BGWM.mean {Branching} | R Documentation |
Calculates the mean matrix of a multi-type Bienayme - Galton - Watson process from its offspring distributions, additionally, it could be obtained the mean matrix in a specific time n and the mean vector of the population in the nth generation, if it is provided the initial population vector.
BGWM.mean(dists, type=c("general","multinomial","independents"), d, n=1, z0=NULL, maxiter = 1e5)
dists |
offspring distributions. Its structure depends on the class of the Bienayme - Galton - Watson process (See details and examples). |
type |
Class or family of the Bienayme - Galton - Watson process (See details and examples). |
d |
positive integer, number of types. |
n |
positive integer, nth generation. |
z0 |
nonnegative integer vector of size d, initial population by type. |
maxiter |
positive integer, size of the simulated sample used to estimate the parameters of univariate distributions that do not have an analytical formula for their exact calculation. |
This function calculates the mean matrix of a multi-type Bienayme - Galton - Watson (BGWM) process from its offspring distributions.
From particular offspring distributions and taking into account a differentiated algorithmic approach, we propose the following classes or types for these processes:
general
This option is for BGWM processes without conditions over
the offspring distributions, in this case, it is required as
input data for each distribution, all d-dimensional vectors with their
respective, greater than zero, probability.
multinomial
This option is for BGMW processes where each offspring
distribution is a multinomial distribution with a random number of
trials, in this case, it is required as input data, d univariate
distributions related to the random number of trials for each
multinomial distribution and a d \times d matrix where each row
contains probabilities of the d possible outcomes for each multinomial
distribution.
independents
This option is for BGMW processes where each offspring
distribution is a joint distribution of d combined independent
discrete random variables, one for each type of individuals, in this
case, it is required as input data d^2 univariate distributions.
The structure need it for each classification is illustrated in the examples.
These are the univariate distributions available:
unif Discrete uniform distribution, parameters min and max. All the non-negative integers between min y max have the same probability.
binom Binomial distribution, parameters n and p.
p(x) = choose(n,x) p^x (1-p)^(n-x)
for x = 0, ..., n.
hyper Hypergeometric distribution, parameters m (the number of white balls in the urn), n (the number of white balls in the urn), k (the number of balls drawn from the urn).
p(x) = choose(m, x) choose(n, k-x) / choose(m+n, k)
for x = 0, ..., k.
geom Geometric distribution, parameter p.
p(x) = p (1-p)^x
for x = 0, 1, 2, ...
nbinom Negative binomial distribution, parameters n and p.
p(x) = Gamma(x+n)/(Gamma(n) x!) p^n (1-p)^x
for x = 0, 1, 2, ...
pois Poisson distribution, parameter lambda.
p(x) = lambda^x exp(-lambda)/x!
for x = 0, 1, 2, ...
norm Normal distribution rounded to integer values and negative values become 0, parameters mu and sigma.
p(x) = \int_{x-0.5}^{x+0.5} 1/(sqrt(2 pi) sigma) e^-((t - mu)^2/(2sigma^2)) dt
for x = 1, 2, ...
p(x) = \int_{-∞}^{0.5} 1/(sqrt(2 pi) sigma) e^-((t - mu)^2/(2sigma^2)) dt
for x = 0
lnorm Lognormal distribution rounded to integer values,
parameters logmean
= mu y logsd
= sigma.
p(x) = \int_{x-0.5}^{x+0.5} 1/(sqrt(2 pi) sigma t) e^-((log t - mu)^2 / (2sigma^2))dt
for x = 1, 2, ...
p(x) = \int_{0}^{0.5} 1/(sqrt(2 pi) sigma t) e^-((log t - mu)^2 / (2sigma^2)) dt
for x = 0
gamma Gamma distribution rounded to integer values,
parameters shape
= a y scale
= s.
p(x)= \int_{x-0.5}^{x+0.5} 1/(s^a Gamma(a)) t^(a-1) e^-(t/s) dt
para x = 1, 2, ...
p(x)= \int_{0}^{0.5} 1/(s^a Gamma(a)) t^(a-1) e^-(t/s) dt
for x = 0
When the offspring distributions used norm
, lnorm
or
gamma
, mean related to these univariate distributions
is estimated by calculating sample mean of maxiter
random
values generated from the corresponding distribution.
A matrix
object with the mean matrix of the process in the nth
generation, or, a vector
object with the mean vector of the
population in the nth generation, in case of provide the initial population
vector (z0).
Camilo Jose Torres-Jimenez cjtorresj@unal.edu.co
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.
Stefanescu, C. (1998), 'Simulation of a multitype Galton-Watson chain', Simulation Practice and Theory 6(7), 657-663.
Athreya, K. & Ney, P. (1972), Branching Processes, Springer-Verlag.
Harris, T. E. (1963), The Theory of Branching Processes, Courier Dover Publications.
rBGWM
, BGWM.covar
, BGWM.mean.estim
, BGWM.covar.estim
## Not run: ## Means of a BGWM process based on a model analyzed in Stefanescu (1998) # Variables and parameters d <- 2 n <- 30 N <- c(90, 10) a <- c(0.2, 0.3) # with independent distributions Dists.i <- data.frame( name=rep( "pois", d*d ), param1=rep( a, rep(d,d) ), stringsAsFactors=FALSE ) # mean matrix of the process I.matriz.m <- BGWM.mean(Dists.i, "independents", d) # mean vector of the population in the nth generation # from vector N representing the initial population I.vector.m.n_N <- BGWM.mean(Dists.i, "independents", d, n, N) # with multinomial distributions dist <- data.frame( name=rep( "pois", d ), param1=a*d, stringsAsFactors=FALSE ) matrix.b <- matrix( rep(0.5, 4), nrow=2 ) Dists.m <- list( dists.eta=dist, matrix.B=matrix.b ) # mean matrix of the process M.matrix.m <- BGWM.mean(Dists.m, "multinomial", d) # mean vector of the population in the nth generation # from vector N representing the initial population M.vector.m.n_N <- BGWM.mean(Dists.m, "multinomial", d, n, N) # with general distributions (approximation) max <- 30 A <- t(expand.grid(c(0:max),c(0:max))) aux1 <- factorial(A) aux1 <- apply(aux1,2,prod) aux2 <- apply(A,2,sum) distp <- function(x,y,z){ exp(-d*x)*(x^y)/z } p <- sapply( a, distp, aux2, aux1 ) prob <- list( dist1=p[,1], dist2=p[,2] ) size <- list( dist1=ncol(A), dist2=ncol(A) ) vect <- list( dist1=t(A), dist2=t(A) ) Dists.g <- list( sizes=size, probs=prob, vects=vect ) # mean matrix of the process G.matrix.m <- BGWM.mean(Dists.g, "general", d) # mean vector of the population in the nth generation # from vector N representing the initial population G.vector.m.n_N <- BGWM.mean(Dists.g, "general", d, n, N) # Comparison of results I.vector.m.n_N I.vector.m.n_N - M.vector.m.n_N M.vector.m.n_N - G.vector.m.n_N G.vector.m.n_N - I.vector.m.n_N ## End(Not run)