The Poisson-Binomial Model for Fish Abundance Estimation

132 Pages·2015·5.77 MB·English

The Poisson-Binomial Model
for Fish Abundance Estimation
With Applications to Northeast Arctic Cod
Kjersti Moss
Master’s Thesis for the degree
Modelling and Data Analysis (MOD5960), November 2015

Abstract
The Institute of Marine Research collects data from different sources for the estimation of
fish abundance. These data can be divided into two groups:
1. Data from research surveys.
2. Fishery based data.
In this thesis, we aim to utilize both data sets to estimate the abundance of fish, along
with the catch. In addition to a point estimate, we wish to assess the uncertainty in these
estimates. More formally, we hope to find quantiles of the simultaneous distribution
π(N,C|DN,DC,parameters),
that is, the distribution of the abundances and catches, given both data sources.
On the way towards this goal, we need to specify a model for the abundance, the catch,
and the data. The model for the abundance and catch is what we call the Poisson-binomial
model. This model is the central theme of the thesis. We explore the properties of the
model, and derive conditions for when it is identifiable. Furthermore, we investigate both
a frequentistic and a Bayesian method to estimate the model parameters. It turns out
that we are not able to describe the simultaneous distribution π(N,C|DN,DC,parameters)
analytically, andwecanneithersampledirectlyfromit. However, wecanobtainMonteCarlo
samples of this distribution through importance sampling techniques, and thereby calculate
approximate quantiles.
The methods we develop are applied to data on Northeast Arctic cod (Skrei in Norwe-
gian), from the years 1985-2003.
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Acknowledgements
First of all, I thank my main supervisor Geir Storvik for his guidance and enthusiasm for
this project. I also thank my co-supervisor Sondre Aanes for sharing his insight into the
fishery science. Further, I must thank the best mathematics teachers from my school years,
Ingve Tofteland and Tor Arne Mjølund, who inspired me to study the subject further. The
data set I analyse was provided by the Norwegian Institute for Marine Research (IMR), and
for that I thank them. I am also grateful to IMR and the Institute of Mathematics at the
University of Oslo, that I was allowed to participate at a one-day seminar at IMR in Bergen.
When it comes to the writing process, I thank my brothers Rolf and Scott who have
proofread much of my material, and my sister Elen who have been babysitting in the last
hectic days.
Further, I thank the Norwegian welfare system, and especially Lånekassen, Husbanken,
and Sogn Studentbarnehage. Without you it would have been both economically and practi-
cally impossible to complete this project.
The past year have been very pleasant socially, as well as academically. For that I thank
my fellow master students, and everyone else on the 8’th floor of Niels Henrik Abels House,
Blindern, Oslo.
Finally, I thank my husband Jonas Moss for a great collaboration through our master
studies, and otherwise.
Kjersti Moss,
Oslo, November 16, 2015
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Dedicated to my children
Richard and Ervin
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Contents
Abstract i
Acknowledgements iii
Notation ix
1 Introduction 1
1.1 Northeast Arctic cod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Current stock assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Introduction to the thesis project . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Explorative data analysis 11
2.1 Data description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Looking at the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Changes made in the data set . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 The Poisson-binomial model 17
3.1 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Properties of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Simultaneous distribution of N|C . . . . . . . . . . . . . . . . . . . . 31
3.4 Model for the data DN,DC . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Identifiability and estimability 45
4.1 Introduction to identifiability and estimability . . . . . . . . . . . . . . . . . 45
4.2 Identifiability and estimability of the Poisson-binomial model . . . . . . . . . 49
5 Inference methods 55
5.1 Method 1: Generalized method of moment estimation (GMM) . . . . . . . . 55
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5.2 Method 2: Priors on model parameters . . . . . . . . . . . . . . . . . . . . . 62
5.3 Importance sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.1 Introduction to importance sampling (IS) . . . . . . . . . . . . . . . . 70
5.3.2 Sequential importance sampling with resampling (SISR) . . . . . . . 73
5.3.3 Finding posterior quantiles . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 The R-functions StockSizeIS and StockSizeISprior . . . . . . . . . . . . 75
6 Simulation experiment 77
6.1 Simulated data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Testing Method 1 and importance sampling . . . . . . . . . . . . . . . . . . 79
6.3 Testing Method 2 and importance sampling . . . . . . . . . . . . . . . . . . 83
6.4 Summary of the testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7 Results 85
7.1 Constant catchability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2 Catchability to match ICES-estimates . . . . . . . . . . . . . . . . . . . . . . 91
7.3 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
8 Concluding remarks 95
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.2 Discussion and suggestions for further work . . . . . . . . . . . . . . . . . . 96
References 102
A Data description 103
B FixingData 105
C ChoosePriors 107
D FindingQuantiles 109
E SimulateDataSet 111
F StockSizeIS 113
G StockSizeISprior 117
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