ibr: Iterative bias reduction smoothing

In ibr: Iterative Bias Reduction

Iterative bias reduction smoothing

Description

Performs iterative bias reduction using kernel, thin plate splines Duchon splines or low rank splines. Missing values are not allowed.

Usage

ibr(formula, data, subset, criterion="gcv", df=1.5, Kmin=1, Kmax=1e+06, smoother="k",
 kernel="g", rank=NULL, control.par=list(), cv.options=list())

Arguments

formula	An object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.
data	An optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which ibr is called.
subset	An optional vector specifying a subset of observations to be used in the fitting process.
criterion	A vector of string. If the number of iterations (iter) is missing or NULL the number of iterations is chosen using the either one criterion (the first coordinate of criterion) or several (see component criterion of argument list control.par). 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" or smoother="ds", the minimum df of 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", Duchon splines "tps" (see Duchon, 1977) 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.
rank	Numeric value that control the rank of low rank splines (denoted as k in mgcv package ; see also choose.k for further details or gam for another smoothing approach with reduced rank smoother.
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), ,s (default to NULL), dftotal (default to FALSE), accuracy (default to 0.01), ddlmaxi (default to 2n/3), fraction (default to c(100, 200, 500, 1000, 5000, 10^4, 5e+04, 1e+05, 5e+05, 1e+06)), scale (default to FALSE), criterion (default to "strict") and aggregfun (default to 10^(floor(log10(x[2]))+2)). 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" or smoother="ds" 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. All criteria for all iterations in the same class (class one: GCV, AIC, corrected AIC, BIC, gMDL ; class two : MAP, RMSE) are returned in argument allcrit. If FALSE the minimum of criterion is searched using optimize between Kmin and Kmax. m: The order of derivatives for the penalty (for thin plate splines it is the order). This integer m must verify 2m+2s/d>1, where d is the number of explanatory variables. The default (for smoother="tps") is to choose the order m as the first integer such that 2m/d>1, where d is the number of explanatory variables. The default (for smoother="ds") is to choose m=2 (p seudo cubic splines). s: the power of weighting function. For thin plate splines s is equal to 0. This real must be strictly smaller than d/2 (where d is the number of explanatory variables) and must verify 2m+2s/d. To get pseudo-cubic splines (the default), choose m=2 and s=(d-1)/2 (See Duchon, 1977).the order of thin plate splines. This integer m must verifies 2m/d>1, where d is the number of explanatory variables. 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). scale: boolean. If TRUE x is scaled (using scale); default to FALSE. criterion Character string. Possible choices are strict, aggregation or recalc. strict allows to select the number of iterations according to the first coordinate of argument criterion. aggregation allows to select the number of iterations by applying the function control.par$aggregfun to the number of iterations selected by all the criteria chosen in argument criterion. recalc allows to select the number of iterations by first calculating the optimal number of the second coordinate of argument criterion, then applying the function control.par$aggregfun (to add some number to it) resulting in a new Kmax and then doing the optimal selction between Kmin and this new Kmax using the first coordinate of argument criterion. ; default to strict. aggregfun function to be applied when control.par$criterion is either recalc or aggregation.
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.
fitted	Vector of fitted values.
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	The matched call
parcall	A list containing several components: p contains the number of explanatory variables and m the order of the splines (if relevant), s the power of weights, scaled boolean which is TRUE when explanatory variables are scaled, mean mean of explanatory variables if scaled=TRUE, sd standard deviation of explanatory variables if scaled=TRUE, critmethod that indicates the method chosen for criteria strict, rank the rank of low rank splines if relevant, criterion the chosen criterion, smoother the chosen smoother, kernel the chosen kernel, smoothobject the smoothobject returned by smoothCon, exhaustive a boolean which indicates if an exhaustive search was chosen
criteria	Value of the chosen criterion at the given iteration, NA is returned when aggregation of criteria is chosen (see component criterion of list control.par). If the number of iterations iter is given by the user, NULL is returned
alliter	Numeric vector giving all the optimal number of iterations selected by the chosen criteria.
allcriteria	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 all the values of criteria at the given optimal iteration if a non exhaustive search is chosen (see also exhaustive component of list control.par).
call	The matched call.
terms	The 'terms' object used.

Author(s)

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

References

Cornillon, P.-A.; Hengartner, N.; Jegou, N. and Matzner-Lober, E. (2012) Iterative bias reduction: a comparative study. Statistics and Computing, 23, 777-791.

Cornillon, P.-A.; Hengartner, N. and Matzner-Lober, E. (2013) Recursive bias estimation for multivariate regression smoothers Recursive bias estimation for multivariate regression smoothers. ESAIM: Probability and Statistics, 18, 483-502.

Cornillon, P.-A.; Hengartner, N. and Matzner-Lober, E. (2017) Iterative Bias Reduction Multivariate Smoothing in R: The ibr Package. Journal of Statistical Software, 77, 1–26.

Wood, S.N. (2003) Thin plate regression splines. J. R. Statist. Soc. B, 65, 95-114.

See Also

predict.ibr, summary.ibr, gam

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.data.frame(x=rep(xf, ngrid),y=rep(xf, rep(ngrid, ngrid)),z=as.vector(zf))
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)))
data <- cbind.data.frame(x=xc,y=yc,z=as.vector(Z))
# fit by thin plate splines (of order 2) ibr
res.ibr <- ibr(z~x+y,data=data,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~.,data=ozone,df=1.1)
summary(res.ibr)
predict(res.ibr)
## End(Not run)

ibr documentation built on Sept. 13, 2023, 5:08 p.m.