aws.gaussian(y, hmax = NULL, hpre = NULL, aws = TRUE, memory = FALSE, varmodel = "Constant", lkern = "Triangle", homogen = TRUE, aggkern = "Uniform", scorr = 0, mask=NULL, ladjust = 1, wghts = NULL, u = NULL, varprop = 0.1, graph = FALSE, demo = FALSE)y contains the observed response data. dim(y) determines the dimensionality and extend of the grid design.hmax specifies the maximal bandwidth. Defaults to
hmax=250, 12, 5 for dd=1, 2, 3, respectively.hpre Bandwidth used for an initial nonadaptive estimate.
The first estimate of variance parameters is obtained from residuals with respect to this estimate. scorr allows to specify a first order correlations of the noise for each coordinate direction,
defaults to 0 (no correlation). mask==TRUE. Defaults to TRUE in all voxel.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=0 varprop times the mean variance.graph=TRUE intermediate results are illustrated after each iteration step. Defaults to graph=FALSE. demo=TRUE the function pauses after each iteration. Defaults to demo=FALSE. aws with slots
length: length(y)aws.gaussianaws 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< code=""> 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.
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
aws, link{awsdata}, aws.irreg