R: vector valued version of function 'paws' with homogeneous...

vpaws {aws} | R Documentation

## vector valued version of function `paws` with homogeneous covariance structure

### Description

The function implements a vector-valued version the propagation separation approach that uses patches instead of individuel voxels for comparisons in parameter space. Functionality is analog to function `vaws`. Using patches allows for an improved handling of locally smooth functions and in 2D and 3D for improved smoothness of discontinuities at the expense of increased computing time.

### Usage

    vpaws(y, kstar = 16, sigma2 = 1, mask = NULL, scorr = 0, spmin = 0.25,
          ladjust = 1, wghts = NULL, u = NULL, maxni = FALSE, patchsize = 1)
    vpawscov(y, kstar = 16, invcov = NULL, mask = NULL, scorr = 0, spmin = 0.25,
          ladjust = 1, wghts = NULL, u = NULL, maxni = FALSE, patchsize = 1)

### Arguments

`y` | `y` contains the observed response data. `dim(y)` determines the dimensionality and extend of the grid design. First component varies over components of the response vector.
`kstar` | maximal number of steps to employ. Determines maximal bandwidth.
`sigma2` | specifies a homogeneous error variance.
`invcov` | array of voxelwise inverse covariance matrixes, first index corresponds to upper diagonal inverse covariance matrix.
`mask` | logical mask. All computations are restrikted to design poins within the mask.
`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`
`maxni` | If TRUE use max_{l<=k}(N_i^{(l)} instead of (N_i^{(k)} in the definition of the statistical penalty.
`patchsize` | positive integer defining the size of patches. Number of grid points within the patch is `(2*patchsize+1)^d` with `d` denoting the dimensionality of the design.

### Details

see `vaws`. The procedure is supposed to produce superior results if the assumption of a local constant image is violated or if smooothness of discontinuities is desired.

### 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)`
`hseq = "numeric"` | sequence of bandwidths employed
`mae = "numeric"` | Mean absolute error for each iteration step if u was specified, numeric(0) else
`psnr = "numeric"` | Peak signal-to-noise ratio 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.Currently also uses factor `1/ni` instead of the correct `sum(wij^2)/ni^2`
`xmin = "numeric"` | numeric(0)
`xmax = "numeric"` | numeric(0)
`wghts = "numeric"` | numeric(0), ratio of distances `wghts[-1]/wghts[1]`
`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`

### Note

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

### Author(s)

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

### References

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.

### See Also

See also `vaws`, `lpaws`, `vawscov`,`link{awsdata}`

### Examples

    ## Not run:
    setCores(2)
    y <- array(rnorm(4*64^3),c(4,64,64,64))
    yhat <- vpaws(y,kstar=20)

    ## End(Not run)

[Package aws version 2.2-1 Index]