Package 'ghcm'

ghcm: A package for Functional Conditional Independence Testing

Description

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

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GHCM simulated data

Description

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

Usage

ghcm_sim_data

ghcm_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

In ghcm_sim_data the functional variables each consists of 101 observations on an equidistant grid on [0, 1].

In ghcm_sim_data_irregular the functional variables X and W are instead only observed on a subsample of the original equidistant grid.

Source

The generation script can be found in the data-raw folder of the package.

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Conditional Independence Test using the GHCM

Description

Test whether X is independent of Y given Z using the Generalised Hilbertian Covariance Measure. The function is applied to residuals from regressing each of X and Y on Z respectively. Its validity is contingent on the performance of the regression methods. For a more in-depth explanation see the package vignette 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 with a suitable regression method. If X (Y) is uni- or multivariate or functional on a constant, fixed grid, the residuals should be supplied as a vector or matrix with no missing values. If instead X (Y) is functional and observed on varying grids or with missing values, the residuals should be 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 is required.
X_limits, Y_limits	The minimum and maximum values of the function argument 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 class ghcm containing:

test_statistic

Numeric, test statistic of the test.

p

Numeric in the unit interval, estimated p-value of the test.

alpha

Numeric in the unit interval, significance level of the test.

reject

TRUE if p < alpha, FALSE otherwise.

References

Please cite the following paper: Anton Rask Lundborg, Rajen D. Shah and Jonas Peters: "Conditional Independence Testing in Hilbert Spaces with Applications to Functional Data Analysis" Journal of the Royal Statistical Society: Series B (Statistical Methodology) 2022 doi: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))
 
}

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

Description

Computes the matrix of L2 inner products of the splines given in list_of_splines as produced by splines::interpSpline. The splines are assumed to be functions 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.

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