Maximum Entropy Theory Research Articles

Relating the Strength of Density Dependence and the Spatial Distribution of Individuals

Spatial patterns in ecology contain useful information about underlying mechanisms and processes. Although there are many summary statistics used to quantify these spatial patterns, there are far fewer models that directly link explicit ecological mechanisms to observed patterns easily derived from available data. We present a model of intraspecific spatial aggregation that quantitatively relates static spatial patterning to negative density dependence. Individuals are placed according to the colonization rule consistent with the Maximum Entropy Theory of Ecology (METE), and die with probability proportional to their abundance raised to a power α, a parameter indicating the degree of density dependence. This model can therefore be interpreted as a hybridization of MaxEnt and mechanism. Our model shows quantitatively and generally that increasing density dependence randomizes spatial patterning. α = 1 recovers the strongly aggregated METE distribution that is consistent with many ecosystems empirically, and as α → 2 our prediction approaches the binomial distribution consistent with random placement. For 1 < α < 2, our model predicts more aggregation than random placement but less than METE. We additionally relate our mechanistic parameter α to the statistical aggregation parameter k in the negative binomial distribution, giving it an ecological interpretation in the context of density dependence. We use our model to analyze two contrasting datasets, a 50 ha tropical forest and a 64 m2 serpentine grassland plot. For each dataset, we infer α for individual species as well as a community α parameter. We find that α is generally larger in the tightly packed forest than the sparse grassland, and the degree of density dependence increases at smaller scales. These results are consistent with current understanding in both ecosystems, and we infer this underlying density dependence using only empirical spatial patterns. Our model can easily be applied to other datasets where spatially explicit data are available.

Integrating metabolic scaling variation into the maximum entropy theory of ecology explains Taylor's law for individual metabolic rate in tropical forests

Individual trait variation has important ecological implications for species populations and communities. In particular, individual variation of metabolic rate links directly with the energy use estimation and scaling patterns in community ecology. Here, we examine the mean-variance relationship of individual metabolic rate by testing Taylor's law (a power-function relationship between mean and variance) for individual rescaled metabolic rate across tree communities in tropical forests. We use a constraint-based model called maximum entropy theory of ecology (METE) to estimate and predict the parameters of Taylor's law. Our results show that, when assuming a universal metabolic scaling between metabolic rate and aboveground biomass, the METE generates the form of Taylor's law but fails to predict its slope. When setting the metabolic scaling exponent as a community-level parameter in the METE model, the estimated and predicted slopes of Taylor's law agree with each other. Our parameterized METE model reveals the positive effect of number of individuals on the metabolic scaling exponent. These results suggest that fluctuation scaling of individual metabolic rate can be explained solely by the macroecological constraints of communities, without relying on the physiological or genetic characters of individual organisms.

Shifting macroecological patterns and static theory failure in a stressed alpine plant community

AbstractAccumulating evidence suggests that ecological communities undergoing change in response to either anthropogenic or natural disturbances exhibit macroecological patterns that differ from those observed in similar types of communities in relatively undisturbed sites. In contrast to such cross‐site comparisons, however, there are few empirical studies of shifts over time in the shapes of macroecological patterns. Here, we provide a dramatic example of a plant community in which the species–area relationship and the species‐abundance distribution change markedly over a period of six years. These patterns increasingly deviate from the predictions of the maximum entropy theory of ecology (METE), which successfully predicts macroecological patterns in relatively static systems. The error in the species–area relationship prediction additionally correlates over time with increased stress measured as mortality minus recruitment, providing a link between demography and the failure of macroecological theory. Information on the dynamic state of an ecosystem inferred from snapshot measurements of macroecological community structure can potentially assist in identifying causes and consequences of disturbance and extending the domain of current theories and models to disturbed ecosystems.

The Shape of Species Abundance Distributions Across Spatial Scales

Species abundance distributions (SADs) describe community structure and are a key component of biodiversity theory and research. Although different distributions have been proposed to represent SADs at different scales, a systematic empirical assessment of how SAD shape varies across wide scale gradients is lacking. Here, we examined 11 empirical large-scale datasets for a wide range of taxa and used maximum likelihood methods to compare the fit of the logseries, lognormal, and multimodal (i.e., with multiple modes of abundance) models to SADs across a scale gradient spanning several orders of magnitude. Overall, there was a higher prevalence of multimodality for larger spatial extents, whereas the logseries was exclusively selected as best fit for smaller areas. For many communities the shape of the SAD at the largest spatial extent (either lognormal or multimodal) was conserved across the scale gradient, despite steep declines in area and taxonomic diversity sampled. Additionally, SAD shape was affected by species richness, but we did not detect a systematic effect of the total number of individuals. Our results reveal clear departures from the predictions of two major macroecological theories of biodiversity for SAD shape. Specifically, neither the Neutral Theory of Biodiversity (NTB) nor the Maximum Entropy Theory of Ecology (METE) are able to accommodate the variability in SAD shape we encountered. This is highlighted by the inadequacy of the logseries distribution at larger scales, contrary to predictions of the NTB, and by departures from METE expectation across scales. Importantly, neither theory accounts for multiple modes in SADs. We suggest our results are underpinned by both inter- and intraspecific spatial aggregation patterns, highlighting the importance of spatial distributions as determinants of biodiversity patterns. Critical developments for macroecological biodiversity theories remain in incorporating the effect of spatial scale, ecological heterogeneity and spatial aggregation patterns in determining SAD shape.

DynaMETE: a hybrid MaxEnt-plus-mechanism theory of dynamic macroecology.

The Maximum Entropy Theory of Ecology (METE) predicts the shapes of macroecological metrics in relatively static ecosystems, across spatial scales, taxonomic categories and habitats, using constraints imposed by static state variables. In disturbed ecosystems, however, with time‐varying state variables, its predictions often fail. We extend macroecological theory from static to dynamic by combining the MaxEnt inference procedure with explicit mechanisms governing disturbance. In the static limit, the resulting theory, DynaMETE, reduces to METE but also predicts a new scaling relationship among static state variables. Under disturbances, expressed as shifts in demographic, ontogenic growth or migration rates, DynaMETE predicts the time trajectories of the state variables as well as the time‐varying shapes of macroecological metrics such as the species abundance distribution and the distribution of metabolic rates over individuals. An iterative procedure for solving the dynamic theory is presented. Characteristic signatures of the deviation from static predictions of macroecological patterns are shown to result from different kinds of disturbance. By combining MaxEnt inference with explicit dynamical mechanisms of disturbance, DynaMETE is a candidate theory of macroecology for ecosystems responding to anthropogenic or natural disturbances.

On the use of the maximum entropy method for reliability evaluation involving stochastic process modeling

From a probabilistic perspective, a stochastic process (or random field) is completely specified by its joint probability distribution, which can be further decomposed into marginal distributions and a multivariate copula function. This paper proposes a non-parametric approach based on the maximum entropy theory to build the two parts of the distribution of stochastic processes. The specification of a stochastic process is interpreted as moment constraints and classified into two categories. The first category describes the uncertainty of the random quantity at an arbitrary point. The second category defines the dependence between the random quantities at any two points. The marginal distribution and copula function are then developed by maximizing the entropy under two classes of constraints, which are formulated as two optimization problems. The proposed method is applied to two reliability problems: a beam reliability evaluation considering time-dependent corrosion, and a tunnel reliability evaluation considering soil spatial variability. The first example illustrates the method when only partial information (marginal distribution and autocorrelation function) is available, while the second example shows the development of a random field model based on data. The comparison of the non-parametric approach with typical parametric models indicates the flexibility of the proposed method in capturing the variation of random quantities in time or space.

Spatial-temporal Evolution and Determinants of the Belt and Road Initiative: A Maximum Entropy Gravity Model Approach

The spatial interaction model is an effective way to explore the geographical disparities inherent in the Belt and Road Initiative (BRI) by simulating spatial flows. The traditional gravity model implies the hypothesis of equilibrium points without any reference to when or how to achieve it. In this paper, a dynamic gravity model was established based on the Maximum Entropy (MaxEnt) theory to estimate and monitor the interconnection intensity and dynamic characters of bilateral relations. In order to detect the determinants of interconnection intensity, a Geodetector method was applied to identify and evaluate the determinants of spatial networks in five dimensions. The empirical study clearly demonstrates a heterogeneous and non-circular spatial structure. The main driving forces of spatial-temporal evolution are foreign direct investment, tourism and railway infrastructure construction, while determinants in different sub-regions show obvious spatial differentiation. Southeast Asian countries are typically multi-island area where aviation infrastructure plays a more important role. North and Central Asian countries regard oil as a pillar industry where power and port facilities have a greater impact on the interconnection. While Western Asian countries are mostly influenced by the railway infrastructure, Eastern European countries already have relatively robust infrastructure where tariff policies provide a greater impetus.

Understanding between-cluster variation in prevalence and limits for how much variation is plausible.

In clinical trials and observational studies of clustered binary data, understanding between-cluster variation is essential: in sample size and power calculations of cluster randomised trials, for example, the intra-cluster correlation coefficient is often specified. However, quantifications of between-cluster variation can be unintuitive, and an intra-cluster correlation coefficient as low as 0.04 may correspond to surprisingly large between-cluster differences. We suggest that understanding is improved through visualising the implied distribution of true cluster prevalences - possibly by assuming they follow a beta distribution - or by calculating their standard deviation, which is more readily interpretable than the intra-cluster correlation coefficient. Even so, the bounded nature of binary data complicates the interpretation of variances as primary measures of uncertainty, and entropy offers an attractive alternative. Appealing to maximum entropy theory, we propose the following rule of thumb: that plausible intra-cluster correlation coefficients and standard deviations of true cluster prevalences are both bounded above by the overall prevalence, its complement, and one third. We also provide corresponding bounds for the coefficient of variation, and for a different standard deviation and intra-cluster correlation defined on the log odds scale. Using previously published data, we observe the quantities defined on the log odds scale to be more transportable between studies with different outcomes with different prevalences than the intra-cluster correlation and coefficient of variation. The latter increase and decrease, respectively, as prevalence increases from 0% to 50%, and the same is true for our bounds. Our work will help clinical trialists better understand between-cluster variation and avoid specifying implausibly high values for the intra-cluster correlation in sample size and power calculations.

A Trust-Game-Based Access Control Model for Cloud Service

In order to promote mutual trust and win-win cooperation between the users and the providers, we propose a trust-game-based access control model for cloud service in the paper. First, we construct a trust evaluation model based on multiple factors, such as the direct trust, feedback trust, reward punishment, and trust risk and further propose a weight method by maximum discrete degree and information entropy theory; second, we combine trust evaluation with the payoff matrix for game analysis and calculate the mixed Nash equilibrium strategy for the users and service providers; third, we give the game control condition based on trust level prediction and payment matrix to encourage participants to make honest strategy. Experimental results show that our research has good effect in terms of acceptance probability, deception probability, accuracy of trust evaluation, and cooperation rate in the cloud service.

Comparing landscape suitability and permeability with and without migration data: the influence of species movement behavior

Maintaining landscape connectivity through identifying movement corridors is the most recommended conservation strategy to reduce the negative impacts of habitat loss and isolation. The basis of most connectivity modelling approaches for modelling corridors is that species choose movement pathways based on the same criteria they used to choose habitats. However, species behave differently in using landscape elements for moving than for selecting habitat. In other words, suitability of a given landscape feature may differ between moving and habitat use stages. In this study, we evaluated how the availability of migration occurrence points for wild sheep (Ovis orientalis) could affect the outputs of distribution models and consequently the strength and extent of landscape connectivity for migratory movements of this species in central Iran. We employed concepts of Maximum entropy and circuit theory and developed 2 groups of habitat suitability and connectivity models with and without migration data. Comparing the results of the developed models showed that the main differences in the outputs of MaxEnt models were associated with suitability values predicted for the unprotected migration habitats. Without migration occurrence points, MaxEnt did not identify the traditionally used migration habitats. In contrast, Circuitscape represented a similar performance in predicting the main migration corridor of the species when using or not using the migration occurrence data. These differences could be associated with wild sheep's different behavior in the selection of habitat during movement and home range stages. Owning to this difference, we suggest using migration data in modelling landscape connectivity as these data may include different environmental conditions to those collected from home range habitats. For wild sheep, we recommend protecting the migration corridor at least during migration time. Maintaining such connectivity would also largely depend on managing the unprotected matrix through preventing expansion of human land uses in the vicinity of the corridor and buffering it to some distance.

A moment-based stochastic edge-based smoothed finite element method for electromagnetic forming process

In this paper, a novel stochastic method named as the moment-based stochastic edge-based finite element method (MSES-FEM) is proposed to deal with the uncertain electromagnetic problems. First, electromagnetic and mechanical field are formulated by smoothed Galerkin Weak Form under edge-based smoothed finite element method (ES-FEM) scheme. The moment analysis is then applied to obtain the first four moments of the responses and to observe the effects of each random variable on electromagnetic field responses. The maximum entropy theory is employed to calculate the probability density functions (PDFs) of the responses. A quasi-static electromagnetic problem and a practical electromagnetic forming problem (EMF) are performed. The proposed method successfully solves stochastic electromagnetic forming analysis under the uncertain parameters. Numerical results obtained by the proposed MSES-FEM are quite satisfactory with the ones by the Monte Carlo simulation (MCS).

A new integrated approach for engineering characteristic prioritization in quality function deployment

As a customer-driven quality improvement tool, quality function deployment (QFD) can convert customer requirements (CRs) into appropriate engineering characteristics (ECs) in product design and development. However, the conventional QFD method has been criticized for a variety of drawbacks, which limit its efficiency and potential applications. In this study, a new QFD approach integrating picture fuzzy linguistic sets (PFLSs) and the evaluation based on distance from average solution (EDAS) method is proposed for the determination of ranking order of ECs. The PFLSs are utilized to express the judgements of experts on the relationships among CRs and ECs. Then, the EDAS method is extended under picture fuzzy linguistic environment for the prioritization of the ECs identified in QFD. Moreover, a combined weighing method based on technique for order of preference by similarity to ideal solution (TOPSIS) and maximum entropy theory is established to calculate the weights of experts objectively. Finally, a product-service system design is provided to illustrate the effectiveness of the proposed QFD approach. The result shows that the manufacturer should pay more attention to “Meantime before failure”, “Warning feature” and “Quality of product manual”. Feedback from domain experts indicates that the integrated approach being proposed in this paper is more suitable for assessing and prioritizing ECs in QFD.

An efficient method for estimating fuzzy failure probability-based global fuzzy reliability sensitivity

The global fuzzy reliability sensitivity can measure the effect of input on fuzzy failure probability (FFP) through the expected difference between the unconditional FFP and the conditional one. Thus, an efficient method is proposed to estimate this sensitivity index. The proposed method includes two steps. First, the unconditional–conditional FFPs are equivalently transformed into the integrals of the unconditional–conditional failure probabilities (FPs) by an introduced variable related to the fuzzy state assumption. The unconditional–conditional FPs can be further expressed as the integrals of the unconditional–conditional probability density functions (PDFs) of the output. Secondly, the unconditional–conditional PDFs are solved using the maximum entropy theory constrained by the unconditional–conditional fractional moments. These fractional moments can be solved simultaneously through the multiplicative dimensional reduction using the same group of Gaussian quadrature points. The computational cost of the proposed method grows linearly with the dimensionality, and the computational efficiency is greatly improved.

Disturbance macroecology: a comparative study of community structure metrics in a high‐severity disturbance regime

AbstractMacroecological studies have established widespread patterns of species diversity and abundance in ecosystems but have generally restricted their scope to relatively steady‐state systems. As a result, how macroecological metrics are expected to scale in ecosystems that experience natural disturbance regimes is unknown. We examine macroecological patterns in a fire‐dependent forest of Bishop pine (Pinus muricata). We target two different‐aged stands in a stand‐replacing fire regime: a mature stand with a diverse understory and with no history of major disturbance for at least 40 yr, and one disturbed by a stand‐replacing fire 17 yr prior to measurement. We compare properties of these stands with macroecological predictions from the Maximum Entropy Theory of Ecology (METE), an information entropy‐based theory that has proven highly successful in predicting macroecological metrics in multiple ecosystems and taxa. Ecological patterns in the mature stand more closely match METE predictions than do data from the more recently disturbed, mid‐seral stage stand. This suggests METE's predictions are more robust in late‐successional, slowly changing, or steady‐state systems than those in rapid flux with respect to species composition, abundances, and organisms’ sizes. Our findings highlight the need for a macroecological theory that incorporates natural disturbance, perturbations, and ecological dynamics into its predictive capabilities, because most natural systems are not in a steady state.

Space-time chlorophyll-a retrieval in optically complex waters that accounts for remote sensing and modeling uncertainties and improves remote estimation accuracy

Remote sensing reflectance (Rrs) values measured by satellite sensors involve large amounts of uncertainty leading to non-negligible noise in remote Chlorophyll-a (Chl-a) concentration estimation. This work distinguished between two main stages in the case of estimating distributions of Chl-a within the Gulf of St. Lawrence (Canada). At the model building stage, the retrieval algorithm used both in-situ Chl-a measurements and the corresponding Moderate Resolution Imaging Spectroradiometer (MODIS) L2-level data estimated Rrs at 412, 443, 469, 488, 531, 547, 555, 645, 667, 678 nm at a 1 km spatial resolution during 2004–2013. Through the training and validation of various models and Rrs combinations of the considered eight techniques (including support vector regression, artificial neural networks, gradient boosting machine, random forests, standard CI-OC3M, multiple linear regression, generalized addictive regression, principal component regression), the support vector regression (SVR) technique was shown to have the best performance in Chl-a concentration estimation using Rrs at 412, 443, 488, 531 and 678 nm. The accuracy indicators for both the training (850) and the validation (213) datasets were found to be very good to excellent (e.g., the R2 value varied between 0.7058 and 0.9068). At the space-time estimation stage, this work took a step forward by using the Bayesian maximum entropy (BME) theory to further process the SVR estimated Chl-a concentrations by incorporating the inherent spatiotemporal dependency of physical Chl-a distribution. A 56% improvement was achieved in the reduction of the mean uncertainty of the validation data decreased considerably (from 1.2222 to 0.5322 mg/m3). Then, this novel BME/SVR framework was employed to estimate the daily Chl-a concentrations in the Gulf of St. Lawrence during Jan 1-Dec 31 of 2017 (1 km spatial resolution). The results showed that the daily mean Chl-a concentration varied from 1.6630 to 3.3431 mg/m3, and that the daily mean Chl-a uncertainty reduction of the composite BME/SVR vs. the SVR estimation had a maximum reduction value of 1.0082 and an average reduction value of 0.6173 mg/m3. The monthly spatial Chl-a distribution covariances showed that the highest Chl-a concentration variability occurred during November and that the spatiotemporal Chl-a concentration pattern changed a lot during the period August to November. In conclusion, the proposed BME/SVR was shown to be a promising remote Chl-a retrieval approach that exhibited a significant ability in reducing the non-negligible uncertainty and improving the accuracy of remote sensing Chl-a concentration estimates.

Renewal Theory as a Universal Quantitative Framework to Characterize Phase Singularity Regeneration in Mammalian Cardiac Fibrillation.

Despite a century of research, no clear quantitative framework exists to model the fundamental processes responsible for the continuous formation and destruction of phase singularities (PS) in cardiac fibrillation. We hypothesized PS formation/destruction in fibrillation could be modeled as self-regenerating Poisson renewal processes, producing exponential distributions of interevent times governed by constant rate parameters defined by the prevailing properties of each system. PS formation/destruction were studied in 5 systems: (1) human persistent atrial fibrillation (n=20), (2) tachypaced sheep atrial fibrillation (n=5), (3) rat atrial fibrillation (n=4), (5) rat ventricular fibrillation (n=11), and (5) computer-simulated fibrillation. PS time-to-event data were fitted by exponential probability distribution functions computed using maximum entropy theory, and rates of PS formation and destruction (λf/λd) determined. A systematic review was conducted to cross-validate with source data from literature. In all systems, PS lifetime and interformation times were consistent with underlying Poisson renewal processes (human: λf, 4.2%/ms±1.1 [95% CI, 4.0-5.0], λd, 4.6%/ms±1.5 [95% CI, 4.3-4.9]; sheep: λf, 4.4%/ms [95% CI, 4.1-4.7], λd, 4.6%/ms±1.4 [95% CI, 4.3-4.8]; rat atrial fibrillation: λf, 33%/ms±8.8 [95% CI, 11-55], λd, 38%/ms [95% CI, 22-55]; rat ventricular fibrillation: λf, 38%/ms±24 [95% CI, 22-55], λf, 46%/ms±21 [95% CI, 31-60]; simulated fibrillation λd, 6.6-8.97%/ms [95% CI, 4.1-6.7]; R2≥0.90 in all cases). All PS distributions identified through systematic review were also consistent with an underlying Poisson renewal process. Poisson renewal theory provides an evolutionarily preserved universal framework to quantify formation and destruction of rotational events in cardiac fibrillation.

The effects of density dependence on spatial aggregation in the maximum entropy theory of ecology

The effects of density dependence on spatial aggregation in the maximum entropy theory of ecology

Derivations of the Core Functions of the Maximum Entropy Theory of Ecology.

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.

Comparison of two maximum entropy models highlights the metabolic structure of metacommunities as a key determinant of local community assembly

The principle of Maximum Entropy (MaxEnt) promises a novel approach for understanding community assembly. Despite reproducing a variety of observed species abundance patterns, MaxEnt models in ecology have been hampered by disparate model assumptions and interpretations. A recurring challenge is that MaxEnt predictions are highly sensitive to the level of detail used to describe the community being modeled, and there seems to be no reason to prefer one level of detail over another. Here we present of formal unification of two previously developed MaxEnt models which differ in their level of detail, but which are otherwise mathematically similar. The less detailed model, “Maximum Entropy Theory of Ecology” (METE), does not resolve species identity or explicitly represent species-specific traits. The more detailed model, “Very Entropic Growth” (VEG), defines each separate species by its per capita metabolic rate ε and assumes a “density of species” function ρ(ε) representing the distribution of ε in the metacommunity. A formal comparison of METE and VEG then highlights ρ(ε) as a key determinant of local community assembly. In particular, appropriate choice of ρ(ε) in VEG can produce more realistic predictions for the metabolic-rank distribution of local communities than METE, which does not explicitly account for metacommunity structure. This opens new avenues of inquiry about what determines metacommunity structure in nature and suggests possible ways to improve METE.

Improved heavy metal mapping and pollution source apportionment in Shanghai City soils using auxiliary information

Soil heavy metal pollution can be a serious threat to human health and the environment. The accurate mapping of the spatial distribution of soil heavy metal pollutant concentrations enables the detection of high pollution areas and facilitates pollution source apportionment and control. To make full use of auxiliary soil properties information and show that they can improve mapping, a synthesis of the Bayesian Maximum Entropy (BME) theory and the Geographically Weighted Regression (GWR) model is proposed and implemented in the study of the Shanghai City soils (China). The results showed that, compared to traditional techniques, the proposed BME-GWR synthesis has certain important advantages: (a) it integrates heavy metal measurements and auxiliary information on a sound theoretical basis, and (b) it performs better in terms of both prediction accuracy and implementation flexibility (including the assimilation of multiple data sources). Based on the heavy metal concentration maps generated by BME-GWR, we found that the As, Cr and Pb concentration levels are high in the eastern part of Shanghai, whereas high Cd concentration levels were observed in the northwestern part of the city. Organic carbon and pH were significantly correlated with most of the heavy metals in Shanghai soils. We concluded that Cd pollution is mainly the result of agricultural activities, and that the Cr pollution is attributed to natural sources, whereas Pb and As have compound pollution sources. Future studies should investigate the implementation of BME-GWR in the case of space-time heavy metal mapping and its ability to integrate human activity information and soil category variables.