lpaws {aws} | R Documentation |

## Local polynomial smoothing by AWS

### Description

The function allows for structural adaptive smoothing using a local polynomial (degree <=2) structural assumption. Response variables are assumed to be observed on a 1 or 2 dimensional regular grid.

### Usage

```
lpaws(y, degree = 1, hmax = NULL, aws = TRUE, memory = FALSE, lkern = "Triangle",
homogen = TRUE, earlystop = TRUE, aggkern = "Uniform", sigma2 = NULL,
hw = NULL, ladjust = 1, u = NULL, graph = FALSE, demo = FALSE)
```

### Arguments

`y` |
Response, either a vector (1D) or matrix (2D). The corresponding design is assumed to be a regular grid in 1D or 2D, respectively. |

`degree` |
Polynomial degree of the local model |

`hmax` |
maximal bandwidth |

`aws` |
logical: if TRUE structural adaptation (AWS) is used. |

`memory` |
logical: if TRUE stagewise aggregation is used as an additional adaptation scheme. |

`lkern` |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian". The default "Triangle" is equivalent to using an Epanechnikov kernel, "Quadratic" and "Cubic" refer to a Bi-weight and Tri-weight kernel, see Fan and Gijbels (1996). "Gaussian" is a truncated (compact support) Gaussian kernel. This is included for comparisons only and should be avoided due to its large computational costs. |

`homogen` |
logical: if TRUE the function tries to determine regions where weights can be fixed to 1. This may increase speed. |

`earlystop` |
logical: if TRUE the function tries to determine points where the homogeneous region is unlikely to change in further steps. This may increase speed. |

`aggkern` |
character: kernel used in stagewise aggregation, either "Triangle" or "Uniform" |

`sigma2` |
Error variance, the value is estimated if not provided. |

`hw` |
Regularisation bandwidth, used to prevent from unidentifiability of local estimates for small bandwidths. |

`ladjust` |
factor to increase the default value of lambda |

`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 |

`graph` |
logical: If TRUE intermediate results are illustrated graphically. May significantly slow down the computations in 2D. Please
avoid using the default |

`demo` |
logical: if TRUE wait after each iteration |

### 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 and derivatives, |

`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), ratio of distances |

`degree = "integer"` |
degree |

`hmax = "numeric"` |
effective hmax |

`sigma2 = "numeric"` |
provided or estimated error variance |

`scorr = "numeric"` |
0 |

`family = "character"` |
"Gaussian" |

`shape = "numeric"` |
numeric(0) |

`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"` |
eralustop |

`varmodel = "character"` |
"Constant" |

`vcoef = "numeric"` |
numeric(0) |

`call = "function"` |
the arguments of the call to |

### Note

If you specify `graph=TRUE`

for 2D problems
avoid using the default `X11()`

on systems build with `cairo`

, use
`X11(type="Xlib")`

instead (faster by a factor of 30).

### Author(s)

Joerg Polzehl polzehl@wias-berlin.de

### References

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06 .

J. Polzehl, V. 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. DOI:10.1007/978-3-540-33037-0_19.

### See Also

### Examples

```
library(aws)
# 1D local polynomial smoothing
## Not run: demo(lpaws_ex1)
# 2D local polynomial smoothing
## Not run: demo(lpaws_ex2)
```

*aws*version 2.5-5 Index]