copulas.multivariate.vine module

VineCopula module.

class copulas.multivariate.vine.VineCopula(vine_type, random_state=None)[source]

Bases: Multivariate

Vine copula model.

A \(vine\) is a graphical representation of one factorization of the n-variate probability distribution in terms of \(n(n − 1)/2\) bivariate copulas by means of the chain rule.

It consists of a sequence of levels and as many levels as variables. Each level consists of a tree (no isolated nodes and no loops) satisfying that if it has \(n\) nodes there must be \(n − 1\) edges.

Each node in tree \(T_1\) is a variable and edges are couplings of variables constructed with bivariate copulas.

Each node in tree \(T_{k+1}\) is a coupling in \(T_{k}\), expressed by the copula of the variables; while edges are couplings between two vertices that must have one variable in common, becoming a conditioning variable in the bivariate copula. Thus, every level has one node less than the former. Once all the trees are drawn, the factorization is the product of all the nodes.

Parameters:

vine_type (str) – type of the vine copula, could be ‘center’,’direct’,’regular’
random_state (int or np.random.RandomState) – Random seed or RandomState to use.

model

Distribution to compute univariates.

Type:: copulas.univariate.Univariate

u_matrix

Univariates.

Type:: numpy.array

n_sample

Number of samples.

Type:: int

n_var

Number of variables.

Type:: int

columns

Names of the variables.

Type:: pandas.Series

tau_mat

Kendall correlation parameters for data.

Type:: numpy.array

truncated

Max level used to build the vine.

Type:: int

depth

Vine depth.

Type:: int

trees

List of trees used by this vine.

Type:: list[Tree]

ppfs

percent point functions from the univariates used by this vine.

Type:: list[callable]

fit(X, truncated=3)[source]

Fit a vine model to the data.

1. Transform all the variables by means of their marginals. In other words, compute

\[u_i = F_i(x_i), i = 1, ..., n\]

and compose the matrix \(u = u_1, ..., u_n,\) where \(u_i\) are their columns.

Parameters:

X (numpy.ndarray) – Data to be fitted to.
truncated (int) – Max level to build the vine.

classmethod from_dict(vine_dict)[source]

Create a new instance from a parameters dictionary.

Parameters:: params (dict) – Parameters of the Vine, in the same format as the one returned by the to_dict method.
Returns:: Instance of the Vine defined on the parameters.
Return type:: Vine

get_likelihood(uni_matrix)[source]: Compute likelihood of the vine.

sample(num_rows)[source]

Sample new rows.

Parameters:: num_rows (int) – Number of rows to sample
Returns:: sampled rows.
Return type:: pandas.DataFrame

to_dict()[source]

Return a dict with the parameters to replicate this Vine.

Returns:: Parameters of this Vine.
Return type:: dict

train_vine(tree_type)[source]

Build the vine.

For the construction of the first tree \(T_1\), assign one node to each variable and then couple them by maximizing the measure of association considered. Different vines impose different constraints on this construction. When those are applied different trees are achieved at this level.
Select the copula that best fits to the pair of variables coupled by each edge in \(T_1\).
Let \(C_{ij}(u_i , u_j )\) be the copula for a given edge \((u_i, u_j)\) in \(T_1\). Then for every edge in \(T_1\), compute either

\[\begin{split}{v^1}_{j|i} = \\frac{\\partial C_{ij}(u_i, u_j)}{\\partial u_j}\end{split}\]

or similarly \({v^1}_{i|j}\), which are conditional cdfs. When finished with all the edges, construct the new matrix with \(v^1\) that has one less column u.
Set k = 2.
Assign one node of \(T_k\) to each edge of \(T_ {k−1}\). The structure of \(T_{k−1}\) imposes a set of constraints on which edges of \(T_k\) are realizable. Hence the next step is to get a linked list of the accesible nodes for every node in \(T_k\).
As in step 1, nodes of \(T_k\) are coupled maximizing the measure of association considered and satisfying the constraints impose by the kind of vine employed plus the set of constraints imposed by tree \(T_{k−1}\).
Select the copula that best fit to each edge created in \(T_k\).
Recompute matrix \(v_k\) as in step 4, but taking \(T_k\) and \(vk−1\) instead of \(T_1\) and u.
Set \(k = k + 1\) and repeat from (5) until all the trees are constructed.

Parameters:: tree_type (str or TreeTypes) – Type of trees to use.