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

## Description

Parameter estimation of the Type B model by using the Palm Log-Likelihood Function.

## Usage

 `1` ``` EstimateTypeB(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 (`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. `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 Type B model is calculated as follows:

For all r >= 0,

λ_o(r) = λ + ν/(4π) * { a/σ1^2 * exp(-r^2/(4σ1^2)) + (1-a)/σ2^2 * exp(-r^2/(4σ2^2)) },

where λ = ν(μ1+μ2) is the total population size and a = μ1/(μ1+μ2) 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(λ, α, β, σ1, σ2)

=∑_{i, j; i < j, r(i, j) <= 1/2} log[ λ + 1/(4π) { α/σ1^2 * exp(-r(i, j)^2/(4σ1^2)) + β/σ2^2 * exp(-r(i, j)^2/(4σ2^2)) } ]

- N(W) [ πλ/4 + α{ 1-exp(-1/(16σ1^2)) } + β{ 1-exp(-1/(16σ2^2)) } ],

where α = aν and β = (1-a)ν.

## 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 (`mu`, `nu`, `a`, `sigma1`, `sigma2`), where `mu` = `mu1`+`mu2` and `a` = `mu1`/(`mu1`+`mu2`). `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 (`mu`, `nu`, `a`, `sigma1`, `sigma2`) as described above.

## 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(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) ## End(Not run) ```

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