superthin: Perform super-thinned residuals method

In stppResid: Perform residual analysis on space-time point process models.

Description

superthin takes a space-time point pattern and conditional intensity model and calculates a set of super-thinned residuals for further analysis.

Usage

superthin(X, cifunction, theta = NULL, k = NULL, lambda = NULL)

Arguments

X	A “stpp” object.
cifunction	A function returning the value of the conditional intensity at all points in X. The function should take arguments X and an optional vector of parameters theta.
theta	Optional: A vector of parameters to be passed to cifunction.
k	The super-thinning rate.
lambda	Optional: A vector of conditional intensities at each point in X.

Details

Super-thinned residuals (Clements et. al. (2012)) is a type of transformation based residuals for space-time point processes based on both thinned residuals (see Schoenberg (2003)) and superposed residuals (see Bremaud (1981)). The residuals consist of a set of points that should be homogeneous Poisson, with rate k, if the model for the conditional intensity is correct. Any patterns or inter-point interaction in the residuals indicates a lack of fit of the model. To test for homogeneity, a commonly used tool is Ripley's K-function, a version of which can be found in the spatstat package.

Super-thinned residuals are found as follows:

1. The super-thinning rate k is specified. This rate determines the amount of thinning and superposition conducted, and also determines the final rate of the super-thinned residual point process.

2. All observed points in X where lambda_hat < k are automatically kept.

3. All points in X where lambda_hat >= k are kept with probability k/lambda_hat.

4. In all space-time locations where λ < k, points are simulated with rate k - lambda_hat.

The result should be a homogeneous Poisson process with rate k if the model is correct.

The conditional intensity function, cifunction, should take X as the first argument, and an optional theta as the second argument, and return a vector of conditional intensity estimates with length equal to the number of points in X, i.e. the length of X$x. cifunction is required, while lambda is optional. lambda eliminates the need for superthin to calculate the conditional intensity at each observed point in X.

If k is not specified, the default is the mean of lambda_hat estimated by the total number of points divided by the volume of the space-time window.

Value

Outputs an object of class “superthin”, which is a list of

X	An object of class “stpp”.
k	The super-thinning rate.
residuals	A data frame consisting of the x, y, and t coordinates of the super-thinned residuals.
super	A data frame consisting of the x, y, and t coordinates of the superposed points.
keep1	A data frame consisting of the x, y, and t coordinates of the automatically retained points.
keep2	A data frame consisting of the x, y, and t coordinates of the points remaining after the thinning has taken place.
deleted	A data frame consisting of the x, y, and t coordinates of the points removed during the thinning process.

Author(s)

Robert Clements

References

Bremaud, P. Point Processes and Queues: Martingale Dynamics. SpringerVerlag, New York, 1981.

Clements, R.A., Schoenberg, F.P., and Veen, A. (2012) Evaluation of space-time point process models using super-thinning. Environmetrics, to appear.

Schoenberg, F.P. (2003) Multi-dimensional residuals analysis of point process models for earthquake occurrences. Journal of the American Statistical Association, 98, 789–795.

Clements, R.A., Schoenberg, F.P., and Schorlemmer, D. (2011) Residual analysis methods for space-time point processes with applications to earthquake forecast models in California. Annals of Applied Statistics, 5, Number 4, 2549–2571.

See Also

stpp, thinresid, supresid, spatstat

Examples

#===> load simulated data <===#
data(simdata)
X <- stpp(simdata$x, simdata$y, simdata$t)

#===> define conditional intensity function <===#
ci1 <- function(X, theta){theta[1]*exp(-theta[2]*X$x - 
theta[3]*X$y - theta[4]*X$t)} #correct model

stresiduals1 <- superthin(X, ci1, theta = c(3000, 2, 2, 2), k = 250)
stresiduals2 <- superthin(X, ci1, theta = c(2500, 5, 5, 10), k = 250)
#===> plot results <===#
par(mfrow = c(1,2))
plot(stresiduals1)
plot(stresiduals2)

summary(stresiduals1)
summary(stresiduals2)

Example output

Loading required package: deldir
deldir 0.1-14
Loading required package: splancs
Loading required package: sp

Spatial Point Pattern Analysis Code in S-Plus
 
 Version 2 - Spatial and Space-Time analysis

Loading required package: cubature
stw missing - using default space-time window [0, 1]x[0, 1]
$k
[1] 250

$n
[1] 230

$n.exp
[1] 250

$p.val
[1] 0.1076399

attr(,"class")
[1] "summary.superthin"
$k
[1] 250

$n
[1] 470

$n.exp
[1] 250

$p.val
[1] 1.015648e-35

attr(,"class")
[1] "summary.superthin"

stppResid documentation built on May 29, 2017, 3:48 p.m.