Branching (version 0.9.7)

BGWM.covar.estim: Estimation of the covariance matrices of a multi-type Bienayme - Galton - Watson process

Description

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

Usage

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

Value

A list object with:

method

method of estimation selected.

V

A matrix object, estimation of the \(d\) covariance matrices of the process, combined by rows.

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-m with empirical estimation of the mean matrix, MLE-m with maximum likelihood estimation of the mean matrix).

d

positive integer, number of types.

n

positive integer, nth generation.

z0

nonnegative integer vector of size d, initial population by type.

Author

Camilo Jose Torres-Jimenez

Details

This function estimates the covariance matrices of a BGWM process using two possible estimators from asymptotic results related with empirical estimator and maximum likelihood estimator of the mean matrix, 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).

References

Torres-Jimenez, C. J. (2010), Relative frequencies and parameter estimation in multi-type Bienayme - 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.

See Also

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

Examples

if (FALSE) {
## Estimation of covariace matrices 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

# covariance matrices of the process from its offspring distributions
V <- BGWM.covar(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

# estimation of covariance matrices using mean matrix empiric estimate
# from generated trajectories of the process 
V.EE.m <- BGWM.covar.estim( simulated.data, "EE-m", d, n, N )$V

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

# Comparison of exact and estimated covariance matrices
V
V - V.EE.m
V - V.MLE.m
}