aws {aws}  R Documentation 
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(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)
y 
array 
hmax 

aws 
logical: if TRUE structural adaptation (AWS) is used. 
mask 
optional logical mask, same dimensionality as 
memory 
logical: if TRUE stagewise aggregation is used as an additional adaptation scheme. 
family 

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 Biweight and Triweight 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 

shape 
Allows to specify an additional shape parameter for certain family models. Currently only used for family="Variance", that is χSquare distributed observations
with 
scorr 
The vector 
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 

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 
graph 
If 
demo 
If 
testprop 
If set this provides diagnostics for testing the propagation condition. The values of 
maxni 
If TRUE use max_{l<=k}(N_i^{(l)} instead of (N_i^{(k)} in the definition of the statistical penalty. 
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 awsestimate 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.
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, 
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 
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 
use setCores='number of threads'
to enable parallel execution.
Joerg Polzehl, polzehl@wiasberlin.de, http://www.wiasberlin.de/people/polzehl/
J. Polzehl, K. Tabelow (2019). Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R. Springer, Use R! series. Appendix A. Doi:10.1007/9783030291846.
J. Polzehl, K. Papafitsoros, K. Tabelow (2020). PatchWise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 127. doi:10.18637/jss.v095.i06 (URL: http://doi.org/10.18637/jss.v095.i06).
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:10.1111/14679868.00235.
J. Polzehl, V. Spokoiny, Propagationseparation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335–362. DOI:10.1007/s0044000504641.
See also paws
, lpaws
, vaws
,link{awsdata}
, aws.irreg
, aws.gaussian
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)