Local polynomial smoothing by AWS

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.

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 u=0
graph	logical: If TRUE intermediate results are illustrated graphically. May significantly slow down the computations in 2D. Please avoid using the default X11() on systems build with cairo, use X11(type="Xlib") instead (faster by a factor of 30).
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, length: length(y)*(degree+1)

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 wghts[-1]/wghts[1]

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 lpaws

References

J. Polzehl, K. Papafitsoros, K. Tabelow. Patch-wise adaptive weights smoothing, Preprint no. 2520, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2520. (to appear in Journal of Statistical Software).

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.

Author

Joerg Polzehl polzehl@wias-berlin.de

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).

See also

link{awsdata},aws, aws.irreg

Examples

library(aws)
# 1D local polynomial smoothing
if (FALSE) demo(lpaws_ex1)
# 2D local polynomial smoothing
if (FALSE) demo(lpaws_ex2)