| Title: | Quantification of Multivariate Dependence |
|---|---|
| Description: | A multivariate copula-based dependence measure. For more information, see Griessenberger, Junker, Trutschnig (2022), On a multivariate copula-based dependence measure and its estimation, Electronic Journal of Statistics, 16, 2206-2251. |
| Authors: | Nicolas Dietrich [aut, cre], Florian Griessenberger [aut], Robert R. Junker [aut], Valentin Petztel [aut], Wolfgang Trutschnig [aut] |
| Maintainer: | Nicolas Dietrich <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.1.2 |
| Built: | 2024-11-23 06:43:50 UTC |
| Source: | CRAN |
Returns the sizes of the adaptive bins used for the adaptive ECBC for one vector.
.adaptive_masses(X, resolution).adaptive_masses(X, resolution)
X |
A vector, representing one sample of one variable |
resolution |
The resolution of the CB approximation |
A numeric vector of bin sizes
Returns a list of reverse cumulative margins of a CB copula. The nth entry is thus the copula of X1,...,Xn
.CB_make_cumulative_df(CB).CB_make_cumulative_df(CB)
CB |
A matrix of CB weights. |
A list of CB weight matrixes of ascending dimension
Calculates an empirical CB approximation with adaptive bin sizes. This will be faster on data with many ties.
.EACBC(X, resolution).EACBC(X, resolution)
X |
A nxrho matrix of n samples of rho variables |
resolution |
The resolution of the CB approximation |
A matrix of dimension resolution^rho
Returns non 0 entries of the EACBC
.EACBC_nonzero(X, resolution).EACBC_nonzero(X, resolution)
X |
A nxrho matrix of n samples of rho variables |
resolution |
The resolution of the CB approximation |
A list of local kernel masses
Calculates the empirical checkerboard approximation to some data.
.ECBC(X, resolution).ECBC(X, resolution)
X |
A nxrho matrix of n samples of rho variables |
resolution |
The resolution of the CB approximation |
A matrix of dimension resolution^rho
Computes the D1-difference of two CB matrizes on a local CB dimension
.local_kernel_integral(k1, k2, y).local_kernel_integral(k1, k2, y)
k1 |
Vector of local CB weights of first matrix |
k2 |
Vector of local CB weights of second matrix |
y |
Vector indicating the bin sizes of the local dimension |
number indicating the difference between k1 and k2
Creates a random CB copula of resolution 2^steps
.random_CB(rho, steps, de, ie).random_CB(rho, steps, de, ie)
rho |
The number of variables |
steps |
Number of iteration steps, the final resolution will be 2^steps |
de |
Exponent to increase dependence |
ie |
Exponent to increase independence |
A matrix of dimension (2^steps)^rho
Generate a sample of some CB copula-
.sample_CB(CB, n).sample_CB(CB, n)
CB |
A weight matrix of a CB copula |
n |
The number of samples to be generated |
Matrix of dimension nxm where m is the dimension of CB
The function ECBC computes the mass distribution of the empirical (checkerboard) copula, given a rho-dimensional sample X. If resolution equals sample size, the bi-linearly extended empirical copula is returned. Note, if there are ties in the sample an adjusted empirical copula is calculated. If bin.size is set to "adaptive" the sizes of the bins will be adjusted to fit the data without overspilling into neighboring bins. This might affects the result, but is more efficient with samples having many ties as no adjustment is needed.
ECBC(X, resolution, bin.size = "fixed")ECBC(X, resolution, bin.size = "fixed")
X |
a numeric matrix of dimension rho indicating a sample of rho variables |
resolution |
an integer indicating the resolution N of the checkerboard copula |
bin.size |
either "fixed" or "adaptive", indicating whether the checkerboard copula may vary its bin sizes (defaults to "fixed") |
array of dimension resolution^rho.
n <- 1000 x1 <- runif(n) x2 <- runif(n) y <- x1 + x2 + rnorm(n) M <- ECBC(X = cbind(x1,x2,y), resolution = 8)n <- 1000 x1 <- runif(n) x2 <- runif(n) y <- x1 + x2 + rnorm(n) M <- ECBC(X = cbind(x1,x2,y), resolution = 8)
Given a d-dimensional random vector X containing the explanatory variables and a uni-variate response variable y, this function uses the qmd-dependence values to select the most relevant (influential) explanatory variables. Two different methods are available and are explained in the section Details.
feature_selection( X, y, method = "combVar", bin.size = "fixed", plot = TRUE, na.exclude = FALSE, max_num_features = NULL, plot.title = NULL, plot.color = "hotpink" )feature_selection( X, y, method = "combVar", bin.size = "fixed", plot = TRUE, na.exclude = FALSE, max_num_features = NULL, plot.title = NULL, plot.color = "hotpink" )
X |
a numeric matrix or data.frame of dimension d containing the explanatory variables |
y |
a numeric vector containing the uni-variate response variable |
method |
possible options are c("combVar", "addVar"), see Details. |
bin.size |
either "fixed", "adaptive" or "sparse.adaptive", indicating whether the checkerboard copula may vary its bin sizes (defaults to "fixed"). Setting this to "adaptive" might affect the results but will be faster if the sample has many ties. |
plot |
logical indicating whether the feature selection plot is printed |
na.exclude |
logical if all rows containing NAs should be removed. |
max_num_features |
maximal number of explanatory variables to be selected |
plot.title |
a label for the title |
plot.color |
a colour for the selected variables |
method 1 (default) - "combVar": computes all qmd-dependence scores, i.e., calculates the dependence of every combination of explanatory variables to the response variable y and selects for each number of explanatory variables the combination with the greatest dependence score. This procedure is computational expensive and is only available up to 15 explanatory variables.
method 2 - "addVar": stepwise procedure which calculates all bi-variate dependence values q(X_i,Y) and selects the variable X_j exhibiting the greatest dependence value. In the next step all three-dimensional combinations q((X_j, X_i), Y) (for every i =1,..., d and i not j) are computed and the variable exhibiting again the greatest dependence score is added. In this manner the procedure works up to dimension d.
a list containing a data.frame (result) and the corresponding plots. The data.frame result contains the number of explanatory variables (nVars), the combination of selected variables (selVars), the dependence measure zeta1 (qmd) of the selected variables to the response y and the resolution of the empirical checkerboard copula (ECBC_resolution). For the method "combVar" the dependence value zeta1 (qmd) is returned for all combinations of explanatory variables and is sorted in decreasing order according to zeta1.
n <- 1000 x1 <- runif(n) x2 <- rexp(n) x3 <- x1 + log(x2) + rnorm(n) x4 <- rnorm(n) x5 <- x4^2 x6 <- x1 + x5 + rnorm(n) x7 <- 1:n y <- x2 + x4*x7 + runif(n) X <- data.frame(x1,x2,x3,x4,x5,x6,x7) fit <- feature_selection(X, y, method = "combVar", plot = TRUE) fit <- feature_selection(X, y, method = "addVar", plot = TRUE)n <- 1000 x1 <- runif(n) x2 <- rexp(n) x3 <- x1 + log(x2) + rnorm(n) x4 <- rnorm(n) x5 <- x4^2 x6 <- x1 + x5 + rnorm(n) x7 <- 1:n y <- x2 + x4*x7 + runif(n) X <- data.frame(x1,x2,x3,x4,x5,x6,x7) fit <- feature_selection(X, y, method = "combVar", plot = TRUE) fit <- feature_selection(X, y, method = "addVar", plot = TRUE)
Function for estimating the non-parametric copula-based multivariate measure of dependence 1.
This measure quantifies the extent of dependence between a d-dimensional random vector X and a uni-variate random variable y (i.e.,
it measures the influence of d explanatory variables X1,...,Xd on a univariate variable y). Further details can be found in the section Details and the corresponding references.
qmd( X, y, ties.correction = FALSE, resolution = NULL, p.value = FALSE, R = 1000, print = TRUE, na.exclude = FALSE )qmd( X, y, ties.correction = FALSE, resolution = NULL, p.value = FALSE, R = 1000, print = TRUE, na.exclude = FALSE )
X |
a numeric matrix or data.frame of dimension d containing the explanatory variables |
y |
a numeric vector containing the uni-variate response variable |
ties.correction |
logical indicating if the measure of dependence should be calculated with ties-correction (experimental version). Default = FALSE. |
resolution |
an integer indicating the resolution N of the checkerboard aggregation. We recommend to use the default configuration (resolution = NULL), which uses the resolution N(n) = floor(n^(1/(d+1))), where d denotes the number of explanatory variables. |
p.value |
logical indicating if a p-value is returned using permutations of Y |
R |
integer indicating the number of repetitions for the calculation of the p-value (default = 1000) |
print |
logical indicating whether the results of the function are printed |
na.exclude |
logical if all rows containing NAs should be removed. |
In the following we will simply write q for the dependence measure 1. Furthermore, X denotes a random vector consisting of d random variables and y denotes a univariate random variable.
Then the theoretical dependence measure q fulfills the following essential properties of a dependence measure:
[N] q(X,y) attains values in [0,1] (normalization).
[I] q(X,y) = 0 if and only if X and y are independent (independence).
[C] q(X,y) = 1 if and only if y is a function of X (complete dependence).
Further properties of q and the exact mathematical definition can be found in Griessenberger et al. (2022). This function qmd() contains the empirical checkerboard-estimator (ECB-estimator), which is strongly consistent and attains always positive values between 0 and 1. Note, that interpretation of low values has to be done with care and always under consideration of the sample size. For instance, values of 0.2 can point towards independence in small sample settings. An additional p-value (testing for independence and being based on permutations of y) helps in order to correctly understand the dependence values. Since independence constitutes the null hypothesis a p-value above the significance level (e.g., 0.05) indicates independence between X and y.
qmd returns a list object containing the following components:
input: data containing the explanatory variables (X)
output: data containing the response (y)
q(X,y): dependence measure indicating the extent of dependence between X and y
results: data.frame containing the dependence measure and the corresponding p-value
resolution: an integer indicating the resolution of the aggregated checkerboard copula
Sample size
Griessenberger, F., Junker, R.R. and Trutschnig, W. (2022). On a multivariate copula-based dependence measure and its estimation, Electronic Journal of Statistics, 16, 2206-2251.
#(complete dependence for dimension 4) n <- 300 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- x1 + x2 + x3 qmd(X = cbind(x1,x2,x3), y = y, p.value = TRUE) #(independence for dimension 4) n <- 500 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- runif(n) qmd(X = cbind(x1,x2,x3), y = y, p.value = TRUE) #(binary output (classification) for dimension 3) n <- 500 x1 <- runif(n) x2 <- runif(n) y <- ifelse(x1 + x2 < 1, 0, 1) qmd(X = cbind(x1,x2), y = y, p.value = TRUE) #(independence) y <- runif(n) qmd(X = cbind(x1,x2), y = y, p.value = TRUE)#(complete dependence for dimension 4) n <- 300 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- x1 + x2 + x3 qmd(X = cbind(x1,x2,x3), y = y, p.value = TRUE) #(independence for dimension 4) n <- 500 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- runif(n) qmd(X = cbind(x1,x2,x3), y = y, p.value = TRUE) #(binary output (classification) for dimension 3) n <- 500 x1 <- runif(n) x2 <- runif(n) y <- ifelse(x1 + x2 < 1, 0, 1) qmd(X = cbind(x1,x2), y = y, p.value = TRUE) #(independence) y <- runif(n) qmd(X = cbind(x1,x2), y = y, p.value = TRUE)
Equivalent to rank(x, ties.method = "max") but not as stupidly slow
qmdrank(x)qmdrank(x)
x |
A numeric vector |
An integer vector specifying for each value in x the rank within x. If one value appears multiple time the maximum is used.
Returns a vector
seq_until_changes(x)seq_until_changes(x)
x |
A usually sorted vector |
A sequence along x. If consecutive values in x are equal the maximal value is used.
Function for estimating the non-parametric copula-based multivariate measure of dependence 1.
This measure quantifies the extent of dependence between a d-dimensional random vector X and a uni-variate random variable y (i.e.,
it measures the influence of d explanatory variables X1,...,Xd on a univariate variable y).
zeta1(X, y, ties.correction = FALSE, bin.size = "fixed", resolution = NULL)zeta1(X, y, ties.correction = FALSE, bin.size = "fixed", resolution = NULL)
X |
a numeric matrix or data.frame of dimension d containing the explanatory variables |
y |
a numeric vector containing the uni-variate response variable |
ties.correction |
logical indicating if the measure of dependence should be calculated with ties-correction (experimental version). Default = FALSE. |
bin.size |
either "fixed", "adaptive" or "sparse.adaptive", indicating whether the checkerboard copula may vary its bin sizes (defaults to "fixed"). Setting this to "adaptive" might affect the results but will be faster if the sample has many ties. |
resolution |
an integer indicating the resolution N of the checkerboard aggregation. We recommend to use the default configuration (resolution = NULL), which uses the resolution N(n) = floor(n^(1/(d+1))), where d denotes the number of explanatory variables. |
see function qmd(...).
A numeric value indicating the extent of dependence between the vector X and the variable y (or, equivalently, the influence of X on y).
Griessenberger, F., Junker, R.R. and Trutschnig, W. (2022). On a multivariate copula-based dependence measure and its estimation, Electronic Journal of Statistics, 16, 2206-2251.
#(complete dependence for dimension 4) n <- 300 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- x1 + x2 + x3 zeta1(X = cbind(x1,x2,x3), y = y) #(independence for dimension 4) n <- 500 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- runif(n) zeta1(X = cbind(x1,x2,x3), y = y) #(binary output for dimension 3) n <- 500 x1 <- runif(n) x2 <- runif(n) y <- ifelse(x1 + x2 < 1, 0, 1) zeta1(X = cbind(x1,x2), y = y)#(complete dependence for dimension 4) n <- 300 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- x1 + x2 + x3 zeta1(X = cbind(x1,x2,x3), y = y) #(independence for dimension 4) n <- 500 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- runif(n) zeta1(X = cbind(x1,x2,x3), y = y) #(binary output for dimension 3) n <- 500 x1 <- runif(n) x2 <- runif(n) y <- ifelse(x1 + x2 < 1, 0, 1) zeta1(X = cbind(x1,x2), y = y)