NScluster (version 1.1.0)

SimplexTypeC: Parameter estimation of Type C model

Description

Parameter estimation of Type C Model by using The Palm Log-Likelihood Function.

Usage

SimplexTypeC(offspring, pa1, pa2, ty=1, eps=0.1e-2, process=0, plot=TRUE)

Arguments

offspring

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

pa1

the initial guesses of the parameters $(mu_1, nu_1, sigma_1)$.

pa2

the initial guesses of the parameters $(mu_2, nu_2, sigma_2)$.

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 Type C model is calculated as follows: For any $r >= 0$,

$$\lambda_0(r) = \lambda + \frac{1}{4 \pi} \left\{ \frac{a\nu_1}{\sigma_1^2} \exp \left( -\frac{r^2}{4 \sigma_1^2} \right) + \frac{(1-a)\nu_2}{\sigma_2^2} \exp \left( -\frac{r^2}{4 \sigma_2^2} \right) \right\},$$

where $lambda=mu_1*nu_1+mu_2*nu_2$ is the total population size and $a = mu_1*nu_1/lambda$ is the ratio of the all offspring points of smaller sized cluster to the total population size.

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

$ log L(lambda, alpha, beta, sigma_1, sigma_2)$ $$= \sum_{\{i,j; i \ne j, r_{ij}

$$-N(W) \left[ \pi R^2\lambda + \alpha \left\{ 1 - \exp \left( -\frac{R^2}{4 \sigma_1^2} \right) \right\} + \beta \left\{ 1- \exp \left( -\frac{R^2}{4 \sigma_2^2} \right) \right\} \right],$$

with $R = 1/2$, where $alpha = a*nu_1$ and $beta = (1-a)nu_2$.

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)
  mu1 <- 5;  nu1 <- 30;  sig1 <- 0.01 
  mu2 <- 9;  nu2 <- 150;  sig2 <- 0.05
  z <- SimulateTypeC(seeds, c(mu1,nu1,sig1), c(mu2,nu2,sig2), pmax=200, omax=300)

  ## Not run: 
#     ## estimation
#     ## need long c.p.u time in the minimization procedure
#     setOmpNumThreads(2)
#     SimplexTypeC(z$offspring, c(mu1,nu1,sig1), c(mu2,nu2,sig2), process=3)
#   ## End(Not run)