aws {aws}R Documentation

AWS for local constant models on a grid

Description

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. For "Gaussian" models, i.e. regression with additive "Gaussian" errors, a homoskedastic or heteroskedastic model is used depending on the content of sigma2

Usage

aws(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)

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.
aws logical: if TRUE structural adaptation (AWS) is used.
memory logical: if TRUE stagewise aggregation is used as an additional adaptation scheme.
family family specifies the probability distribution. Default is family="Gaussian", also implemented are "Bernoulli", "Poisson", "Exponential", "Volatility" and "Variance". family="Volatility" specifies a Gaussian distribution with expectation 0 and unknown variance. family="Volatility" specifies that p*y/theta is distributed as \Chi^2 with p=shape degrees of freedom.
lkern character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian"
homogen logical: if TRUE the function tries to determine regions where weights can be fixed to 1. This may increase speed.
aggkern character: kernel used in stagewise aggregation, either "Triangle" or "Uniform"
sigma2 sigma2 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(sigma2)==length(y) then sigma2 is assumed to contain the pointwise variance of y and a heteroscedastic variance model is used.
shape Allows to specify an additional shape parameter for certain family models. Currently only used for family="Variance", that is \Chi-Square distributed observations with shape degrees of freedom.
scorr The vector scorr allows to specify a first order correlations of the noise for each coordinate direction, defaults to 0 (no correlation).
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
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.
testprop If set this provides diagnostics for testing the propagation condition. The values of y should correspond to the specified family and a global model.

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. For "Gaussian" models, i.e. regression with additive "Gaussian" errors, a homoskedastic or heteroskedastic model is used depending on the content of sigma2. 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=\infty). 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)
degree = "integer" 0
hmax = "numeric" effective hmax
sigma2 = "numeric" provided or estimated error variance
scorr = "numeric" scorr
family = "character" family
shape = "numeric" shape
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" "Constant"
vcoef = "numeric" numeric(0)
call = "function" the arguments of the call to aws

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, http://www.wias-berlin.de/project-areas/stat/projects/adaptive-image-processing.html

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

See Also

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

Examples

require(aws)
# 1D local constant smoothing
## Not run: demo(aws_ex1)
## Not run: demo(aws_ex2)
# 2D local constant smoothing
## Not run: demo(aws_ex3)

[Package aws version 1.6 Index]