Package 'cdcsis'

Conditional Distance Correlation Based Feature Screening and Conditional Independence Inference

Description

Conditional distance correlation <doi:10.1080/01621459.2014.993081> is a novel conditional dependence measurement of two multivariate random variables given a confounding variable. This package provides conditional distance correlation, performs the conditional distance correlation sure independence screening procedure for ultrahigh dimensional data <doi:10.5705/ss.202014.0117>, and conducts conditional distance covariance test for conditional independence assumption of two multivariate variable.

Author(s)

Wenhao Hu, Mian Huang, Wenliang Pan, Xueqin Wang, Canhong Wen, Yuan Tian, Heping Zhang, Jin Zhu Maintainer: Canhong Wen <[email protected]>

References

Wang, X., Pan, W., Hu, W., Tian, Y. and Zhang, H., 2015. Conditional distance correlation. Journal of the American Statistical Association, 110(512), pp.1726-1734.

Wen, C., Pan, W., Huang, M. and Wang, X., 2018. Sure independence screening adjusted for confounding covariates with ultrahigh-dimensional data. Statistica Sinica, 28, pp.293-317. URL http://www3.stat.sinica.edu.tw/statistica/J28N1/28-1.html

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Conditional Distance Covariance/Correlation Statistics

Description

Computes conditional distance covariance and conditional distance correlation statistics, which are multivariate measures of conditional dependence.

Usage

cdcov(x, y, z, width, index = 1, distance = FALSE)

cdcor(x, y, z, width, index = 1, distance = FALSE)

Arguments

x	a numeric vector, matrix, or dist object
y	a numeric vector, matrix, or dist object
z	z is a numeric vector or matrix. It is the variable being conditioned.
width	a user-specified positive value (univariate conditional variable) or vector (multivariate conditional variable) for gaussian kernel bandwidth. Its default value is relies on stats::bw.nrd0.
index	exponent on Euclidean distance, in $(0,2]$
distance	if distance = TRUE, x and y will be considered as distance matrices. Default: distance = FALSE.

Details

cdcov and cdcor compute conditional distance covariance and conditional distance correlation statistics. The sample sizes (number of rows or length of the vector) of the two variables must agree, and samples must not contain missing values. If we set distance = TRUE, arguments x, y can be a dist object recording distance between samples; otherwise, these arguments are treated as multivariate data.

Value

cdcov	conditional distance covariance test statistic.
cdcor	conditional distance correlation statistic.
cdc	conditional distance covariance/correlation vector.

Author(s)

Canhong Wen, Wenliang Pan, and Xueqin Wang

References

Wang, X., Pan, W., Hu, W., Tian, Y. and Zhang, H., 2015. Conditional distance correlation. Journal of the American Statistical Association, 110(512), pp.1726-1734.

See Also

cdcor

Examples

library(cdcsis)

############# Conditional Distance Covariance #############
set.seed(1)
x <- rnorm(25)
y <- rnorm(25)
z <- rnorm(25)
cdcov(x, y, z)
############# Conditional Distance Correlation #############
num <- 25
set.seed(1)
x <- rnorm(num)
y <- rnorm(num)
z <- rnorm(num)
cdcor(x, y, z)

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Conditional Distance Covariance Independence Test

Description

Performs the nonparametric conditional distance covariance test for conditional independence assumption

Usage

cdcov.test(
  x,
  y,
  z,
  num.bootstrap = 99,
  width,
  distance = FALSE,
  index = 1,
  seed = 1,
  num.threads = 1
)

Arguments

x	a numeric vector, matrix, or dist object
y	a numeric vector, matrix, or dist object
z	z is a numeric vector or matrix. It is the variable being conditioned.
num.bootstrap	the number of local bootstrap procedure replications. Default: num.bootstrap = 99.
width	a user-specified positive value (univariate conditional variable) or vector (multivariate conditional variable) for gaussian kernel bandwidth. Its default value is relies on stats::bw.nrd0 function when conditional variable is univariate, ks::Hpi.diag when conditional variable with at most trivariate, and stats::bw.nrd on the other cases.
distance	if distance = TRUE, x and y will be considered as distance matrices. Default: distance = FALSE.
index	exponent on Euclidean distance, in $(0,2]$
seed	the random seed
num.threads	number of threads. Default num.threads = 1.

Value

cdcov.test returns a list with class "htest" containing the following components:

statistic	conditional distance covariance statistic.
p.value	the $p$-value for the test.
replicates	the number of local bootstrap procedure replications.
size	sample sizes.
alternative	a character string describing the alternative hypothesis.
method	a character string indicating what type of test was performed.
data.name	description of data.

References

Wang, X., Pan, W., Hu, W., Tian, Y. and Zhang, H., 2015. Conditional distance correlation. Journal of the American Statistical Association, 110(512), pp.1726-1734.

See Also

cdcov

Examples

library(cdcsis)
set.seed(1)
num <- 50
################# Conditional Independent #################
## Case 1:
cov_mat <- matrix(c(1, 0.36, 0.6, 0.36, 1, 0.6, 0.6, 0.6, 1), nrow = 3)
dat <- mvtnorm::rmvnorm(n = num, sigma = cov_mat)
x <- dat[, 1]
y <- dat[, 2]
z <- dat[, 3]
cdcov.test(x, y, z)
## Case 2:
z <- rnorm(num)
x <- 0.5 * (z^3 / 7 + z / 2) + tanh(rnorm(num))
x <- x + x^3 / 3
y <- (z^3 + z) / 3 + rnorm(num)
y <- y + tanh(y / 3)
cdcov.test(x, y, z, num.bootstrap = 99)

################# Conditional Dependent #################
## Case 3:
cov_mat <- matrix(c(1, 0.7, 0.6, 0.7, 1, 0.6, 0.6, 0.6, 1), nrow = 3)
dat <- mvtnorm::rmvnorm(n = num, sigma = cov_mat)
x <- dat[, 1]
y <- dat[, 2]
z <- dat[, 3]
cdcov.test(x, y, z, width = 0.5)
## Case 4:
z <- matrix(rt(num * 4, df = 2), nrow = num)
x <- z
y <- cbind(sin(z[, 1]) + cos(z[, 2]) + (z[, 3])^2 + (z[, 4])^2, 
           (z[, 1])^2 + (z[, 2])^2 + z[, 3] + z[, 4])
z <- z[, 1:2]
cdcov.test(x, y, z, seed = 2)

################# Distance Matrix Input #################
x <- dist(x)
y <- dist(y)
cdcov.test(x, y, z, seed = 2, distance = TRUE)

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Conditional Distance Correlation Sure Independence Screening (CDC-SIS)

Description

Performs conditional distance correlation sure independence screening (CDC-SIS).

Usage

cdcsis(
  x,
  y,
  z = NULL,
  width,
  threshold = nrow(y),
  distance = FALSE,
  index = 1,
  num.threads = 1
)

Arguments

x	a numeric matrix, or a list which contains multiple numeric matrix
y	a numeric vector, matrix, or dist object
z	z is a numeric vector or matrix. It is the variable being conditioned.
width	a user-specified positive value (univariate conditional variable) or vector (multivariate conditional variable) for gaussian kernel bandwidth. Its default value is relies on stats::bw.nrd0 function when conditional variable is univariate, ks::Hpi.diag when conditional variable with at most trivariate, and stats::bw.nrd on the other cases.
threshold	the threshold of the number of predictors recuited by CDC-SIS. Should be less than or equal than the number of column of x. Default value threshold is sample size.
distance	if distance = TRUE, only y will be considered as distance matrices. Default: distance = FALSE
index	exponent on Euclidean distance, in $(0,2]$
num.threads	number of threads. Default num.threads = 1.

Value

ix	the vector of indices selected by CDC-SIS
cdcor	the conditional distance correlation for each univariate/multivariate variable in x

Author(s)

Canhong Wen, Wenliang Pan, Mian Huang, and Xueqin Wang

References

Wen, C., Pan, W., Huang, M. and Wang, X., 2018. Sure independence screening adjusted for confounding covariates with ultrahigh-dimensional data. Statistica Sinica, 28, pp.293-317. URL http://www3.stat.sinica.edu.tw/statistica/J28N1/J28N114/J28N114.html

See Also

cdcor

Examples

## Not run: 

library(cdcsis)

########## univariate explanative variables ##########
set.seed(1)
num <- 100
p <- 150
x <- matrix(rnorm(num * p), nrow = num)
z <- rnorm(num)
y <- 3 * x[, 1] + 1.5 * x[, 2] + 4 * z * x[, 5] + rnorm(num)
res <- cdcsis(x, y, z)
head(res[["ix"]], n = 10)

########## multivariate explanative variables ##########
x <- as.list(as.data.frame(x))
x <- lapply(x, as.matrix)
x[[1]] <- cbind(x[[1]], x[[2]])
x[[2]] <- NULL
res <- cdcsis(x, y, z)
head(res[["ix"]], n = 10)

########## multivariate response variables ##########
num <- 100
p <- 150
x <- matrix(rnorm(num * p), nrow = num)
z <- rnorm(num)
y1 <- 3 * x[, 1] + 5 * z * x[, 4] + rnorm(num)
y2 <- 3 * x[, 2] + 5 * x[, 3] + 2 * z + rnorm(num)
y <- cbind(y1, y2)
res <- cdcsis(x, y, z)
head(res[["ix"]], n = 10)

## End(Not run)

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