eulermultinom {pomp} | R Documentation |

## 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).

BretÃ³ & 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 |

`deulermultinom` |
Returns a vector (of length equal to the number of columns of |

`rgammawn` |
Returns a vector of length |

### 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

C. BretÃ³ 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. 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. doi:10.1098/rsif.2009.0151.

### 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*version 5.10.0.1 Index]