distributions {pomp} | R Documentation |
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.
reulermultinom(n = 1, size, rate, dt) deulermultinom(x, size, rate, dt, log = FALSE) rgammawn(n = 1, sigma, dt)
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. |
If N individuals face constant hazards of death in k ways at rates r1,r2,…,rk, then in an interval of duration dt, the number of individuals remaining alive and dying in each way is multinomially distributed:
(N-∑(dni), dn1, …, dnk) ~ multinomial(N;p0,p1,…,pk),
where dni is the number of individuals dying in way i over the interval, the probability of remaining alive is p0=exp(-∑(ri dt)), and the probability of dying in way j is
pj=(1-exp(-sum(ri dt))) rj/(∑(ri)).
In this case, we say that
(dn1,…,dnk)~eulermultinom(N,r,dt),
where r=(r1,…,rk). 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=(x1,…,xk) are the numbers of individuals who have died in each of the k ways over the interval dt
,
by doing
deulermultinom(x=x,size=N,rate=r,dt=dt).
Breto & 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 {Δ W}/{Δ t}, where Δ W \sim \mathrm{Gamma}(Δ t/σ^2,σ^2) is the increment of an integrated Gamma white noise process with intensity σ. That is, Δ W has mean Δ t and variance σ^2 Δ t. The resulting process is overdispersed and converges (as Δ 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.
For all of the functions described here, access to the underlying C routines is available: see below.
reulermultinom |
Returns a |
deulermultinom |
Returns a vector (of length equal to the number of columns of |
rgammawn |
Returns a vector of length |
An interface for C codes using these functions is provided by the package. Visit the package homepage to view the pomp C API document.
Aaron A. King
C. Breto & E. L. Ionides, Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems. Stoch. Proc. Appl., 121:2571–2591, 2011.
D. He, E. L. Ionides, & A. A. King, Plug-and-play inference for disease dynamics: measles in large and small populations as a case study. J. R. Soc. Interface, 7:271–283, 2010.
Other information on model implementation: Csnippet
,
accumulators
,
covariate_table
,
dmeasure_spec
, dprocess_spec
,
parameter_trans
,
pomp-package
, prior_spec
,
rinit_spec
, rmeasure_spec
,
rprocess_spec
, skeleton_spec
,
transformations
, userdata
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))