mple.cppm {NScluster} | R Documentation |
MPLE of the five cluster point process models.
mple.cppm(model = "Thomas", xy.points, pars = NULL, eps = 0.001, uplimit = 0.3, skip = 1) ## S3 method for class 'mple' coef(object, ...) ## S3 method for class 'mple' summary(object, ...)
model |
a character string indicating each cluster point process model:
|
xy.points |
a matrix containing the coordinates |
pars |
a named vector containing a given initial guess of each
parameter. If |
eps |
the sufficiently small number to implement the optimization
procedure for the log-Palm likelihood function. The procedure is iterated
at most 1000 times until the |
uplimit |
upper limit in place of ∞ of the integral in the
probability distribution function relative to the random distance between
two descendant points within the same cluster. The |
skip |
the variable enables one to obtain speedily the initial MPLEs, but
rough approximation. The |
object |
an object of class |
... |
ignored. |
"Thomas"
(Thomas model)
The Palm intensity function is given as follows:
For all r >= 0,
λ_o(r) = μν + ν/(4πσ^2) * exp(-r^2/(4σ^2)).
The log-Palm likelihood function is given by
log L(μ,ν,σ) = ∑_{i, j; i < j, r(i, j) <= 1/2} logν { μ + 1/(4πσ^2) * exp(-r(i, j)^2/(4σ^2)) }
- N(W) ν { πμ/4 + 1 - exp(-1/(16σ^2)) }.
"TypeB"
(Type B model)
The Palm intensity function is given as follows:
For all r >= 0,
λ_o(r) = λ + ν/(4π) * { a/σ1^2 * exp(-r^2/(4σ1^2)) + (1-a)/σ2^2 * exp(-r^2/(4σ2^2)) },
where λ = ν(μ1+μ2) and a = μ1/(μ1+μ2) are the total intensity and the ratio of the intensity of the parent points of the smaller cluster to the total one, respectively.
The log-Palm likelihood function is given by
log L(λ, α, β, σ1, σ2)
=∑_{i, j; i < j, r(i, j) <= 1/2} log[ λ + 1/(4π) { α/σ1^2 * exp(-r(i, j)^2/(4σ1^2)) + β/σ2^2 * exp(-r(i, j)^2/(4σ2^2))}]
- N(W) [ πλ/4 + α{ 1-exp(-1/(16σ1^2)) } + β{ 1-exp(-1/(16σ2^2)) } ],
where α = aν and β = (1-a)ν.
"TypeC"
(Type C model)
The Palm intensity function is given as follows:
For all r >= 0,
λ_o(r) = λ + 1/(4π) * { aν1/σ1^2 * exp(-r^2/(4σ1^2)) + (1-a)ν2/σ2^2 * exp(-r^2/(4σ2^2))},
where λ = μ1ν1 + μ2ν2 and a = μ1ν1/λ are the total intensity and the ratio of the intensity of the smaller cluster to the total one, respectively.
The log-Palm likelihood function is given by
log L(λ, α, β, σ1, σ2)
=∑_{i, j; i < j, r(i, j) <= 1/2} log[ λ + 1/(4π) * { α/σ1^2 * exp(-r(i, j)^2/(4σ1^2)) + β/σ2^2 * exp(-r(i, j)^2/(4σ2^2)) } ]
-N(W) [ πλ/4 + α{ 1-exp(-1/(16σ1^2)) } + β{ 1-exp(-1/(16σ2^2)) } ],
where α = aν1 and β = (1-a)ν2.
For the inverse-power model and the Type A models, we need to take the alternative form without explicit representation of the Palm intensity function. See the second reference below for details.
mple.cppm
returns an object of class "mple"
containing the
following main components:
mple |
MPLE (maximum Palm likelihood estimate). |
log.mpl |
the log maximum Palm likelihood. |
aic |
AIC. |
process1 |
a list with following components.
|
process2 |
a list with following components.
|
There are other methods plot.mple
and print.mple
for this class.
Tanaka, U., Ogata, Y. and Katsura, K. (2008) Simulation and estimation of the Neyman-Scott type spatial cluster models. Computer Science Monographs 34, 1-44. The Institute of Statistical Mathematics, Tokyo. https://www.ism.ac.jp/editsec/csm/.
Tanaka, U., Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott point processes. Biometrical Journal 50, 43-57.
## Not run: # The computation of MPLEs takes a long CPU time in the minimization procedure, # especially for the Inverse-power type and the Type A models. ### Thomas Model # simulation pars <- c(mu = 50.0, nu = 30.0, sigma = 0.03) t.sim <- sim.cppm("Thomas", pars, seed = 117) ## estimation init.pars <- c(mu = 40.0, nu = 40.0, sigma = 0.05) t.mple <- mple.cppm("Thomas", t.sim$offspring$xy, init.pars) coef(t.mple) ### Inverse-Power Type Model # simulation pars <- c(mu = 50.0, nu = 30.0, p = 1.5, c = 0.005) ip.sim <- sim.cppm("IP", pars, seed = 353) ## estimation init.pars <- c(mu = 55.0, nu = 35.0, p = 1.0, c = 0.01) ip.mple <- mple.cppm("IP", ip.sim$offspring$xy, init.pars, skip = 100) coef(ip.mple) ### Type A Model # simulation pars <- c(mu = 50.0, nu = 30.0, a = 0.3, sigma1 = 0.005, sigma2 = 0.1) a.sim <- sim.cppm("TypeA", pars, seed = 575) ## estimation init.pars <- c(mu = 60.0, nu = 40.0, a = 0.5, sigma1 = 0.01, sigma2 = 0.1) a.mple <- mple.cppm("TypeA", a.sim$offspring$xy, init.pars, skip = 100) coef(a.mple) ### Type B Model # simulation pars <- c(mu1 = 10.0, mu2 = 40.0, nu = 30.0, sigma1 = 0.01, sigma2 = 0.03) b.sim <- sim.cppm("TypeB", pars, seed = 257) ## estimation init.pars <- c(mu1 = 20.0, mu2 = 30.0, nu = 30.0, sigma1 = 0.02, sigma2 = 0.02) b.mple <- mple.cppm("TypeB", b.sim$offspring$xy, init.pars) coef(b.mple) ### Type C Model # simulation pars <- c(mu1 = 5.0, mu2 = 9.0, nu1 = 30.0, nu2 = 150.0, sigma1 = 0.01, sigma2 = 0.05) c.sim <- sim.cppm("TypeC", pars, seed = 555) ## estimation init.pars <- c(mu1 = 10.0, mu2 = 10.0, nu1 = 30.0, nu2 = 120.0, sigma1 = 0.03, sigma2 = 0.03) c.mple <- mple.cppm("TypeC", c.sim$offspring$xy, init.pars) coef(c.mple) ## End(Not run)