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Ball Correlation based Sure Independence Screening (BCor-SIS)

Source: R/bcorsis.R
bcorsis.Rd

Generic non-parametric sure independence screening (SIS) procedure based on Ball Correlation. Ball correlation is a generic measure of dependence in Banach spaces.

Usage

bcorsis(
  x,
  y,
  d = "small",
  weight = c("constant", "probability", "chisquare"),
  method = "standard",
  distance = FALSE,
  category = FALSE,
  parms = list(d1 = 5, d2 = 5, df = 3),
  num.threads = 0
)

Arguments

x

a numeric matrix or data.frame included \(n\) rows and \(p\) columns. Each row is an observation vector and each column corresponding to a explanatory variable, generally \(p >> n\).

y

a numeric vector, matrix, or data.frame.

d

the hard cutoff rule suggests selecting \(d\) variables. Setting d = "large" or d = "small" means n - 1 or floor(n/log(n)) variables are selected. If d is a integer, d variables are selected. Default: d = "small".

weight

a logical or character string used to choose the weight form of Ball Covariance statistic.. If input is a character string, it must be one of "constant", "probability", "chisquare", "rbf". Any unambiguous substring can be given. If input is a logical value, it is equivalent to weight = "probability" if weight = TRUE while equivalent to weight = "constant" if weight = FALSE. Default: weight = FALSE.

method

specific method for the BCor-SIS procedure. It must be one of "standard", "lm", "gam", "interaction", or "survival". Setting method = "standard" means performing standard SIS procedure while the options "lm" and "gam" mean carrying out iterative SIS procedure with ordinary linear regression and generalized additive models, respectively. The options "interaction" and "survival" are designed for detecting variables with potential linear interaction and associated with left censored responses, respectively. Any unambiguous substring can be given. Default: method = "standard".

distance

if distance = TRUE, y will be considered as a distance matrix. Arguments only available when method = "standard" and method = "interaction". Default: distance = FALSE.

category

a logical value or integer vector indicating columns to be selected as categorical variables. If category is an integer vector, the positive/negative integers select/discard the corresponding columns; If category is a logical value, category = TRUE select all columns, category = FALSE select none column. Default: category = FALSE.

parms

parameters list only available when method = "lm" or "gam". It contains three parameters: d1, d2, and df. d1 is the number of initially selected variables, d2 is the number of variables added in each iteration. df is a degree freedom of basis in generalized additive models playing a role only when method = "gam". Default: parms = list(d1 = 5, d2 = 5, df = 3).

num.threads

number of threads. If num.threads = 0, then all of available cores will be used. Default num.threads = 0.

Value

ix

the indices vector corresponding to variables selected by BCor-SIS.

method

the method used.

weight

the weight used.

complete.info

a list mainly containing a \(p \times 3\) matrix, where each row is a variable and each column is a weight Ball Correlation statistic. If method = "gam" or method = "lm", complete.info is an empty list.

Details

bcorsis performs a model-free generic sure independence screening procedure, BCor-SIS, to pick out variables from x which are potentially associated with y. BCor-SIS relies on Ball correlation, a universal dependence measure in Banach spaces. Ball correlation (BCor) ranges from 0 to 1. A larger BCor implies they are likely to be associated while Bcor is equal to 0 implies they are unassociated. (See bcor for details.) Consequently, BCor-SIS pick out variables with larger Bcor values with y.

Theory and numerical result indicate that BCor-SIS has following advantages:

  • BCor-SIS can retain the efficient variables even when the dimensionality (i.e., ncol(x)) is an exponential order of the sample size (i.e., exp(nrow(x)));

  • It is distribution-free and model-free;

  • It is very robust;

  • It is works well for complex data, such as shape and survival data;

If x is a matrix, the sample sizes of x and y must agree. If x is a list object, each element of this list must with the same sample size. x and y must not contain missing or infinite values.

When method = "survival", the matrix or data.frame pass to y must have exactly two columns, where the first column is event (failure) time while the second column is a dichotomous censored status.

Note

bcorsis simultaneously computing Ball Correlation statistics with "constant", "probability", and "chisquare" weights. Users can get other Ball Correlation statistics with different weight in the complete.info element of output. We give a quick example below to illustrate.

References

Wenliang Pan, Xueqin Wang, Weinan Xiao & Hongtu Zhu (2018) A Generic Sure Independence Screening Procedure, Journal of the American Statistical Association, DOI: 10.1080/01621459.2018.1462709

See also

bcor

Author

Wenliang Pan, Weinan Xiao, Xueqin Wang, Hongtu Zhu, Jin Zhu

Examples

if (FALSE) { # \dontrun{

############### Quick Start for bcorsis function ###############
set.seed(1)
n <- 150
p <- 3000
x <- matrix(rnorm(n * p), nrow = n)
eps <- rnorm(n)
y <- 3 * x[, 1] + 5 * (x[, 3])^2 + eps
res <- bcorsis(y = y, x = x)
head(res[["ix"]])
head(res[["complete.info"]][["statistic"]])

############### BCor-SIS: Censored Data Example ###############
data("genlung")
result <- bcorsis(x = genlung[["covariate"]], y = genlung[["survival"]], 
                  method = "survival")
index <- result[["ix"]]
top_gene <- colnames(genlung[["covariate"]])[index]
head(top_gene, n = 1)


############### BCor-SIS: Interaction Pursuing ###############
set.seed(1)
n <- 150
p <- 3000
x <- matrix(rnorm(n * p), nrow = n)
eps <- rnorm(n)
y <- 3 * x[, 1] * x[, 5] * x[, 10] + eps
res <- bcorsis(y = y, x = x, method = "interaction")
head(res[["ix"]])

############### BCor-SIS: Iterative Method ###############
library(mvtnorm)
set.seed(1)
n <- 150
p <- 3000
sigma_mat <- matrix(0.5, nrow = p, ncol = p)
diag(sigma_mat) <- 1
x <- rmvnorm(n = n, sigma = sigma_mat)
eps <- rnorm(n)
rm(sigma_mat); gc(reset = TRUE)
y <- 3 * (x[, 1])^2 + 5 * (x[, 2])^2 + 5 * x[, 8] - 8 * x[, 16] + eps
res <- bcorsis(y = y, x = x, method = "lm", d = 15)
res <- bcorsis(y = y, x = x, method = "gam", d = 15)
res[["ix"]]

############### Weighted BCor-SIS: Probability weight ###############
set.seed(1)
n <- 150
p <- 3000
x <- matrix(rnorm(n * p), nrow = n)
eps <- rnorm(n)
y <- 3 * x[, 1] + 5 * (x[, 3])^2 + eps
res <- bcorsis(y = y, x = x, weight = "prob")
head(res[["ix"]])
# Alternative, chisq weight:
res <- bcorsis(y = y, x = x, weight = "chisq")
head(res[["ix"]])

############### BCor-SIS: GWAS data ###############
set.seed(1)
n <- 150
p <- 3000
x <- sapply(1:p, function(i) {
  sample(0:2, size = n, replace = TRUE)
})
eps <- rnorm(n)
y <- 6 * x[, 1] - 7 * x[, 2] + 5 * x[, 3] + eps
res <- bcorsis(x = x, y = y, category = TRUE)
head(res[["ix"]])
head(res[["complete.info"]][["statistic"]])

x <- cbind(matrix(rnorm(n * 2), ncol = 2), x)
# remove the first two columns:
res <- bcorsis(x = x, y = y, category = c(-1, -2))
head(res[["ix"]])

x <- cbind(x[, 3:5], matrix(rnorm(n * p), ncol = p))
res <- bcorsis(x = x, y = y, category = 1:3)
head(res[["ix"]], n = 10)
} # }