OPG: Outer Product of Gradients Regression

In abnormally-distributed/cvreg: Cross Validation and Robust Estimation Utilities

Description

Unlike most other sufficient dimension reduction methods in this package the outer product of gradients (OPG) method does not utilize the inverse regression of E[X | Y] to estimate the central subspace. Rather, it uses a kernel method to calculate the central mean subspace of E[Y | X] directly.

1. The algorithm procedes by using a kernel function K to calculate two matrices, A and B:

\hat { A } = ∑ _ { j = 1 } ^ { n }≤ft[\begin{bmatrix}1 \\ x_{j}-X_{i}\end{bmatrix} \begin{bmatrix}1 \\ x_{j}-X_{i}\end{bmatrix}^{T} κ ≤ft( \frac{X_{j} - X_{i}}{h} \right) \right]

\hat { B } = ∑ _ { j = 1 } ^ { n } ≤ft[ \begin{bmatrix}1 \\ x_{j}-X_{i}\end{bmatrix}^{T} Y_{j} κ ≤ft(\frac{x_{j} - X_{i}}{ h } \right) \right]

2. Next, calculate β, which consists of the 2nd, . . . , p+1-th entries of the vector A^{-1} \cdot B. Then sum β β^{T}.

3. Perform the eigendecomposition of ∑{β β^{T}}. The eigenvectors are the estimate of the central mean subspace.

Usage

OPG(
  formula,
  data,
  rank = "all",
  kern = c("rbf", "gt", "cauchy", "laplace", "bessel", "anova"),
  order = 1,
  df = 3
)

Arguments

formula	a model formula
data	a data frame
rank	the desired number of sufficient predictors to return. the default is "all".
kern	the kernel function to be used. must be one of "rbf" (the default), "gt", "cauchy, "laplace", "bessel", or "anova".
order	the order to be used for the bessel function.
df	the degrees of freedom to be used for the bessel kernel and anova kernel. defaults to 3.

Value

an sdr object

References

Xia, Y., Tong, H., Li, W., & Zhu, L. (2002). An Adaptive Estimation of Dimension Reduction Space. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 64(3), 363-410.

abnormally-distributed/cvreg documentation built on May 3, 2020, 3:45 p.m.