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