### Post provided by Edgar J. González

In demography, a set of processes (survival, growth, fecundity, etc.) interacts to produce observable patterns (population size, structure, growth rate, etc.) that change over time. With traditional approaches you follow the individuals of a population over some timespan and track all of these processes.

However, depending on the organism, some processes may be very hard to quantify (e.g. mortality or recruitment in animals or plants with long lifespans). You may have observed the patterns for the organism that you’re studying and, even better, measured some, but not all, of the processes. The question is: can we use this limited information to estimate the processes we couldn’t measure?

# Integral Projection Models

To estimate processes from patterns, we need a model to connect them. Integral Projection Models (IPMs) are becoming the go-to approach to model the dynamics of structured populations (see Merow et al. and Rees, Childs & Ellner for some examples). In these populations, there is a trait (e.g., size, sex, experienced environment) important enough so that individuals with different variations of this trait have different probabilities of surviving, growing, reproducing, etc.

IPMs model each process separately as a function of that trait, and mathematically connect the processes with the patterns. The usual approach to implement IPMs is to gather information on all the processes (green in the figure below) and use the model to obtain estimated patterns over time (orange, below).

# The Problem of Incomplete Information

However, demographers often have to deal with the problem that not all processes are easy to observe. This kind of scenario occurs when you’re trying to study the change of a population over a large time span or the individuals are hard to track because, for example, they’re too small or they become untraceable during part of their life cycle. In extreme cases only the patterns can be observed over time, not the processes. A nicer scenario is one where you have the patterns and partial information on the processes. From either of these starting points, we want to infer the unobserved processes. Here’s where inverse modelling becomes really useful.

# Inverse Modelling as a Solution

Essentially, inverse modelling involves taking a model that uses processes as inputs and produces patterns as an output and reversing it, so that it uses the output as an input. This then gives us data on the unobserved processes that we were looking for as though the patterns were our starting set of information. Amazing!

Inverse modelling has been used in many areas of research (astronomy, remote sensing, hydrology, etc.) and in demography to infer unobserved birth and mortality rates in human populations. For IPMs, inverse modelling was first proposed as a means to align observed population-level patterns with estimated ones. At the same time, and as part of my PhD, I used inverse modelling in IPMs to infer unobserved processes that change through time. This Special Feature also presents two cases where unobserved processes – mortality and fecundity – are estimated based on existing data on the dynamics.

My contribution to the Special Feature Demography Beyond the Population shows how this can be done with IPMs in a more general setting, regardless of which processes are missing. My co-authors (Carlos Martorell and Benjamin M. Bolker) and I consider cases where all the processes are missing and we only have information on the demographic patterns (scenario 1 on the right) and easier scenarios where partial information on the processes is also available (e.g. fecundity, scenario 2 on the right).

We are able to estimate unobserved processes by aligning observed patterns with estimated ones through maximum likelihood. As Sean McMahon & Jess Metcalf mentioned in their blog post on Monday the idea is to propose potential estimates on the unobserved processes, iterate the IPM to obtain estimated patterns and compare them with the observed ones. Through repetition we can find the best estimates. Although it’s a simple idea, its implementation can be challenging…

# The Challenge of Inverse Modelling

1. Identify the vital rates that integrate the population dynamics
2. Perform regressions between each rate and the structuring variable using data informing this relationship
3. Combine each regression function into a single function, the kernel, and repeat it over time to obtain estimated patterns

With inverse modelling, steps 2) and 3) have to be performed simultaneously. As we want to estimate some vital rates, the regressions for all of them have to be performed all at once as the model is iterated and the patterns are obtained. This represents a more challenging endeavour as no pre-existing easy-to-use regression software can be used. Instead, the kernel and its iteration have to be coded and regressed against the pattern data.

Fortunately, R packages have been developed that can accommodate complex non-linear functions to be regressed against data. Two examples are TMB and NIMBLE. The first stems from ADMB, which was the package I used for my analyses. However, as I show, the problem is so difficult that ADMB isn’t capable of estimating the processes on its own and more exploratory tools are needed… but that’s going into too many details for a blog post.

In our paper we develop a method that will allow you to estimate unobserved processes from patterns changing over time. We demonstrate its performance with simulated and real data from a 15-year long dataset of a chamaephyte plant, Cryptantha flava. We also provide the R code used to inverse model the simulated data.

To find out more about inverse modelling and IPMs, read Inverse estimation of integral projection model parameters using time series of population-level data by González et al. (2016).

This article is part of the British Ecological Society’s Cross Journal Special Feature, Demography Beyond the Population. All articles in the Special Feature are freely available for a limited time.