Maximum Entropy

2020 ◽  
pp. 161-184
Author(s):  
John Harte
Keyword(s):  
Maximum Entropy ◽  
Rapid Change ◽  
Entropy Theory ◽  
State Variables ◽  
Total Abundance ◽  
Inference Procedure ◽  
Functional Forms ◽  
Natural Historian ◽  
Parameter Values ◽  
Maximum Entropy Inference

A major goal of ecology is to predict patterns and changes in the abundance, distribution, and energetics of individuals and species in ecosystems. The maximum entropy theory of ecology (METE) predicts the functional forms and parameter values describing the central metrics of macroecology, including the distribution of abundances over all the species, metabolic rates over all individuals, spatial aggregation of individuals within species, and the dependence of species diversity on areas of habitat. In METE, the maximum entropy inference procedure is implemented using the constraints imposed by a few macroscopic state variables, including the number of species, total abundance, and total metabolic rate in an ecological community. Although the theory adequately predicts pervasive empirical patterns in relatively static ecosystems, there is mounting evidence that in ecosystems in which the state variables are changing rapidly, many of the predictions of METE systematically fail. Here we discuss the underlying logic and predictions of the static theory and then describe progress toward achieving a dynamic theory (DynaMETE) of macroecology capable of describing ecosystems undergoing rapid change as a result of disturbance. An emphasis throughout is on the tension between, and reconciliation of, two legitimate perspectives on ecology: that of the natural historian who studies the uniqueness of every ecosystem and the theorist seeking unification and generality.

scholarly journals Land use, macroecology, and the accuracy of the Maximum Entropy Theory of Ecology: A case study of Azorean arthropods

2021 ◽  
Author(s):  
Micah Brush ◽  
Thomas J. Matthews ◽  
Paulo A.V. Borges ◽  
John Harte
Keyword(s):  
Land Use ◽  
Land Use Change ◽  
Metabolic Rate ◽  
Maximum Entropy ◽  
Management Practices ◽  
Entropy Theory ◽  
State Variables ◽  
Arthropod Communities ◽  
Natural Pasture ◽  
Maximum Entropy Theory

AbstractHuman activity and land management practices, in particular land use change, have resulted in the global loss of biodiversity. These types of disturbances affect the shape of macroecological patterns, and analyzing these patterns can provide insights into how ecosystems are affected by land use change. The Maximum Entropy Theory of Ecology (METE) simultaneously predicts many of these patterns using a set of ecological state variables: the number of species, the number of individuals, and the total metabolic rate. The theory’s predictions have been shown to be successful across habitats and taxa in undisturbed natural ecosystems, although previous tests of METE in relation to disturbance have focused primarily on systems where the state variables are changing relatively quickly. Here, we assess predictions of METE applied to a different type of disturbance: land use change. We use METE to simultaneously predict the species abundance distribution (SAD), the metabolic rate distribution of individuals (MRDI), and the species–area relationship (SAR) and compare these predictions to arthropod data from 96 sites at Terceira Island in the Azores archipelago across four different land uses of increasing management intensity: 1. native forest, 2. exotic forest, 3. semi-natural pasture, and 4. intensive pasture. Across these patterns, we find that the forest habitats are the best fit by METE predictions, while the semi-natural pasture consistently provided the worst fit. The intensive pasture is intermediately well fit for the SAD and MRDI, and comparatively well fit for the SAR, though the residuals are not normally distributed. The direction of failure of the METE predictions at the pasture sites is likely due to the hyperdominance of introduced spider species present there. We hypothesize that the particularly poor fit for the semi-natural pasture is due to the mix of arthropod communities out of equilibrium and the changing management practices throughout the year, leading to greater heterogeneity in composition and complex dynamics that violate METE’s assumption of static state variables. The comparative better fit for the intensive pasture could then result from more homogeneous arthropod communities that are well adapted to intensive management, and thus whose state variables are less in flux.

scholarly journals Derivations of the Core Functions of the Maximum Entropy Theory of Ecology

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 712 ◽  
Author(s):  
Alexander Brummer ◽  
Erica Newman
Keyword(s):  
Species Richness ◽  
Structure Function ◽  
Maximum Entropy ◽  
Information Entropy ◽  
Lagrange Multipliers ◽  
Metabolic Energy ◽  
Entropy Theory ◽  
State Variables ◽  
The Core ◽  
Maximum Entropy Theory

The Maximum Entropy Theory of Ecology (METE), is a theoretical framework of macroecology that makes a variety of realistic ecological predictions about how species richness, abundance of species, metabolic rate distributions, and spatial aggregation of species interrelate in a given region. In the METE framework, “ecological state variables” (representing total area, total species richness, total abundance, and total metabolic energy) describe macroecological properties of an ecosystem. METE incorporates these state variables into constraints on underlying probability distributions. The method of Lagrange multipliers and maximization of information entropy (MaxEnt) lead to predicted functional forms of distributions of interest. We demonstrate how information entropy is maximized for the general case of a distribution, which has empirical information that provides constraints on the overall predictions. We then show how METE’s two core functions are derived. These functions, called the “Spatial Structure Function” and the “Ecosystem Structure Function” are the core pieces of the theory, from which all the predictions of METE follow (including the Species Area Relationship, the Species Abundance Distribution, and various metabolic distributions). Primarily, we consider the discrete distributions predicted by METE. We also explore the parameter space defined by the METE’s state variables and Lagrange multipliers. We aim to provide a comprehensive resource for ecologists who want to understand the derivations and assumptions of the basic mathematical structure of METE.

scholarly journals Derivations of the Core Functions of the Maximum Entropy Theory of Ecology

2019 ◽  
Author(s):  
Alexander Brummer ◽  
Erica Newman
Keyword(s):  
Species Richness ◽  
Structure Function ◽  
Maximum Entropy ◽  
Information Entropy ◽  
Lagrange Multipliers ◽  
Metabolic Energy ◽  
Entropy Theory ◽  
State Variables ◽  
The Core ◽  
Maximum Entropy Theory

The Maximum Entropy Theory of Ecology, or METE, is a theoretical framework of macroecology that makes a variety of realistic ecological predictions about how species richness, abundance of species, metabolic rate distributions, and spatial aggregation of species interrelate in a given region. In the METE framework, "ecological state variables" (representing total area, total species richness, total abundance, and total metabolic energy) describe macroecological properties of an ecosystem. METE incorporates these state variables into constraints on underlying probability distributions. The method of Lagrange multipliers and maximization of information entropy (MaxEnt) lead to predicted functional forms of distributions of interest. We demonstrate how information entropy is maximized for the general case of a distribution, which has empirical information that provides constraints on the overall predictions. We then show how METE’s two core functions are derived. These functions, called the "Spatial Structure Function" and the "Ecosystem Structure Function" are the core pieces of the theory, from which all the predictions of METE follow (including the Species Area Distribution, the Species Abundance Distribution, and various metabolic distributions). Primarily, we consider the discrete distributions predicted by METE.We also explore the parameter space defined by the METE’s state variables and Lagrange multipliers. We aim to provide a comprehensive resource for ecologists who want to understand the derivations and assumptions of basic mathematical structure of METE.

Maximum entropy theory

1985 ◽  
Vol 41 (2) ◽  
pp. 113-122 ◽  
Author(s):  
A. K. Livesey ◽  
J. Skilling
Keyword(s):  
Maximum Entropy ◽  
Entropy Theory ◽  
Maximum Entropy Theory

scholarly journals Representation Dependence in Probabilistic Inference

2004 ◽  
Vol 21 ◽  
pp. 319-356 ◽  
Author(s):  
J. Y. Halpern ◽  
D. Koller
Keyword(s):  
Maximum Entropy ◽  
Deductive Reasoning ◽  
Probabilistic Inference ◽  
Irrelevant Information ◽  
Restricted Class ◽  
Inference Procedure ◽  
Reasoning Systems ◽  
Restricted Representation ◽  
Default Assumption ◽  
Principle Of Maximum

Non-deductive reasoning systems are often representation dependent: representing the same situation in two different ways may cause such a system to return two different answers. Some have viewed this as a significant problem. For example, the principle of maximum entropyhas been subjected to much criticism due to its representation dependence. There has, however, been almost no work investigating representation dependence. In this paper, we formalize this notion and show that it is not a problem specific to maximum entropy. In fact, we show that any representation-independent probabilistic inference procedure that ignores irrelevant information is essentially entailment, in a precise sense. Moreover, we show that representation independence is incompatible with even a weak default assumption of independence. We then show that invariance under a restricted class of representation changes can form a reasonable compromise between representation independence and other desiderata, and provide a construction of a family of inference procedures that provides such restricted representation independence, using relative entropy.

scholarly journals An Entropy-Based Machine Learning Algorithm for Combining Macroeconomic Forecasts

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 1015 ◽  
Author(s):  
Carles Bretó ◽  
Priscila Espinosa ◽  
Penélope Hernández ◽  
Jose M. Pavía
Keyword(s):  
Machine Learning ◽  
Gross Domestic Product ◽  
Maximum Entropy ◽  
Simulation Study ◽  
Learning Algorithm ◽  
Predictive Ability ◽  
Learning Approach ◽  
Machine Learning Algorithm ◽  
Machine Learning Approach ◽  
Maximum Entropy Inference

This paper applies a Machine Learning approach with the aim of providing a single aggregated prediction from a set of individual predictions. Departing from the well-known maximum-entropy inference methodology, a new factor capturing the distance between the true and the estimated aggregated predictions presents a new problem. Algorithms such as ridge, lasso or elastic net help in finding a new methodology to tackle this issue. We carry out a simulation study to evaluate the performance of such a procedure and apply it in order to forecast and measure predictive ability using a dataset of predictions on Spanish gross domestic product.

scholarly journals Observability and Structural Identifiability of Nonlinear Biological Systems

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Alejandro F. Villaverde
Keyword(s):  
Differential Geometry ◽  
Internal State ◽  
Biological Processes ◽  
State Variables ◽  
Related Property ◽  
Theoretical Possibility ◽  
Structural Identifiability ◽  
Open Problems ◽  
Constant State ◽  
Parameter Values

Observability is a modelling property that describes the possibility of inferring the internal state of a system from observations of its output. A related property, structural identifiability, refers to the theoretical possibility of determining the parameter values from the output. In fact, structural identifiability becomes a particular case of observability if the parameters are considered as constant state variables. It is possible to simultaneously analyse the observability and structural identifiability of a model using the conceptual tools of differential geometry. Many complex biological processes can be described by systems of nonlinear ordinary differential equations and can therefore be analysed with this approach. The purpose of this review article is threefold: (I) to serve as a tutorial on observability and structural identifiability of nonlinear systems, using the differential geometry approach for their analysis; (II) to review recent advances in the field; and (III) to identify open problems and suggest new avenues for research in this area.

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