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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The study of ecological systems is often impeded by components that escape perfect observation, such as the trajectories of moving animals or the status of plant seed banks. These hidden components can be efficiently handled with statistical modeling by using hidden variables, which are often called latent variables. Notably, the hidden variables framework enables us to model an underlying interaction structure between variables (including random effects in regression models) and perform data clustering, which are useful tools in the analysis of ecological data.
This book provides an introduction to hidden variables in ecology, through recent works on statistical modeling as well as on estimation in models with latent variables. All models are illustrated with ecological examples involving different types of latent variables at different scales of organization, from individuals to ecosystems. Readers have access to the data and R codes to facilitate understanding of the model and to adapt inference tools to their own data.
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Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Title Page
Copyright
Introduction
I.1. Hidden variables in ecology
I.2. Hidden variables in statistical modeling
I.3. Statistical methods
I.4. Approach and structure of our work
I.5. Directions for further perspectives
I.6. References
1 Trajectory Reconstruction and Behavior Identification Using Geolocation Data
1.1. Introduction
1.2. Hierarchical models of movement
1.3. Case study: masked booby, Sula dactylatra (originals)
1.4. References
2 Detection of Eco-Evolutionary Processes in the Wild: Evolutionary Trade-Offs Between Life History Traits
2.1. Context
2.2. The correlative approach to detecting evolutionary trade-offs in natural settings: problems
2.3. Case study
2.4. References
3 Studying Species Demography and Distribution in Natural Conditions: Hidden Markov Models
3.1. Introduction
3.2. Overview of HMMs
3.3. HMM and demography
3.4. HMM and species distribution
3.5. Discussion
3.6. Acknowledgments
3.7. References
4 Inferring Mechanistic Models in Spatial Ecology Using a Mechanistic-Statistical Approach
4.1. Introduction
4.2. Dynamic systems in ecology
4.3. Estimation
4.4. Examples
4.5. References
5 Using Coupled Hidden Markov Chains to Estimate Colonization and Seed Bank Survival in a Metapopulation of Annual Plants
5.1. Introduction
5.2. Metapopulation model for plants: introduction of a dormant state
5.3. Dynamics of weed species in cultivated parcels
5.4. Discussion and conclusion
5.5. Acknowledgments
5.6. References
6 Using Latent Block Models to Detect Structure in Ecological Networks
6.1 Introduction
6.2. Formalism
6.3. Probabilistic mixture models for networks
6.4. Statistical inference
6.5. Application
6.6. Conclusion
6.7. References
7 Latent Factor Models: A Tool for Dimension Reduction in Joint Species Distribution Models
7.1. Introduction
7.2. Joint species distribution models
7.3. Dimension reduction with latent factors
7.4. Inference
7.5. Ecological interpretation of latent factors
7.6. On the interpretation of JSDMs
7.7. Case study
7.8. Conclusion
7.9. References
8 The Poisson Log-Normal Model: A Generic Framework for Analyzing Joint Abundance Distributions
8.1. Introduction
8.2. The Poisson log-normal model
8.3. Data analysis: marine species
8.4. Discussion
8.5. Acknowledgments
8.6. References
9 Supervised Component-Based Generalized Linear Regression: Method and Extensions
9.1. Introduction
9.2. Models and methods
9.3. Case study: predicting the abundance of 15 common tree species in the forests of Central Africa
9.4. Discussion
9.5. References
10 Structural Equation Models for the Study of Ecosystems and Socio-Ecosystems
10.1. Introduction
10.2. Structural equation model
10.3. Case study: biodiversity in managed forests
10.4. Discussion
10.5. Acknowledgments
10.6. References
List of Authors
Index
End User License Agreement
List of Tables
Introduction
Table I.1. Chapters and contents
Chapter 1
Table 1.1. Evolution of model selection criteria (AIC and ICL) as a function of ...
Chapter 3
Table 3.1. Prevalence of hybrids: observed and estimated using the Viterbi algor...
Chapter 4
Table 4.1. Parameters estimated by maximum likelihood
Chapter 5
Table 5.1. Notation of variables in the MHMM-DF
Table 5.2. Interpretation of parameters of the MHMM-DF based on binomial distrib...
Table 5.3. Boundaries of the five abundance classes for the seven weed species i...
Table 5.4. Probabilities of survival, colonization and germination from a dorman...
Table 5.5. BIC selection values for models with and without taking account of cr...
Table 5.6. Probabilities of survival and emergence from a dormant state in the M...
Chapter 8
Table 8.1. Glossary of species codes, scientific names and common names for spec...
Chapter 9
Table 9.1. Spearman correlations between observed and predicted abundances obtai...
Chapter 10
Table 10.1. Variables used in the case study
Table 10.2. Latent variables based on prior knowledge
List of Illustrations
Chapter 1
Figure 1.1. The map at the top shows the tracking data for a male Cape dolphin (...
Figure 1.2. Figure extracted from Figure 4 in Lopez et al. (2015). The black lin...
Figure 1.3. Graphical model. For a color version of this figure, see www.iste.co...
Figure 1.4. Illustration of the quantities present in equations [1.5]–[1.8]. Pt ...
Figure 1.5. Masked Booby (Sula dactylatra) Photo: Sophie Bertrand. For a color v...
Figure 1.6. Area of study (shown in red on the map) and three trajectories obtai...
Figure 1.7. Result of Kalman smoothing on part of the booby trajectories. Smooth...
Figure 1.8. Representation of states along trajectories estimated using two diff...
Figure 1.9. Distribution of our chosen metrics for the states estimated using ou...
Figure 1.10. Contingencies of estimated states for our two models. For a color v...
Figure 1.11. Evolution of the probability of being in state 1 or state 2 over ti...
Figure 1.12. Evolution of estimated transition probabilities as a function of di...
Figure 1.13. Study zone (red dot on the map) and three trajectories of three dif...
Chapter 2
Figure 2.1. Illustration of van Noordwijk and de Jong’s (1986) “Y” model. Exampl...
Figure 2.2. Directed acyclic graph of the model. The squares represent observabl...
Figure 2.3. A posteriori distributions of parameters in the latent model (logari...
Figure 2.4. Comparison of observed and simulated ring width from 1989 to 2015 an...
Figure 2.5. Boxplot of resource (net primary productivity, in gC.m−2.year−1) sim...
Figure 2.6. Illustration of the developed Bayesian model, with process and data ...
Figure 2.7. Correlation between sinks and probabilities. a) Correlation density ...
Chapter 3
Figure 3.1. Schematic illustration of a hidden Markov model
Figure 3.2. Two-state capture–recapture model expressed in HMM form
Figure 3.3. Multi-state capture–recapture model expressed in HMM form
Figure 3.4. Diagram of a dynamic occupancy model expressed as an HMM
Figure 3.5. Identification of local minima in the deviance of an HMM. Numerica...
Figure 3.6. Visualization of heterogeneity: map of the heterogeneity class to wh...
Chapter 4
Figure 4.1. Expectation of the total number of cases associated with the posteri...
Figure 4.2. Joint posterior distributions of couples (α, κ), (t0, α) and (t0, κ)...
Figure 4.3. Map of wolf detections in southeastern France (black dots) and the a...
Figure 4.4. Estimated response curves. Estimated relations between individual de...
Figure 4.5. Predicted occupation probability map for 2016, obtained using the mo...
Figure 4.6. Proportion of plants susceptible to WMV across the area of study. Fo...
Figure 4.7. Proportions of classic and invasive variants in a landscape: data an...
Chapter 5
Figure 5.1. Illustration of dependency relationships in an MHMM-DF. Case of two ...
Figure 5.2. Layout of the 10 fields and 90 patches in the experimental farm at E...
Chapter 6
Figure 6.1. The Chilean network: 1,362 trophic interactions observed in the inte...
Figure 6.2. Adjacency matrix and corresponding representation of the non-directe...
Figure 6.3. Incidence matrix and corresponding representation of the bipartite b...
Figure 6.4. Simulation of a modular network for the parameters shown on the left...
Figure 6.5. Simulation of a food web for the parameters shown on the left. A rea...
Figure 6.6. Simulation of a nested bipartite network for the parameters shown on...
Figure 6.7. Simulation of a bipartite network with a modular and nested structur...
Figure 6.8. Schematic representation (based on Picard et al. 2009) of the estima...
Figure 6.9. Classic representation obtained using the R bipartite package. Diffe...
Chapter 7
Figure 7.1. The three factors that determine the actual distribution of a specie...
Figure 7.2. Localization and names of the 18 gradients of ORCHAMP. For a color v...
Figure 7.3. Effective sample size (ESS, top panels) and potential scale reductio...
Figure 7.4. Distribution of TSS and RMSE score across species for in-sample pred...
Figure 7.5. Posterior support values for species regression coefficients. Red if...
Figure 7.6. The residual correlation matrix. Only significant values (i.e. 95% c...
Figure 7.7. Model-based ordination analysis. The two latent variables can be see...
Figure 7.8. Model-based ordination analysis, as above, but when we include the h...
Figure 7.9. Cross-validation predicted probability of (a) presence and (b) cross...
Chapter 8
Figure 8.1. Illustration of dependency in the PLN model. Random variables are ci...
Figure 8.2. PLN: geometric view of the model for two species. A) Positions in th...
Figure 8.3. Poisson log-normal model with the “site” covariate. Representation o...
Figure 8.4. Poisson log-normal model with the “period” covariate. Representation...
Figure 8.5. Dimension reduction for (left to right): a model without covariates;...
Figure 8.6. Representation of samples on the AEI islet in the first principal pl...
Figure 8.7. Effect of parameter λ on edge density (left) and model fitting (righ...
Figure 8.8. Stability of edges selected for the model including the effects of t...
Figure 8.9. The selected interaction network: visualization using the PLNmodels ...
Figure 8.10. The selected interaction network: visualization of the partial corr...
Figure 8.11. Interactions between two sea urchin species: red (STRFRAAD, Strongy...
Chapter 9
Figure 9.1. Polar representation of VPI (equation 9.3) by values of ℓ for four c...
Figure 9.2. Geometric mean of the square mean quadratic prediction errors over r...
Figure 9.3. Circles of correlations resulting from the first three components. F...
Figure 9.4. Spatial representation of the first three components. For a color ve...
Figure 9.5. Geometric mean of the square roots of the mean quadratic prediction ...
Figure 9.6. Geometric mean of the RMSE as a function of the number of components...
Figure 9.7. Factorial planes (1,2), (1,3) and (2,3) obtained using mixed-SCGLR o...
Figure 9.8. Observed and predicted abundance maps for the species with the highe...
Chapter 10
Figure 10.1. Conventions used in representing an SEM. Latent variables are shown...
Figure 10.2. SEM defined a priori for our case study. The exogenous latent varia...
Figure 10.3. Main steps in SEM analysis. The diagrams follow the SEM presentatio...
Figure 10.4. Estimators of the parameters of the SEM in Figure 10.2. Section A: ...
Figure 10.5. SEM resulting from correction of the model shown in Figure 10.2, af...
Figure 10.6. Divergences between the correlation matrices of the latent variable...
Figure 10.7. SEM resulting from correction of the model in Figure 10.5, limiting...
Figure 10.8. Estimated relationships within the relational model of the SEM from...
Figure 10.9. Test of causal relations in the relational model shown in Figure 10...
Guide
Cover
Table of Contents
Title Page
Copyright
Introduction
Begin Reading
List of Authors
Index
End User License Agreement
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SCIENCES
Statistics, Field Directors – Nikolaos Limnios, Kerrie Mengersen
Statistics and Ecology, Subject Head – Nathalie Peyrard
Statistical Approaches for Hidden Variables in Ecology
Coordinated by
Nathalie Peyrard
Olivier Gimenez
First published 2022 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2022
The rights of Nathalie Peyrard and Olivier Gimenez to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2021949076
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78945-047-7
ERC code:
PE1 MathematicsPE1_14 Statistics
LS8 Ecology, Evolution and Environmental Biology
Introduction
Nathalie PEYRARD1, Stéphane ROBIN2 and Olivier GIMENEZ3
1University of Toulouse, INRAE, UR MIAT, Castanet-Tolosan, France
2Paris-Saclay University, AgroParisTech, INRAE, UMR MIA-Paris, France
3CEFE, University of Montpellier, CNRS, EPHE, IRD, Paul Valéry Montpellier 3 University, France
I.1. Hidden variables in ecology
Ecology is the study of living organisms in interaction with their environment. These interactions occur at individual level (an animal, a plant), at the level of groups of individuals (a population, a species) or across several species (a community). Statistics provides us with tools to study these interactions, enabling us to collect, organize, present, analyze and draw conclusions from data collected on ecological systems. However, some components of these ecological systems may escape observation: these are known as hidden variables. This book is devoted to models incorporating hidden variables in ecology and to the statistical inference for these models.
The hidden variables studied throughout this book can be grouped into three classes corresponding to three types of questions that can be posed concerning an ecological system. We may consider the identification of groups of individuals or species, such as groups of individuals with the same behavior or similar genetic profiles, or groups of species that interact with the same species or with their environment in a similar way. Alternatively, we may wish to study variables which can only be observed in a “noisy” form, often called a “proxy”. For example, the presence of certain species may be missed as a result of detection difficulties or errors (confusion with another species), or as a result of “noisy” data resulting from technology-related measurement errors. Finally, in the context of data analysis, we may wish to reduce the dimension of the information contained in data sets to a small number of explanatory variables. Note the shift from the notion of a variable which escapes observation, in the first cases, to a more generalized notion of hidden variables.
All three of these problems can be translated into questions of inference concerning variables which, in statistical terms, are said to be latent. Inference poses statistical problems that require specific methods, described in detail here. The ecological interpretation of these variables will also be discussed at length. As we shall see, while the statistical treatment of these variables may be complex, their inclusion in models is essential in providing us with a better understanding of ecological systems.
I.2. Hidden variables in statistical modeling
The term “hidden variable”, widely used in ecology, finds its translation in the more general notion of latent variables in statistical modeling. This notion encompasses several situations and goes beyond the idea of unobservable physical variables alone. In statistics, a latent variable is generally defined as a variable of interest, which is not observable and does not necessarily have a physical meaning, the value of which must be deduced from observations. More precisely, latent variables are characterized by the following two specificities: (i) in terms of number, they are comparable to the number of data items, unlike parameters that are fewer in number. Consider, for example, the case of a hidden Markov chain, where the number of observed variables and latent variables is equal to the number of observation time steps; (ii) if their value were known, then model parameter estimation would be easier. For example, consider the estimation of parameters of a mixture model where the groups of individuals are known.
In practice, if a latent variable has a physical reality but cannot be observed in the field (e.g. the precise trajectory of an animal, or the abundance of a seedbank), it is often referred to as a hidden variable (although both terms are often used interchangeably). In other cases, the latent variable naturally plays a role in the description of a given process or system, but has no physical existence. This is the case, for example, of latent variables corresponding to a classification of observations into different groups. We will refer to them as fictitious variables. Finally, latent variables may also play an instrumental role in describing a source of variability in observations that cannot be explained by known covariates, or in establishing a concise description of a dependency structure. They may result from a dimension reduction operation applied to a group of explanatory variables in the context of regression, as we see in the case of the principal components of a principal component analysis.
The notion of latent variables is connected to that of hierarchical models: if they are not parameters, the elements in the higher levels of the model are latent variables. It is important to note that the notion of latent variables may be extended to cover the case of determinist quantities (represented by a constant in a model). For example, this holds true in cases where the latent variable is the trajectory of an ordinary differential equation (ODE) for which only noisy observations are available.
I.3. Statistical methods
Some of the most common examples of statistical models featuring latent variables are described here.
Mixture models are used to define a small number of groups into which a set of observations may be sorted. In this case, the latent variables are discrete variables indicating which group each observation belongs to. Stochastic block models (SBMs) or latent block models (LBMs, or bipartite SBM) are specific forms of mixture models used in cases where the observations take the form of a network. Hidden Markov models (HMMs) are often used to analyze data collected over a period of time (such as the trajectory of an animal, observed over a series of dates) and take account of a subjacent process (such as the activity of the tracked animal: sleep, movement, hunting, etc.), which affects observations (the animal’s position or trajectory). In this case, the latent variables are discrete and represent the activity of the animal at every instant. In other models, the hidden process itself may be continuous. Mixed (generalized) linear models are one of the key tools used in ecology to describe the effects of a set of conditions (environmental or otherwise) on a population or community. These models include random effects which are, in essence, latent variables, used to account for higher than expected dispersions or dependency relationships between variables. In most cases, these latent variables are continuous and essentially instrumental in nature. Joint species distribution models (JSDMs) are a multidimensional version of generalized linear models, used to describe the composition of a community as a function of both environmental variables and of the interactions between constituent species. Many JSDMs use a multidimensionsal (e.g. Gaussian) latent variable, the dependency structure of which is used to describe inter-species interactions.
In ecology, models are often used to describe the effect of experimental conditions or environmental variables on the response or behavior of one or more species. Explanatory variables of this kind are often known as covariates. These effects are typically accounted for using a regression term, as in the case of generalized linear models. A regression term of this type may also be used in latent variable models, in which case the distribution of the response variable in question is considered to depend on both the observed covariates and non-observable latent variables.
Many methods have been proposed for estimating the parameters of a model featuring latent variables. From a frequentist perspective, the oldest and most widespread means of computing the maximum likelihood estimator is the expectation–maximization (EM) algorithm, which draws on the fact that the parameters for many of these models would be easy to estimate if the latent variables could be observed. The EM algorithm alternates between two steps: in step E, all of the quantities involving latent variables are calculated in order to update the estimation of parameters in the second step, M. Step E focuses on determining the conditional distribution of latent variables given the observed data. This calculation may be immediate (as in the case of mixture models and certain mixed models) or possible but costly (as in the case of HMMs); alternatively, it may be impossible for combinatorial or formal reasons.
The estimation problem is even more striking in the context of Bayesian inference, as a conditional distribution must be established not only for the latent variables, but also for parameters. Once again, except in very specific circumstances, precise determination of this joint conditional law (latent variables and parameters) is usually impossible.
The inference methods used in models with a non-calculable conditional law fall into two broad categories: sampling methods and approximation methods. Sampling methods use a sample of data relating to the non-calculable law to obtain precise estimations of all relevant quantities. This category includes the Monte Carlo, the Markov chain Monte Carlo (MCMC) and the sequential Monte Carlo (SMC) methods. These algorithms are inherently random, and are notably used in Bayesian inference. Methods in the second category are used to determine an approximation of the conditional law of the latent variables (and, in the Bayesian case, of parameters) based on observations. This category includes variational methods and their extensions. These approaches vary in terms of the measure of proximity between the approximated law and the actual conditional law, and in terms of the distribution family used when searching for the approximation.
I.4. Approach and structure of our work
This book provides an overview of recent work on statistical modeling and estimation in latent variable models for ecology. The different chapters illustrate the main principles described above. In some cases, they present statistical methods based on classical models and algorithms; in others, the focus is on developments from recent research in others. Each chapter addresses a specific ecological issue and a modeling approach to solving the problem, illustrated using one or more case studies.
Readers may also access the R code1 in order to make use of the tools presented here, applied to their own data.
Most of the questions associated with the case studies presented here relate to the comprehension or description of systems. While the issue of forecasting and prediction is touched upon in some chapters, this subject lies outside the main scope of our work. The issue of missing data (i.e. values not observed in samples) is also not addressed either. Finally, note that this work is not an exhaustive summary of latent variable models, or of the inference methods and algorithms used with these models. Each chapter touches on the question of inference in relation to the selected model; readers wishing to explore the subject in greater depth may wish to consult the references provided.
This book is not designed to be read from front to back, but rather as a resource on which ecologists working with models or statisticians working in the field of ecology may draw. Chapters are arranged in order of ecological scale, from individuals up to ecosystems, providing an initial interpretive framework. Another approach would be to consider the nature of the hidden variable being modeled. One final approach would be to examine different statistical models: some models are used in several chapters, in connection with questions on different scales, and using different estimation methods.
Table I.1 gives a summary of the contents of the different chapters and is designed to help readers identify material which is of interest to them.
I.5. Directions for further perspectives
The examples described above, along with those presented in the following chapters, highlight the immense flexibility of latent variables models. These models, involving one or more latent layers, provide a rich framework for the description of complex dependency structures, and/or for the approximation of a mechanistic description of the phenomena involved.
However, it is important to note that the most sophisticated models are almost always the most complex in terms of inference. It would be wrong to assume that inference simply “happens”, whatever the statistical approach (frequentist, Bayesian, etc.). At the time of writing, there is no fully generic approach suitable for use with all models, and this is unlikely to change in the near future. Even the best-established algorithms (EM, MCMC, etc.) require users to have a good understanding of the underlying principles in order to guide and control their behavior, and/or to adjust the algorithm as needed. This need for adjustment is clearly visible in the chapters of this book.
To conclude this introduction, we wish to highlight two areas for further research in ecology, drawing on statistical modeling of hidden variables, which are not covered in this book but which show promise: namely the integration (or combination) of data from multiple sources, and the use of participative scientific data.
Table I.1.Chapters and contents
Several works have recently been published on the integration of data from multiple sources in the field of ecology (Miller et al. 2019; Isaac et al. 2020). The aim of the authors is to systematically improve the precision of estimated data, potentially decreasing sample size, and to enable the estimation of parameters that cannot be approximated by any other means. Data integration generally involves a hierarchical modeling approach in which the hidden variable is present in all of the sources used in its estimation.
Data from participative scientific activity has also received increasing attention in the literature in recent years (Dickinson et al. 2012; McKinley et al. 2017). This is due to the increasing availability of the data, and to the fact that information can now be collected across an increasingly broad spatial and temporal scale. Participative data sources are a fascinating subject of study in statistical ecology, raising a number of new challenges in terms of spatial bias in sampling, or variations in participant expertise. Once again, a clear distinction between the ecological processes embodied by the hidden variables and the associated observation methods is essential in order to develop a full response to the ecological question.
I.6. References
Dickinson, J.L., Shirk, J., Bonter, D., Bonney, R., Crain, R.L., Martin, J., Phillips, T., Purcell, K. (2012). The current state of citizen science as a tool for ecological research and public engagement. Frontiers in Ecology and the Environment, 10(6), 291–297.
Isaac, N.J., Jarzyna, M.A., Keil, P., Dambly, L.I., Boersch-Supan, P.H., Browning, E., Freeman, S.N., Golding, N., Guillera-Arroita, G., Henrys, P.A., Jarvis, S., Lahoz-Monfort, J., Pagel, J., Pescott, O.L., Schmucki, R., Simmonds, E.G., O’Hara, R.B. (2020). Data integration for large-scale models of species distributions. Trends in Ecology & Evolution, 35(1), 56–67.
McKinley, D.C., Miller-Rushing, A.J., Ballard, H.L., Bonney, R., Brown, H., Cook-Patton, S.C., Evans, D.M., French, R.A., Parrish, J.K., Phillips, T.B., Ryan, S.F., Shanley, L.A., Shirk, J.L., Stepenuck, K.F., Weltzin, J.F., Wiggins, A., Boyle, O.D., Briggs, R.D., Chapin, S.F., Hewitt, D.A., Preuss, P.W., Soukup, M.A. (2017). Citizen science can improve conservation science, natural resource management, and environmental protection. Biological Conservation, 208, 15–28.
Miller, D.A.W., Pacifici, K., Sanderlin, J.S., Reich, B.J. (2019). The recent past and promising future for data integration methods to estimate species’ distributions. Methods in Ecology and Evolution, 10(1), 22–37.
1
https://oliviergimenez.github.io/code_livre_variables_cachees/
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