Parameter estimation of the Type B model by using the Palm Log-Likelihood Function.
EstimateTypeB(xy.points, pars, eps = 0.001, process.report = 0, plot = TRUE)a matrix containing the coordinates (x,y) of points in
a unit square: \(W=[0,1]\times[0,1]\).
a named vector of containing the initial guess of the model
parameters (mu1, mu2, nu, sigma1, sigma2),
where (mu\(i\), nu, sigma\(i\)) is an intensity of
parents, an expected number of descendants, a parameter of the dispersal
kernel for superposed component \(i\) (\(i = 1,2\)), respectively.
the optimization procedure is iterated at most 1000 times until
process2$stderr becomes smaller than eps.
the level of reporting the process of minimizing. Allowed values are as follows:
no report (default).
output the process of minimizing the negative Palm log-likelihood
function until the values converge to MPLEs. (process1)
output the process of optimizing by the simplex with
the normalized parameters. (process2)
output both processes.
logical. If TRUE (default), the process of optimizing by
the simplex with the normalized parameters is plotted.
MPLE (maximum Palm likelihood estimate).
a list with following components.
(Only returned if process.report = 1 or 3.)
1 (="update") or -1 (="testfn"), where "update" indicates that -log L value has attained the minimum so far, otherwise not.
the minimized -log L in the process of minimizing the negative Palm log-likelihood function.
corresponding MPLEs (mu, nu, a,
sigma1, sigma2), where mu = mu1+mu2
and a = mu1/(mu1+mu2).
a list with following components.
(Only returned if process.report = 2 or 3.)
the minimized -log L by the simplex method.
the standard deviations.
the normalized variables (mu, nu, a,
sigma1, sigma2) as described above.
The Palm intensity function of the Type B model is calculated as follows:
For all \(r \ge 0\),
$$\lambda_{\bm{o}}(r) = \lambda + \frac{\nu}{4 \pi} \left\{ \frac{a}{{\sigma_1}^2} \exp \left( -\frac{r^2}{4{\sigma_1}^2} \right)+ \frac{(1-a)}{{\sigma_2}^2} \exp \left( -\frac{r^2}{4{\sigma_2}^2} \right) \right\},$$
where \(\lambda = \nu(\mu_1+\mu_2)\) is the total population size and \(a = \mu_1/(\mu_1+\mu_2)\) is the ratio of the parent points of the smaller sized cluster to the total ones.
The Palm log-likelihood function of the Type B model on \(W\) is given by
\(\log L(\lambda, \alpha, \beta, \sigma_1, \sigma_2)\) $$=\sum_{\{i,j; i<j, r_{ij} \le 1/2\}} \log \left[ \lambda + \frac{1}{4 \pi} \left\{ \frac{\alpha}{{\sigma_1}^2} \exp \left( -\frac{{r_{ij}}^2}{4{\sigma_1}^2} \right) + \frac{\beta}{{\sigma_2}^2} \exp \left( -\frac{{r_{ij}}^2}{4{\sigma_2}^2} \right) \right\} \right]$$ $$- N(W) \left[ \frac{\pi \lambda}{4} + \alpha \left\{ 1 - \exp \left( -\frac{1}{16{\sigma_1}^2} \right) \right\} + \beta \left\{ 1- \exp \left( -\frac{1}{16{\sigma_2}^2} \right) \right\} \right],$$
where \(\alpha = a\nu\) and \(\beta = (1-a)\nu\).
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.
# NOT RUN {
## simulation
pars <- c(mu1 = 10.0, mu2 = 40.0, nu = 30.0, sigma1 = 0.01, sigma2 = 0.03)
z <- SimulateTypeB(pars, seed = 257)
## estimation
## need very long c.p.u time in the minimization procedure
# }
# NOT RUN {
init.pars <- c(mu1 = 20.0, mu2 = 30.0, nu = 30.0, sigma1 = 0.02, sigma2 = 0.02)
EstimateTypeB(z$offspring$xy, init.pars)
# }
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