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Huxley’s model of simple allometry provides a parsimonious scheme for examining scaling relationships in scientific research, resource management, and species conservation endeavors. Factors including biological error, analysis method, sample size, and overall data quality can undermine the reliability of a fit of Huxley’s model. Customary amendments enhance the complexity of the power function-conveyed systematic term while keeping the usual normality-borne error structure. The resulting protocols bear multiple-parameter complex allometry forms that could pose interpretative shortcomings and parameter estimation difficulties, and even being empirically pertinent, they could potentially bear overfitting. A subsequent heavy-tailed Q-Q normal spread often remains undetected since the adequacy of a normally distributed error term remains unexplored. Previously, we promoted the advantages of keeping Huxley’s model-driven systematic part while switching to a logistically distributed error term to improve fit quality. Here, we analyzed eelgrass leaf biomass and area data exhibiting a marked size-related heterogeneity, perhaps explaining a lack of systematization at data gathering. Overdispersion precluded adequacy of the logistically adapted protocol, thereby suggesting processing data through a median absolute deviation scheme aimed to remove unduly replicates. Nevertheless, achieving regularity to Huxley’s power function-like trend required the removal of many replicates, thereby questioning the integrity of a data cleaning approach. But, we managed to adapt the complexity of the error term to reliably identify Huxley’s model-like systematic part masked by variability in data. Achieving this relied on an error term conforming to a normal mixture distribution which successfully managed overdispersion in data. Compared to normal-complex allometry and data cleaning composites present arrangement delivered a coherent Q-Q normal mixture spread and a remarkable reproducibility strength of derived proxies. By keeping the analysis within Huxley’s original theory, the present approach enables substantiating nondestructive allometric proxies aimed at eelgrass conservation. The viewpoint endorsed here could also make data cleaning unnecessary.

Customary approaches in allometric examination include linear regression in geometrical space, as well as, nonlinear regression in the original scale of data. These protocols could not produce consistent results in a circumstance in which the allometric response manifest heterogeneity as the covariate changes. The paradigm of log-nonlinear allometry offers a mechanism for the analysis of heterogeneity in geometric space. However, the use of a logarithmic transformation in allometry is controversial. In this contribution, we present a fuzzy approach aimed to examination of allometric heterogeneity in direct arithmetical space. Offered construct relies on a hybrid procedure integrating crisp cluster analysis and a fuzzy inference system of Mamdani type. Calibration aims depended on an extensive data set composing measurements of eelgrass leaf biomass and their corresponding areas. Results on raw data suggest heterogeneity more clearly manifest in the normalization constant than in the allometric exponent. Nevertheless, differences in normalization constant values among clusters are only slight for data remaining after removal of inconsistent replicates. This suggests heterogeneity produced by intrinsic factors of leaf growth.

Current anthropogenic influences threaten the permanence of eelgrass, a relevant macrophyte that brings about important ecological benefits including nursery for waterfowl and fish species, shoreline stabilization, nutrient recycling and carbon sequestration. Eelgrass restoration normally involves transplanted plots and monitoring success requires noninvasive assessments of standing stock and productivity. Allometric scaling of eelgrass leaf biomass and length can provide proxies for these assessments, but accuracy of allometric projections is mainly resultant of uncertainty propagation of parameters, so for the sake of suitability it is very important ensuring the most accurate estimates. The traditional approach for producing estimates of allometric parameters considers a linear regression model involving logarithms of the original response and explanatory variables along with a normally distributed additive error. The suitability of this method has been questioned on the ground of biased results raising nonlinear regression as a necessary amendment. Here we demonstrate that this controversy can be surpassed allowing for a logistic error structure and heteroscedasticity in the traditional method. The present arrangement delivered parameter estimates from raw data surpassing inconveniences of the traditional fitting procedure. Moreover, associated allometric proxies for average leaf biomass in shoots entailed similar reproducibilities than produced by using nonlinear regression and quality controlled data. In achieving suitable accuracy levels, the present approach required a sample of raw data of about 9% of involved in prior estimations based on quality controlled data. The improvements associated to the present approach grant highly consistent non-destructive assessments for the sake of eelgrass conservation.

Background The effects of current anthropogenic influences on eelgrass (Zostera marina) meadows are noticeable. Eelgrass ecological services grant important benefits for mankind. Preservation of eelgrass meadows include several transplantation methods. Evaluation of establishing success relies on the estimation of standing stock and productivity. Average leaf biomass in shoots is a fundamental component of standing stock. Existing methods of leaf biomass measurement are destructive and time consuming. These assessments could alter shoot density in developing transplants. Allometric methods offer convenient indirect assessments of individual leaf biomass. Aggregation of single leaf projections produce surrogates for average leaf biomass in shoots. Involved parameters are time invariant, then derived proxies yield simplified nondestructive approximations. On spite of time invariance local factors induce relative variability of parameter estimates. This influences accuracy of surrogates. And factors like analysis method, sample size and data quality also impact precision. Besides, scaling projections are sensitive to parameter fluctuation. Thus the suitability of the addressed allometric approximations requires clarification. Results The considered proxies produced accurate indirect assessments of observed values. Only parameter estimates fitted from raw data using nonlinear regression, produced robust approximations. Data quality influenced sensitivity and sample size for an optimal precision. Conclusions Allometric surrogates of average leaf biomass in eelgrass shoots offer convenient nondestructive assessments. But analysis method and sample size can influence accuracy in a direct manner. Standardized routines for data quality are crucial on granting cost-effectiveness of the method.

Individual growth of the brown crab (Callinectes bellicosus) is analyzed using 24-month experimental molt data, information theory and Bayes theorem to infer the best model and parameter values and their uncertainties. The von Bertalanffy model fit best experimental data than the general von Bertalanffy and Gompertz models. No evidence was found of different growth for separate sexes; molting does not seem to be related to ambient temperature. An exploration was done to show the effect of using informative vs. uninformative priors with Bayes theorem. Two years of experimental results indicate that the brown crab might not be a good candidate to produce soft crab for commercial purposes as compared to its blue crab congeners.

Conservation of eelgrass relies on transplants and evaluation of success depends on nondestructive measurements of average leaf biomass in shoots among other variables. Allometric proxies offer a convenient way to assessments. Identifying surrogates via log transformation and linear regression can set biased results. Views conceive this approach to be meaningful, asserting that curvature in geometrical space explains bias. Inappropriateness of correction factor of retransformation bias could also explain inconsistencies. Accounting for nonlinearity of the log transformed response relied on a generalized allometric model. Scaling parameters depend continuously on the descriptor. Joining correction factor is conceived as the partial sum of series expansion of mean retransformed residuals leading to highest reproducibility strength. Fits of particular characterizations of the generalized curvature model conveyed outstanding reproducibility of average eelgrass leaf biomass in shoots. Although nonlinear heteroscedastic regression resulted also to be suitable, only log transformation approaches can unmask a size related differentiation in growth form of the leaf. Generally, whenever structure of regression error is undetermined, choosing a suitable form of retransformation correction factor becomes elusive. Compared to customary nonparametric characterizations of this correction factor, present form proved more efficient. We expect that offered generalized allometric model along with proposed correction factor form provides a suitable analytical arrangement for the general settings of allometric examination.

Current genome-enabled prediction models assumed errors normally distributed, which are sensitive to outliers. We propose a model with errors assumed to follow a Laplace distribution to deal better with outliers. Current genome-enabled prediction models use regressions that fit the expected value (mean) of a response variable with errors assumed normally distributed, which are often sensitive to outliers, either genetic or environmental. For this reason, we propose a robust Bayesian genome median regression (BGMR) model that fits regressions to the medians of a distribution, with errors assumed to follow a Laplace distribution to deal better with outliers. The BGMR model was evaluated under a Bayesian framework with Markov Chain Monte Carlo sampling using a location–scale mixture representation of the Laplace distribution. The BGMR was implemented with two simulated and two real genomic data sets, and we compared its prediction performance with that of a conventional genomic best linear unbiased prediction (GBLUP) model and the Laplace maximum a posteriori (LMAP) method. The prediction accuracies of BGMR were higher than those of the GBLUP and LMAP methods when there were outliers. The BGMR model could be useful to breeders who need to predict and select genotypes based on data with unknown outliers.

Background The traditional allometric analysis relies on log- transformation to contemplate linear regression in geometrical space then retransforming to get Huxley’s model of simple allometry. Views assert this induces bias endorsing multi-parameter complex allometry forms and nonlinear regression in arithmetical scales. Defenders of traditional approach deem it necessary since generally organismal growth is essentially multiplicative. Then keeping allometry as originally envisioned by Huxley requires a paradigm of polyphasic loglinear allometry. A Takagi-Sugeno-Kang fuzzy model assembles a mixture of weighted sub models. This allows direct identification of break points for transition between phases. Then, this paradigm is seamlessly appropriate for efficient allometric examination of polyphasic loglinear allometry patterns. Here, we explore its suitability. Methods Present fuzzy model embraces firing strength weights from Gaussian membership functions and linear consequents. Weights are identified by subtractive clustering and consequents through recursive least squares or maximum likelihood. Intersection of firing strength factors set criterion to estimate breakpoints. A multi-parameter complex allometry model follows by adapting firing strengths by composite membership functions and linear consequents in arithmetical space. Results Takagi-Sugeno-Kang surrogates adapted complexity depending on analyzed data set. Retransformation results conveyed reproducibility strength of similar proxies identified in arithmetical space. Breakpoints were straightforwardly identified. Retransformed form implies complex allometry as a generalization of Huxley’s power model involving covariate depending parameters. Huxley reported a breakpoint in the log–log plot of chela mass vs. body mass of fiddler crabs (Uca pugnax), attributed to a sudden change in relative growth of the chela approximately when crabs reach sexual maturity. G.C. Packard implied this breakpoint as putative. However, according to present fuzzy methods existence of a break point in Huxley’s data could be validated. Conclusions Offered scheme bears reliable analysis of zero intercept allometries based on geometrical space protocols. Endorsed affine structure accommodates either polyphasic or simple allometry if whatever turns required. Interpretation of break points characterizing heterogeneity is intuitive. Analysis can be achieved in an interactive way. This could not have been obtained by relying on customary approaches. Besides, identification of break points in arithmetical scale is straightforward. Present Takagi-Sugeno-Kang arrangement offers a way to overcome the controversy between a school considering a log-transformation necessary and their critics claiming that consistent results can be only obtained through complex allometry models fitted by direct nonlinear regression in the original scales.

(1) Background: We previously demonstrated that customary regression protocols for curvature in geometrical space all derive from a generalized model of complex allometry combining scaling parameters expressing as continuous functions of covariate. Results highlighted the relevance of addressing suitable complexity in enhancing the accuracy of allometric surrogates of plant biomass units. Nevertheless, examination was circumscribed to particular characterizations of the generalized model. Here we address the general identification problem. (2) Methods: We first suggest a log-scales protocol composing a mixture of linear models weighted by exponential powers. Alternatively, adopting an operating regime-based modeling slant we offer mixture regression or Takagi&ndash, Sugeno&ndash, Kang arrangements. This last approach allows polyphasic identification in direct scales. A derived index measures the extent on what complexity in arithmetic space drives curvature in arithmetical space. (3) Results: Fits on real and simulated data produced proxies of outstanding reproducibility strength indistinctly of data scales. (4) Conclusions: Presented analytical constructs are expected to grant efficient allometric projection of plant biomass units and also for the general settings of allometric examination. A traditional perspective deems log-transformation and allometry inseparable. Recent views assert that this leads to biased results. The present examination suggests this controversy can be resolved by addressing adequately the complexity of geometrical space protocols

This paper presents the use of graphical models and copula functions in Estimation of Distribution Algorithms (EDAs) for solving multivariate optimization problems. It is shown in this work how the incorporation of copula functions and graphical models for modeling the dependencies among variables provides some theoretical advantages over traditional EDAs. By means of copula functions and two well known graphical models, this paper presents a novel approach for defining new EDAs. Either dependence is modeled by a copula function chosen from a predefined set of six functions that aim to cover a wide range of inter-relations. It is also shown how the use of mutual information in the learning of graphical models implies a natural way of employing copula entropies. The experimental results on separable and non-separable functions show that the two new EDAs, which adopt copula functions to model dependencies, perform better than their original version with Gaussian variables.

The article Improved allometric proxies for eelgrass conservation, written by A. Montesinos-Lopez, E. Villa-Diharce, H. Echavarria-Heras, and C. Leal-Ramirez, was originally published electronically on the publisher’s internet portal (currently SpringerLink) on July 30, 2018 without open access.

Background Eelgrass grants important ecological benefits including a nursery for waterfowl and fish species, shoreline stabilization, nutrient recycling and carbon sequestration. Upon the exacerbation of deleterious anthropogenic influences, re-establishment of eelgrass beds has mainly depended on transplantation. Productivity estimations provide valuable information for the appraisal of the restoration of ecological functions of natural populations. Assessments over early stages of transplants should preferably be nondestructive. Allometric scaling of eelgrass leaf biomass in terms of matching length provides a proxy that reduces leaf biomass and productivity estimations to simple measurements of leaf length and its elongation over a period. We examine how parameter variability impacts the accuracy of the considered proxy and the extent on what data quality and sample size influence the uncertainties of the involved allometric parameters. Methods We adapted a Median Absolute Deviation data quality control procedure to remove inconsistencies in the crude data. For evaluating the effect of parametric uncertainty we performed both a formal exploration and an analysis of the sensitivity of the allometric projection method to parameter changes. We used parameter estimates obtained by means of nonlinear regression from crude as well as processed data. Results We obtained reference leaf growth rates by allometric projection using parameter estimates produced by the crude data, and then considered changes in fitted parameters bounded by the modulus of the vector of the linked standard errors, we found absolute deviations up to 10 % of reference values. After data quality control, the equivalent maximum deviation was under 7 % of corresponding reference rates. Therefore, the addressed allometric method is robust. Even the smaller sized samples in the quality controlled dataset produced better accuracy levels than the whole set of crude data. Conclusions We propose quality control of data as a highly recommended step in the overall procedure that leads to reliable allometric surrogates of eelgrass leaf growth rates. The proliferation of inconsistent replicates in the crude data points towards the importance of discarding incomplete leaves. We also recommend avoiding errors in estimating the biomass of small leaves for which precision of the used analytical scale might be an issue.

Background Eelgrass is a cosmopolitan seagrass species that provides important ecological services in coastal and near-shore environments. Despite its relevance, loss of eelgrass habitats is noted worldwide. Restoration by replanting plays an important role, and accurate measurements of the standing crop and productivity of transplants are important for evaluating restoration of the ecological functions of natural populations. Traditional assessments are destructive, and although they do not harm natural populations, in transplants the destruction of shoots might cause undesirable alterations. Non-destructive assessments of the aforementioned variables are obtained through allometric proxies expressed in terms of measurements of the lengths or areas of leaves. Digital imagery could produce measurements of leaf attributes without the removal of shoots, but sediment attachments, damage infringed by drag forces or humidity contents induce noise-effects, reducing precision. Available techniques for dealing with noise caused by humidity contents on leaves use the concepts of adjacency, vicinity, connectivity and tolerance of similarity between pixels. Selection of an interval of tolerance of similarity for efficient measurements requires extended computational routines with tied statistical inferences making concomitant tasks complicated and time consuming. The present approach proposes a simplified and cost-effective alternative, and also a general tool aimed to deal with any sort of noise modifying eelgrass leaves images. Moreover, this selection criterion relies only on a single statistics; the calculation of the maximum value of the Concordance Correlation Coefficient for reproducibility of observed areas of leaves through proxies obtained from digital images. Results Available data reveals that the present method delivers simplified, consistent estimations of areas of eelgrass leaves taken from noisy digital images. Moreover, the proposed procedure is robust because both the optimal interval of tolerance of similarity and the reproducibility of observed leaf areas through digital image surrogates were independent of sample size. Conclusion The present method provides simplified, unbiased and non-destructive measurements of eelgrass leaf area. These measurements, in conjunction with allometric methods, can predict the dynamics of eelgrass biomass and leaf growth through indirect techniques, reducing the destructive effect of sampling, fundamental to the evaluation of eelgrass restoration projects thereby contributing to the conservation of this important seagrass species.

This chapter presents the incorporation and use of regular vines into Estimation of Distribution Algorithms for solving numerical optimization problems. Several kinds of statistical dependencies among continuous variables can be taken into account by using regular vines. This work presents a procedure for selecting the most important dependencies in EDAs by truncating regular vines. Moreover, this chapter also shows how the use of mutual information in the learning of graphical models implies a natural way of employing copula functions.

The main objective of this doctoral research is to study Estimation of Distribution Algorithms (EDAs) based on copula functions. This new class of EDAs has shown that it is possible to incorporate successfully copula functions in EDAs.

In this paper, a new Estimation of Distribution Algorithm (EDA) is presented. The proposed algorithm employs a dependency tree as a graphical model and bivariate copula functions for modeling relationships between pairwise variables. By selecting copula functions it is possible to build a very flexible joint distribution as a probabilistic model. The experimental results show that the proposed algorithm has a better performance than EDAs based on Gaussian assumptions.

This paper explains how to construct Gaussian polytrees and their application to estimation of distribution algorithms in continuous variables.

A new Estimation of Distribution Algorithm is presented. The proposed algorithm, called D-vine EDA, uses a graphical model which is based on pair copula decomposition. By means of copula functions it is possible to model the dependence structure in a joint distribution with marginals of different type. Thus, this paper introduces the D-vine EDA and performs experiments and statistical tests to assess the best algorithm. The set of experiments shows the potential of the D-vine EDA

This chapter introduces copula functions and the use of the Gaussian copula function to model probabilistic dependencies in supervised classification tasks. A copula is a distribution function with the implicit capacity to model non linear dependencies via concordance measures, such as Kendall’s τ. Hence, this chapter studies the performance of a simple probabilistic classifier based on the Gaussian copula function. Without additional preprocessing of the source data, a supervised pixel classifier is tested with a 50-images benchmark; the experiments show this simple classifier has an excellent performance.

This paper introduces copula functions and the use of the Gaussian copula function to model probabilistic dependencies in supervised classification tasks. A copula is a distribution function with the implicit capacity to model non linear dependencies via concordance measures, such as Kendall's τ. Hence, this work studies the performance of a simple probabilistic classifier based on the Gaussian copula function. Without additional preprocessing of the source data, a supervised pixel classifier is tested with a 50-images benchmark; the experiments show this simple classifier has an excellent performance.

The Vehicle Routing Problem with Time Windows (VRPTW), is an extension to the standard vehicle routing problem. VRPTW includes an additional constraint that restricts every customer to be served within a given time window. An approach for the VRPTW with the next three objectives is presented: 1)total distance (or time), 2)total waiting time, 3)number of vehicles. A data mining strategy, namely space partitioning, is adopted in this work. Optimal routes are extracted as features hidden in variable size regions where depots and customers are located. This chapter proposes the sector model for partitioning the space into regions. A new hybrid Particle Swarm Optimization algorithm (PSO), and combinatorial operators ad-hoc with space partitioning are described. A set of well-known benchmark functions in VRPTW are used to compare the effectiveness of the proposed method. The results show the importance of examining characteristics of a set of non-dominated solutions, that fairly consider the three dimensions, when a user should select only one solution according to problem conditions.

A new way of modeling probabilistic dependencies in Estimation of Distribution Algorithm (EDAs) is presented. By means of copulas it is possible to separate the structure of dependence from marginal distributions in a joint distribution. The use of copulas as a mechanism for modeling joint distributions and its application to EDAs is illustrated on several benchmark examples.

This article arguments that rank correlation coefficients are powerful association measures and how can they be adopted by EDAs. A new EDA implements the proposed ideas: the Non-Parametric Real-valued Estimation Distribution Algorithm (NOPREDA). The paper fully describes the rank correlation coefficient, and the procedure to build a non parametric model for the probability distribution of the source data. A benchmark of global optimization problems is solved with NOPREDA.