.

## Methods

### VineCopula (VineCopulaObject)

The "function" VineCopula can be used to construct members of the VineCopula class. At least the properties dimension, type and simplified have to be specified.

### Fit (VineCopulaObject)

The method Fit can be used to estimate a vine copula model, specified as an object of the VineCopula class, using a dataset by maximizing the log-likelihood of the vine copula.

### Sim (VineCopulaObject)

The method Sim can be used to simulate from a vine copula, specified as an object of the VineCopula class.

### StructureSelect (VineCopulaObject)

The method StructureSelect can be used to find an "adequate" structure and pair-copula families (and parameters) for a given data set.

### GetPseudoObsFromVine

The method GetPseudoObsFromVine can be used to obtain pseudo- observations from the conditional copulas being part of the specified PCC.

### SeqTestOnSimplified (VineCopulaObject)

The method SeqTestOnSimplified can be used to sequentially test on the simplifying assumption by applying a vectorial independence test.

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 Nikoloulopoulos, A. K., H. Joe, H. Li (2012), "Vine copulas with asymmetric tail dependence and applications to financial return data", Computational Statistics & Data Analysis 56(11), pp. 3659-3673.

 Schepsmeier, U., J. Stöber and E. C. Brechmann (2013), VineCopula: Statistical inference of vine copulas, R package version 1.2, url: http://CRAN.R-project.org/package=VineCopula.

 Spanhel, F. and M. Kurz (2014), "Simplified vine copula approximations -- Properties and consequences", submitted for publication.

 Stöber, J., H. Joe and C. Czado (2013), "Simplified pair copula constructions -- Limitations and extensions", Journal of Multivariate Analysis 119, pp. 101-118.