ABC {hySAINT}

ABC Evaluation

Description

Gives ABC score for each fitted model. For a model I, the ABC is defined as

ABC(I)=\sum\limits_{i=1}^n\bigg(Y_i-\hat{Y}_i^{I}\bigg)^2+2r_I\sigma^2+\lambda\sigma^2C_I.

When comparing ABC of fitted models to the same dataset, the smaller the ABC, the better fit.

Usage

ABC(
  X,
  y,
  heredity = "Strong",
  sigma,
  varind = NULL,
  interaction.ind = NULL,
  lambda = 10
)

Arguments

X

Input data. An optional data frame, or numeric matrix of dimension n observations by p main effects.

y

Response variable. A n-dimensional vector.

heredity

Whether to enforce Strong, Weak, or No heredity. Default is "Strong".

sigma

The standard deviation of the noise term. In practice, sigma is usually unknown. Users can estimate sigma from function selectiveInference::estimateSigma, then use the output as the sigma value. See examples for details.

varind

A numeric vector that specifies the indices of variables to be extracted from X. Default is "No".

interaction.ind

A two-column numeric matrix. Each row represents a unique interaction pair, with the columns indicating the index numbers of the variables involved in each interaction. Note that interaction.ind must be generated outside of this function using t(utils::combn(p,2)). See Example section for details.

lambda

A numeric value defined by users. The number needs to satisfy the condition: \lambda\geq 5.1/log(2). Default is 10.

Value

A numeric value is returned. It represents the ABC score of the fitted model.

References

Ye, C. and Yang, Y., 2019. High-dimensional adaptive minimax sparse estimation with interactions.

Examples

# When sigma is known
set.seed(0)
interaction.ind <- t(combn(4,2))
X <- matrix(rnorm(50*4,1,0.1), 50, 4)
epl <- rnorm(50,0,0.01)
y <- 1+X[,1]+X[,2]+X[,1]*X[,2] + epl
 ABC(X, y, sigma = 0.01, varind = c(1,2,5), interaction.ind = interaction.ind)

# When sigma is not known
full <- Extract(X, varind = c(1:(dim(X)[2]+dim(interaction.ind)[1])), interaction.ind)
sigma <- selectiveInference::estimateSigma(full, y)$sigmahat # Estimate sigma