cve.call {CVarE}R Documentation

Conditional Variance Estimator (CVE).

Description

This is the main function in the CVE package. It creates objects of class "cve" to estimate the mean subspace. Helper functions that require a "cve" object can then be applied to the output from this function.

Conditional Variance Estimation (CVE) is a sufficient dimension reduction (SDR) method for regressions studying E(Y|X), the conditional expectation of a response Y given a set of predictors X. This function provides methods for estimating the dimension and the subspace spanned by the columns of a p x k matrix B of minimal rank k such that

E(Y|X) = E(Y|B'X)

or, equivalently,

Y = g(B'X) + ε

where X is independent of ε with positive definite variance-covariance matrix Var(X) = Σ_X. ε is a mean zero random variable with finite Var(ε) = E(ε^2), g is an unknown, continuous non-constant function, and B = (b_1,..., b_k) is a real p x k matrix of rank k <= p.

Both the dimension k and the subspace span(B) are unknown. The CVE method makes very few assumptions.

A kernel matrix Bhat is estimated such that the column space of Bhat should be close to the mean subspace span(B). The primary output from this method is a set of orthonormal vectors, Bhat, whose span estimates span(B).

The method central implements the Ensemble Conditional Variance Estimation (ECVE) as described in [2]. It augments the CVE method by applying an ensemble of functions (parameter func_list) to the response to estimate the central subspace. This corresponds to the generalization

F(Y|X) = F(Y|B'X)

or, equivalently,

Y = g(B'X, ε)

where F is the conditional cumulative distribution function.

Usage

cve.call(
  X,
  Y,
  method = c("mean", "weighted.mean", "central", "weighted.central"),
  func_list = NULL,
  nObs = sqrt(nrow(X)),
  h = NULL,
  min.dim = 1L,
  max.dim = 10L,
  k = NULL,
  momentum = 0,
  tau = 1,
  tol = 0.001,
  slack = 0,
  gamma = 0.5,
  V.init = NULL,
  max.iter = 50L,
  attempts = 10L,
  nr.proj = 1L,
  logger = NULL
)

Arguments

X

Design predictor matrix.

Y

n-dimensional vector of responses.

method

This character string specifies the method of fitting. The options are

  • "mean" method to estimate the mean subspace, see [1].

  • "central" ensemble method to estimate the central subspace, see [2].

  • "weighted.mean" variation of "mean" method with adaptive weighting of slices, see [1].

  • "weighted.central" variation of "central" method with adaptive weighting of slices, see [2].

func_list

a list of functions applied to Y used by ECVE (see [2]) for central subspace estimation. The default ensemble are indicator functions of the [0, 10], (10, 20], ..., (90, 100] percent response quantiles. (only relevant if method is "central" or "weighted.central", ignored otherwise)

nObs

parameter for choosing bandwidth h using estimate.bandwidth (ignored if h is supplied).

h

bandwidth or function to estimate bandwidth, defaults to internaly estimated bandwidth.

min.dim

lower bounds for k, (ignored if k is supplied).

max.dim

upper bounds for k, (ignored if k is supplied).

k

Dimension of lower dimensional projection, if k is given only the specified dimension B matrix is estimated.

momentum

number of [0, 1) giving the ration of momentum for eucledian gradient update with a momentum term. momentum = 0 corresponds to normal gradient descend.

tau

Initial step-size.

tol

Tolerance for break condition.

slack

Positive scaling to allow small increases of the loss while optimizing, i.e. slack = 0.1 allows the target function to increase up to 10 \% in one optimization step.

gamma

step-size reduction multiple. If gradient step with step size tau is not accepted gamma * tau is set to the next step size.

V.init

Semi-orthogonal matrix of dimensions '(ncol(X), ncol(X) - k) used as starting value in the optimization. (If supplied, attempts is set to 0 and k to match dimension).

max.iter

maximum number of optimization steps.

attempts

If V.init not supplied, the optimization is carried out attempts times with starting values drawn from the invariant measure on the Stiefel manifold (see rStiefel).

nr.proj

The number of projection used for projective resampling for multivariate response Y (under active development, ignored for univariate response).

logger

a logger function (only for advanced users, slows down the computation).

Value

an S3 object of class cve with components:

X

design matrix of predictor vector used for calculating cve-estimate,

Y

n-dimensional vector of responses used for calculating cve-estimate,

method

Name of used method,

call

the matched call,

res

list of components V, L, B, loss, h for each k = min.dim, ..., max.dim. If k was supplied in the call min.dim = max.dim = k.

  • B is the cve-estimate with dimension p x k.

  • V is the orthogonal complement of B.

  • L is the loss for each sample seperatels such that it's mean is loss.

  • loss is the value of the target function that is minimized, evaluated at V.

  • h bandwidth parameter used to calculate B, V, loss, L.

References

[1] Fertl, L. and Bura, E. (2021) "Conditional Variance Estimation for Sufficient Dimension Reduction" <arXiv:2102.08782>

[2] Fertl, L. and Bura, E. (2021) "Ensemble Conditional Variance Estimation for Sufficient Dimension Reduction" <arXiv:2102.13435>

Examples

# create B for simulation (k = 1)
B <- rep(1, 5) / sqrt(5)

set.seed(21)
# creat predictor data X ~ N(0, I_p)
X <- matrix(rnorm(500), 100, 5)
# simulate response variable
#     Y = f(B'X) + err
# with f(x1) = x1 and err ~ N(0, 0.25^2)
Y <- X %*% B + 0.25 * rnorm(100)

# calculate cve with method 'simple' for k = 1
set.seed(21)
cve.obj.simple1 <- cve(Y ~ X, k = 1)

# same as
set.seed(21)
cve.obj.simple2 <- cve.call(X, Y, k = 1)

# extract estimated B's.
coef(cve.obj.simple1, k = 1)
coef(cve.obj.simple2, k = 1)

[Package CVarE version 1.1 Index]