# EstimateThomas: Parameter Estimation of the Thomas Model In NScluster: Simulation and Estimation of the Neyman-Scott Type Spatial Cluster Models

## Description

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

## Usage

 `1` ``` 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]*[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: 0no report (default). 1output the process of minimizing the negative Palm log-likelihood function until the values converge to MPLEs. (`process1`) 2output the process of optimizing by the simplex with the normalized parameters. (`process2`) 3output both processes. `plot` logical. If `TRUE` (default), the process of optimizing by the simplex with the normalized parameters is plotted.

## Details

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

For all r >= 0,

λ_o(r) = μν + ν/(4πσ^2) * exp(-r^2/(4σ^2)).

The Palm log-likelihood function of the Thomas model on W 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)) }.

## Value

 `mple` MPLE (maximum Palm likelihood estimate). `process1` a list with following components. (Only returned if `process.report` = 1 or 3.) cflg1 (="update") or -1 (="testfn"), where "update" indicates that -log L value has attained the minimum so far, otherwise not. logl.palmthe minimized -log L in the process of minimizing the negative Palm log-likelihood function. mplescorresponding MPLEs. `process2` a list with following components. (Only returned if `process.report` = 2 or 3.) logl.simplexthe minimized -log L by the simplex method. stderrthe standard deviations. pa.normalthe normalized variables corresponding the initial estimates.

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```## 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) ## End(Not run) ```

NScluster documentation built on March 19, 2018, 9:03 a.m.