Using R for Modelling and Quantitative Methods in Fisheries

353 Pages·10.021 MB·English

by  Malcolm Haddon

Using R for Modelling
and Quantitative
Methods in Fisheries
Chapman & Hall/CRC
The R Series
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Using R for Modelling
and Quantitative
Methods in Fisheries
Malcolm Haddon
CSIRO, Oceans and Atmosphere, Hobart
IMAS, University of Tasmania
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Names:Haddon,Malcolm,author.
Title: Using R for modelling and quantitative methods in fisheries / Malcolm
Haddon.
Description:BocaRaton:CRCPress,[2021]|Summary:“Thebookhasevolved
and adapted from an earlier book by the same author and provides a detailed
introduction to analytical methods commonly used by fishery scientists, ecolo-
gists,andadvancedstudentsusingtheopensourcesoftwareRasaprogramming
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Identifiers: LCCN 2020014277 (print) | LCCN 2020014278 (ebook) | ISBN
9780367469894 (hardback) | ISBN 9780367469887 (paperback) | ISBN
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Subjects: LCSH: Fisheries--Mathematical models. | R (Computer program
language)
Classification: LCC SH331.5.M48 H343 2021 (print) | LCC SH331.5.M48
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Contents
Preface xi
1 On Modelling 1
1.1 Characteristics of Mathematical Models . . . . . . . . . . . . 1
1.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Model Design or Selection . . . . . . . . . . . . . . . . 1
1.1.3 Constraints Due to the Model Type . . . . . . . . . . 3
1.1.4 Mathematical Models . . . . . . . . . . . . . . . . . . 3
1.1.5 Parameters and Variables . . . . . . . . . . . . . . . . 4
1.2 Mathematical Model Properties . . . . . . . . . . . . . . . . 5
1.2.1 Deterministic vs Stochastic . . . . . . . . . . . . . . . 5
1.2.2 Continuous vs Discrete Models . . . . . . . . . . . . . 6
1.2.3 Descriptive vs Explanatory . . . . . . . . . . . . . . . 7
1.2.4 Testing Explanatory Models . . . . . . . . . . . . . . . 8
1.2.5 Realism vs Generality . . . . . . . . . . . . . . . . . . 9
1.2.6 When Is a Model a Theory? . . . . . . . . . . . . . . . 10
1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 11
2 A Non-Introduction to R 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Programming in R . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Getting Started with R . . . . . . . . . . . . . . . . . 14
2.2.2 R Packages . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Getting Started with MQMF . . . . . . . . . . . . . 16
2.2.4 Examining Code within Functions . . . . . . . . . . . 17
2.2.5 Using Functions . . . . . . . . . . . . . . . . . . . . . 18
2.2.6 Random Number Generation . . . . . . . . . . . . . . 19
2.2.7 Printing in R . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.8 Plotting in R . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.9 Dealing with Factors . . . . . . . . . . . . . . . . . . . 22
2.2.10 Inputting Data . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Writing Functions . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Simple Functions . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Function Input Values . . . . . . . . . . . . . . . . . . 27
2.3.3 R Objects . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.4 Scoping of Objects . . . . . . . . . . . . . . . . . . . . 28
v
vi Contents
2.3.5 Function Inputs and Outputs . . . . . . . . . . . . . . 29
2.4 Appendix: Less-Traveled Functions . . . . . . . . . . . . . . 33
2.5 Appendix: Additional Learning Resources . . . . . . . . . . . 33
3 Simple Population Models 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 The Discrete Logistic Model. . . . . . . . . . . . . . . 35
3.1.2 Dynamic Behaviour . . . . . . . . . . . . . . . . . . . 37
3.1.3 Finding Boundaries between Behaviours . . . . . . . . 40
3.1.4 Classical Bifurcation Diagram of Chaos . . . . . . . . 42
3.1.5 The Effect of Fishing on Dynamics . . . . . . . . . . . 43
3.1.6 Determinism . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Age-Structured Modelling Concepts . . . . . . . . . . . . . . 48
3.2.1 Survivorship in a Cohort. . . . . . . . . . . . . . . . . 48
3.2.2 Instantaneous vs Annual Mortality Rates . . . . . . . 49
3.3 Simple Yield per Recruit . . . . . . . . . . . . . . . . . . . . 52
3.3.1 Selectivity in Yield-per-Recruit . . . . . . . . . . . . . 55
3.3.2 The Baranov Catch Equation . . . . . . . . . . . . . . 57
3.3.3 Growth and Weight-at-Age . . . . . . . . . . . . . . . 59
3.4 Full Yield-per-Recruit . . . . . . . . . . . . . . . . . . . . . . 60
3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 62
4 Model Parameter Estimation 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Criteria of Best Fit . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Model Fitting in R . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.1 Model Requirements . . . . . . . . . . . . . . . . . . . 70
4.3.2 A Length-at-Age Example . . . . . . . . . . . . . . . . 71
4.3.3 Alternative Models of Growth . . . . . . . . . . . . . . 72
4.4 Sum of Squared Residual Deviations . . . . . . . . . . . . . . 73
4.4.1 Assumptions of Least-Squares . . . . . . . . . . . . . . 74
4.4.2 Numerical Solutions . . . . . . . . . . . . . . . . . . . 74
4.4.3 Passing Functions as Arguments to Other Functions . 75
4.4.4 Fitting the Models . . . . . . . . . . . . . . . . . . . . 77
4.4.5 Objective Model Selection . . . . . . . . . . . . . . . . 83
4.4.6 The Influence of Residual Error Choice on Model Fit . 84
4.4.7 Remarks on Initial Model Fitting . . . . . . . . . . . . 86
4.5 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . 87
4.5.1 Introductory Examples. . . . . . . . . . . . . . . . . . 87
4.6 Likelihoods from the Normal Distribution . . . . . . . . . . . 90
4.6.1 Equivalence with Sum-of-Squares . . . . . . . . . . . . 92
4.6.2 Fitting a Model to Data Using Normal Likelihoods . . 94
4.7 Log-Normal Likelihoods . . . . . . . . . . . . . . . . . . . . . 98
4.7.1 Simplification of Log-Normal Likelihoods . . . . . . . 99
Contents vii
4.7.2 Log-Normal Properties . . . . . . . . . . . . . . . . . . 99
4.7.3 Fitting a Curve Using Log-Normal Likelihoods . . . . 102
4.7.4 Fitting a Dynamic Model Using Log-Normal Errors . 105
4.8 Likelihoods from the Binomial Distribution . . . . . . . . . . 108
4.8.1 An Example Using Binomial Likelihoods . . . . . . . . 109
4.8.2 Open Bay Juvenile Fur Seal Population Size. . . . . . 112
4.8.3 Using Multiple Independent Samples . . . . . . . . . . 114
4.8.4 Analytical Approaches . . . . . . . . . . . . . . . . . . 116
4.9 Other Distributions . . . . . . . . . . . . . . . . . . . . . . . 117
4.10 Likelihoods from the Multinomial Distribution . . . . . . . . 117
4.10.1 Using the Multinomial Distribution. . . . . . . . . . . 119
4.11 Likelihoods from the Gamma Distribution . . . . . . . . . . 125
4.12 Likelihoods from the Beta Distribution . . . . . . . . . . . . 127
4.13 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 128
4.13.2 Bayesian Methods . . . . . . . . . . . . . . . . . . . . 130
4.13.3 Prior Probabilities . . . . . . . . . . . . . . . . . . . . 131
4.14 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 134
5 Static Models 135
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 Productivity Parameters . . . . . . . . . . . . . . . . . . . . 136
5.3 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.1 Seasonal Growth Curves . . . . . . . . . . . . . . . . . 137
5.3.2 Fabens Method with Tagging Data . . . . . . . . . . . 142
5.3.3 Fitting Models to Tagging Data . . . . . . . . . . . . 144
5.3.4 A Closer Look at the Fabens Methods . . . . . . . . . 146
5.3.5 Implementation of Non-Constant Variances . . . . . . 148
5.4 Objective Model Selection . . . . . . . . . . . . . . . . . . . 151
5.4.1 Akiake’s Information Criterion . . . . . . . . . . . . . 151
5.4.2 Likelihood Ratio Test . . . . . . . . . . . . . . . . . . 153
5.4.3 Caveats on Likelihood Ratio Tests . . . . . . . . . . . 154
5.5 Remarks on Growth . . . . . . . . . . . . . . . . . . . . . . . 154
5.6 Maturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 155
5.6.2 Alternative Maturity Ogives. . . . . . . . . . . . . . . 157
5.6.3 The Assumption of Symmetry . . . . . . . . . . . . . 161
5.7 Recruitment . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 165
5.7.2 Properties of “Good” Stock Recruitment Relationships 166
5.7.3 Recruitment Overfishing . . . . . . . . . . . . . . . . . 167
5.7.4 Beverton and Holt Recruitment . . . . . . . . . . . . . 168
5.7.5 Ricker Recruitment. . . . . . . . . . . . . . . . . . . . 169
5.7.6 Deriso’s Generalized Model . . . . . . . . . . . . . . . 171
5.7.7 Re-Parameterized Beverton-Holt Equation . . . . . . . 172
viii Contents
5.7.8 Re-Parameterized Ricker Equation . . . . . . . . . . . 175
5.8 Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 175
5.8.2 Logistic Selection . . . . . . . . . . . . . . . . . . . . . 176
5.8.3 Dome-Shaped Selection . . . . . . . . . . . . . . . . . 177
5.9 Concluding Remarks for Static Models . . . . . . . . . . . . 180
5.10 Appendix: Derivation of Fabens Transformation . . . . . . . 181
5.11 Appendix: Reparameterization of Beverton-Holt . . . . . . . 182
6 On Uncertainty 185
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.1.1 Types of Uncertainty. . . . . . . . . . . . . . . . . . . 185
6.1.2 The Example Model . . . . . . . . . . . . . . . . . . . 188
6.2 Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2.1 Empirical Probability Density Distributions . . . . . . 191
6.3 A Simple Bootstrap Example . . . . . . . . . . . . . . . . . . 192
6.4 Bootstrapping Time-Series Data . . . . . . . . . . . . . . . . 195
6.4.1 Parameter Correlation . . . . . . . . . . . . . . . . . . 199
6.5 Asymptotic Errors . . . . . . . . . . . . . . . . . . . . . . . . 200
6.5.1 Uncertainty about the Model Outputs . . . . . . . . . 203
6.5.2 Sampling from a Multivariate Normal Distribution . . 204
6.6 Likelihood Profiles . . . . . . . . . . . . . . . . . . . . . . . . 209
6.6.1 Likelihood Ratio-Based Confidence Intervals. . . . . . 212
6.6.2 -ve Log-Likelihoods or Likelihoods . . . . . . . . . . . 214
6.6.3 Percentile Likelihood Profiles for Model Outputs . . . 216
6.7 Bayesian Posterior Distributions . . . . . . . . . . . . . . . . 219
6.7.1 Generating the Markov Chain . . . . . . . . . . . . . . 221
6.7.2 The Starting Point . . . . . . . . . . . . . . . . . . . . 222
6.7.3 The Burn-In Period . . . . . . . . . . . . . . . . . . . 223
6.7.4 Convergence to the Stationary Distribution . . . . . . 223
6.7.5 The Jumping Distribution . . . . . . . . . . . . . . . . 224
6.7.6 Application of MCMC to the Example . . . . . . . . . 225
6.7.7 Markov Chain Monte Carlo . . . . . . . . . . . . . . . 225
6.7.8 A First Example of an MCMC . . . . . . . . . . . . . 228
6.7.9 Marginal Distributions . . . . . . . . . . . . . . . . . . 235
6.8 The Use of Rcpp . . . . . . . . . . . . . . . . . . . . . . . . . 236
6.8.1 Addressing Vectors and Matrices . . . . . . . . . . . . 239
6.8.2 Replacement for simpspm() . . . . . . . . . . . . . . . 240
6.8.3 Multiple Independent Chains . . . . . . . . . . . . . . 243
6.8.4 Replicates Required to Avoid Serial Correlation . . . . 247
6.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 251
7 Surplus Production Models 253
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.1.1 Data Needs . . . . . . . . . . . . . . . . . . . . . . . . 254
Contents ix
7.1.2 The Need for Contrast . . . . . . . . . . . . . . . . . . 254
7.1.3 When Are Catch-Rates Informative? . . . . . . . . . . 255
7.2 Some Equations . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.2.1 Production Functions . . . . . . . . . . . . . . . . . . 261
7.2.2 The Schaefer Model . . . . . . . . . . . . . . . . . . . 263
7.2.3 Sum of Squared Residuals . . . . . . . . . . . . . . . . 264
7.2.4 Estimating Management Statistics . . . . . . . . . . . 265
7.2.5 The Trouble with Equilibria . . . . . . . . . . . . . . . 266
7.3 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 267
7.3.1 A Possible Workflow for Stock Assessment . . . . . . . 268
7.3.2 Is the Analysis Robust? . . . . . . . . . . . . . . . . . 274
7.3.3 Using Different Data . . . . . . . . . . . . . . . . . . . 277
7.4 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7.4.1 Likelihood Profiles . . . . . . . . . . . . . . . . . . . . 280
7.4.2 Bootstrap Confidence Intervals . . . . . . . . . . . . . 283
7.4.3 Parameter Correlations . . . . . . . . . . . . . . . . . 290
7.4.4 Asymptotic Errors . . . . . . . . . . . . . . . . . . . . 290
7.4.5 Sometimes Asymptotic Errors Work . . . . . . . . . . 297
7.4.6 Bayesian Posteriors. . . . . . . . . . . . . . . . . . . . 299
7.5 Management Advice . . . . . . . . . . . . . . . . . . . . . . . 308
7.5.1 Two Views of Risk . . . . . . . . . . . . . . . . . . . . 308
7.5.2 Harvest Strategies . . . . . . . . . . . . . . . . . . . . 309
7.6 Risk Assessment Projections . . . . . . . . . . . . . . . . . . 310
7.6.1 Deterministic Projections . . . . . . . . . . . . . . . . 310
7.6.2 Accounting for Uncertainty . . . . . . . . . . . . . . . 313
7.6.3 Using Asymptotic Errors . . . . . . . . . . . . . . . . 314
7.6.4 Using Bootstrap Parameter Vectors . . . . . . . . . . 316
7.6.5 Using Samples from a Bayesian Posterior . . . . . . . 316
7.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 319
7.8 Appendix: The Use of Rcpp to Replace simpspm . . . . . . . 321
References 323
Index 335