awsh {aws} | R Documentation |

This function implements a univariate local polynomial Adaptive Weights Smoothing procedure for regression problems with heteroskedastic additive errors as described in Polzehl & Spokoiny (2003).

awsh(y, x = NULL, p = 0, sigma2 = NULL, qlambda = NULL, eta = 0.5, tau = NULL, lkern = "Triangle", hinit = NULL, hincr = NULL, hmax = 100, hmaxs= 2*hmax, u = NULL, graph = FALSE, demo = FALSE, symmetric = NULL, conf = FALSE, qconf = 0.95, alpha = 2)

`y` |
`y` contains the observed values (regression function plus errors).
In case of `x=NULL` (second parameter) `y` is assumed to be
observed on a one-dimensional grid. |

`x` |
`x` is either `NULL` , in this case `y` is assumed
to be observed on a grid, or is a vector determining the design. |

`p` |
`p` is the degree of the polynomial model to use. For univariate
regression `p` can be an nonnegative integer less or equal than 5. |

`sigma2` |
`sigma2` can be used to provide an estimate for the error
variance. If `is.null(sigma2)` a homoskedastic model is assumed and
a variance estimate is generated from the data. If `length(sigma2)==length(y)`
this parameter provides variances at the design points. |

`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 Chi-square-distribution with number of parameters in the polynomial
model as degrees of freedom. If `is.null(qlambda)` a standard value
depending on `p` is choosen. |

`eta` |
`eta` is a memory parameter used to stabilize the procedure.
`eta` has to be between `0` and `1` , with
`eta=.5` being the default. |

`tau` |
`tau` is used in case of a the scale parameter polynomial
degree `p!=0` only. It is the scale parameter in the extention
penalty used to prevent from leverage problems. The default value
for `tau` depends on `p` and on the number of parameters
in the local model. |

`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. |

`hinit` |
`hinit` Initial bandwidth for the location penalty.
Appropriate value is choosen in case of `hinit=NULL` |

`hincr` |
`hincr` `hincr` is used as a factor to increase the
bandwidth between iterations. Defauts to `hincr=1.25` |

`hmax` |
`hmax` Maximal bandwidth to be used. Determines the
number of iterations and is used as the stopping rule. |

`hmaxs` |
`hmaxs` Maximal bandwidth to be used when estimating the
heterogenous variance from consequtive differences of `y` by function
`laws` . Determines the number of iterations of `laws` . |

`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.
`symmetric==FALSE` is forced if `p!=0` |

`conf` |
`conf` if `TRUE` conditional (on weights) confidence intervals are provided
at each design point. |

`qconf` |
`qconf` determines the level of the conditional (on weights) confidence intervals |

`alpha` |
`alpha` Parameter used for a penalized MSE estimate for `p=0` . This is
experimental to try to select `hmax` . |

This function implements an local polynomial adaptive weights smoothing (AWS) procedure for a univariate heteroskedastic regression model. 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 `p`

and attributes
of `x`

and `y`

) are implemented:

- p=0, x=NULL
- local constant univariate AWS on a grid. Measurement unit for
`hinit`

and`hmax`

is 1 (number grid points within distance`h`

) - p=0, !is.null(x)
- local constant AWS for arbitrary design given in
`x`

. `p>2`

- univariate local polynomial AWS for arbitrary univariate design
given in
`x`

.`y`

has to be a vector. If`x`

is provided it has to be a vector of the same length as`y`

. If`is.null(x)`

an equidistant design (grid) with grid-step 1 is assumed.

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 `p`

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 |

`theta ` |
Standard deviations (conditional on weights) of 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@wias-berlin.de, http://www.wias-berlin.de/project-areas/stat/projects/adaptive-image-processing.html

##---- Should be DIRECTLY executable !! ---- ##-- ==> Define data, use random, ##-- or do help(data=index) for the standard data sets.

[Package *aws* version 1.3-0 Index]