- Introduction
- Partially observed Markov process (POMP) models
- Preliminaries
- Specifying a continuous-time process model
- Specifying the measurement model
- A continuous-time deterministic skeleton
- Specifying a discrete-time process model and skeleton
- Parameter transformations
- Trajectory matching
- Maximizing the likelihood by iterated filtering
- Sampling the posterior using particle Markov chain Monte Carlo
**R**codes for this document- References

Licensed under the Creative Commons attribution-noncommercial license. Please share and remix noncommercially, mentioning its origin.

This document was produced using **pomp** version 1.13.4.2 and **R** version 3.4.1.

This tutorial aims to help you get started using **pomp** as a suite of tools for analysis of time series data based on dynamical systems models. First, we give some conceptual background regarding the class of models—partially observed Markov processes—that **pomp** handles. We then discuss some preliminaries: installing the package and so on. Next, using a basic question about ecological population regulation as an example, we load some data and implement some models as `R`

objects of class `pomp`

. Finally, we illustrate some of the package’s capabilities by using its algorithms to fit and compare the models using various inference methods.

As its name implies **pomp** is focused on partially observed Markov process models. These are also known as state-space model, hidden Markov models, and stochastic dynamical systems models. Such models consist of an unobserved stochastic *state process*, connected to the data via an explicit model of the observation process. We refer to the former as the *process model* and the latter as the *measurement model*. Each model is, in fact, a probability distribution. Suppose we have a sequence of measurements, \(y^*_{n}\), made at times \(t_n\), \(n=1,\dots,N\). Let \(Y_n\) denote the measurement process and \(X_n\) the state process, then by definition, the process model is determined by the density \(f_{X_n|X_{n-1}}\) and the initial density \(f_{X_0}\). The measurement process is determined by the density \(f_{Y_n|X_n}\). We think of the data, \(y^*_n\), as being a single realization of the \(Y\) process. To implement a POMP model in **pomp**, we have to specify the measurement and process distributions.

Note however, that, for each of the process and the measurement model there are two distinct operations that we might wish to perform:

- we might wish to
*simulate*, i.e., to draw a (pseudo)random sample from the distribution, or - we might wish to
*evaluate the density*itself at given values of \(X_n\) and/or \(Y_n\).

Following the **R** convention, we refer to the simulation of \(f_{X_n|X_{n-1}}\) as the *rprocess* component of the POMP model and the evaluation of \(f_{X_n|X_{n-1}}\) as the *dprocess* component. Similarly, the simulation of \(f_{Y_n|X_n}\) is the *rmeasure* component while the evaluation of \(f_{Y_n|X_n}\) is the *dmeasure* component. Methods that make no use of the *dprocess* component are called “plug-and-play” methods. At present, **pomp** is focused on such methods, so there is no reason to discuss the dprocess component further. In the following, we will illustrate and explain how one specifies the rprocess, rmeasure, and dmeasure components of a model in **pomp**. We will illustrate this using some simple examples.

The following is a schematic of the structure of a POMP showing causal relations between the process model, the measurement model, and the data. **The key perspective to keep in mind is that the model is to be viewed as the process that generated the data.**

Here is another POMP model schematic, showing dependence among model variables.

The state process, \(X_n\), is Markovian, i.e., \[\mathrm{Prob}[X_n|X_0,\dots,X_{n-1},Y_1,\dots,Y_{n-1}]=\mathrm{Prob}[X_n|X_{n-1}].\] Moreover, the measurements, \(Y_n\), depend only on the state at that time: \[\mathrm{Prob}[Y_n|X_0,\dots,X_{n},Y_1,\dots,Y_{n-1}]=\mathrm{Prob}[Y_n|X_{n}],\] for all \(n=1,\dots,N\).

To get started, we must install **pomp**, if it is not already installed. The package can be downloaded from CRAN, but the latest version is always available at the package homepage on Github. See the package website for installation instructions.

In this document, we will implement **pomp** models using the package’s `Csnippet`

facility. This allows the user to write model components using snippets of C code, which is then compiled and linked into a running `R`

session. This typically affords a manyfold increase in computation time. It is possible to avoid `Csnippet`

s entirely by writing model components as `R`

functions, but the resulting implementations are typically too slow to be of practical use. To use `Csnippet`

s, you must be able to compile C codes. Compilers are not by default installed on Windows or Mac systems, so users of such systems must do a bit more work to make use of **pomp**’s facilities. The installation instructions on the package website give details.

We will illustrate **pomp** by performing a limited data analysis on a set of bird abundance data. The data are from annual censuses of a population of *Parus major* (Great Tit) in Wytham Wood, Oxfordshire. They were retrieved as dataset #10163 from the Global Population Dynamics Database version 2 (NERC Centre for Population Biology, Imperial College, 2010). The original source is McCleery and Perrins (1991).

We load the data by doing

```
parus.dat <- read.csv(text="
year,P
1960,148
1961,258
1962,185
1963,170
1964,267
1965,239
1966,196
1967,132
1968,167
1969,186
1970,128
1971,227
1972,174
1973,177
1974,137
1975,172
1976,119
1977,226
1978,166
1979,161
1980,199
1981,306
1982,206
1983,350
1984,214
1985,175
1986,211"
)
```

Here is a plot of these data:

```
library(ggplot2)
ggplot(data=parus.dat,mapping=aes(x=year,y=P))+
geom_line()+geom_point()+
expand_limits(y=0)+
theme_bw()
```