Predictive models in ecology: Comparison of performances and assessment of applicability

Ecological systems are governed by complex interactions which are mainly nonlinear. In order to capture the inherent complexity and nonlinearity of ecological, and in general biological systems, empirical models recently gained popularity. However, although these models, particularly connectionist approaches such as multilayered backpropagation networks, are commonly applied as predictive models in ecology to a wide variety of ecosystems and questions, there are no studies to date aiming to assess the performance, both in terms of data fitting and generalizability, and applicability of empirical models in ecology. Our aim is hence to provide an overview for nature of the wide range of the data sets and predictive variables, from both aquatic and terrestrial ecosystems with different scales of time-dependent dynamics, and the applicability and robustness of predictive modeling methods on such data sets by comparing different empirical modeling approaches. The models used in this study range from predicting the occurrence of submerged plants in shallow lakes to predicting nest occurrence of bird species from environmental variables and satellite images. The methods considered include k-nearest neighbor (k-NN), linear and quadratic discriminant analysis (LDA and QDA), generalized linear models (GLM) feedforward multilayer backpropagation networks and pseudo-supervised network ARTMAP.Our results show that the predictive performances of the models on training data could be misleading, and one should consider the predictive performance of a given model on an independent test set for assessing its predictive power. Moreover, our results suggest that for ecosystems involving time-dependent dynamics and periodicities whose frequency are possibly less than the time scale of the data considered, GLM and connectionist neural network models appear to be most suitable and robust, provided that a predictive variable reflecting these time-dependent dynamics included in the model either implicitly or explicitly. For spatial data, which does not include any time-dependence comparable to the time scale covered by the data, on the other hand, neighborhood based methods such as k-NN and ARTMAP proved to be more robust than other methods considered in this study. In addition, for predictive modeling purposes, first a suitable, computationally inexpensive method should be applied to the problem at hand a good predictive performance of which would render the computational cost and efforts associated with complex variants unnecessary.     

Inference for nonlinear epidemiological models using genealogies and time series

Phylodynamics - the field aiming to quantitatively integrate the ecological and evolutionary dynamics of rapidly evolving populations like those of RNA viruses - increasingly relies upon coalescent approaches to infer past population dynamics from reconstructed genealogies. As sequence data have become more abundant, these approaches are beginning to be used on populations undergoing rapid and rather complex dynamics. In such cases, the simple demographic models that current phylodynamic methods employ can be limiting. First, these models are not ideal for yielding biological insight into the processes that drive the dynamics of the populations of interest. Second, these models differ in form from mechanistic and often stochastic population dynamic models that are currently widely used when fitting models to time series data. As such, their use does not allow for both genealogical data and time series data to be considered in tandem when conducting inference. Here, we present a flexible statistical framework for phylodynamic inference that goes beyond these current limitations. The framework we present employs a recently developed method known as particle MCMC to fit stochastic, nonlinear mechanistic models for complex population dynamics to gene genealogies and time series data in a Bayesian framework. We demonstrate our approach using a nonlinear Susceptible-Infected-Recovered (SIR) model for the transmission dynamics of an infectious disease and show through simulations that it provides accurate estimates of past disease dynamics and key epidemiological parameters from genealogies with or without accompanying time series data.     

Forcing Versus Feedback: Epidemic Malaria and Monsoon Rains in Northwest India

Malaria epidemics in regions with seasonal windows of transmission can vary greatly in size from year to year. A central question has been whether these interannual cycles are driven by climate, are instead generated by the intrinsic dynamics of the disease, or result from the resonance of these two mechanisms. This corresponds to the more general inverse problem of identifying the respective roles of external forcings vs. internal feedbacks from time series for nonlinear and noisy systems. We propose here a quantitative approach to formally compare rival hypotheses on climate vs. disease dynamics, or external forcings vs. internal feedbacks, that combines dynamical models with recently developed, computational inference methods. The interannual patterns of epidemic malaria are investigated here for desert regions of northwest India, with extensive epidemiological records for Plasmodium falciparum malaria for the past two decades. We formulate a dynamical model of malaria transmission that explicitly incorporates rainfall, and we rely on recent advances on parameter estimation for nonlinear and stochastic dynamical systems based on sequential Monte Carlo methods. Results show a significant effect of rainfall in the inter-annual variability of epidemic malaria that involves a threshold in the disease response. The model exhibits high prediction skill for yearly cases in the malaria transmission season following the monsoonal rains. Consideration of a more complex model with clinical immunity demonstrates the robustness of the findings and suggests a role of infected individuals that lack clinical symptoms as a reservoir for transmission. Our results indicate that the nonlinear dynamics of the disease itself play a role at the seasonal, but not the interannual, time scales. They illustrate the feasibility of forecasting malaria epidemics in desert and semi-arid regions of India based on climate variability. This approach should be applicable to malaria in other locations, to other infectious diseases, and to other nonlinear systems under forcing.     

Inferring the Relative Resilience of Alternative States

Ecological systems may occur in alternative states that differ in ecological structures, functions and processes. Resilience is the measure of disturbance an ecological system can absorb before changing states. However, how the intrinsic structures and processes of systems that characterize their states affects their resilience remains unclear. We analyzed time series of phytoplankton communities at three sites in a floodplain in central Spain to assess the dominant frequencies or “temporal scales” in community dynamics and compared the patterns between a wet and a dry alternative state. The identified frequencies and cross-scale structures are expected to arise from positive feedbacks that are thought to reinforce processes in alternative states of ecological systems and regulate emergent phenomena such as resilience. Our analyses show a higher species richness and diversity but lower evenness in the dry state. Time series modeling revealed a decrease in the importance of short-term variability in the communities, suggesting that community dynamics slowed down in the dry relative to the wet state. The number of temporal scales at which community dynamics manifested, and the explanatory power of time series models, was lower in the dry state. The higher diversity, reduced number of temporal scales and the lower explanatory power of time series models suggest that species dynamics tended to be more stochastic in the dry state. From a resilience perspective our results highlight a paradox: increasing species richness may not necessarily enhance resilience. The loss of cross-scale structure (i.e. the lower number of temporal scales) in community dynamics across sites suggests that resilience erodes during drought. Phytoplankton communities in the dry state are therefore likely less resilient than in the wet state. Our case study demonstrates the potential of time series modeling to assess attributes that mediate resilience. The approach is useful for assessing resilience of alternative states across ecological and other complex systems.     

Modeling of dynamical systems through deep learning

This review presents a modern perspective on dynamical systems in the context of current goals and open challenges. In particular, our review focuses on the key challenges of discovering dynamics from data and finding data-driven representations that make nonlinear systems amenable to linear analysis. We explore various challenges in modern dynamical systems, along with emerging techniques in data science and machine learning to tackle them. The two chief challenges are (1) nonlinear dynamics and (2) unknown or partially known dynamics. Machine learning is providing new and powerful techniques for both challenges. Dimensionality reduction methods are used for projecting dynamical methods in reduced form, and these methods perform computational efficiency on real-world data. Data-driven models drive to discover the governing equations and give laws of physics. The identification of dynamical systems through deep learning techniques succeeds in inferring physical systems. Machine learning provides advanced new and powerful algorithms for nonlinear dynamics. Advanced deep learning methods like autoencoders, recurrent neural networks, convolutional neural networks, and reinforcement learning are used in modeling of dynamical systems.     

How complex should models be? Comparing correlative and mechanistic range dynamics models

Criticism has been levelled at climate‐change‐induced forecasts of species range shifts that do not account explicitly for complex population dynamics. The relative importance of such dynamics under climate change is, however, undetermined because direct tests comparing the performance of demographic models vs. simpler ecological niche models are still lacking owing to difficulties in evaluating forecasts using real‐world data. We provide the first comparison of the skill of coupled ecological‐niche‐population models and ecological niche models in predicting documented shifts in the ranges of 20 British breeding bird species across a 40‐year period. Forecasts from models calibrated with data centred on 1970 were evaluated using data centred on 2010. We found that more complex coupled ecological‐niche‐population models (that account for dispersal and metapopulation dynamics) tend to have higher predictive accuracy in forecasting species range shifts than structurally simpler models that only account for variation in climate. However, these better forecasts are achieved only if ecological responses to climate change are simulated without static snapshots of historic land use, taken at a single point in time. In contrast, including both static land use and dynamic climate variables in simpler ecological niche models improve forecasts of observed range shifts. Despite being less skilful at predicting range changes at the grid‐cell level, ecological niche models do as well, or better, than more complex models at predicting the magnitude of relative change in range size. Therefore, ecological niche models can provide a reasonable first approximation of the magnitude of species' potential range shifts, especially when more detailed data are lacking on dispersal dynamics, demographic processes underpinning population performance, and change in land cover.     

Temporal ecology in the Anthropocene

Two fundamental axes – space and time – shape ecological systems. Over the last 30 years spatial ecology has developed as an integrative, multidisciplinary science that has improved our understanding of the ecological consequences of habitat fragmentation and loss. We argue that accelerating climate change – the effective manipulation of time by humans – has generated a current need to build an equivalent framework for temporal ecology. Climate change has at once pressed ecologists to understand and predict ecological dynamics in non‐stationary environments, while also challenged fundamental assumptions of many concepts, models and approaches. However, similarities between space and time, especially related issues of scaling, provide an outline for improving ecological models and forecasting of temporal dynamics, while the unique attributes of time, particularly its emphasis on events and its singular direction, highlight where new approaches are needed. We emphasise how a renewed, interdisciplinary focus on time would coalesce related concepts, help develop new theories and methods and guide further data collection. The next challenge will be to unite predictive frameworks from spatial and temporal ecology to build robust forecasts of when and where environmental change will pose the largest threats to species and ecosystems, as well as identifying the best opportunities for conservation.     

The presence of both stochastic and deterministic dynamics in electroretinograms measured under identical experimental conditions

We have studied whether living things investigated under the same measuring conditions can generate signals with different types of dynamics. We also wanted to detect the possible effects of low-intensity microwaves using parameters of deterministic chaos. For this purpose, two sets of electroretinograms were analysed by methods aimed at recognizing different types of dynamics. Both sets included the time series recorded from objects exposed to low-intensity microwaves and those that were not exposed. The analytical methods are based on nonlinear forecasting and a “surrogate data” technique. Although the experimental conditions were identical for the two sets, we have shown that both have time series with deterministic and stochastic dynamics. We also found that the use of parameters of deterministic dynamics is insufficient to distinguish between the sets. Received: 18 January 1994 / Accepted: 13 June 1997     

Assessing temporal scales and patterns in time series: Comparing methods based on redundancy analysis

Time-series modelling techniques are powerful tools for studying temporal scaling structures and dynamics present in ecological and other complex systems and are gaining popularity for assessing resilience quantitatively. Among other methods, canonical ordinations based on redundancy analysis are increasingly used for determining temporal scaling patterns that are inherent in ecological data. However, modelling outcomes and thus inference about ecological dynamics and resilience may vary depending on the approaches used. In this study, we compare the statistical performance, logical consistency and information content of two approaches: (i) asymmetric eigenvector maps (AEM) that account for linear trends and (ii) symmetric distance-based Moran's eigenvector maps (MEM), which requires detrending of raw data to remove linear trends prior to analysis. Our comparison is done using long-term water quality data (25 years) from three Swedish lakes. This data set therefore provides the opportunity for assessing how the modelling approach used affects performance and inference in time series modelling. We found that AEM models had consistently more explanatory power than MEM, and in two out of three lakes AEM extracted one more temporal scale than MEM. The scale-specific patterns detected by AEM and MEM were uncorrelated. Also individual water quality variables explaining these patterns differed between methods, suggesting that inferences about systems dynamics are dependent on modelling approach. These findings suggest that AEM might be more suitable for assessing dynamics in time series analysis compared to MEM when temporal trends are relevant. The AEM approach is logically consistent with temporal autocorrelation where earlier conditions can influence later conditions but not vice versa. The symmetric MEM approach, which ignores the asymmetric nature of time, might be suitable for addressing specific questions about the importance of correlations in fluctuation patterns where there are no confounding elements of linear trends or a need to assess causality.     

A general framework for modeling population abundance data

Time-series data resulting from surveying wild animals are often described using state-space population dynamics models, in particular with Gompertz, Beverton-Holt, or Moran-Ricker latent processes. We show how hidden Markov model methodology provides a flexible framework for fitting a wide range of models to such data. This general approach makes it possible to model abundance on the natural or log scale, include multiple observations at each sampling occasion and compare alternative models using information criteria. It also easily accommodates unequal sampling time intervals, should that possibility occur, and allows testing for density dependence using the bootstrap. The paper is illustrated by replicated time series of red kangaroo abundances, and a univariate time series of ibex counts which are an order of magnitude larger. In the analyses carried out, we fit different latent process and observation models using the hidden Markov framework. Results are robust with regard to the necessary discretization of the state variable. We find no effective difference between the three latent models of the paper in terms of maximized likelihood value for the two applications presented, and also others analyzed. Simulations suggest that ecological time series are not sufficiently informative to distinguish between alternative latent processes for modeling population survey data when data do not indicate strong density dependence.     

Robustness of early warning signals of regime shifts in time-delayed ecological models

Various ecological and other complex dynamical systems may exhibit abrupt regime shifts or critical transitions, wherein they reorganize from one stable state to another over relatively short time scales. Because of potential losses to ecosystem services, forecasting such unexpected shifts would be valuable. Using mathematical models of regime shifts, ecologists have proposed various early warning signals of imminent shifts. However, their generality and applicability to real ecosystems remain unclear because these mathematical models are considered too simplistic. Here, we investigate the robustness of recently proposed early warning signals of regime shifts in two well-studied ecological models, but with the inclusion of time-delayed processes. We find that the average variance may either increase or decrease prior to a regime shift and, thus, may not be a robust leading indicator in time-delayed ecological systems. In contrast, changing average skewness, increasing autocorrelation at short time lags, and reddening power spectra of time series of the ecological state variable all show trends consistent with those of models with no time delays. Our results provide insights into the robustness of early warning signals of regime shifts in a broader class of ecological systems.     

城市生态系统的动力学演化模型研究进展

从系统分析出发,对城市生态系统的动力学演化模型的发展历程、建模的方法和步骤过程、软件开发方法和目前的模型软件等进行了总结。归纳了城市生态系统的动力学演化建模的方法,主要包括模型定义、模拟、实现、验证、分析和应用等六大步骤。目前国内外用于城市生态系统动力学演化模型的主要方法有：基于数理模型的方法、生态控制论和灵敏度模型、系统动力学模型、多目标规划法等。已经开发的用于城市生态系统的动力学模拟的软件可以划分为两类：基于土地利用和交通规划的专业模型和基于系统动力学和灵敏度模型的一般软件。总结了常用的城市演化模型软件,讨论了模型的研究对象和应用范围。分析了城市生态系统的动力学演化模型建模的不确定性的来源,并指出：向宏观和微观两极化发展是城市生态系统动力学演化模型的发展趋势之一,而与人工智能和地理信息系统等新方法的集成是发展的另一趋势。城市生态系统动力学演化模型的开发前景在于对不确定性问题的定性、定量分析,而多模型的耦合和集成是发展的必然趋势。     

Complexity is costly: a meta‐analysis of parametric and non‐parametric methods for short‐term population forecasting

Short‐term forecasts based on time series of counts or survey data are widely used in population biology to provide advice concerning the management, harvest and conservation of natural populations. A common approach to produce these forecasts uses time‐series models, of different types, fit to time series of counts. Similar time‐series models are used in many other disciplines, however relative to the data available in these other disciplines, population data are often unusually short and noisy and models that perform well for data from other disciplines may not be appropriate for population data. In order to study the performance of time‐series forecasting models for natural animal population data, we assembled 2379 time series of vertebrate population indices from actual surveys. Our data were comprised of three vastly different types: highly variable (marine fish productivity), strongly cyclic (adult salmon counts), and small variance but long‐memory (bird and mammal counts). We tested the predictive performance of 49 different forecasting models grouped into three broad classes: autoregressive time‐series models, non‐linear regression‐type models and non‐parametric time‐series models. Low‐dimensional parametric autoregressive models gave the most accurate forecasts across a wide range of taxa; the most accurate model was one that simply treated the most recent observation as the forecast. More complex parametric and non‐parametric models performed worse, except when applied to highly cyclic species. Across taxa, certain life history characteristics were correlated with lower forecast error; specifically, we found that better forecasts were correlated with attributes of slow growing species: large maximum age and size for fishes and high trophic level for birds. Synthesis Evaluating the data support for multiple plausible models has been an integral focus of many ecological analyses. However, the most commonly used tools to quantify support have weighted models’ hindcasting and forecasting abilities. For many applications, predicting the past may be of little interest. Concentrating only on the future predictive performance of time series models, we performed a forecasting competition among many different kinds of statistical models, applying each to many different kinds of vertebrate time series of population abundance. Low‐dimensional (simple) models performed well overall, but more complex models did slightly better when applied to time series of cyclic species (e.g. salmon).     

What kind of spatial and temporal details are required in models of heterogeneous systems?

Spatial and temporal heterogeneity can make ecological systems hard to understand and model. We propose a simple classification of the types of spatial and temporal complexity contained in ecological systems, and describe the kinds of data and models needed to account for each. We classify ecological systems by the presence of heterogeneity at the scale of study, the nature of their dynamics (linear vs non-linear), attributes of the patches that constitute the heterogeneous system, and the presence and directionality of interactions among patches. Heterogeneity in space and time are nearly equivalent in our framework. Advanced modeling skills are necessary to create appropriate mathematical representations of highly complex systems (with non-linear dynamics, patches with more than one kind of important attribute, or interactive patches). Simple models can work well when the scale of heterogeneity is much finer than the scale of observation, when low precision is sufficient, when patches interact only weakly, or when empirical approaches are used to fit functions and constants. Having a way to classify complexity in space and time in ecological systems should help ecologists to select modeling approaches consistent with their abilities and goals.     

Detecting Causality by Combined Use of Multiple Methods: Climate and Brain Examples

Identifying causal relations from time series is the first step to understanding the behavior of complex systems. Although many methods have been proposed, few papers have applied multiple methods together to detect causal relations based on time series generated from coupled nonlinear systems with some unobserved parts. Here we propose the combined use of three methods and a majority vote to infer causality under such circumstances. Two of these methods are proposed here for the first time, and all of the three methods can be applied even if the underlying dynamics is nonlinear and there are hidden common causes. We test our methods with coupled logistic maps, coupled Rössler models, and coupled Lorenz models. In addition, we show from ice core data how the causal relations among the temperature, the CH₄ level, and the CO₂ level in the atmosphere changed in the last 800,000 years, a conclusion also supported by irregularly sampled data analysis. Moreover, these methods show how three regions of the brain interact with each other during the visually cued, two-choice arm reaching task. Especially, we demonstrate that this is due to bottom up influences at the beginning of the task, while there exist mutual influences between the posterior medial prefrontal cortex and the presupplementary motor area. Based on our results, we conclude that identifying causality with an appropriate ensemble of multiple methods ensures the validity of the obtained results more firmly.     

区域气候变化统计降尺度研究进展

统计降尺度方法(the Statistical Downscaling Methods, SDM)是为合理预测区域尺度的气候变化情景而提出的新型研究方法。统计降尺度法利用多年大气环流的观测资料建立大尺度气候要素和区域气候要素之间的统计关系,并用独立的观测资料检验这种关系的合理性。把这种关系应用于大气环流模式(Global atmospheric general circulation models, GCMs)中输出大尺度气候信息,来预估区域未来的气候变化情景(如气温和降水)。同时,10a来降尺度方法在生态过程模拟以及气候变化与生态预报关系拟合研究方面也取得一定进展。对统计降尺度方法概念的内涵和外延、基本原理和操作步骤的创新研究方面进行了综述,归纳了该方法在模拟区域气候变化中的应用进展、研究热点及发展趋势,介绍了降尺度在生态预报中的相关应用,为相关研究提供参考。     

When can noise induce chaos and why does it matter: a critique

Noise‐induced chaos illustrates how small amounts of exogenous noise can have disproportionate qualitative impacts on the long term dynamics of a nonlinear system. This property is particularly clear in chaotic systems but is also important for the majority of ecological systems which are nonchaotic, and has direct implications for analyzing ecological time series and testing models against field data. Dennis et al. point out that a definition of chaos which we advocated allows a noise‐dominated system to be classified as chaotic when its Lyapunov exponent λ is positive, which misses what is really going on. As a solution, they propose to eliminate the concept of noise‐induced chaos: chaos “should retain its strictly deterministic definition”, hence “ecological populations cannot be strictly chaotic”. Instead, they suggest that ecologists ask whether ecological systems are strongly influenced by “underlying skeletons with chaotic dynamics or whatever other dynamics”– the skeleton being the hypothetical system that would result if all external and internal noise sources were eliminated. We agree with Dennis et al. about the problem – noise‐dominated systems should not be called chaotic – but not the solution. Even when an estimated skeleton predicts a system's short term dynamics with extremely high accuracy, the skeleton's long term dynamics and attractor may be very different from those of the actual noisy system. Using theoretical models and empirical data on microtine rodent cycles and laboratory populations of Tribolium, we illustrate how data analyses focusing on attributes of the skeleton and its attractor – such as the “deterministic Lyapunov exponent”λ₀ that Dennis et al. have used as their primary indicator of chaos – will frequently give misleading results. In contrast, quantitative measures of the actual noisy system, such as λ, provide useful information for characterizing observed dynamics and for testing proposed mechanistic explanations.     

A stochastic limit cycle oscillator model of the EEG

We present an empirical model of the electroencephalogram (EEG) signal based on the construction of a stochastic limit cycle oscillator using Itô calculus. This formulation, where the noise influences actually interact with the dynamics, is substantially different from the usual definition of measurement noise. Analysis of model data is compared with actual EEG data using both traditional methods and modern techniques from nonlinear time series analysis. The model demonstrates visually displayed patterns and statistics that are similar to actual EEG data. In addition, the nonlinear mechanisms underlying the dynamics of the model do not manifest themselves in nonlinear time series analysis, paralleling the situation with real, non-pathological EEG data. This modeling exercise suggests that the EEG is optimally described by stochastic limit cycle behavior.     

Dynamics extraction in multivariate biomedical time series

A nonlinear analysis of the underlying dynamics of a biomedical time series is proposed by means of a multi-dimensional testing of nonlinear Markovian hypotheses in the observed time series. The observed dynamics of the original N-dimensional biomedical time series is tested against a hierarchy of null hypotheses corresponding to N-dimensional nonlinear Markov processes of increasing order, whose conditional probability densities are estimated using neural networks. For each of the N time series, a measure based on higher order cumulants quantifies the independence between the past of the N-dimensional time series, and its value r steps ahead. This cumulant-based measure is used as a discriminating statistic for testing the null hypotheses. Experiments performed on artificial and real world examples, including autoregressive models, noisy chaos, and nonchaotic nonlinear processes, show the effectiveness of the proposed approach in modeling multivariate systems, predicting multidimensional time series, and characterizing the structure of biological systems. Electroencephalogram (EEG) time series and heart rate variability trends are tested as biomedical signal examples. Received: 2 July 1997 / Accepted in revised form: 26 March 1998