laws {aws}  R Documentation 
This function implements a Adaptive Weights Smoothing procedure for local constant Poisson, Bernoulli, Exponential, Weibull, Volatility and Gaussian models as described in Polzehl & Spokoiny (2002).
laws(y, x = NULL, qlambda = NULL, eta = 0.5, lkern = "Triangle", model = "Poisson", shape = NULL, hinit = NULL, hincr = NULL, hmax = 10, NN = FALSE, u = NULL, graph = FALSE, demo = FALSE, symmetric = FALSE, wghts=NULL)
y 
y contains the observed values at location x .
In case of x==NULL (second parameter) y is assumed to be
observed on a one, two or threedimensional grid. The dimension of
y determines if one, two or threedimensional AWS is used. 
x 
x is either NULL , in this case y is assumed
to be observed on a grid, or is a matrix, with rows corresponding to variables,
containing the design points where y is observed. 
qlambda 
qlambda determines the scale parameter qlambda
for the stochastic penalty. The scaling parameter in the stochastic
penalty lambda is choosen as the qlambda quantile
of a Chisquaredistribution with number of parameters in the polynomial
model as degrees of freedom. If qlambda==NULL a standard value
depending on model and symmetric is choosen. 
eta 
eta is a memory parameter used to stabilize the procedure.
eta has to be between 0 and 1 , with
eta==0.5 being the default. 
lkern 
lkern determines the location kernel to be used. Options
are "Uniform" , "Triangle" , "Quadratic" ,
"Cubic" and "Exponential" . Default is "Triangle" .
The Kernel operates on the squared distance, so "Triangle"
corresponds to the use of an Epanechnikov kernel in kernel smoothing.
"Exponential" requires larger values of hmax and
therefore more iterations to reach comparable results. 
model 
model determines the distribution type of y .
Currently implemented models are "Poisson" , "Bernoulli" ,
"Gaussian" , "Exponential" , "Weibull" , "Volatility"
(Estimation of the scale parameter of a Gaussian distribution).
Defaults to "Poisson" . 
shape 
used for additional parameters of the specified distribution if needed,
i.e. variance if model=="Gaussian"

hinit 
hinit Initial bandwidth for the location penalty.
Appropriate value is choosen in case of hinit==NULL 
hincr 
hincr hincr^(1/d) , with d the
dimensionality of the design, is used as a factor to increase the
bandwidth between iterations. Defauts to hincr==1.2 
hmax 
hmax Maximal bandwidth to be used. Determines the
number of iterations and is used as the stopping rule. 
NN 
If NN==TRUE use nearest neighborrules instead of
distances in the location term. 
u 
u used to supply values of the true regression function
for test purposes to calculate Mean Squared Error (MSE) and
Mean Absolute Error (MAE) 
graph 
graph if TRUE results are displayed after each
iteration step. 
demo 
demo if TRUE after each iteration step results
are displayed and the process waits for user interaction. 
symmetric 
If symmetric==TRUE the stochastic penalty is
symmetrized, i.e. (sij + sji)/lambda is used instead of
sij/lambda . See references for details. 
wghts 
Specifies wghts for distance evaluation on a bi or trivariate grid.
Allows for anisotropic local neighborhoods. If wghts=NULL
isotropic neighborhoods are used. 
This function implements an adaptive weights smoothing (AWS) procedure for a several classes of distributions for the dependent variable in local constant regression models. The approach generalizes the original AWS procedure from Polzehl and Spokoiny (2000).
Adaptive weights smoothing is an iterative data adaptive smoothing technique that
is designed for smoothing in regression problems with discontinuous regression
function. The basic assumption is that the regression function can be approximated
by a simple local, e.g. local constant or local polynomial, model.
The estimate of the regression function, i.e. the conditional expectation of y
given x
is computed as a weighted maximum likelihood estimate, with weights choosen
in a completely data adaptive way. The procedure is edge preserving. If the assumed local
model is globally valid, almost all weights used will be 1, i.e. the resulting estimate
almost is the global estimate.
Currently implemented are the following models (specified by parameter model
):
model="Bernoulli"
model="Poisson"
This model allows e.g. for
density estimation or for the analysis of poisson count data on a grid (e.g. Positron
emission tomography (PET)).model="Exponential"
Applications of this model
include e.g. test for constant (over time) failure rates and estimation of tail indicies.model="Gaussian"
This essentially coincides with the
local constant regression model with additive subgaussian errors as described in
Polzehl and Spokoiny (2000, 2003).model="Weibull"
Applications in reliability analysis.model="Volatility"
.
The essential parameter in the procedure is qlambda
. 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 model
and the value of symmetric
(determines the use of an asymmetric or a symmetrized
test). The optimal values only slightly depend on the model parameters, i.e. the
default parameters should work in most situations. Larger values of qlambda
may lead to oversmoothing, small values of qlambda
lead to a random segmentation
of homogeneous regions. A good value of qlambda
can be obtained by the propagation
condition, requiring that in case of global validity of the local model the
estimate for large hmax
should be equal to the global estimate.
The numerical complexity of the procedure is mainly determined by hmax
. The number
of iterations is d*log(hmax/hinit)/log(hincr)
with d
being the dimension
of y
. 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.
theta 
Parameter estimates, first dimension corresponds to parameter components 
y 
values provided in y 
x 
values provided in x 
call 
actual function call 
Joerg Polzehl, polzehl@wiasberlin.de, http://www.wiasberlin.de/projectareas/stat/projects/adaptiveimageprocessing.html
SEE ALSO aws
, awsdens
, awstindex
### ### Artificial PET data ### x < 1:128 f12 < function(x,y){ 2*((1.5*(x64)^2+(y64)^2<3025)) + ((x64)^2+(y104)^2<16)((x92)^2+(y84)^2<25)+ ((x78)^2+(y84)^2<25)((x64)^2+(y84)^2<36)+ ((x50)^2+(y84)^2<36)((x36)^2+(y84)^2<25)+ ((x92)^2+(y64)^2<25)((x78)^2+(y64)^2<16)+ ((x64)^2+(y64)^2<16)((x50)^2+(y64)^2<25)+ ((x36)^2+(y64)^2<25)((x92)^2+(y44)^2<36)+ ((x78)^2+(y44)^2<36)((x64)^2+(y44)^2<25)+ ((x50)^2+(y44)^2<25)((x36)^2+(y44)^2<16)+ ((x64)^2+(y24)^2<16) } u0 < 2*outer(x,x,"f12") set.seed(1) y < matrix(rpois(u0,u0),128,128) # use larger hmax for good results yhat < laws(y,model="Poisson",hmax=4)$theta par(mfrow=c(1,3),mar=c(3,3,3,.5),mgp=c(2,1,0)) image(y,col=gray((0:255)/255)) title("Observed image") image(yhat,col=gray((0:255)/255)) title("AWSReconstruction") image(u0,col=gray((0:255)/255)) title("True image") rm(u0,y,yhat,x) ### ### VOLATITILTY ESTIMATION ### ### artificial example ### sigma < c(rep(1,125),rep(2,125),rep(.5,125),rep(1,125)) set.seed(1) x < rnorm(sigma,0,sigma) tmpa < laws(x,model="Volatility",u=sigma,graph=TRUE,hmax=250) tmps < laws(x,model="Volatility",u=sigma,hmax=250,symmetric=TRUE) par(mfrow=c(1,1),mar=c(3,3,3,1)) plot(abs(x),col=3,xlab="time t",ylab=expression(abs( R[t] ))) lines(tmpa$theta,col=1,lwd=2) lines(tmps$theta,col=1,lwd=2,lty=2) lines(sigma,col=2,lwd=2,lty=3) legend(350,5.5,c("asymmetric AWS","symmetric AWS","true sigma"), lwd=c(2,2,2),lty=c(1,2,3),col=c(1,1,2)) title(expression(paste("Estimates of ",sigma(t)))) rm(tmpa,tmps,sigma,x)