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
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", "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"
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 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
f{E} |hat{θ}^k - θ| ≤ (1+α) f{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)
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"
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
See also lpaws, 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)
providing 
.