Adaptive weights smoothing for Gaussian data with variance depending on the mean.

The function implements an semiparametric adaptive weights smoothing algorithm designed for regression with additive heteroskedastic Gaussian noise. The noise variance is assumed to depend on the value of the regression function. This dependence is modeled by a global parametric (polynomial) model.

aws.gaussian(y, hmax = NULL, hpre = NULL, aws = TRUE, memory = FALSE,
             varmodel = "Constant", lkern = "Triangle",
             aggkern = "Uniform", scorr = 0, mask=NULL, ladjust = 1,
             wghts = NULL, u = NULL, varprop = 0.1, graph = FALSE, demo = FALSE)

Arguments

y	y contains the observed response data. dim(y) determines the dimensionality and extend of the grid design.
hmax	hmax specifies the maximal bandwidth. Defaults to hmax=250, 12, 5 for dd=1, 2, 3, respectively.
hpre	Describe hpre Bandwidth used for an initial nonadaptive estimate. The first estimate of variance parameters is obtained from residuals with respect to this estimate.
aws	logical: if TRUE structural adaptation (AWS) is used.
memory	logical: if TRUE stagewise aggregation is used as an additional adaptation scheme.
varmodel	Implemented are "Constant", "Linear" and "Quadratic" refering to a polynomial model of degree 0 to 2.
lkern	character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian". The default "Triangle" is equivalent to using an Epanechnikov kernel, "Quadratic" and "Cubic" refer to a Bi-weight and Tri-weight kernel, see Fan and Gijbels (1996). "Gaussian" is a truncated (compact support) Gaussian kernel. This is included for comparisons only and should be avoided due to its large computational costs.
aggkern	character: kernel used in stagewise aggregation, either "Triangle" or "Uniform"
scorr	The vector scorr allows to specify a first order correlations of the noise for each coordinate direction, defaults to 0 (no correlation).
mask	Restrict smoothing to points where mask==TRUE. Defaults to TRUE in all voxel.
ladjust	factor to increase the default value of lambda
wghts	wghts specifies the diagonal elements of a weight matrix to adjust for different distances between grid-points in different coordinate directions, i.e. allows to define a more appropriate metric in the design space.
u	a "true" value of the regression function, may be provided to report risks at each iteration. This can be used to test the propagation condition with u=0
varprop	Small variance estimates are replaced by varprop times the mean variance.
graph	If graph=TRUE intermediate results are illustrated after each iteration step. Defaults to graph=FALSE.
demo	If demo=TRUE the function pauses after each iteration. Defaults to demo=FALSE.

Details

The function implements the propagation separation approach to nonparametric smoothing (formerly introduced as Adaptive weights smoothing) for varying coefficient likelihood models on a 1D, 2D or 3D grid. In contrast to function aws observations are assumed to follow a Gaussian distribution with variance depending on the mean according to a specified global variance model. aws==FALSE provides the stagewise aggregation procedure from Belomestny and Spokoiny (2004). memory==FALSE provides Adaptive weights smoothing without control by stagewise aggregation.

The essential parameter in the procedure is a critical value lambda. This parameter has an interpretation as a significance level of a test for equivalence of two local parameter estimates. Values set internally are choosen to fulfil a propagation condition, i.e. in case of a constant (global) parameter value and large hmax the procedure provides, with a high probability, the global (parametric) estimate. More formally we require the parameter lambda to be specified such that \(\bf{E} |\hat{\theta}^k - \theta| \le (1+\alpha) \bf{E} |\tilde{\theta}^k - \theta|\) where \(\hat{\theta}^k\) is the aws-estimate in step k and \(\tilde{\theta}^k\) is corresponding nonadaptive estimate using the same bandwidth (lambda=Inf). The value of lambda can be adjusted by specifying the factor ladjust. Values ladjust>1 lead to an less effective adaptation while ladjust<<1 may lead to random segmentation of, with respect to a constant model, homogeneous regions.

The numerical complexity of the procedure is mainly determined by hmax. The number of iterations is approximately Const*d*log(hmax)/log(1.25) with d being the dimension of y and the constant depending on the kernel lkern. Comlexity in each iteration step is Const*hakt*n with hakt being the actual bandwith in the iteration step and n the number of design points. hmax determines the maximal possible variance reduction.

Value

returns anobject of class aws with slots

y = "numeric"

y

dy = "numeric"

dim(y)

x = "numeric"

numeric(0)

ni = "integer"

integer(0)

mask = "logical"

logical(0)

theta = "numeric"

Estimates of regression function, length: length(y)

mae = "numeric"

Mean absolute error for each iteration step if u was specified, numeric(0) else

var = "numeric"

approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights.

xmin = "numeric"

numeric(0)

xmax = "numeric"

numeric(0)

wghts = "numeric"

numeric(0), ratio of distances wghts[-1]/wghts[1]

degree = "integer"

0

hmax = "numeric"

effective hmax

sigma2 = "numeric"

provided or estimated error variance

scorr = "numeric"

scorr

family = "character"

"Gaussian"

shape = "numeric"

NULL

lkern = "integer"

integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian"

lambda = "numeric"

effective value of lambda

ladjust = "numeric"

effective value of ladjust

aws = "logical"

aws

memory = "logical"

memory

homogen = "logical"

homogen

earlystop = "logical"

FALSE

varmodel = "character"

varmodel

vcoef = "numeric"

estimated parameters of the variance model

call = "function"

the arguments of the call to aws.gaussian

References

Joerg Polzehl, Vladimir Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335--354

Joerg Polzehl, Vladimir Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335--362.

Joerg Polzehl, Vladimir Spokoiny, in V. Chen, C.; Haerdle, W. and Unwin, A. (ed.) Handbook of Data Visualization Structural adaptive smoothing by propagation-separation methods Springer-Verlag, 2008, 471-492

Author

Joerg Polzehl, polzehl@wias-berlin.de, http://www.wias-berlin.de/people/polzehl/

See also

See also aws, link{awsdata}, aws.irreg

Examples

require(aws)