edr: Estimation of the effective dimension reduction (EDR) space:...

In EDR: Estimation of the Effective Dimension Reduction ('EDR') Space

Description

This function implements the algorithms, proposed in M. Hristache, A. Juditsky, J. Polzehl and V. Spokoiny (2001) and ... (2006), for estimation of the effective dimension reduction (EDR) space in multi-index regression models

y=f(x)+\varepsilon=g(B_m^T x) + \varepsilon.

Usage

edr(x, y, m = 2, rho0 = 1, h0 = NULL, ch = exp(0.5/max(4, (dim(x)[2]))), 
crhomin = 1, cm = 4, method = "Penalized", basis = "Quadratic", cw = NULL, 
graph = FALSE, show = 1, trace = FALSE, fx = NULL, R = NULL)

Arguments

x	x specifies the design matrix, dimension (n,d)
y	y specifies the response, length n.
m	Rank of matrix M in case of method="Penalized", not used for the other methods.
rho0	Initial value for the regularization parameter ρ.
h0	Initial bandwidth.
ch	Factor for indecreasing h with iterations.
crhomin	Factor to in(de)crease the default value of rhomin. This is just added to explore properties of the algorithms. Defaults to 1.
cm	Factor in the definition of Π_k=C_m*ρ_k^2 I_L + \hat{M}_{k-1}. Only used if method="Penalized".
method	Secifies the algoritm to use. The default method="Penalized" corresponds to the algoritm proposed in ... (2006). method="HJPS" corresponds to the original algorithm from Hristache et.al. (2001) while method="HJPS2" specifies a modifification (correction) of this algoritm.
basis	Specifies the set of basis functions. Options are basis="Quadratic" (default) and basis="Linear".
cw	cw another regularization parameter, secures identifiability of a minimum number of local gradient directions. Defaults to 1/d . Has to be positive or NULL.
graph	If graph==TRUE intermediate results are plotted.
show	If graph==TRUE the parameter show determines the dimension of the EDR that is to be used when plotting intermediate results. If trace=TRUE and !is.null(R) it determines the dimension of the EDR when computing the risk values.
trace	trace=TRUE additional diagnostics are provided for each iteration. This includes current, at iteration k, values of the regularization parameter ρ_k and bandwidth h_k, normalized cimmulative sums of eigenvalues of \hat{B} and if !is.null(R) two distances between the true, specified in R and estimated EDR.
fx	True values of f(x). This is just added to explore properties of the algorithms and not used in the algorithms.
R	True matrix R. This is just added to explore properties of the algorithms and not used in the algorithms.

Details

See reference for details.

Value

Object of class "edr" with components.

x	The design matrix.
y	The values of the response.
bhat	Matrix \hat{B} characterizing the effective dimension space. For a specified dimension m \hat{B}_m = \hat{B} O_m, with \hat{B}^T \hat{B}= O Λ O^T being the eigenvalue decomposition of \hat{B}^T \hat{B}, specifies the projection to the m-dimensional subspace that provides the best approximation.
fhat	an highly oversmoothed estimate of the values of the regression function at the design points. This is provided as a backup only for the case that package sm is not installed.
cumlam	Cummulative amount of information explained by the first components of \hat{B}.
nmean	Mean numbers of observations used in each iteration.
h	Final bandwidth
rho	Final value of ρ
h0	Initial bandwidth
rho0	Initial value of ρ
cm	The factor cm
call	Arguments of the call to edr

Author(s)

Joerg Polzehl, [e-mail]

References

M. Hristache, A. Juditsky, J. Polzehl and V. Spokoiny (2001). Structure adaptive approach for dimension reduction, The Annals of Statistics. Vol.29, pp. 1537-1566. \ J. Polzehl, S. Sperlich (2009). A note on structural adaptive dimension reduction, J. Stat. Comput. Simul.. Vol. 79 (6), pp. 805–818.

See Also

edrcv,plot.edr, summary.edr, print.edr, edr.R

Examples

require(EDR)
demo(edr_ex1)
demo(edr_ex2)

EDR documentation built on March 26, 2020, 8:42 p.m.