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 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.

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

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))
 
}

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

Value

matrix of inner products.