ipc.estK {ecespa} | R Documentation |
Fits the (In)homogeneous Poisson Cluster point process to a point pattern dataset by the Method of Minimum Contrast.
ipc.estK(mippp, lambda = NULL, correction = "iso", r = NULL, sigma2 = NULL, rho = NULL, q = 1/4, p = 2) ## S3 method for class 'ecespa.minconfit': plot(x, type="L", add=FALSE, xlim=NULL, ylim=NULL, lwd=c(1,1), lty=c(1,2), col=c(1,2), main=NULL, ...)
mippp |
Point pattern to which the (I)PCP will be fitted. A point pattern with the ppp format of spatstat . |
lambda |
Optional. Values of the estimated intensity function as a pixel image (object of class "im " of spatstat) giving the intensity values at all locations of mippp . |
correction |
A character item selecting any of the options "border", "bord.modif", "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied in the computation of the K(r) function. |
r |
Numeric vector. The values of the argument r at which the K(r) functions should be evaluated. |
sigma2 |
Optional. Starting value for the parameter sigma2 of the Poisson Cluster process. |
rho |
Optional. Starting value for the parameter rho of the Poisson Cluster process. |
q |
q exponent of the contrast criterion (see mincontrast ). |
p |
p exponent of the contrast criterion (see mincontrast ). |
x |
An object of class 'ecespa.minconfit', resulting of applying ipc.estK to fit a Poisson Cluster Process. |
type |
Type of function to be ploted. If type="L", function L(r) [= sqrt(K(r)/pi)-r] is ploted. Otherwise, function K(r) is ploted. |
add |
Logical. Should the curves be added to another plot? |
xlim |
Vector setting the limits of the x-axis. |
ylim |
Vector setting the limits of the y-axis. |
lwd |
Vector (length=2) setting the line width for ploting the two functions. |
lty |
Vector (length=2) setting the line type for ploting the two functions. |
col |
Vector (length=2) setting the line color for ploting the two functions. |
main |
Optional. Text to appear as a title of the plot. |
... |
Additional graphical parameters passed to link{plot} . |
The algorithm fits the (inhomogeneous) Poisson cluster point process (PCP) to a point pattern, by finding the parameters of the (inhomogeneous) Poisson cluster model
which give the closest match between the theoretical K function of the Poisson cluster process and the observed
K function. For a concise explanation of the PCP see pc.estK
. For a more detailed explanation of the Method of Minimum Contrast, see mincontrast
in spatstat or Diggle (2003: 86).
The inhomogeneous PCP can be thought of as a thinned process of an homogeneous PCP, where the spatially varying thinning probability f(s) is related to the spatially varying intensity function lambda(s) as f(s) = lambda(s)/max{lambda(s)} (Waagepetersen, 2007). As the inhomogeneous K function for the IPCP coincides with the (homogeneous) K function for the corresponding homogeneous PCP, the parameters of the underlying homomgeneous PCP can be estimated as those that give the closest match between the theoretical K function for the homogeneous PCP and the empirical inhomogeneous K function for the observed IPCP.
This Poisson cluster process can be simulated with rIPCP
.
ipc.estK
gives an object of class 'ecespa.minconfit
', basically a list with the following components:
sigma2 |
Parameter sigma^2. |
rho |
Parameter rho. |
d.theta |
Minimized value of the contrast criterion D(theta). |
Kobs |
Values of the observed K-function. |
Kfit |
Values of the fitted K-function. |
r |
Sequence of distances at which Kobs and Kfit have been estimated. |
data |
Original point pattern. |
lambda |
Original intensity function. |
dataname |
Name of the original point pattern. |
lambdaname |
Name of the original intensity function image. |
q |
q exponent of the contrast criterion. |
p |
p exponent of the contrast criterion. |
Marcelino de la Cruz Rot marcelino.delacruz@upm.es, inspired by some code of Philip M. Dixon http://www.public.iastate.edu/~pdixon/
Diggle, P. J. 2003. Statistical analysis of spatial point patterns. Arnold, London.
Waagepetersen, R. P. 2007. An estimating function approach to inference for inhomogeneous Neymann-Scott processes. Biometrics 63: 252-258.
some functions in spatstat: mincontrast
for a more general implementation of the method of mimimum contrast;
matclust.estK
and lgcp.estK
fit other appropriate processes for clustered patterns.
## Not run: ##################### ## Same example as in pc.estK data(gypsophylous) ## Estimate K function ("Kobs"). gyps.env <- envelope(gypsophylous, Kest, correction="iso", nsim=99) plot(gyps.env, sqrt(./pi)-r~r) ## Fit Poisson Cluster Process. The limits of integration ## rmin and rmax are setup to 0 and 60, respectively. cosa.pc2 <- ipc.estK(gypsophylous, r = gyps.env$r[gyps.env$r<=60]) ## Add fitted Kclust function to the plot. plot(cosa.pc2, add=T, lwd=c(3,3)) ## A kind of pointwise test of the gypsophylous pattern been a realisation ## of the fitted model, simulating with sim.poissonc and using function J (Jest). gyps.env.sim2 <- envelope(gypsophylous, Jest, nsim=99, simulate=expression(rIPCP(cosa.pc2))) plot(gyps.env.sim2, main="") ##################### ## Inhomogeneous example data(urkiola) # get univariate pp I.ppp <- split.ppp(urkiola)$birch #estimate inhomogeneous intensity function I.lam <- predict (ppm(I.ppp, ~polynom(x,y,2)), type="trend", ngrid=200) # Compute and plot envelopes to Kinhom, simulating from an Inhomogeneous Poisson Process: I2.env <- envelope( I.ppp,Kinhom, lambda=I.lam, correction="trans", nsim=99, simulate=expression(rpoispp(I.lam))) plot(I2.env, sqrt(./pi)-r~r) # It seems that there is short scale clustering; lets fit an IPCP: I.ki <- ipc.estK(mippp=I.ppp, lambda=I.lam, correction="trans") # Compute and plot envelopes to Kinhom, simulating from the fitted IPCP: Ipc.env= Ki(I.ki, correction="trans", nsim=99, ngrid=200) plot (Ipc.env) ## End(Not run)