Description Usage Arguments Details Value Author(s) References See Also Examples
supresid
takes a space-time point pattern and conditional intensity model and calculates a set of superposed residuals for further analysis.
1 |
X |
A “ |
cifunction |
A function returning the value of the conditional intensity at all points in |
theta |
Optional: A vector of parameters to be passed to |
k |
The superposition rate. |
lambda |
Optional: A vector of conditional intensities at each point in |
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$x
. cifunction
is required, while lambda
is optional. lambda
eliminates the need for supresid
to calculate the conditional intensity at each observed point in X
.
If k
is not specified, the default is the maximum lambda_hat estimated at the points.
Outputs an object of class “supresid
”, which is a list of
X |
An object of class “ |
k |
The superposition rate. |
residuals |
A data frame consisting of the x, y, and t coordinates of the superposed residuals. |
super |
A data frame consisting of the x, y, and t coordinates of the superposed points. |
Robert Clements
Bremaud, P. Point Processes and Queues: Martingale Dynamics. SpringerVerlag, New York, 1981.
Clements, R.A., Schoenberg, F.P., and Schorlemmer, D. (2011) Residual analysis methods for space-time point processes with applications to earthquake forecast models in California. Annals of Applied Statistics, 5, Number 4, 2549–2571.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | #===> load simulated data <===#
data(simdata)
X <- stpp(simdata$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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