edr {EDR} | R Documentation |
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.
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)
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 rho. |
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 Pi_k=C_m*rho_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 rho_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. |
See reference for details.
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 Lambda 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 rho |
h0 |
Initial bandwidth |
rho0 |
Initial value of rho |
cm |
The factor cm |
call |
Arguments of the call to edr |
Joerg Polzehl, polzehl@wias-berlin.de
edrcv
,plot.edr
, summary.edr
, print.edr
, edr.R
require(EDR) demo(edr_ex1) demo(edr_ex2)