bw.abram.temp: Abramson's adaptive temporal bandwidths

In kernstadapt: Adaptive Kernel Estimators for Point Process Intensities on Linear Networks

Abramson's adaptive temporal bandwidths

Description

Computes adaptive smoothing bandwidth in the temporal case according to the inverse-square-root rule of Abramson (1982).

Usage

## S3 method for class 'temp'
bw.abram(X, h0 = NULL, ..., nt = 128, trim = NULL, at = "points")

Arguments

X	A vector (a temporal point pattern) from which the bandwidths should be computed.
h0	The global smoothing bandwidth. The default is Silverman's rule of thumb (bw.nrd0).
...	Additional arguments passed to smoother to control the type of smoothing.
nt	The number of equally spaced points at which the temporal density is to be estimated.
trim	A trimming value to cut extreme large bandwidths.
at	Character string specifying whether to compute bandwidths at the points (at = "points", the default) or to compute bandwidths at every bin in a bin grid (at = "bins").

Details

This function returns a set of temporal adaptive smoothing bandwidths driven by the methods of Abramson (1982) and Hall and Marron (1988). The bandwidth at location v is given by

\delta(v) = h0 * \mbox{min}\left[ \frac{1}{\gamma} \sqrt{\frac{n}{\lambda^{\mbox{t}}(v)}}, \mbox{\texttt{trim}} \right]

where \lambda^{\mbox{t}}(v) is a pilot estimate of the temporally varying intensity and \gamma is a scaling constant depending on the pilot estimate.

Value

If at = "points" (the default), the result is a numeric vector with one bandwidth for each data point in X. If at = "bins", the output is an object with class "density" where y component is a vector with the estimated intensity values (see density).

Author(s)

Jonatan A. González

References

Abramson, I. (1982) On bandwidth variation in kernel estimates — a square root law. Annals of Statistics, 10(4), 1217-1223.

Davies, T.M. and Baddeley, A. (2018) Fast computation of spatially adaptive kernel estimates. Statistics and Computing, 28(4), 937-956.

Davies, T.M., Marshall, J.C., and Hazelton, M.L. (2018) Tutorial on kernel estimation of continuous spatial and spatiotemporal relative risk. Statistics in Medicine, 37(7), 1191-1221.

Hall, P. and Marron, J.S. (1988) Variable window width kernel density estimates of probability densities. Probability Theory and Related Fields, 80, 37-49.

Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York.

González J.A. and Moraga P. (2018) An adaptive kernel estimator for the intensity function of spatio-temporal point processes https://arxiv.org/pdf/2208.12026

Examples

t <- 2 * rbeta(100, 1.5, 5.5, 0.2) #Simulated temporal point pattern
bw.abram.temp(t)

kernstadapt documentation built on Sept. 30, 2024, 9:44 a.m.