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Type:

Package

Title:

Functional Conditional Independence Testing with the GHCM

Version:

3.0.1

Description:

A statistical hypothesis test for conditional independence. Given residuals from a sufficiently powerful regression, it tests whether the covariance of the residuals is vanishing. It can be applied to both discretely-observed functional data and multivariate data. Details of the method can be found in Anton Rask Lundborg, Rajen D. Shah and Jonas Peters (2022) <doi:10.1111/rssb.12544>.

License:

MIT + file LICENSE

Encoding:

UTF-8

LazyData:

true

Imports:

CompQuadForm, Rcpp, splines

Depends:

R (≥ 4.0.0)

RoxygenNote:

7.2.3

Suggests:

graphics, stats, utils, refund, testthat, knitr, rmarkdown,bookdown, ggplot2, reshape2, dplyr, tidyr

URL:

https://github.com/arlundborg/ghcm

BugReports:

https://github.com/arlundborg/ghcm/issues

VignetteBuilder:

knitr

LinkingTo:

Rcpp

NeedsCompilation:

yes

Packaged:

2023-11-02 10:57:37 UTC; lundborg

Author:

Anton Rask Lundborg [aut, cre], Rajen D. Shah [aut], Jonas Peters [aut]

Maintainer:

Anton Rask Lundborg <arl@math.ku.dk>

Repository:

CRAN

Date/Publication:

2023-11-02 13:20:03 UTC

ghcm: A package for Functional Conditional Independence Testing

Description

To learn more about ghcm, start with the vignette:'browseVignettes(package = "ghcm")'

GHCM simulated data

Description

A simulated dataset containing a combination of functional and scalarvariables. Y_1 and Y_2 are scalar random variables and are both functionsof Z. X, Z and W are functional, Z is a function of X and W is a function of Z.

Usage

ghcm_sim_dataghcm_sim_data_irregular

Format

ghcm_sim_data is a data frame with 500 rows of 5 variables:

Y_1: Numeric vector.
Y_2: Numeric vector.
Z: 500 x 101 matrix.
X: 500 x 101 matrix.
W: 500 x 101 matrix.

ghcm_sim_data_irregular is a list with 5 elements:

Y_1

Numeric vector.

Y_2

Numeric vector.

Z

500 x 101 matrix.

X

A data frame with

.obs: Integer between 1 and 500 indicating which curve the row corresponds to.
.index: Function argument that the curve is evaluated at.
.value: Value of the function.

W

A data frame with

.obs: Integer between 1 and 500 indicating which curve the row corresponds to.
.index: Function argument that the curve is evaluated at.
.value: Value of the function.

Details

Inghcm_sim_data the functional variables each consists of 101 observations onan equidistant grid on [0, 1].

Inghcm_sim_data_irregular the functional variables X and W are instead only observed ona subsample of the original equidistant grid.

Source

The generation script can be found in thedata-raw folder ofthe package.

Conditional Independence Test using the GHCM

Description

Test whether X is independent of Y given Z using the Generalised HilbertianCovariance Measure. The function is applied to residuals from regressingeach of X and Y on Z respectively. Its validity is contingent on the performanceof the regression methods. For a more in-depth explanation see the packagevignette or the paper mentioned in the references.

Usage

ghcm_test(  resid_X_on_Z,  resid_Y_on_Z,  X_limits = NULL,  Y_limits = NULL,  alpha = 0.05)

Arguments

resid_X_on_Z,resid_Y_on_Z

Residuals from regressing X (Y) on Z witha suitable regression method. If X (Y) is uni- or multivariate orfunctional on a constant, fixed grid, the residuals should be supplied asa vector or matrix with no missing values. If instead X (Y) is functionaland observed on varying grids or with missing values, the residuals shouldbe supplied as a "melted" data frame with

.obs: Integer indicating which curve the row corresponds to.
.index: Function argument that the curve is evaluated at.
.value: Value of the function.

Note that in the irregular case, a minimum of 4 observations per curve isrequired.

X_limits,Y_limits

The minimum and maximum values of the functionargument of the X (Y) curves. Ignored if X (Y) is not functional.

alpha

Numeric in the unit interval. Significance level of the test.

Value

An object of classghcm containing:

test_statistic: Numeric, test statistic of the test.
p: Numeric in the unit interval, estimated p-value ofthe test.
alpha: Numeric in the unit interval, significance levelof the test.
reject: TRUE ifp <alpha,FALSE otherwise.

References

Please cite the following paper: Anton Rask Lundborg, Rajen D. Shah andJonas Peters: "Conditional Independence Testing in Hilbert Spaces withApplications to Functional Data Analysis" Journal of the Royal StatisticalSociety: Series B (Statistical Methodology) 2022doi:10.1111/rssb.12544.

Examples

if (require(refund)) {  set.seed(1)  data(ghcm_sim_data)  grid <- seq(0, 1, length.out = 101)# Test independence of two scalars given a functional variable  m_1 <- pfr(Y_1 ~ lf(Z), data=ghcm_sim_data)  m_2 <- pfr(Y_2 ~ lf(Z), data=ghcm_sim_data)  ghcm_test(resid(m_1), resid(m_2))# Test independence of a regularly observed functional variable and a# scalar variable given a functional variable      m_X <- pffr(X ~ ff(Z), data=ghcm_sim_data, chunk.size=31000)    ghcm_test(resid(m_X), resid(m_1))  # Test independence of two regularly observed functional variables given# a functional variable       m_W <- pffr(W ~ ff(Z), data=ghcm_sim_data, chunk.size=31000)    ghcm_test(resid(m_X), resid(m_W))    data(ghcm_sim_data_irregular)  n <- length(ghcm_sim_data_irregular$Y_1)  Z_df <- data.frame(.obs=1:n)  Z_df$Z <- ghcm_sim_data_irregular$Z# Test independence of an irregularly observed functional variable and a# scalar variable given a functional variable      m_1 <- pfr(Y_1 ~ lf(Z), data=ghcm_sim_data_irregular)    m_X <- pffr(X ~ ff(Z), ydata = ghcm_sim_data_irregular$X,    data=Z_df, chunk.size=31000)    ghcm_test(resid(m_X), resid(m_1), X_limits=c(0, 1)) # Test independence of two irregularly observed functional variables given# a functional variable      m_W <- pffr(W ~ ff(Z), ydata = ghcm_sim_data_irregular$W,    data=Z_df, chunk.size=31000)    ghcm_test(resid(m_X), resid(m_W), X_limits=c(0, 1), Y_limits=c(0, 1)) }

Computes the matrix of L2 inner products of the splines given in list_of_splinesas produced by splines::interpSpline. The splines are assumed to befunctions on the interval [from, to].

Description

Computes the matrix of L2 inner products of the splines given in list_of_splinesas produced by splines::interpSpline. The splines are assumed to befunctions on the interval [from, to].

Usage

inner_product_matrix_splines(list_of_splines, from, to)

Arguments

list_of_splines

list of interpSpline objects.

from,to

limits of integration.

Value

matrix of inner products.