BGWM.mean.estim {Branching} R Documentation

## Estimation of the mean matrix of a multi-type Bienayme - Galton - Watson process

### Description

Calculates a estimation of the mean matrix of a multi-type Bienayme - Galton - Watson process from experimental observed data that can be modeled by this kind of process.

### Usage

BGWM.mean.estim(sample, method=c("EE","MLE"), d, n, z0)


### Arguments

 sample  nonnegative integer matrix with d columns and dn rows, trajectory of the process with the number of individuals for every combination parent type - descendent type (observed data). method  methods of estimation (EE Empirical estimacion, MLE Maximum likelihood estimation). d  positive integer, number of types. n  positive integer, nth generation. z0  nonnegative integer vector of size d, initial population by type.

### Details

This function estimates the mean matrix of a BGWM process using two possible estimators, empirical estimator and maximum likelihood estimator, they both require the so-called full sample associated with the process, ie, it is required to have the trajectory of the process with the number of individuals for every combination parent type - descendent type. For more details see Torres-Jimenez (2010) or Maaouia & Touati (2005).

### Value

A list object with:

 method  method of estimation selected. m  A matrix object, estimation of the d \times d mean matrix of the process.

### Author(s)

Camilo Jose Torres-Jimenez cjtorresj@unal.edu.co

### References

Torres-Jimenez, C. J. (2010), Relative frequencies and parameter estimation in multi-type Bienaym? - Galton - Watson processes, Master's Thesis, Master of Science in Statistics. Universidad Nacional de Colombia. Bogota, Colombia.

Maaouia, F. & Touati, A. (2005), 'Identification of Multitype Branching Processes', The Annals of Statistics 33(6), 2655-2694.

BGWM.mean, BGWM.covar, rBGWM, BGWM.covar.estim

### Examples

## Not run:
## Estimation of mean matrix from simulated data

# Variables and parameters
d <- 3
n <- 30
N <- c(10,10,10)
LeslieMatrix <- matrix( c(0.08, 1.06, 0.07,
0.99, 0, 0,
0, 0.98, 0), 3, 3 )

# offspring distributions from the Leslie matrix
# (with independent distributions)
Dists.pois <- data.frame( name=rep( "pois", d ),
param1=LeslieMatrix[,1],
param2=NA,
stringsAsFactors=FALSE )
Dists.binom <- data.frame( name=rep( "binom", 2*d ),
param1=rep( 1, 2*d ),
param2=c(t(LeslieMatrix[,-1])),
stringsAsFactors=FALSE )
Dists.i <- rbind(Dists.pois,Dists.binom)
Dists.i <- Dists.i[c(1,4,5,2,6,7,3,8,9),]
Dists.i

# mean matrix of the process from its offspring distributions
m <- BGWM.mean(Dists.i,"independents",d)

# generated trajectories of the process from its offspring distributions
simulated.data <- rBGWM(Dists.i, "independents", d, n, N,
TRUE, FALSE, FALSE)$o.c.s # mean matrix empiric estimate from generated trajectories of the process m.EE <- BGWM.mean.estim( simulated.data, "EE", d, n, N )$m

# mean matrix maximum likelihood estimate from generated trajectories
# of the process
m.MLE <- BGWM.mean.estim( simulated.data, "MLE", d, n, N )\$m

# Comparison of exact and estimated mean matrices
m
m - m.EE
m - m.MLE

## End(Not run)


[Package Branching version 0.9.4 Index]