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After stressing the need to keep separated the concept of variability and/or inequality and dissimilarity from that of diversity, it is suggested that diversity of a system should be measured primarily by the number of different classes (K) we can define in it (richness) by classification or identification processes. An index d, ranging between 0 and 1, that summarizes the similarity pattern within the system, can be used if necessary to transform K to a “fuzzy” diversity number, according to the idea that the higher is the similarity within the system the lower should be its diversity. Another index, r, is proposed to measure the “loss” of diversity due to similarity within the system, an index that fits the concept of “redundancy”. Since every diversity vector may be interpreted as a crisp symmetric similarity matrix, of which the Gini-Simpson’s index is the average dissimilarity, while the index of Shannon is the entropy of its eigenvalues, the index d can be chosen to quantify one among the following similarities: a) the overall average similarity of the classes considering the within classes similarity equal to 1 and the between classes similarity equal to 0 (crisp similarity pattern): this is coincident with the evenness of the proportion of importance of the classes, b) the average similarity between the classes without considering evenness, or c) the combination of the two similarities (similarity between the classes and evenness). In these last two cases, the similarity between the classes is characterizing the similarity pattern of a system in a fuzzy way (fuzzy diversity). It is stressed that the diversity of vegetation systems may be of two complementary types: plant individual-based diversity and plant community-based diversity. If we assume that each plant community type corresponds to one habitat then habitat diversity (or niche width) can be calculated for each class of plant individuals according to the number of classes of plant communities in which we can find it. Habitat diversity can be used to measure the indicator value of species or other classes of plant individuals and of plant communities. In this last case, we have to consider the distribution of plant communities in classes defined by environmental factors. It is suggested that the terminology alpha, beta, gamma diversity can be useful only if used to distinguish types of diversity in vegetation systems: alpha diversity = plant individual based diversity, gamma diversity = the union of alpha diversities, beta diversity = plant community based diversity. Thanks to the availability of mathematical tools, it is concluded that rather than being worried about measuring diversity it would be more fruitful to worry about why we are willing to measure it.

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With this paper we want to stress that, on the basis of some matrix algebraic theorems, eigenvectors of similarity matrices are strictly related with clusters that we can obtain with clustering procedures applied to the same similarity matrices and that the fuzzy sets obtained by cluster analysis can be efficiently used as ordination axes and also as tools to measure the diagnostic value (or the indicator value) of attributes (species or other characters) of the ecological systems.

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It is shown that a similarity theory (ST) formulated in the context of plant community research can lead to methodological developments based on similarity functions. ST can explain and predict ecosystem states, discover links between the physical-chemical environment and the plant communities at different scales of generalization. ST is compared with fuzzy systems theory, which supports similar developments, and it is concluded that fuzzy set theory could be considered as an extension of similarity theory.

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We propose a simple method to validate vegetation types in phytosociological tables obtained with the Braun-Blanquet approach. The method is based on fuzzy set theory and it measures the sharpness of a classification of relevés based on degrees of belonging calculated by similarity functions. The idea animating this work is that the validated vegetation types offer non-random species combinations that can be used to model environmental changes taking biodiversity into account. Phytosociological tables are widely available today and the information they contain can be very useful in studying environmental changes at different scales, from local to global.

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A method for studying the response of vegetation to environmental gradients, based on the community niche and fuzzy set theory, is presented. The approach is illustrated using an example from perennial halophilous vegetation along the Northern Adriatic coast of Italy. Compatibility curves are obtained by fuzzy set theoretical methods, and are used tomodel the response functions of plant associations to environmental gradients, including soil and ground water salinity, soil pH, soil and ground water temperature, percentage of sand, and variations in the ground water level. The compatibility curves summarize the similarity of a given plant community, with a particular value of an environmental variable, to the species combination of a given plant association. Compatibility curves offer an alternative approach to non-linear regression and best fit analyses normally used to model single species responses to environmental gradients. The approach is particularly useful given there is no singlemechanisticmodel that can capture the exact shape of the functional response along environmental gradients, and given that environmental data are commonly affected by high levels of noise.

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We propose a method that has general relevance to the digital representation of spatial variation of multivariate landscape data. It is based on the average similarity that operational geographic units (OGU) have with the adjacent ones according to characters relevant understanding landscape patterns and dynamics. The method is flexible and easily executable within the technological framework of geographic information systems (GIS) that today is available even free of charge or at very low cost. An example shows how the method, applied to spatial data of a floristic project for the urban area of Trieste (NE-Italy), can identify floristically homogeneous patches and can quantify the heterogeneity of the transition zones between such patches.

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The 2013 reform of the European Common Agricultural Policy tries to support farmers willing to follow environment- friendly rural practices, by the so called “green payment”. Within this framework, it is suggested that governments and regions should maintain a certain ratio of the area of permanent grasslands to the total Utilized Agricultural Area according to the greening rules of the reform. However, the weak economic performance of permanent grasslands does not encourage farmers to invest into their conservation. This fact persuaded us to revisit our old unpublished data, obtained by experiments on the use of chemical fertilizers in permanent grasslands. By this reanalysis we hope to further support the new European policy with the perspective to find a trade-off between the conservation of the biodiversity and the economic productivity of permanent grasslands. Of the many possibilities we have chosen to present the results of two experiments, one in Italy and the other in Slovakia. The main reason for this choice was that these two studies followed complementary strategies of fertilization that appeared useful to detect both the single and the synergistic effects of N, P, and K on the relationships between yield and species diversity. The results of cluster and diversity analysis suggest that chemical fertilization should be carefully planned according to soil conditions, since different treatments may have the same effect on the floristic and vegetation patterns of grasslands. These results, according to similarity theory, allow to choose the least expensive and polluting combination of N, P and K from those that, according to the species combinations, are assigned to the same cluster.

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Community Ecology
Authors: J. Podani, L. Orlóci, E. Feoli, and L. Mucina
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We suggest a classificatory approach for land cover analysis that integrates fuzzy set theory with permutation techniques. It represents a non parametric alternative and/or a complement of traditional multivariate statistics when data are scarce, missing, burdened with high degree of uncertainty and originated from different sources and/or times. According to this approach, the Operational Geographic Units (OGUs) in which landscape is subdivided and sampled are classified with hierarchical clustering methods. The clusters of a classification which are significantly sharp are used to define fuzzy sets. In this way, the original data scores are transformed by degrees of belonging. We introduce the concepts of endogenous and exogenous fuzzy sets and we suggest to apply the Mantel test between the similarity matrices of these fuzzy sets to test the predictivity of internal variables with respect to external variables. The approach is applied to OGUs corresponding to the smallest administrative units (kebeles) of the Ethiopian Rift Valley, a degrading area with high risk of further degradation. We found that: 1) there is a high correlation between geo-physical features of the landscape (geology, rainfall and elevation) and some indicators of the human pressure such as land use/cover, land management for livestock breeding and human, household and livestock densities, 2) there is a high correlation between land degradation, measured with relative loss of Normalized Difference Vegetation Index (NDVI) and the human pressure. However, the correlation is higher when the human pressure is considered in the geo-physical context of the landscape. The approach can be easily applied to produce maps useful for planning purposes thanks to geographical information system (GIS) technology that is becoming available at low cost even to small administrative units of developing countries.

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Abstract

In this paper, we want to support the idea of using a family of indices of similarity, that we call the Simpson's family indices or nestedness-based similarity functions (NBSF) for comparing operational geographic units (OGUs) (phytosociological relevés, animal traps, watersheds, administrative units, industrial areas, islands etc.). In these cases, similarity-dissimilarity depends, in addition to factors that induce replacement, also on factors that produce reduction or increment in the number of features within the same typology of OGUs (e.g., extent, reduction of fertility, anthropogenic pressure etc.). To keep into consideration this aspect, the indices are defined to be equal to 1 when the OGUs are completely nested. The results of the application to four simulated data sets prove that, when the data set does not show clear nested pattern, the use of NBSF produces results similar to the nestedness-free similarity functions, however since NBSF clearly detect nested situations, we should prefer their use in the circumstances where we think important to put in evidence nestedness. In conclusion, we support the idea of using both types of indices in order to improve the knowledge about the structure of any data set.

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