EstimateTypeB: Parameter Estimation of the Type B Model

Description Usage Arguments Details Value References Examples

View source: R/NScluster.R

Description

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

Usage

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  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 (mui, nu, sigmai) 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:

0

no report (default).

1

output the process of minimizing the negative Palm log-likelihood function until the values converge to MPLEs. (process1)

2

output the process of optimizing by the simplex with the normalized parameters. (process2)

3

output 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.)

cflg

1 (="update") or -1 (="testfn"), where "update" indicates that -log L value has attained the minimum so far, otherwise not.

logl.palm

the minimized -log L in the process of minimizing the negative Palm log-likelihood function.

mples

corresponding 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.simplex

the minimized -log L by the simplex method.

stderr

the standard deviations.

pa.normal

the 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

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## 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.