aws: AWS for local constant models on a grid

In neuroconductor-releases/aws: Adaptive Weights Smoothing

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, mask=NULL, aws=TRUE, memory=FALSE, family="Gaussian",
                lkern="Triangle", aggkern="Uniform",
                sigma2=NULL, shape=NULL, scorr=0, spmin=0.25,
		            ladjust=1,wghts=NULL,u=NULL,graph=FALSE,demo=FALSE,
                testprop=FALSE,maxni=FALSE)

Arguments

y	array y containing the observe response (image intensity) 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 1D, 2D, 3D images, respectively. In case of lkern="Gaussian" the bandwidth is assumed to be given in full width half maximum (FWHM) units, i.e., 0.42466 times gridsize.
aws	logical: if TRUE structural adaptation (AWS) is used.
mask	optional logical mask, same dimensionality as y
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", "Variance" and "NCchi". family="Volatility" specifies a Gaussian distribution with expectation 0 and unknown variance. family="Volatility" specifies that p*y/theta is distributed as χ^2 with p=shape degrees of freedom. family="NCchi" uses a noncentral Chi distribution with p=shape degrees of freedom and noncentrality parameter theta
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"
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 χ-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).
spmin	Determines the form (size of the plateau) in the adaptation kernel. Not to be changed by the user.
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.
maxni	If TRUE use max_{l<=k}(N_i^{(l)} instead of (N_i^{(k)} in the definition of the statistical penalty.

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{θ}^k - θ| ≤ (1+α) \bf{E} |\tilde{θ}^k - θ| where \hat{θ}^k is the aws-estimate in step k and \tilde{θ}^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"	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

Note

use setCores='number of threads' to enable parallel execution.

Author(s)

Joerg Polzehl, [e-mail], http://www.wias-berlin.de/people/polzehl/

References

J. Polzehl, K. Tabelow (2019). Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R. Springer, Use R! series. Appendix A. Doi:10.1007/978-3-030-29184-6.

J. Polzehl, K. Papafitsoros, K. Tabelow. Patch-wise adaptive weights smoothing, Preprint no. 2520, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2520. (to appear in Journal of Statistical Software).

J. Polzehl, V. Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335–354. DOI:10.1111/1467-9868.00235.

J. Polzehl, V. Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335–362. DOI:10.1007/s00440-005-0464-1.

See Also

See also paws, lpaws, vaws,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)

neuroconductor-releases/aws documentation built on Oct. 30, 2020, 11:05 p.m.