Nathalie Peyrard - Statistical Approaches for Hidden Variables in Ecology

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2 Title Page SCIENCES Statistics, Field Directors – Nikolaos Limnios, Kerrie Mengersen Statistics and Ecology , Subject Head – Nathalie Peyrard

3 Copyright 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 Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.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 Mathematics PE1_14 Statistics LS8 Ecology, Evolution and Environmental Biology

4 Introduction I.1. Hidden variables in ecology 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 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 I.4. Approach and structure of our work I.5. Directions for further perspectives I.6. References

5 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

6 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

7 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

8 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

9 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