NScluster (version 1.2.0)

EstimateThomas: Parameter Estimation of the Thomas Model

Description

Parameter estimation of the Thomas model by using the Palm log-likelihood function.

Usage

EstimateThomas(xy.points, pars, eps = 0.001, process.report = 0, plot = TRUE)

Arguments

xy.points

a matrix containing the coordinates (x,y) of points in a unit square: \(W=[0,1]\times[0,1]\).

pars

a named vector of containing the initial guess of the model parameters (mu, nu, sigma), where mu is an intensity of parents, nu is an expected number of descendants for each parent and sigma is a parameter of the dispersal kernel.

eps

the optimization procedure is iterated at most 1000 times until process2$stderr becomes smaller than eps.

process.report

the level of reporting the process of minimizing. Allowed values are as follows:

0

no report (default).

1

output the process of minimizing the negative Palm log-likelihood function until the values converge to MPLEs. (process1)

2

output the process of optimizing by the simplex with the normalized parameters. (process2)

3

output both processes.

plot

logical. If TRUE (default), the process of optimizing by the simplex with the normalized parameters is plotted.

Value

mple

MPLE (maximum Palm likelihood estimate).

process1

a list with following components. (Only returned if process.report = 1 or 3.)

cflg

1 (="update") or -1 (="testfn"), where "update" indicates that -log L value has attained the minimum so far, otherwise not.

logl.palm

the minimized -log L in the process of minimizing the negative Palm log-likelihood function.

mples

corresponding MPLEs.

process2

a list with following components. (Only returned if process.report = 2 or 3.)

logl.simplex

the minimized -log L by the simplex method.

stderr

the standard deviations.

pa.normal

the normalized variables corresponding the initial estimates.

Details

The Palm intensity function of the Thomas model is calculated as follows:

For all \(r \ge 0\),

$$\lambda_{\bm{o}}(r) = \mu\nu + \frac{\nu}{4\pi \sigma^2} \exp \left( -\frac{r^2}{4 \sigma^2} \right).$$

The Palm log-likelihood function of the Thomas model on \(W\) is given by

$$\log L(\mu,\nu,\sigma) = \sum_{\{i,j; i<j, r_{ij} \le 1/2\}} \log \nu \left\{ \mu + \frac{1}{4 \pi \sigma^2} \exp \left( -\frac{{r_{ij}}^2}{4 \sigma^2} \right) \right\}$$

$$- N(W)\nu \left\{ \frac{\pi \mu}{4} + 1 - \exp \left( -\frac{1}{16 \sigma^2} \right) \right\}.$$

References

U. Tanaka, Y. Ogata and K. Katsura, Simulation and estimation of the Neyman-Scott type spatial cluster models, Computer Science Monographs No.34, 2008, 1-44. The Institute of Statistical Mathematics.

Examples

# NOT RUN {
## simulation
pars <- c(mu = 50.0, nu = 30.0, sigma = 0.03)
z <- SimulateThomas(pars, seed = 117)

## estimation
## need long c.p.u time in the minimization procedure
# }
# NOT RUN {
init.pars <- c(mu = 40.0, nu = 40.0, sigma = 0.05)
EstimateThomas(z$offspring$xy, init.pars)
# }