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 specifies the design matrix, dimension (n,d)
y specifies the response, length n.
Rank of matrix M in case of method="Penalized", not used for the other methods.
Initial value for the regularization parameter \(\rho\).
Initial bandwidth.
Factor for indecreasing \(h\) with iterations.
Factor to in(de)crease the default value of rhomin. This is just added to explore properties of the algorithms. Defaults to 1.
Factor in the definition of \(\Pi_k=C_m*\rho_k^2 I_L + \hat{M}_{k-1}\). Only used if method="Penalized".
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.
Specifies the set of basis functions. Options are basis="Quadratic" (default) and basis="Linear".
cw another regularization parameter, secures identifiability of a minimum number of local gradient directions. Defaults to 1/d . Has to be positive or NULL.
If graph==TRUE intermediate results are plotted.
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=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.
True values of \(f(x)\). This is just added to explore properties of the algorithms and not used in the algorithms.
True matrix R. This is just added to explore properties of the algorithms and not used in the algorithms.
Object of class "edr" with components.
The design matrix.
The values of the response.
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.
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.
Cummulative amount of information explained by the first components of \(\hat{B}\).
Mean numbers of observations used in each iteration.
Final bandwidth
Final value of \(\rho\)
Initial bandwidth
Initial value of \(\rho\)
The factor cm
Arguments of the call to edr
See reference for details.
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.
# NOT RUN {
require(EDR)
demo(edr_ex1)
demo(edr_ex2)
# }
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