sigma2aws(y,hmax=NULL,aws=TRUE,memory=FALSE,family="Gaussian",
lkern="Triangle",homogen=TRUE,aggkern="Uniform",
sigma2=NULL,shape=NULL,scorr=0,
ladjust=1,wghts=NULL,u=NULL,graph=FALSE,demo=FALSE,
testprop=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.family specifies the probability distribution. Default is family="Gaussian", also implemented
are "Bernoulli", "Poisson", "Exponential", "Volatility" and "Variance". family="Volatility" specifies a Gaussian distrisigma2 allows to specify the variance in case of family="Gaussian". Not used if family!="Gaussian".
Defaults to NULL. In this case a homoskedastic variance estimate is generated. If length(sigmashape degrees of freedom.scorr allows to specify a first order correlations of the noise for each coordinate direction,
defaults to 0 (no correlation).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=0graph=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.y should correspond to the specified
family and a global model.aws with slotslength: length(y)awssigma2.
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. Optimal values mainly depend on the choosen family.
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
lpaws, link{awsdata}, aws.irreg, aws.gaussianrequire(aws)
# 1D local constant smoothing
demo(aws_ex1)
demo(aws_ex2)
# 2D local constant smoothing
demo(aws_ex3)Run the code above in your browser using DataLab