Moments for Markovian arrival pcess (MAP)

Description

Moments for MAP.

Usage

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Arguments

map

an object of S4 class of MAP (map, gmmpp).

k

an integer of dgrees of moments.

lag

an integer of time lag for corrleation.

Details

MAP parameters are α, D_0 and D_1;

P = (-D_0)^{-1} D_1

and

s P = s.

Then the moments for MAP are marginal moment;

m_k = k! s (-D_0)^{-k} 1,

joint moment;

s_{ij}(lag) = i! j! s (-D_0)^{-i} P^{lag} (-D_0)^{-j} 1,

k-lag correlation (autocorrelation);

rho(lag) = (s_{11}(lag) - m_1^2)/(m_2 - m_1^2)

Value

map.mmoment gives a vector of up to k moments. map.jmoment gives a matrix of s_{ij}(lag), i=1,..,n, j=1,..,n where n is the size of phases. map.acf gives a vector of up to n-lag correlation, where n is the size of phases.

See Also

map, gmmpp, erhmm

Examples

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## create an MAP with specific parameters
(param1 <- map(alpha=c(1,0,0),
	D0=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-4)),
	D1=rbind(c(1,1,0),c(1,0,1),c(2,0,1))))

## create an ER-HMM with specific parameters
(param2 <- erhmm(shape=c(2,3), alpha=c(0.3,0.7),
                rate=c(1.0,10.0),
                P=rbind(c(0.3, 0.7), c(0.1, 0.9))))

## marginal moments of MAP
map.mmoment(k=3, map=param1)
map.mmoment(k=3, map=as(param2, "map"))

## joint moments of MAP
map.jmoment(lag=1, map=param1)
map.jmoment(lag=1, map=as(param2, "map"))

## k-lag correlation
map.acf(map=param1)
map.acf(map=as(param2, "map"))