supresid: Perform superposed residuals method

Description

supresid takes a space-time point pattern and conditional intensity model and calculates a set of superposed residuals for further analysis.

Usage

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

Arguments

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.

The superposition rate.

lambda

Optional: A vector of conditional intensities at each point in X.

Value

X: An object of class “stpp”.
k: The superposition rate.
residuals: A data frame consisting of the x, y, and t coordinates of the superposed residuals.
super: A data frame consisting of the x, y, and t coordinates of the superposed points.

Details

Superposed residuals is a type of transformation based residuals for space-time point processes (see Bremaud (1981)) which consists of superimposing a point process with rate k - $lambda_hat$ onto the observed point process. k should be the maximum conditional intensity over the entire space-time window. If the model for the conditional intensity is correct, the residuals should be homogeneous Poisson with rate k. 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.

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 supresid to calculate the conditional intensity at each observed point in X.

If k is not specified, the default is the maximum $lambda_hat$ estimated at the points.

References

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

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.

Examples

Run this code

#===> 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

sresiduals1 <- supresid(X, ci1, theta = c(3000, 2, 2, 2))
sresiduals2 <- supresid(X, ci1, theta = c(2500, 5, 5, 10))
#===> plot results <===#
par(mfrow = c(1,2))
plot(sresiduals1)
plot(sresiduals2)

summary(sresiduals1)
summary(sresiduals2)

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