Tis function, additionally to estimating the
effective dimension reduction space (EDR), see also function edr, estimates the Mean Squared Error of Prediction (MSEP) and the Mean Absolute Error of Prediction (MAEP) when using the estimated EDR by Cross-Validation. Estimates of the regression function are produced using function sm.regression from package sm.
edrcv(x, y, m = 2, rho0 = 1, h0 = NULL, ch = exp(0.5/max(4, (dim(x)[2]))), crhomin = 1,
cm = 4, method = "Penalized",fit = "sm" , basis = "Quadratic", cw = NULL,
graph = FALSE, show = 1, trace = FALSE, seed = 1, cvsize = 1, m0 = min(m, 2),
hsm = 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 method for estimating and predicting values of the link function. This can either be fit="sm"
specifying use of the sm package or fit="direct" specifying the use of a local linear smoother. In case of m0>2
fit="direct" is used due to restrictions in the sm package.
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.
Seed for generating random groups for CV
Groupsize k in leave-k-out CV
Dimension of the dimension reduction space to use when fitting the data. Should be either 1 or 2.
If is.null(hsm) the bandwidth used by sm.regression for smoothing within the EDR is chosen by cross-validation within sm.regression when needed. Alternatively a grid of bandwidths may be specified. In that case a bandwidth for sm.regression is chosen from the grid that minimizes the extimated mean absolute error of prediction.
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 edrcv
Residuals from cross-validation.
Estimates of MSEP for bandwidths hsm
Estimates of MAEP for bandwidths hsm
Estimate of MSEP
Estimate of MAEP
Set of bandwidths specified for use with sm.regression
Bandwidth selected for use with sm.regression if hsm was specified.
This function performs a leave-k-out cross-validation to estimate the risk
in terms of Mean Squared Error of Prediction (MSEP) and Mean Absolute Error of Prediction (MAEP) when using function edr to estimate an
effective dimension reduction space of dimension m0 and using this estimated space to predict values of the response. Smoothing within the dimension reduction space is performed using the function sm.regression from package sm. The bandwidth for sm.regression is
chosen by Cross-Validation.
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 (2008). A Note on Stuctural Adaptive Dimension Reduction, Journal of Statistical Computation and Simulation, DOI: 10.1080/00949650801959699
# NOT RUN {
require(EDR)
# }
# NOT RUN {
demo(edr_ex4)
# }
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