eulermultinom: Eulermultinomial and gamma-whitenoise distributions

In pomp: Statistical Inference for Partially Observed Markov Processes

Eulermultinomial and gamma-whitenoise distributions

Description

pomp provides a number of probability distributions that have proved useful in modeling partially observed Markov processes. These include the Euler-multinomial family of distributions and the the Gamma white-noise processes.

Usage

reulermultinom(n = 1, size, rate, dt)

deulermultinom(x, size, rate, dt, log = FALSE)

rgammawn(n = 1, sigma, dt)

Arguments

n	integer; number of random variates to generate.
size	scalar integer; number of individuals at risk.
rate	numeric vector of hazard rates.
dt	numeric scalar; duration of Euler step.
x	matrix or vector containing number of individuals that have succumbed to each death process.
log	logical; if TRUE, return logarithm(s) of probabilities.
sigma	numeric scalar; intensity of the Gamma white noise process.

Details

If N individuals face constant hazards of death in K ways at rates r_1, r_2, \dots, r_K, then in an interval of duration \Delta{t}, the number of individuals remaining alive and dying in each way is multinomially distributed:

(\Delta{n_0}, \Delta{n_1}, \dots, \Delta{n_K}) \sim \mathrm{Multinomial}(N;p_0,p_1,\dots,p_K),

where \Delta{n_0}=N-\sum_{k=1}^K \Delta{n_k} is the number of individuals remaining alive and \Delta{n_k} is the number of individuals dying in way k over the interval. Here, the probability of remaining alive is

p_0=\exp(-\sum_k r_k \Delta{t})

and the probability of dying in way k is

p_k=\frac{r_k}{\sum_j r_j} (1-p_0).

In this case, we say that

(\Delta{n_1},\dots,\Delta{n_K})\sim\mathrm{Eulermultinom}(N,r,\Delta t),

where r=(r_1,\dots,r_K). Draw m random samples from this distribution by doing

    dn <- reulermultinom(n=m,size=N,rate=r,dt=dt),

where r is the vector of rates. Evaluate the probability that x=(x_1,\dots,x_K) are the numbers of individuals who have died in each of the K ways over the interval \Delta t=dt, by doing

    deulermultinom(x=x,size=N,rate=r,dt=dt).

Bretó & Ionides (2011) discuss how an infinitesimally overdispersed death process can be constructed by compounding a multinomial process with a Gamma white noise process. The Euler approximation of the resulting process can be obtained as follows. Let the increments of the equidispersed process be given by

    reulermultinom(size=N,rate=r,dt=dt).

In this expression, replace the rate r with r\,{\Delta{W}}/{\Delta t}, where \Delta{W} \sim \mathrm{Gamma}(\Delta{t}/\sigma^2,\sigma^2) is the increment of an integrated Gamma white noise process with intensity \sigma. That is, \Delta{W} has mean \Delta{t} and variance \sigma^2 \Delta{t}. The resulting process is overdispersed and converges (as \Delta{t} goes to zero) to a well-defined process. The following lines of code accomplish this:

    dW <- rgammawn(sigma=sigma,dt=dt)

    dn <- reulermultinom(size=N,rate=r,dt=dW)

or

    dn <- reulermultinom(size=N,rate=r*dW/dt,dt=dt).

He et al. (2010) use such overdispersed death processes in modeling measles and the "Simulation-based Inference" course discusses the value of allowing for overdispersion more generally.

For all of the functions described here, access to the underlying C routines is available: see below.

Value

reulermultinom	Returns a length(rate) by n matrix. Each column is a different random draw. Each row contains the numbers of individuals that have succumbed to the corresponding process.
deulermultinom	Returns a vector (of length equal to the number of columns of x) containing the probabilities of observing each column of x given the specified parameters (size, rate, dt).
rgammawn	Returns a vector of length n containing random increments of the integrated Gamma white noise process with intensity sigma.

C API

An interface for C codes using these functions is provided by the package. Visit the package homepage to view the pomp C API document.

Author(s)

Aaron A. King

References

\Breto

2011

\He

2010

See Also

More on implementing POMP models: Csnippet, accumvars, basic_components, betabinomial, covariates, dinit_spec, dmeasure_spec, dprocess_spec, emeasure_spec, parameter_trans(), pomp-package, pomp_constructor, prior_spec, rinit_spec, rmeasure_spec, rprocess_spec, skeleton_spec, transformations, userdata, vmeasure_spec

Examples

print(dn <- reulermultinom(5,size=100,rate=c(a=1,b=2,c=3),dt=0.1))
deulermultinom(x=dn,size=100,rate=c(1,2,3),dt=0.1)
## an Euler-multinomial with overdispersed transitions:
dt <- 0.1
dW <- rgammawn(sigma=0.1,dt=dt)
print(dn <- reulermultinom(5,size=100,rate=c(a=1,b=2,c=3),dt=dW))

pomp documentation built on Sept. 13, 2024, 1:08 a.m.