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)
[Package
ibr version 2.0-4
Index]