edrcv: Risk assessment by Cross-Validation
In EDR: Estimation of the Effective Dimension Reduction ('EDR') Space

Description Usage Arguments Details Value Note Author(s) References See Also Examples

View source: R/edrcv.r

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`	`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_mρ_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.
`fit`	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.
`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.
`seed`	Seed for generating random groups for CV
`cvsize`	Groupsize k in leave-k-out CV
`m0`	Dimension of the dimension reduction space to use when fitting the data. Should be either 1 or 2.
`hsm`	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.

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.

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 edrcv
`cvres`	Residuals from cross-validation.
`cvmseofh`	Estimates of MSEP for bandwidths `hsm`
`cvmaeofh`	Estimates of MAEP for bandwidths `hsm`
`cvmse`	Estimate of MSEP
`cvmae`	Estimate of MAEP
`hsm`	Set of bandwidths specified for use with `sm.regression`
`hsmopt`	Bandwidth selected for use with `sm.regression` if `hsm` was specified.

This function requires package sm if fit="sm".

Joerg Polzehl, polzehl@wias-berlin.de

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

edr,plot.edr, summary.edr, print.edr, edr.R, predict.edr