A fuzzy logic controller.

This paper describes a fuzzy sets method which is very useful for handling uncertainties and essential for knowledge acquisition of a human expert. Kinetics of a reactor is often complex and not trivial to describe by mathematical equations. Reactor control by traditional control technology is therefore difficult. A novel technology is presented. In the following a fuzzy inference (approximate reasoning) is used for decision making in analogy to human thinking, facilitating a more sophisticated control. Readers of this paper do not need any advanced mathematics beyond the four basic operations in arithmetic (+, -, x, divided by) and using the maximum and minimum values. This fuzzy inference is introduced to construct a fuzzy logic controller which is suitable for a nonlinear, multivariable and time variant system applied to a bioreactor.     

Niche width and niche overlap: a method based on type-2 fuzzy sets

Complicated ecosystems and the high non-linearity of evolution has made biology more adaptable to a variety of environments. The relationship between life and environment demands a dynamic definition of niche and its measurement. In this paper we propose a model of niche with dynamic character based on the “broad band” effect in type-2 fuzzy sets. The niche in this definition is an interval in each ecological dimension which is dynamic in character and depends on the actual environment. We also give formulas for niche width and niche overlap. We compute the niche width and overlap for plants and animals and then compare these results with previous results. The results for niche width in this paper reflect the diversity of resources used by species or communities. The results for niche overlap demonstrate overlap under different environmental conditions. The results are, moreover, intervals, which could provide more information. The model in this paper could therefore be used to describe the state of every resource comprehensively, reflecting the interaction between species and environment.     

Rigorous proof of fuzzy error propagation with matrix-based LCI

Background, aim, and scope  

Propagation of parametric uncertainty in life cycle inventory (LCI) models is usually performed based on probabilistic Monte Carlo techniques. However, alternative approaches using interval or fuzzy numbers have been proposed based on the argument that these provide a better reflection of epistemological uncertainties inherent in some process data. Recent progress has been made to integrate fuzzy arithmetic into matrix-based LCI using decomposition into α-cut intervals. However, the proposed technique implicitly assumes that the lower bounds of the technology matrix elements give the highest inventory results, and vice versa, without providing rigorous proof.     

Unanticipated consequences of logarithmic transformation in bivariate allometry

Parameters in the two-parameter allometric equation are commonly estimated by fitting a straight line to logarithmic transformations of the original data and by back-transforming the resulting model to the arithmetic scale. However, log transformation distorts the relationship between the predictor and response variables, and this distortion may be sufficient to lead unsuspecting investigators to analyze data that actually are unsuited for allometric research. Two data sets from the current literature are re-examined here to illustrate instances in which log transformation caused ugly data to look deceptively good. One of the investigations focused on the scaling of metabolism to body mass in evolutionary transitions from prokaryotic to protistan to metazoan levels of organization whereas the other addressed the scaling of intestines to body size in rodents. In both instances investigators were led to conclusions that are not supported by the original data. Problems of the sort described here can readily be avoided simply by performing preliminary graphical analysis of observations expressed in the original units and by validating the final model in the arithmetic domain.     

Growing of a Fuzzy Recurrent Artificial Neural Network (FRANN) for pattern classification

This paper describes a method for growing a recurrent neural network of fuzzy threshold units for the classification of feature vectors. Fuzzy networks seem natural for performing classification, since classification is concerned with set membership and objects generally belonging to sets of various degrees. A fuzzy unit in the architecture proposed here determines the degree to which the input vector lies in the fuzzy set associated with the fuzzy unit. This is in contrast to perceptrons that determine the correlation between input vector and a weighting vector. The resulting membership value, in the case of the fuzzy unit, is compared with a threshold, which is interpreted as a membership value. Training of a fuzzy unit is based on an algorithm for linear inequalities similar to Ho-Kashyap recording. These fuzzy threshold units are fully connected in a recurrent network. The network grows as it is trained. The advantages of the network and its training method are: (1) Allowing the network to grow to the required size which is generally much smaller than the size of the network which would be obtained otherwise, implying better generalization, smaller storage requirements and fewer calculations during classification; (2) The training time is extremely short; (3) Recurrent networks such as this one are generally readily implemented in hardware; (4) Classification accuracy obtained on several standard data sets is better than that obtained by the majority of other standard methods; and (5) The use of fuzzy logic is very intuitive since class membership is generally fuzzy.     

Ordination on the basis of fuzzy set theory

Fuzzy set theory is an extension of classical set theory where elements of a set have grades of membership ranging from zero for non-membership to one for full membership. Exactly as for classical sets, there exist operators, relations, and mappings appropriate for these fuzzy sets. This paper presents the concepts of fuzzy sets, operations, relations, and mappings in an ecological context. Fuzzy set theory is then established as a theoretical basis for ordination, and is employed in a sequence of examples in an analysis of forest vegetation of western Montana, U.S.A. The example ordinations show how site characteristics can be analyzed for their effect on vegetation composition, and how different site factors can be synthesized into complex environmental factors using the calculus of fuzzy set theory.In contrast to current ordination methods, ordinations based on fuzzy set theory require the investigator to hypothesize an ecological relationship between vegetation and environment, or between different vegatation compositions, before constructing the ordination. The plotted ordination is then viewed as evidence to corroborate or discredit the hypothesis.I am grateful to Dr R. D. Pfister (formerly USDA Forest Service) for kind permission to publish data from a Forest Service study.I would like to gratefully acknowledge the helpful comments and criticisms of Drs. G. Cottam, J. D. Aber, T. F. H. Allen, E. W. Beals, I. C. Prentice, C. G. Lorimer, and two anonymous reviewers.Taxonomic nomenclature follows Hitchcock & Cronquist (1973).I would like to thank the Dean of the College of Letters and Sciences, University of Wisconsin—Madison, for a fellowship which supported this research, and the Department of Botany for computer funds to perform the analyses.     

APPLICATION OF FUZZY SETS AND NEURAL NETWORKS IN SENSORY ANALYSIS

This paper describes an initial but fundamental attempt to lay some groundwork for a fuzzy-set-based paradigm for sensory analysis and to demonstrate how fuzzy set and neural network techniques may lead to a natural way for sensory data interpretation. Sensory scales are described as fuzzy sets, sensory attributes as fuzzy variables, and sensory responses as sample membership grades. Multi-judge responses are formulated as a fuzzy membership vector or fuzzy histogram of response, which gives an overall panel response free of the unverifiable assumptions implied in conventional approaches. Neural networks are used to provide an effective tool for modeling and analysis of sensory responses in their naturally fuzzy and complex forms. A maximum method of defuzzification is proposed to give a crisp grade of the majority opinion. Two applications in meat quality evaluation are used to demonstrate the use of the paradigm and procedure. It is hoped that this work will bring up some new ideas and generate interest in research on application of fuzzy sets and neural networks in sensory analysis.     

Basic math in monkeys and college students

Adult humans possess a sophisticated repertoire of mathematical faculties. Many of these capacities are rooted in symbolic language and are therefore unlikely to be shared with nonhuman animals. However, a subset of these skills is shared with other animals, and this set is considered a cognitive vestige of our common evolutionary history. Current evidence indicates that humans and nonhuman animals share a core set of abilities for representing and comparing approximate numerosities nonverbally; however, it remains unclear whether nonhuman animals can perform approximate mental arithmetic. Here we show that monkeys can mentally add the numerical values of two sets of objects and choose a visual array that roughly corresponds to the arithmetic sum of these two sets. Furthermore, monkeys' performance during these calculations adheres to the same pattern as humans tested on the same nonverbal addition task. Our data demonstrate that nonverbal arithmetic is not unique to humans but is instead part of an evolutionarily primitive system for mathematical thinking shared by monkeys.     

Basic Math in Monkeys and College Students

Adult humans possess a sophisticated repertoire of mathematical faculties. Many of these capacities are rooted in symbolic language and are therefore unlikely to be shared with nonhuman animals. However, a subset of these skills is shared with other animals, and this set is considered a cognitive vestige of our common evolutionary history. Current evidence indicates that humans and nonhuman animals share a core set of abilities for representing and comparing approximate numerosities nonverbally; however, it remains unclear whether nonhuman animals can perform approximate mental arithmetic. Here we show that monkeys can mentally add the numerical values of two sets of objects and choose a visual array that roughly corresponds to the arithmetic sum of these two sets. Furthermore, monkeys' performance during these calculations adheres to the same pattern as humans tested on the same nonverbal addition task. Our data demonstrate that nonverbal arithmetic is not unique to humans but is instead part of an evolutionarily primitive system for mathematical thinking shared by monkeys.     

Fuzzy Stochastic Petri Nets for Modeling Biological Systems with Uncertain Kinetic Parameters

Stochastic Petri nets (SPNs) have been widely used to model randomness which is an inherent feature of biological systems. However, for many biological systems, some kinetic parameters may be uncertain due to incomplete, vague or missing kinetic data (often called fuzzy uncertainty), or naturally vary, e.g., between different individuals, experimental conditions, etc. (often called variability), which has prevented a wider application of SPNs that require accurate parameters. Considering the strength of fuzzy sets to deal with uncertain information, we apply a specific type of stochastic Petri nets, fuzzy stochastic Petri nets (FSPNs), to model and analyze biological systems with uncertain kinetic parameters. FSPNs combine SPNs and fuzzy sets, thereby taking into account both randomness and fuzziness of biological systems. For a biological system, SPNs model the randomness, while fuzzy sets model kinetic parameters with fuzzy uncertainty or variability by associating each parameter with a fuzzy number instead of a crisp real value. We introduce a simulation-based analysis method for FSPNs to explore the uncertainties of outputs resulting from the uncertainties associated with input parameters, which works equally well for bounded and unbounded models. We illustrate our approach using a yeast polarization model having an infinite state space, which shows the appropriateness of FSPNs in combination with simulation-based analysis for modeling and analyzing biological systems with uncertain information.     

基于CVM的海涂湿地生态服务价值的模糊评估模型

正确评估海涂湿地生态服务价值有助于加强人们保护海涂湿地的意识,为海涂湿地围垦生态补偿标准的制定提供依据。视价格"便宜"、"适中"和"昂贵"为模糊集,基于CVM法区间型数据建立了传统模糊统计模型、赋权模糊统计模型和三相划分模型,并以此评价了杭州湾国家湿地公园单位面积年生态系统服务价值。结果显示:3种方法得到的单价分别为10.28、10.38、9.76元m~(-2)a~(-1),三者一致程度较高,与国内沿海湿地价值比较接近。分析了"生态价值"概念客观上的"模糊性"引致模糊数学模型的合理性;采用"橄榄球"式赋权建立赋权模糊统计模型以克服传统模糊统计模型中对区间数据均匀赋权的不合理性,结果在隶属度函数的光滑性和拟合度上好于后者;第三,三相划分模型拟合出3个模糊集的隶属度函数,可以比较相同价格在不同模糊集中隶属度差异。建立的模型对于基于CVM法的生态资源价值评估具有借鉴意义。     

Evaluations of corporate sustainability indicators based on fuzzy similarity graphs

The paper deals with inconsistencies of composite sustainability indicators and their different subsets (economic, environmental, social, and corporate governance). Corporate sustainability performance is usually highly nonlinear, vague, partially inconsistent and multidimensional. The resulting models are often oversimplified. The key reason is an information shortage which eliminates the unsophisticated applications of classical statistical methods. Numbers are accurate and information intensive. Verbal quantifications are less accurate and therefore not that information intensive. Fuzzy sets and fuzzy reasoning are used to make verbal quantifiers suitable for computer applications. A fuzzy similarity graph is defined. A team of experts identified 17 relevant variables (e.g. Environmental costs, Occupational diseases, Number of complaints received from stakeholders) and 12 company data sets are available. Each company is presented as a fuzzy conditional statement. A set of fuzzy pairwise similarities is generated and used to evaluate five similarity graphs: a Total Graph (based on all 17 variables) and graphs based on relevant specific subsets of variables, Economic, Environmental, Social and Corporate Governance graphs. The topologies of these graphs are significantly different. No prior knowledge of fuzzy reasoning is required.     

Measurable prediction for the single patient and the results of large double blind controlled randomized trials

BACKGROUND: It has been shown that the clinical state of one patient can be represented by known measured variables of interest, each of which then form the element of a fuzzy set as point in the unit hypercube. We hypothesized that precise comparison of a single patient with the average patient of a large double blind controlled randomized study is possible using fuzzy theory. METHODS/PRINCIPLE FINDINGS: The sets as points unit hypercube geometry allows fuzzy subsethood to define in measures of fuzzy cardinality different conditions, similarity and comparison between fuzzy sets. A fuzzy measure of prediction is defined from fuzzy measures of similarity and comparison. It is a measure of the degree to which fuzzy set A is similar to fuzzy set B when different conditions are taken into account and removed from the comparison. When represented as a fuzzy set as point in the unit hypercube, a clinical patient can be compared to an average patient of a large group study in a precise manner. This comparison is expressed by the fuzzy prediction measure. This measure in itself is not a probability. Once thus precisely matched to the average patient of a large group study, risk reduction is calculated by multiplying the measured similarity of the clinical patient to the risk of the average trial patient. CONCLUSION/SIGNIFICANCE: Otherwise not precisely translatable to the single case, the result of group statistics can be applied to the single case through the use of fuzzy subsethood and measured in fuzzy cardinality. This measure is an alternative to a Bayesian or other probability based statistical approach.     

Estimating parameters for generalized mass action models using constraint propagation

As modern molecular biology moves towards the analysis of biological systems as opposed to their individual components, the need for appropriate mathematical and computational techniques for understanding the dynamics and structure of such systems is becoming more pressing. For example, the modeling of biochemical systems using ordinary differential equations (ODEs) based on high-throughput, time-dense profiles is becoming more common-place, which is necessitating the development of improved techniques to estimate model parameters from such data. Due to the high dimensionality of this estimation problem, straight-forward optimization strategies rarely produce correct parameter values, and hence current methods tend to utilize genetic/evolutionary algorithms to perform non-linear parameter fitting. Here, we describe a completely deterministic approach, which is based on interval analysis. This allows us to examine entire sets of parameters, and thus to exhaust the global search within a finite number of steps. In particular, we show how our method may be applied to a generic class of ODEs used for modeling biochemical systems called Generalized Mass Action Models (GMAs). In addition, we show that for GMAs our method is amenable to the technique in interval arithmetic called constraint propagation, which allows great improvement of its efficiency. To illustrate the applicability of our method we apply it to some networks of biochemical reactions appearing in the literature, showing in particular that, in addition to estimating system parameters in the absence of noise, our method may also be used to recover the topology of these networks.     

A brief review: factors affecting the length of the rest interval between resistance exercise sets

Research has indicated that multiple sets are superior to single sets for maximal strength development. However, whether maximal strength gains are achieved may depend on the ability to sustain a consistent number of repetitions over consecutive sets. A key factor that determines the ability to sustain repetitions is the length of rest interval between sets. The length of the rest interval is commonly prescribed based on the training goal, but may vary based on several other factors. The purpose of this review was to discuss these factors in the context of different training goals. When training for muscular strength, the magnitude of the load lifted is a key determinant of the rest interval prescribed between sets. For loads less than 90% of 1 repetition maximum, 3-5 minutes rest between sets allows for greater strength increases through the maintenance of training intensity. However, when testing for maximal strength, 1-2 minutes rest between sets might be sufficient between repeated attempts. When training for muscular power, a minimum of 3 minutes rest should be prescribed between sets of repeated maximal effort movements (e.g., plyometric jumps). When training for muscular hypertrophy, consecutive sets should be performed prior to when full recovery has taken place. Shorter rest intervals of 30-60 seconds between sets have been associated with higher acute increases in growth hormone, which may contribute to the hypertrophic effect. When training for muscular endurance, an ideal strategy might be to perform resistance exercises in a circuit, with shorter rest intervals (e.g., 30 seconds) between exercises that involve dissimilar muscle groups, and longer rest intervals (e.g., 3 minutes) between exercises that involve similar muscle groups. In summary, the length of the rest interval between sets is only 1 component of a resistance exercise program directed toward different training goals. Prescribing the appropriate rest interval does not ensure a desired outcome if other components such as intensity and volume are not prescribed appropriately.     

秋茄幼苗的形态特征及其生物量

对秋茄一年生幼苗的主要形态指标及其生物量进行了测定。结果表明,株高,茎高,基径,根高,根长,叶面积以及胚,胚轴,茎,叶和全株生物量等计量指标,呈现算术平均值〉中值〉众值组段中点的规律,符合正偏态分布。     

Partitioning an elevation gradient of vegetation from southeastern Ethiopia by probabilistic methods

Vegetation and environment have been analyzed along an altitudinal gradient in Harena Forest, Bale Mountains National Park, southeastern Ethiopia. Vegetation data include numbers of each tree and shrub species and cover-abundance values of each herbaceous species. Environmental data comprise edaphic factors, altitude and topography. The two vegetation layers data were analysed separately.Probabilistic similarity coefficients were computed between the relevés, and these values were used in subsequent computations for classification and ordination. Two sets of stratocoena, comprising 6 types each, derived on the basis of separate analyses of tree-shrub and herb layers of the forest were recognised. A combination of the two sets of stratocoena produced a total of 11 vegetation types. Environmental fuzzy set analysis was applied to determine the strength of the relationship of the relevés to the environmental factors. Autocorrelation analysis was applied to the eigenvectors of probabilistic similarity matrices and environmental data. Altitude appears to be more important thant the other environmental factors in controlling the zonation of the forest. Other important environmental influences on the vegetation include pH, organic matter content and texture of the soil. It is suggested that the whole forest be included in the National Park to create suitable conditions for adequate protection.Abbreviations EFS Environmental fuzzy sets - PROSIM Probability similarity index - PCA Principal components analysis     

Using fuzzy numbers to propagate uncertainty in matrix-based LCI

Background, aim, and scope  Analysis of uncertainties plays a vital role in the interpretation of life cycle assessment findings. Some of these uncertainties arise from parametric data variability in life cycle inventory analysis. For instance, the efficiencies of manufacturing processes may vary among different industrial sites or geographic regions; or, in the case of new and unproven technologies, it is possible that prospective performance levels can only be estimated. Although such data variability is usually treated using a probabilistic framework, some recent work on the use of fuzzy sets or possibility theory has appeared in the literature. The latter school of thought is based on the notion that not all data variability can be properly described in terms of frequency of occurrence. In many cases, it is necessary to model the uncertainty associated with the subjective degree of plausibility of parameter values. Fuzzy set theory is appropriate for such uncertainties. However, the computations required for handling fuzzy quantities has not been fully integrated with the formal matrix-based life cycle inventory analysis (LCI) described by Heijungs and Suh (2002). Materials and methods  This paper integrates computations with fuzzy numbers into the matrix-based LCI computational model described in the literature. The approach uses fuzzy numbers to propagate the data variability in LCI calculations, and results in fuzzy distributions of the inventory results. The approach is developed based on similarities with the fuzzy economic input–output (EIO) model proposed by Buckley (Eur J Oper Res 39:54–60, 1989). Results  The matrix-based fuzzy LCI model is illustrated using three simple case studies. The first case shows how fuzzy inventory results arise in simple systems with variability in industrial efficiency and emissions data. The second case study illustrates how the model applies for life cycle systems with co-products, and thus requires the inclusion of displaced processes. The third case study demonstrates the use of the method in the context of comparing different carbon sequestration technologies. Discussion  These simple case studies illustrate the important features of the model, including possible computational issues that can arise with larger and more complex life cycle systems. Conclusions  A fuzzy matrix-based LCI model has been proposed. The model extends the conventional matrix-based LCI model to allow for computations with parametric data variability represented as fuzzy numbers. This approach is an alternative or complementary approach to interval analysis, probabilistic or Monte Carlo techniques. Recommendations and perspectives  Potential further work in this area includes extension of the fuzzy model to EIO-LCA models and to life cycle impact assessment (LCIA); development of hybrid fuzzy-probabilistic approaches; and integration with life cycle-based optimization or decision analysis. Additional theoretical work is needed for modeling correlations of the variability of parameters using interacting or correlated fuzzy numbers, which remains an unresolved computational issue. Furthermore, integration of the fuzzy model into LCA software can also be investigated.     

Fuzzy chorotypes as a conceptual tool to improve insight into biogeographic patterns

Chorotypes--statistically significant groups of coincident distribution areas--constitute biogeographic units that are fuzzy by nature. This quality has been referred to in the literature but has not been analyzed in depth or methodologically developed. The present work redefines chorotypes as fuzzy sets from a pragmatic perspective and basically focuses on the methodological and interpretative implications of this approach. The amphibian fauna in the Iberian Peninsula was used as an example to explore the fuzzy nature of chorotypes. The method on which this article is based is a widely used technique to define chorotypes. This method involves the fuzziness that is inherent to the identification between degree of similarity and degree of membership and includes a probabilistic analysis of the classification for the objective delimitation of chorotypes. The main innovation of this paper is a procedure to analyze chorotypes as fuzzy biogeographic units. A set of fuzzy parameters to deal with the biogeographic interpretation of fuzzy chorotypes is also described. A computer program has been developed and is freely available. History may be related to the degree of fuzziness of chorotypes. In our example, with amphibian distributions in Iberia, less fuzzy chorotypes could have a historical explanation, and the internal fuzziness of chorotypes increases with their distance to hypothetical Pleistocene refugia.