Using R for Modelling and Quantitative Methods in Fisheries

Malcolm Haddon

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9780367469887
• eBook ISBN: 9781000079272
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Statistics for Business & Economics
• Index: Technology & Engineering

Using R for Modelling and Quantitative Methods in Fisheries has evolved and been adapted from an earlier book by the same author and provides a detailed introduction to analytical methods commonly used by fishery scientists, ecologists, and advanced students using the open-source software R as a programming tool. Some knowledge of R is assumed, as this is a book about using R, but an introduction to the development and working of functions, and how one can explore the contents of R functions and packages, is provided.

The example analyses proceed step-by-step using code listed in the book and from the book's companion R package, MQMF, available from GitHub and the standard archive, CRAN. The examples are designed to be simple to modify so the reader can quickly adapt the methods described to use with their own data. A primary aim of the book is to be a useful resource to natural resource practitioners and students.

Featured Chapters:

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• Model Parameter Estimation provides a detailed explanation of the requirements and steps involved in fitting models to data, using R and, mainly, maximum likelihood methods.

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• On Uncertainty uses R to implement bootstrapping, likelihood profiles, asymptotic errors, and Bayesian posteriors to characterize any uncertainty in an analysis. The use of the Monte Carlo Markov Chain methodology is examined in some detail.

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• Surplus Production Models applies all the methods examined in the earlier parts of the book to conducting a stock assessment. This included fitting alternative models to the available data, characterizing the uncertainty in different ways, and projecting the optimum models forward in time as the basis for providing useful management advice.

1

On Modelling

1.1Characteristics of Mathematical Models

1.1.1General

Formal fishery stock assessments are generally based upon mathematical models of population production processes and of the population dynamics of the stock being fished. The positive production processes include the recruitment dynamics (which add numbers) and individual growth (which adds biomass), while the negative production processes include natural mortality and fishing mortality (that includes selectivity), which both reduce numbers and hence biomass. The population dynamics includes details such as the time-steps used in the modelled dynamics, whether biomass or numbers are modelled (at age or at size, or both), details of spatial structuring, and other specifics to the case in hand. Such a plethora of potential details means there is a vast potential array of diverse mathematical models of biological populations and processes. Nevertheless, it is still possible to make some general statements regarding such models.

All models constitute an abstraction or simulation by the modeller of what is currently known about the process or phenomenon being modelled. Mathematical models are only a subset of the class of all models, and models may take many forms, ranging from a physical representation of whatever is being modelled (think of ball and stick models of DNA, as produced by Watson and Crick, 1953), diagrammatic models (such as a geographical map), and the more abstract mathematical representations being discussed here. We can impose some sort of conceptual order on this diversity by focussing on different properties of the models and on some of the constraints imposed by decisions made by the modellers.

1.1.2Model Design or Selection

As abstractions, models are never perfect copies of what is known about the modelled subject, so there must be some degree of selection of what the modeller considers to be a system’s essential properties or components. This notion of “essential properties or components” is making the assumption that all parts of a system are not all equally important. For example, in a model of the human blood circulation system a superficial vein somewhere in the skin would not be as important as the renal artery. If that assumption is accepted then a fundamental idea behind modelling is to select the properties to be included in order that the behaviour of the model may be expected to exhibit a close approximation to the observable behaviour of the modelled system. This selection of what are considered to be the important properties of a system permits, or even forces, the modeller to emphasize particular aspects of the system being modelled. A road map shows roads greatly magnified in true geographical scale because that is the point of the map. A topological map emphasizes different things, so the purpose for which a model is to be used is also important when determining what structure to use.

The selection of what aspects of a system to include in a model is what determines whether a model will be generally applicable to a class of systems, or is so specialized that it is attempting to simulate the detailed behaviour of a particular system (for system one might read a fished stock or population). However, by selecting particular parts of a natural system the model is also being constrained in what it can describe. The assumption is that despite not being complete it will provide an adequate description of the process of interest and that those aspects not included will not unexpectedly distort the representation of the whole (Haddon, 1980).

Of course, in order to make an abstraction one first needs to understand the whole, but unfortunately, in the real world, there is a great deal that remains unknown or misunderstood. Hence it is quite possible that a model becomes what is known as “misspecified”. This is where the model’s dynamics or behaviour, fails to capture the full dynamics of the system being studied. In some of the examples illustrated later in this book we will see the average predicted biomass trajectory for a stock fail to account for what appear to be oscillations in stock size that exhibit an approximate 10-year cycle (see, for example, the surplus production model fitted to the dataspm data-set in the Bootstrap Confidence Intervals section of the chapter on Surplus Production Models). In such a case there is an influence (or influences) acting with an unknown mechanism on the stock size in what appears to be a patterned or repeatable manner. Making the assumption that the pattern is meaningful, because the mechanism behind it is not included in the model structure then, obviously, the model cannot account for its influence. This is classical misspecification, although not all misspecifications are so clear or have such clear patterns.

Model design, or model selection, is complex because the decisions made when putting a model together will depend on what is already known and the use to which the model is to be put.

1.1.3Constraints Due to the Model Type

A model can be physical, verbal, graphical, or mathematical, however, the particular form chosen for a model imposes limits on what it can describe. For example, a verbal description of a dynamic population process would be a challenge for anyone as invariably there is a limit to how well one can capture or express the dynamic properties of a population using words. Words appear to be better suited to the description of static objects. This limitation is not necessarily due to any lack of language skills on the part of the speaker. Rather, it is because spoken languages (at least those of which I am aware) do not seem well designed for describing dynamic processes, especially where more than one variable or aspect of a system is changing through time or relative to other variables. Happily, we can consider mathematics to be an alternative language that provides excellent ways of describing dynamic systems. But even with mathematics as the basis of our descriptions there are many decisions that need to be made.

1.1.4Mathematical Models

There are many types of mathematical models. They can be characterized as descriptive, explanatory, realistic, idealistic, general, or particular; they can also be deterministic, stochastic, continuous, and discrete. Sometimes they can be combinations of some or all of these things. With all these possibilities, there is a great potential for confusion over exactly what role mathematical models can play in scientific investigations. To gain a better understanding of the potential limitations of particular models, we will attempt to explain the meaning of some of these terms.

Mathematical population models are termed dynamic because they can represent the present state of a population/fishery in terms of its past state or states, with the potential to describe future states. For example, the Schaefer model (Schaefer, 1957) of stock biomass dynamics (of which we will be hearing more) can be partly represented as:

Bt+1=Bt+rBt(1−BtK)−Ct	(1.1)

where the variable C_t is the catch taken during time t, and B_t is the stock biomass at the start of time t (B_t is also an output of the model). The model parameters are r, representing the population growth rate of biomass (or numbers, depending on what interpretation is given to B_t, perhaps = N_t), and K, the maximum biomass (or numbers) that the system can attain (these parameters come from the logistic model from early mathematical ecology; see the Simple Population Models chapter). By examining this relatively simple model one can see that expected biomass levels at one time (t + 1) are directly related to catches and the earlier biomass (time = t; the values are serially correlated). The influence of the earlier biomass on population growth is controlled by the combination of the two parameters r and K. By accounting for the serial correlations between variables from time period to time period, such dynamic state models differ markedly from traditional statistical analyses. Serial correlation means that if we were to sample a population each year then, strictly, the samples would not be independent, which is a requirement of more classical statistical analyses. For example, in a closed population the number of two-year old fish in one year cannot be greater than the number of one-year fish the year before; they are not independent.

1.1.5Parameters and Variables

At the most primitive level, mathematical models are made up of variables and parameters. A model’s variables must represent something definable or measurable in nature (at least in principle). Parameters modify the impact or contribution of a variable to the model’s outputs, or are concerned with the relationships between the variables within the model. Parameters are the things that dictate quantitatively how the variables interact. They differ from a model’s variables because the parameters are the things estimated when a model is fitted to observed data. In Equ(1.1), B_t and C_t are the variables and r and K are the parameters. There can be overlap as, for example, one might estimate the very first value in the B_t series, perhaps B_init and hence the series would be made up of one parameter with the rest a direct function of theB_init, r and K parameters and the C_t variable.

In any model, such as Equ(1.1), we must either estimate or provide constant values for the parameters. With the variables, either one provides observed values for them (e.g., a time-series of catches, C_t) or they are an output from the model (with the exception of B_init as described above). Thus, in Equ(1.1), given a time-series of observed catches plus estimates of parameter values for B_init, r, and K, then a time-series of biomass values, B_t, is implied by the model as an output. As long as one is aware of the possibilities for confusion that can arise over the terms observe, estimate, variable, parameter, and model output, one can be more clear about exactly what on...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1. On Modelling
• 2. A Non-Introduction to R
• 3. Simple Population Models
• 4. Model Parameter Estimation
• 5. Static Models
• 6. On Uncertainty
• 7. Surplus Production Models
• References
• Index