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)
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.
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).
Returns a vector of length n containing random increments of the integrated Gamma white noise process with intensity sigma.
integer; number of random variates to generate.
scalar integer; number of individuals at risk.
numeric vector of hazard rates.
numeric scalar; duration of Euler step.
matrix or vector containing number of individuals that have succumbed to each death process.
logical; if TRUE, return logarithm(s) of probabilities.
numeric scalar; intensity of the Gamma white noise process.
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
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.
C. Bretó and E. L. Ionides. Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems. Stochastic Processes and their Applications 121, 2571--2591, 2011. tools:::Rd_expr_doi("10.1016/j.spa.2011.07.005").
D. He, E.L. Ionides, and A.A. King. Plug-and-play inference for disease dynamics: measles in large and small populations as a case study. Journal of the Royal Society Interface 7, 271--283, 2010. tools:::Rd_expr_doi("10.1098/rsif.2009.0151").
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
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))
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