Line transect abundance estimation with uncertain detection on the trackline

236 Pages·2015·4.62 MB·English

Line Transect Abundance Estimation with Uncertain Detection on the
Trackline
D.L. Borchers
Thesis presented for the degree of
DOCTOR OF PHILOSOPHY ...
in the Department of Mathematical Statistics
UNIVERSITY OF CAPE TOWN
February, 1996
" ·- ".:" .: .. _:: . ·-~'''->··· :_
The copyright of this thesis vests in the author. No
quotation from it or information derived from it is to be
published without full acknowledgement of the source.
The thesis is to be used for private study or non-
commercial research purposes only.

Published by the University of Cape Town (UCT) in terms
of the non-exclusive license granted to UCT by the author.
0.1 Abstract
After critically reviewing developments in line transect estimation theory to date, general likelihood
functions are derived for the case in which detection probabilities are modelled as functions of any
number of explanatory variables and detection of animals on the trackline (i.e. directly in the observer's
path) is not certain.
Existing models are shown to correspond to special cases of the general models. Maximum likelihood esti
mators are derived for some special cases of the general model and some existing line transect estimators
are shown to correspond to maximum likelihood estimators for other special cases. The likelihoods are
shown to be extensions of existing mark-recapture likelihoods as well as being generalizations of existing
line transect likelihoods.
Two new abundance estimators are developed. The first is a Horvitz-Thompson-like estimator which
utilizes the fact that for point estimation of abundance the density of perpendicular distances in the
population can be treated as known in appropriately designed line transect surveys. The second is based
on modelling the probability density function of detection probabilities in the population. Existing line
transect estimators are shown to correspond to special cases of the new Horvitz-Thompson-like estimator,
so that this estimator, together with the general likelihoods, provides a unifying framework for estimating
abundance from line transect surveys.
The performance of the new estimators and an existing mark-recapture Horvitz-Thompson like estimator
are compared in a simulation study. The new Horvitz- Thompson like estimator is shown to perform
best on statistical criteria. It also has a number of other advantages over the other estimators, not least
of which is the fact that it is readily adapted to provide an estimator which is able to correct for random
or responsive animal movement.
Unresolved issues in line transect theory are mentioned and suggestions for future reserarch are presented.
1
Acknowledgements
Firstly to Carol for her support through the long process of completing this thesis, despite the weekends
and evenings of my time it has taken from her and Alice.
To Walter Zucchini for the encouragement and inspiration he provided while he was at the University of
Cape Town. To Doug Butterworth for reading and commenting on the thesis draft. To Steve Buckland
for making me feel I was doing something worthwhile.
Finally, to my mother for her support, financial and otherwise.
Contents
0.1 Abstract .
1
0.2 Summary
2
1 Introduction
7
1.1 Thesis Outline ...........................................
11
2 Conventional Radial Distance Based LT Models 13
2.1 Introduction ................ . 13
2.2 Independent Radial Distances and Angles 17
2.2.1 Hayne-like Models ........ . 18
2.2.2 . Generalizations of Hayne-like Models . 21
2.3 Other Radial Distance Models . 21
2.3.1 Elliptic Hayne Models .
21
2.4 A Horvitz-Thompson Perspective. 23
2.5 The Failure of Models Based on Radial Distance 25
2.5.1 Summary 27
2.6 Appendices . . . 28
2.6.1 Some results from a Hazard-rate representation of the Detection Process 28
3 Conventional LT Models Based on Perpendicular Distance 31
3.1 Introduction .................... . 31
3.2 Assumptions about the Detection Function, g(x) 32
3.3 The Effect of Ignoring Other Explanatory Variables 33
3.4 Two Approaches to Model Robustness ....... .
34
3.4.1 Estimation via a Specific Functional Form for g(x)
34
3.4.2 Estimation via Series Representations of f(x) 36
3.5 The Performance of the Estimators .
38
3.6 A Unified Approach . . .
40
3. 7 An Alternative Approach 41
3.8 Interval Estimation ... 42
3.9 Putting It All Together 43
3.10 Summary: The Conventional LT Problem Solved 44
4 Incorporating Covariates into Conventional LT Models 47
4.1 Introduction ..... 47
4.2 Some New Notation 48
4.3 Approaches to Estimating /(0) 49
4.4 Estimation by Modelling f ( ) directly 50
4.5 Estimation Conditional on the Observed Additional Variables . 50'
4.5.1 Truncated Series Estimation .............. . 51
4.5.2 Estimators based on Modelling the Detection Function . 51
4.6 Estimating Var[f (O)] ....... . 52
4.7 A Horvitz-Thompson Perspective 54
4.8 Summary and Discussion . . . . . 56
5 Univariate "g(O)" Estimation Methods 59
5.1 Introduction . . . . . . . . . . . . . . . . 59
5.1.1 A Short Note on Aerial Survey "Visibility Bias" 60
5.2 A Little More Notation . . 61
5.3 Variable Effort Estimators . 62
5.3.1 The Estimators of Butterworth et al. and Cooke 63
5.3.2 Zahl's Estimator . . . . . . . . . . 65
5.3.3 A Robust {but biased) Estimator . 66
5.3.4 The Performance of VS Estimators . 67
5.4 Independent Observer Estimators .. 69
5.4.1 Discrete Availability Models. 70
5.4.2 The "Product" Method Estimator 71
ii
5.4.3 The ,;Direct" Method Estimator .... 73
5.4.4 The Method of Buckland and Turnock . 74
5.4.5 The Performance of IO ;Estimators 76
5.5 Summary and Discussion 83
5.6 Appendices . . . . . . . . 86
5.6.1 A Note on the Similarity of Cooke's and Zahl's Estimators 86
5.6.2 Continuous vs Poisson Availability LT Models 88
6 Line Transect as Mark-Recapture: General Models 91
6 .1 Overview . . 91
6.2 Introduction . 92
6.2.1 Line Transect as an Experiment 92
6.2.2 Types of LT Models . . . . . . . 93
6.3 The Elements of the Model and the Notation 94
6.3.1 The Data ......... . 96
6.3.2 The Detection Probabilities 96
6.3.3 Numbers of Animals Observed 97
6.3.4 The Density of the Explanatory Variables 98
6.3.5 Mean Detection Probabilities and Effective Strip Width 98
6.4 The General Likelihoods . . . . . 100
6.5 The Likelihood for Binned Data . 101
6.5.1 LB from a MR Perspective 102
6.5.2 LB from the LT Perspective . 104
6.6 Some Illustrative Examples . . . . . 105
6.6.1 Conventional Two-Sample Mark-Recapture 105
6.6.2 Two-Sample Mark-Recapture 'with Detectability Categories 106
6.6.3 Independent Observer Line Transect with homogeneous targets 107
6.7 The Unbinned Likelihood ..... 110
6.7.l Lu from a MR Perspective 111
6.7.2 Lu from the LT Perspective . 112
6.8 Modelling the Densities of Explanatory Variables 113
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6,9 Summary . 114
6.10 Appendices 116
6.10.1 The effect of ignoring heterogeneity in MRLT models 116
6.10.2 Derivation of the Likelihood for Binned Observations 120
6.10.3 Derivation of the Likelihood for Unbinned Observations 122
6.10.4 MLE's for Binned Data with constant detectability in each bin, known 7rk's , and
.n o structure on Pik 123
7 Estimators for MRLT Models 125
7.1 Overview .......... . 125
7.2 Existing LT Estimators and some Generalization~. 126
7.2.1 Univariate Conventional Models . 126
7.2.2 Multivariate Conventional Models 127
7.2.3 Constant-Shape, Constant-G, Multivariate MRLT Models 128
7.3 Conditional Likelihood Estimation 133
7.4 Detection Probability Estimators . 135
7.4.1 Finding Suitable Multivariate Detection Function Forms . 136
7.5 Abundance Estimators for Multivariate MRLT Models 138
7.5.1 NH: A Horvitz-Thompson-like Estimator . 139
7.5.2 . Nw: Estimation with 7r(x) ~ w-1 144
7.5.3 Nµ: Estimation by Modelling 7r .. 145
7.6 Estimating pdf's and Functions of Explanatory Variables 147
7. 7 An Estimator for Dealing with Animal Movement . 150
7.8 Estimation When Animals Cluster 151
7.9 · Conclusion 152
7.10 Appendices 155
7.10.1 Derivation of the "Constant G" estimators from hP 155
7.10.2 An Estimator of Effective Strip Width when G is Constant 157
7,10.3 The "Modified Logistic Regression" Method 159
8 The Simulation Experiment Design 161
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10 Conclusions
215
10.1 Overview ........................ . 215
10.2 The Main Developments and Results of the Thesis 216
10.2.1 The Development of a General Theory for MRLT Models 216
10.2.2 The Development of MRLT Detection Functions 216
10.2.3 The Development and Testing of Estimators. 217
10.3 Future Developments . . . . . . . . . . . . . . . 219
10.3.1 Generalized MRLT' Detection Functions 219
10.3.2 Unifying MRLT and Conventional LT Estimation Methods 220
10.3.3 Variable Coverage Probability Designs 221
10.3.4 Duplicate Identification . . . 223
10.3.5 Discrete Animal Availability 223
11 References 225
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8.1 Introduction ........ .
161
8.2 The Explanatory Variables
162
8.2.1 A Note on (x; z) and Truncation at Perpendicular Distance 162
8.2.2 The Joint Density of x and z
163
8.3 The Detection Functions ...... .
164
8.3.1 The Form of the Detection Functions.
164
8.3.2 The Values of the Detection Function Parameters . 164
8.4 The Simulated Abundance, N .
170
8.5 The Responses
170
8.6 The Estimators
172
8.6.1 Point Estimators of N
172
8.6.2 Interval Estimators .. 173
9 Simulation Results .
175
9.1 Overview ..
175
9.1.1 Point Estimation of N . 175
9.1.2 Interval Estimation of N . 176
9.2 Point Estimation Results .. 177
9.2.1 The %Bias and %cv of NH 177
9.2.2 The "w-Estimators": Nw, Nwt, Nwr ., . 183
9.2.3 Nµ. .. 184
9.2.4 NwT vs Nwt vs Nµ 186
9.3 Interval Estimation of N . 186
9.3.1 Analytic Estimation and Neglect of Variance due to Estimating P.(x, z) 186
9.3.2 Estimated 95% Coverage of NH vs Nw: Analytic ..... 187
9.3.3 Estimated 95% Coverage of Nwt: Analytic and Bootstrap 187
9.3.4 Estimated 95% Coverage of Nµ: Bootstrap 187
9.4 Conclusion 187
9.5 Appendices 190
9.5.i Tables Comparing the Estimator Properties . 190
9.5.2 Figures Comparing the Estimator Properties 196
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