BGWM.covar.estim {Branching}

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)

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.

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).

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.

Author(s)

Camilo Jose Torres-Jimenez [e-mail]

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

## Not run: 
## 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

## End(Not run)