NScluster (version 1.1.1)

SimplexThomas: Parameter estimation of The Thomas Model

Description

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

Usage

SimplexThomas(offspring, pa, ty=1, eps=0.1e-2, process=0, plot=TRUE)

Arguments

offspring

the list of coordinates $(x,y)$ of offspring points.

pa

the initial guesses of the parameters $(mu, nu, sigma)$.

ty

the variable $Ty$ for the standardized coordinates of points in the rectangular region $[0,1]*[0,Ty]$ without loss of generality except for the scaling.

eps

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

process

repot the process of minimizing. Allowed values are

0 :

no report.

1 :

output the process of minimizing the negative Palm log-likelihood function until the values converge to

the MPLE values for given data.

2 :

output the process of optimizing by the simplex with the normalized parameters depending on pa.

The actual estimates are obtained by the indicated x-values times pa.

plot

plot the process of optimizing by the simplex with the normalized parameters depending on pa.

Value

Details

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

For any $r >= 0$,

$$\lambda_0(r) = \mu\nu + \frac{\nu}{4\pi \sigma^2} \exp \left( -\frac{r^2}{4 \sigma^2} \right).$$

The Palm log-likelihood function of the Thomas model is analytically calculated as follows:

$$\log L(\mu,\nu,\sigma) = \sum_{\{i,j; i \ne j, r_{ij}

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

with $R = 1/2$ which means the half of the $t_x$ (TX) in side length of the normalized rectangular.

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

  ## simulation
  seeds <- c(822, 913, 905)
  mu <- 50;  nu <- 30;  sig <- 0.03
  z <- SimulateThomas(seeds, c(mu,nu,sig))

  ## Not run: 
#     ## estimation
#     ## need long c.p.u time in the minimization procedure
#     SimplexThomas(z$offspring, c(mu,nu,sig), process=3)
#   ## End(Not run)