AWS for local constant models on a grid

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 \(\chi^2\) 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 \(\chi\)-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).
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 \(\bf{E} |\hat{\theta}^k - \theta| \le (1+\alpha) \bf{E} |\tilde{\theta}^k - \theta|\) 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"

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"

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

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.

Author

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

Note

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

Examples

require(aws)
# 1D local constant smoothing
if (FALSE) demo(aws_ex1)
if (FALSE) demo(aws_ex2)
# 2D local constant smoothing
if (FALSE) demo(aws_ex3)

AWS for local constant models on a grid

Arguments

Details

Value

References

Author

Note

See also

Examples