phiellip: phiellip
In VecDep: Measuring Copula-Based Dependence Between Random Vectors
phiellip R Documentation
phiellip
Description
Given a q-dimensional random vector \mathbf{X} = (\mathbf{X}_{1},...,\mathbf{X}_{k}) with \mathbf{X}_{i} a d_{i}-dimensional random vector, i.e., q = d_{1} + ... + d_{k},
this function estimates the \Phi-dependence between \mathbf{X}_{1},...,\mathbf{X}_{k} by estimating the joint and marginal
meta-elliptical copula generators via the MECIP.
Usage
phiellip(sample, dim, phi, grid, params, normalize = 1)
Arguments
sample
A sample from a q-dimensional random vector \mathbf{X} (n \times q matrix with observations in rows, variables in columns).
dim
The vector of dimensions (d_{1},...,d_{k}).
phi
The function \Phi.
grid
The grid of values on which to estimate the density generators.
params
The tuning parameters to be used when estimating the density generators.
normalize
A value in \{1,2\} indicating the normalization procedure that is applied to the estimated generator (default = 1).
Details
When \mathbf{X} = (\mathbf{X}_{1}, \dots, \mathbf{X}_{k}) has a meta-elliptical copula with generator g_{\mathcal{R}}, marginal generators g_{\mathcal{R}_{i}} of \mathbf{X}_{i} for i = 1, \dots, k, and scale matrix
\mathbf{R} = \begin{pmatrix} \mathbf{R}_{11} & \mathbf{R}_{12} & \cdots & \mathbf{R}_{1k} \\
\mathbf{R}_{12}^{\text{T}} & \mathbf{R}_{22} & \cdots & \mathbf{R}_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
\mathbf{R}_{1k}^{\text{T}} & \mathbf{R}_{2k}^{\text{T}} & \cdots & \mathbf{R}_{kk} \end{pmatrix},
the \Phi-dependence between \mathbf{X}_{1}, \dots, \mathbf{X}_{k} equals
\mathcal{D}_{\Phi}\left (\mathbf{X}_{1}, \dots, \mathbf{X}_{k} \right ) = \mathbb{E} \left \{\frac{\prod_{i = 1}^{k} g_{\mathcal{R}_{i}}\left (\mathbf{Z}_{i}^{\text{T}} \mathbf{R}_{ii}^{-1} \mathbf{Z}_{i} \right ) \left | \mathbf{R} \right |^{1/2}}{g_{\mathcal{R}}\left (\mathbf{Z}^{\text{T}} \mathbf{R}^{-1} \mathbf{Z} \right ) \prod_{i = 1}^{k} \left | \mathbf{R}_{ii} \right |^{1/2}} \Phi \left (\frac{g_{\mathcal{R}} \left (\mathbf{Z}^{\text{T}} \mathbf{R}^{-1} \mathbf{Z} \right ) \prod_{i = 1}^{k} \left |\mathbf{R}_{ii} \right |^{1/2}}{\prod_{i = 1}^{k} g_{\mathcal{R}_{i}} \left (\mathbf{Z}_{i}^{\text{T}}\mathbf{R}_{ii}^{-1} \mathbf{Z}_{i} \right ) \left |\mathbf{R} \right |^{1/2} } \right )\right \},
where (recall that \mathbf{X}_{i} = (X_{i1}, \dots, X_{id_{i}}) for i = 1, \dots, k)
\mathbf{Z}_{i} = (Z_{i1}, \dots, Z_{id_{i}}) = \left(\left (Q \circ F_{i1} \right ) \left (X_{i1} \right ), \dots, \left (Q \circ F_{id_{i}} \right ) \left (X_{id_{i}} \right ) \right ),
and \mathbf{Z} = (\mathbf{Z}_{1}, \dots, \mathbf{Z}_{k}), with Q the quantile function corresponding to g_{\mathcal{R}}.
The expectation \mathbb{E} is replaced by the empirical mean using the estimated sample \widehat{\mathbf{Z}}^{(1)}, \dots, \widehat{\mathbf{Z}}^{(n)} with \widehat{\mathbf{Z}}^{(\ell)} = (\widehat{\mathbf{Z}}_{1}^{(\ell)}, \dots, \widehat{\mathbf{Z}}_{k}^{(\ell)}) for \ell = 1, \dots, n, where
\widehat{\mathbf{Z}}_{i}^{(\ell)} = \left (\widehat{Z}_{i1}^{(\ell)}, \dots, \widehat{Z}_{id_{i}}^{(\ell)} \right ) = \left ( \left (\widehat{Q} \circ \widehat{F}_{i1} \right ) \left (X_{i1}^{(\ell)} \right ), \dots, \left (\widehat{Q} \circ \widehat{F}_{id_{i}} \right ) \left (X_{id_{i}}^{(\ell)} \right ) \right ),
for i = 1, \dots, k.
Here, \widehat{Q} will be the quantile function corresponding to the final estimator for g_{\mathcal{R}}, and
\widehat{F}_{ij}(x_{ij}) = \frac{1}{n+1} \sum_{\ell = 1}^{n} 1 \left (X_{ij}^{(\ell)} \leq x_{ij} \right )
is the (rescaled) empirical cdf of X_{ij} based on a sample X_{ij}^{(1)}, \dots, X_{ij}^{(n)} for i = 1, \dots, k and j = 1, \dots, d_{i}.
The estimation of \mathbf{R} is done via its relation with the Kendall's tau matrix, see the function “KTMatrixEst.R” in
the R package ‘ElliptCopulas’ of Derumigny et al. (2024).
For estimating g_{\mathcal{R}} and g_{\mathcal{R}_{i}} for i = 1, \dots, k, the function ellcopest is used.
This function requires certain tuning parameters (a bandwidth h, a parameter a, and a parameter \delta for the shrinkage function). Suppose that there are
m marginal random vectors (among \mathbf{X}_{1}, \dots, \mathbf{X}_{k}) that are of dimension strictly larger than one.
Then, all tuning parameters should be given as
\text{params} = \text{list}(\text{"h"} = (h,h_{1},\dots,h_{m}), \text{"a"} = (a,a_{1}, \dots, a_{m}), \text{"p"} = (\delta, \delta_{1}, \dots, \delta_{m})),
i.e., (h,a,\delta) will be used for estimating g_{\mathcal{R}}, and (h_{i},a_{i},\delta_{i}) will be used for estimating g_{\mathcal{R}_{i}} for i = 1, \dots, k.
When d_{i} = 1 for a certain i \in \{1, \dots, k \}, the function “Convert_gd_To_g1.R” from the R package ‘ElliptCopulas’ is used to estimate g_{\mathcal{R}_{i}}.
In order to make g_{\mathcal{R}} identifiable, an extra normalization procedure is implemented
in line with an extra constraint on g_{\mathcal{R}}.
When normalize = 1, this corresponds to \mathbf{R} being the correlation matrix of \mathbf{Z}.
When normalize = 2, this corresponds to the identifiability condition of Derumigny & Fermanian (2022).
Value
The estimated \Phi-dependence between \mathbf{X}_{1}, \dots, \mathbf{X}_{k}.
References
Derumigny, A., Fermanian, J.-D., Ryan, V., van der Spek, R. (2024).
ElliptCopulas, R package version 0.1.4.1.
url: https://CRAN.R-project.org/package=ElliptCopulas.
De Keyser, S. & Gijbels, I. (2024).
Hierarchical variable clustering via copula-based divergence measures between random vectors.
International Journal of Approximate Reasoning 165:109090.
doi: https://doi.org/10.1016/j.ijar.2023.109090.
See Also
elldistrest for improved kernel estimation of the elliptical generator of an elliptical distribution,
ellcopest for improved kernel estimation of the elliptical generator of a meta-elliptical copula,
elliptselect for selecting optimal tuning parameters for the improved kernel estimator of the elliptical generator.
Examples
q = 4
dim = c(2,2)
# Sample size
n = 1000
# Grid on which to evaluate the elliptical generator
grid = seq(0.005,100,by = 0.005)
# Degrees of freedom
nu = 7
# Student-t generator with 7 degrees of freedom
g_q = ((nu/(nu-2))^(q/2))*(gamma((q+nu)/2)/(((pi*nu)^(q/2))*gamma(nu/2))) *
((1+(grid/(nu-2)))^(-(q+nu)/2))
# Density of squared radius
R2 = function(t,q){(gamma((q+nu)/2)/(((nu-2)^(q/2))*gamma(nu/2)*gamma(q/2))) *
(t^((q/2)-1)) * ((1+(t/(nu-2)))^(-(q+nu)/2))}
# Sample from 4-dimensional Student-t distribution with 7 degrees of freedom
# and identity covariance matrix
sample = ElliptCopulas::EllDistrSim(n,q,diag(q),density_R2 = function(t){R2(t,q)})
# Tuning parameter selection for g_R
opt_parameters_joint = elliptselect(n,q,seq((3/4)-(1/q)+0.01,1-0.01,len = 200),
seq(0.01,2,len = 200))
# Optimal tuning parameters for g_R
a = opt_parameters_joint$Opta ; p = opt_parameters_joint$Optp ;
h = opt_parameters_joint$Opth
# Tuning parameter selection for g_R_1 (same for g_R_2)
opt_parameters_marg = elliptselect(n,2,seq((3/4)-(1/2)+0.01,1-0.01,len = 200),
seq(0.01,2,len = 200))
# Optimal tuning parameters for g_R_1 (same for g_R_2)
a1 = opt_parameters_marg$Opta ; p1 = opt_parameters_marg$Optp ;
h1 = opt_parameters_marg$Opth
a2 = a1 ; p2 = p1 ; h2 = h1
params = list("h" = c(h,h1,h2), "a" = c(a,a1,a2), "p" = c(p,p1,p2))
# Mutual information between two random vectors of size 2
est_phi = phiellip(sample, dim, function(t){t * log(t)}, grid, params)
VecDep documentation built on April 4, 2025, 5:14 a.m.
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| phiellip | R Documentation |
phiellip
Description
Given a q-dimensional random vector \mathbf{X} = (\mathbf{X}_{1},...,\mathbf{X}_{k}) with \mathbf{X}_{i} a d_{i}-dimensional random vector, i.e., q = d_{1} + ... + d_{k},
this function estimates the \Phi-dependence between \mathbf{X}_{1},...,\mathbf{X}_{k} by estimating the joint and marginal
meta-elliptical copula generators via the MECIP.
Usage
phiellip(sample, dim, phi, grid, params, normalize = 1)
Arguments
sample |
A sample from a |
dim |
The vector of dimensions |
phi |
The function |
grid |
The grid of values on which to estimate the density generators. |
params |
The tuning parameters to be used when estimating the density generators. |
normalize |
A value in |
Details
When \mathbf{X} = (\mathbf{X}_{1}, \dots, \mathbf{X}_{k}) has a meta-elliptical copula with generator g_{\mathcal{R}}, marginal generators g_{\mathcal{R}_{i}} of \mathbf{X}_{i} for i = 1, \dots, k, and scale matrix
\mathbf{R} = \begin{pmatrix} \mathbf{R}_{11} & \mathbf{R}_{12} & \cdots & \mathbf{R}_{1k} \\
\mathbf{R}_{12}^{\text{T}} & \mathbf{R}_{22} & \cdots & \mathbf{R}_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
\mathbf{R}_{1k}^{\text{T}} & \mathbf{R}_{2k}^{\text{T}} & \cdots & \mathbf{R}_{kk} \end{pmatrix},
the \Phi-dependence between \mathbf{X}_{1}, \dots, \mathbf{X}_{k} equals
\mathcal{D}_{\Phi}\left (\mathbf{X}_{1}, \dots, \mathbf{X}_{k} \right ) = \mathbb{E} \left \{\frac{\prod_{i = 1}^{k} g_{\mathcal{R}_{i}}\left (\mathbf{Z}_{i}^{\text{T}} \mathbf{R}_{ii}^{-1} \mathbf{Z}_{i} \right ) \left | \mathbf{R} \right |^{1/2}}{g_{\mathcal{R}}\left (\mathbf{Z}^{\text{T}} \mathbf{R}^{-1} \mathbf{Z} \right ) \prod_{i = 1}^{k} \left | \mathbf{R}_{ii} \right |^{1/2}} \Phi \left (\frac{g_{\mathcal{R}} \left (\mathbf{Z}^{\text{T}} \mathbf{R}^{-1} \mathbf{Z} \right ) \prod_{i = 1}^{k} \left |\mathbf{R}_{ii} \right |^{1/2}}{\prod_{i = 1}^{k} g_{\mathcal{R}_{i}} \left (\mathbf{Z}_{i}^{\text{T}}\mathbf{R}_{ii}^{-1} \mathbf{Z}_{i} \right ) \left |\mathbf{R} \right |^{1/2} } \right )\right \},
where (recall that \mathbf{X}_{i} = (X_{i1}, \dots, X_{id_{i}}) for i = 1, \dots, k)
\mathbf{Z}_{i} = (Z_{i1}, \dots, Z_{id_{i}}) = \left(\left (Q \circ F_{i1} \right ) \left (X_{i1} \right ), \dots, \left (Q \circ F_{id_{i}} \right ) \left (X_{id_{i}} \right ) \right ),
and \mathbf{Z} = (\mathbf{Z}_{1}, \dots, \mathbf{Z}_{k}), with Q the quantile function corresponding to g_{\mathcal{R}}.
The expectation \mathbb{E} is replaced by the empirical mean using the estimated sample \widehat{\mathbf{Z}}^{(1)}, \dots, \widehat{\mathbf{Z}}^{(n)} with \widehat{\mathbf{Z}}^{(\ell)} = (\widehat{\mathbf{Z}}_{1}^{(\ell)}, \dots, \widehat{\mathbf{Z}}_{k}^{(\ell)}) for \ell = 1, \dots, n, where
\widehat{\mathbf{Z}}_{i}^{(\ell)} = \left (\widehat{Z}_{i1}^{(\ell)}, \dots, \widehat{Z}_{id_{i}}^{(\ell)} \right ) = \left ( \left (\widehat{Q} \circ \widehat{F}_{i1} \right ) \left (X_{i1}^{(\ell)} \right ), \dots, \left (\widehat{Q} \circ \widehat{F}_{id_{i}} \right ) \left (X_{id_{i}}^{(\ell)} \right ) \right ),
for i = 1, \dots, k.
Here, \widehat{Q} will be the quantile function corresponding to the final estimator for g_{\mathcal{R}}, and
\widehat{F}_{ij}(x_{ij}) = \frac{1}{n+1} \sum_{\ell = 1}^{n} 1 \left (X_{ij}^{(\ell)} \leq x_{ij} \right )
is the (rescaled) empirical cdf of X_{ij} based on a sample X_{ij}^{(1)}, \dots, X_{ij}^{(n)} for i = 1, \dots, k and j = 1, \dots, d_{i}.
The estimation of \mathbf{R} is done via its relation with the Kendall's tau matrix, see the function “KTMatrixEst.R” in
the R package ‘ElliptCopulas’ of Derumigny et al. (2024).
For estimating g_{\mathcal{R}} and g_{\mathcal{R}_{i}} for i = 1, \dots, k, the function ellcopest is used.
This function requires certain tuning parameters (a bandwidth h, a parameter a, and a parameter \delta for the shrinkage function). Suppose that there are
m marginal random vectors (among \mathbf{X}_{1}, \dots, \mathbf{X}_{k}) that are of dimension strictly larger than one.
Then, all tuning parameters should be given as
\text{params} = \text{list}(\text{"h"} = (h,h_{1},\dots,h_{m}), \text{"a"} = (a,a_{1}, \dots, a_{m}), \text{"p"} = (\delta, \delta_{1}, \dots, \delta_{m})),
i.e., (h,a,\delta) will be used for estimating g_{\mathcal{R}}, and (h_{i},a_{i},\delta_{i}) will be used for estimating g_{\mathcal{R}_{i}} for i = 1, \dots, k.
When d_{i} = 1 for a certain i \in \{1, \dots, k \}, the function “Convert_gd_To_g1.R” from the R package ‘ElliptCopulas’ is used to estimate g_{\mathcal{R}_{i}}.
In order to make g_{\mathcal{R}} identifiable, an extra normalization procedure is implemented
in line with an extra constraint on g_{\mathcal{R}}.
When normalize = 1, this corresponds to \mathbf{R} being the correlation matrix of \mathbf{Z}.
When normalize = 2, this corresponds to the identifiability condition of Derumigny & Fermanian (2022).
Value
The estimated \Phi-dependence between \mathbf{X}_{1}, \dots, \mathbf{X}_{k}.
References
Derumigny, A., Fermanian, J.-D., Ryan, V., van der Spek, R. (2024). ElliptCopulas, R package version 0.1.4.1. url: https://CRAN.R-project.org/package=ElliptCopulas.
De Keyser, S. & Gijbels, I. (2024). Hierarchical variable clustering via copula-based divergence measures between random vectors. International Journal of Approximate Reasoning 165:109090. doi: https://doi.org/10.1016/j.ijar.2023.109090.
See Also
elldistrest for improved kernel estimation of the elliptical generator of an elliptical distribution,
ellcopest for improved kernel estimation of the elliptical generator of a meta-elliptical copula,
elliptselect for selecting optimal tuning parameters for the improved kernel estimator of the elliptical generator.
Examples
q = 4
dim = c(2,2)
# Sample size
n = 1000
# Grid on which to evaluate the elliptical generator
grid = seq(0.005,100,by = 0.005)
# Degrees of freedom
nu = 7
# Student-t generator with 7 degrees of freedom
g_q = ((nu/(nu-2))^(q/2))*(gamma((q+nu)/2)/(((pi*nu)^(q/2))*gamma(nu/2))) *
((1+(grid/(nu-2)))^(-(q+nu)/2))
# Density of squared radius
R2 = function(t,q){(gamma((q+nu)/2)/(((nu-2)^(q/2))*gamma(nu/2)*gamma(q/2))) *
(t^((q/2)-1)) * ((1+(t/(nu-2)))^(-(q+nu)/2))}
# Sample from 4-dimensional Student-t distribution with 7 degrees of freedom
# and identity covariance matrix
sample = ElliptCopulas::EllDistrSim(n,q,diag(q),density_R2 = function(t){R2(t,q)})
# Tuning parameter selection for g_R
opt_parameters_joint = elliptselect(n,q,seq((3/4)-(1/q)+0.01,1-0.01,len = 200),
seq(0.01,2,len = 200))
# Optimal tuning parameters for g_R
a = opt_parameters_joint$Opta ; p = opt_parameters_joint$Optp ;
h = opt_parameters_joint$Opth
# Tuning parameter selection for g_R_1 (same for g_R_2)
opt_parameters_marg = elliptselect(n,2,seq((3/4)-(1/2)+0.01,1-0.01,len = 200),
seq(0.01,2,len = 200))
# Optimal tuning parameters for g_R_1 (same for g_R_2)
a1 = opt_parameters_marg$Opta ; p1 = opt_parameters_marg$Optp ;
h1 = opt_parameters_marg$Opth
a2 = a1 ; p2 = p1 ; h2 = h1
params = list("h" = c(h,h1,h2), "a" = c(a,a1,a2), "p" = c(p,p1,p2))
# Mutual information between two random vectors of size 2
est_phi = phiellip(sample, dim, function(t){t * log(t)}, grid, params)
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