ibr {ibr}R Documentation

Iterative bias reduction smoothing

Description

Performs iterative bias reduction using kernel or thin plate splines. In the latter case, the order m is chosen as the first integer such that 2m/d>1, where d is the number of explanatory variables.
Missing values are not allowed.

Usage

ibr(x, y, criterion="gcv", df=1.5, Kmin=1, Kmax=10000, smoother="k",
 kernel="g", control.par=list(), cv.options=list())

Arguments

x A numeric matrix of explanatory variables, with n rows and p columns.
y A numeric vector of variable to be explained of length n.
criterion Character string. If the number of iterations (iter) is missing or NULL the number of iterations is chosen using criterion. The criteria available are GCV (default, "gcv"), AIC ("aic"), corrected AIC ("aicc"), BIC ("bic"), gMDL ("gmdl"), map ("map") or rmse ("rmse"). The last two are designed for cross-validation.
df A numeric vector of either length 1 or length equal to the number of columns of x. If smoother="k", it indicates the desired effective degree of freedom (trace) of the smoothing matrix for each variable or for the initial smoother (see contr.sp$dftotal); df is repeated when the length of vector df is 1. If smoother="tps", the minimum df of thin plate splines is multiplied by df. This argument is useless if bandwidth is supplied (non null).
Kmin The minimum number of bias correction iterations of the search grid considered by the model selection procedure for selecting the optimal number of iterations.
Kmax The maximum number of bias correction iterations of the search grid considered by the model selection procedure for selecting the optimal number of iterations.
smoother Character string which allows to choose between thin plate splines "tps" or kernel ("k").
kernel Character string which allows to choose between gaussian kernel ("g"), Epanechnikov ("e"), uniform ("u"), quartic ("q"). The default (gaussian kernel) is strongly advised.
control.par A named list that control optional parameters. The components are bandwidth (default to NULL), iter (default to NULL), really.big (default to FALSE), dftobwitmax (default to 1000), exhaustive (default to FALSE),m (default to NULL), dftotal (default to FALSE), accuracy (default to 0.01), ddlmaxi (default to 2n/3) and fraction (default to c(100, 200, 500, 1000, 5000, 10^4, 5e+04, 1e+05, 5e+05, 1e+06)).
bandwidth: a vector of either length 1 or length equal to the number of columns of x. If smoother="k", it indicates the bandwidth used for each variable, bandwidth is repeated when the length of vector bandwidth is 1. If smoother="tps", it indicates the amount of penalty (coefficient lambda). The default (missing) indicates, for smoother="k", that bandwidth for each variable is chosen such that each univariate kernel smoother (for each explanatory variable) has df effective degrees of freedom and for smoother="tps" that lambda is chosen such that the df of the smoothing matrix is df times the minimum df.
iter: the number of iterations. If null or missing, an optimal number of iterations is chosen from the search grid (integer from Kmin to Kmax) to minimize the criterion.
really.big: a boolean: if TRUE it overides the limitation at 500 observations. Expect long computation times if TRUE.
dftobwitmax: When bandwidth is chosen by specifying the effective degree of freedom (see df) a search is done by uniroot. This argument specifies the maximum number of iterations transmitted to uniroot function.
exhaustive: boolean, if TRUE an exhaustive search of optimal number of iteration on the grid Kmin:Kmax is performed. If FALSE the minimum of criterion is searched using optimize between Kmin and Kmax.
m: the order of thin plate splines. This integer m must verifies 2m/d>1, where d is the number of explanatory variables. The missing default to choose the order m as the first integer such that 2m/d>1, where d is the number of explanatory variables (same for NULL).
dftotal: a boolean wich indicates when FAlSE that the argument df is the objective df for each univariate kernel (the default) calculated for each explanatory variable or for the overall (product) kernel, that is the base smoother (when TRUE).
accuracy: tolerance when searching bandwidths which lead to a chosen overall intial df.
dfmaxi: the maximum effective degree of freedom allowed for iterated biased reduction smoother.
fraction: the subdivision of interval Kmin,Kmax if non exhaustive search is performed (see also iterchoiceA or iterchoiceS1).
cv.options A named list which controls the way to do cross validation with component bwchange, ntest, ntrain, Kfold, type, seed, method and npermut. bwchange is a boolean (default to FALSE) which indicates if bandwidth have to be recomputed each time. ntest is the number of observations in test set and ntrain is the number of observations in training set. Actually, only one of these is needed the other can be NULL or missing. Kfold a boolean or an integer. If Kfold is TRUE then the number of fold is deduced from ntest (or ntrain). type is a character string in random,timeseries,consecutive, interleaved and give the type of segments. seed controls the seed of random generator. method is either "inmemory" or "outmemory"; "inmemory" induces some calculations outside the loop saving computational time but leading to an increase of the required memory. npermut is the number of random draws. If cv.options is list(), then component ntest is set to floor(nrow(x)/10), type is random, npermut is 20 and method is "inmemory", and the other components are NULL

Value

Returns an object of class ibr which is a list including:

beta Vector of coefficients.
residuals Vector of residuals.
iter The number of iterations used.
initialdf The initial effective degree of freedom of the pilot (or base) smoother.
finaldf The effective degree of freedom of the iterated bias reduction smoother at the iter iterations.
bandwidth Vector of bandwith for each explanatory variable
call A list containing four components: x contains the initial explanatory variables, y contains the initial dependant variables, criterion contains the chosen criterion, kernel the kernel, p the number of explanatory variables and m the order of the splines (if relevant).
criteria either a list containing all the criteria evaluated on the grid Kmin:Kmax (along with the effective degree of freedom of the smoother and the sigma squared on this grid) if an exhaustive search is chosen (see the value of function iterchoiceAe or iterchoiceS1e) or the value of the chosen criterion at the given iteration if a non exhaustive search is chosen (see exhaustive). If the number of iterations iter is given by the user NULL is returned

Author(s)

Pierre-Andre Cornillon, Nicolas Hengartner and Eric Matzner-Lober.

References

Cornillon, P. A., Hengartner, N. and Matzner-Lober, E. (2009) Recursive Bias Estimation for high dimensional regression smoothers. submitted.

See Also

predict.ibr, summary.ibr

Examples

f <- function(x, y) { .75*exp(-((9*x-2)^2 + (9*y-2)^2)/4) +
                      .75*exp(-((9*x+1)^2/49 + (9*y+1)^2/10)) +
                      .50*exp(-((9*x-7)^2 + (9*y-3)^2)/4) -
                      .20*exp(-((9*x-4)^2 + (9*y-7)^2)) }
# define a (fine) x-y grid and calculate the function values on the grid
ngrid <- 50; xf <- seq(0,1, length=ngrid+2)[-c(1,ngrid+2)]
yf <- xf ; zf <- outer(xf, yf, f)
grid <- cbind(rep(xf, ngrid), rep(xf, rep(ngrid, ngrid)))
persp(xf, yf, zf, theta=130, phi=20, expand=0.45,main="True Function")
#generate a data set with function f and noise to signal ratio 5
noise <- .2 ; N <- 100 
xr <- seq(0.05,0.95,by=0.1) ; yr <- xr ; zr <- outer(xr,yr,f) ; set.seed(25)
std <- sqrt(noise*var(as.vector(zr))) ; noise <- rnorm(length(zr),0,std)
Z <- zr + matrix(noise,sqrt(N),sqrt(N))
# transpose the data to a column format 
xc <- rep(xr, sqrt(N)) ; yc <- rep(yr, rep(sqrt(N),sqrt(N)))
X <- cbind(xc, yc) ; Zc <- as.vector(Z)
# fit by thin plate splines (of order 2) ibr
res.ibr <- ibr(X,Zc,df=1.1,smoother="tps")
fit <- matrix(predict(res.ibr,grid),ngrid,ngrid)
persp(xf, yf, fit ,theta=130,phi=20,expand=0.45,main="Fit",zlab="fit")

## Not run: 
data(ozone, package = "ibr")
res.ibr <- ibr(ozone[,-1],ozone[,1],df=1.1)
summary(res.ibr)
predict(res.ibr)
## End(Not run)

[Package ibr version 1.2 Index]