Returns a list of reverse cumulative margins of a CB copula. The nth entry is thus the copula of X1,...,Xn

Description

Returns a list of reverse cumulative margins of a CB copula. The nth entry is thus the copula of X1,...,Xn

Usage

.CB_make_cumulative_df(CB)

Arguments

Value

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.

Description

Calculates an empirical CB approximation with adaptive bin sizes. This will be faster on data with many ties.

Usage

.EACBC(X, resolution)

Arguments

Value

A matrix of dimension resolution^rho

Returns non 0 entries of the EACBC

Description

Returns non 0 entries of the EACBC

Usage

.EACBC_nonzero(X, resolution)

Arguments

Value

A list of local kernel masses

Calculates the empirical checkerboard approximation to some data.

Description

Calculates the empirical checkerboard approximation to some data.

Usage

.ECBC(X, resolution)

Arguments

Value

A matrix of dimension resolution^rho

Returns the sizes of the adaptive bins used for the adaptive ECBC for one vector.

Description

Returns the sizes of the adaptive bins used for the adaptive ECBC for one vector.

Usage

.adaptive_masses(X, resolution)

Arguments

Value

A numeric vector of bin sizes

Computes the D1-difference of two CB matrizes on a local CB dimension

Description

Computes the D1-difference of two CB matrizes on a local CB dimension

Usage

.local_kernel_integral(k1, k2, y)

Arguments

Value

number indicating the difference between k1 and k2

Creates a random CB copula of resolution 2^steps

Description

Creates a random CB copula of resolution 2^steps

Usage

.random_CB(rho, steps, de, ie)

Arguments

Value

A matrix of dimension (2^steps)^rho

Generate a sample of some CB copula-

Description

Generate a sample of some CB copula-

Usage

.sample_CB(CB, n)

Arguments

Value

Matrix of dimension nxm where m is the dimension of CB

Compute empirical checkerboard copula in arbitrary dimension

Description

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.

Usage

ECBC(X, resolution, bin.size = "fixed")

Arguments

Value

array of dimension resolution^rho.

Examples

n <- 1000
x1 <- runif(n)
x2 <- runif(n)
y <- x1 + x2 + rnorm(n)
M <- ECBC(X = cbind(x1,x2,y), resolution = 8)

Variable selection using the qmd-dependence values

Description

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.

Usage

feature_selection(
  X,
  y,
  method = "combVar",
  bin.size = "fixed",
  plot = TRUE,
  na.exclude = FALSE,
  max_num_features = NULL,
  plot.title = NULL,
  plot.color = "hotpink"
)

Arguments

Details

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.

Value

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.

Examples

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)

Quantification of Multivariate Dependence

Description

Function for estimating the non-parametric copula-based multivariate measure of dependence \zeta1. 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.

Usage

qmd(
  X,
  y,
  ties.correction = FALSE,
  resolution = NULL,
  p.value = FALSE,
  R = 1000,
  print = TRUE,
  na.exclude = FALSE
)

Arguments

Details

In the following we will simply write q for the dependence measure \zeta1. 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.

Value

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

References

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.

Examples

#(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

Description

Equivalent to rank(x, ties.method = "max") but not as stupidly slow

Usage

qmdrank(x)

Arguments

Value

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

Description

Returns a vector

Usage

seq_until_changes(x)

Arguments

Value

A sequence along x. If consecutive values in x are equal the maximal value is used.

Multivariate dependence measure

Description

Function for estimating the non-parametric copula-based multivariate measure of dependence \zeta1. 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).

Usage

zeta1(X, y, ties.correction = FALSE, bin.size = "fixed", resolution = NULL)

Arguments

Details

see function qmd(...).

Value

A numeric value indicating the extent of dependence between the vector X and the variable y (or, equivalently, the influence of X on y).

References

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.

Examples

#(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)