Generalized Mahalanobis Distance Research Articles

Future-ready crops, genetic variability in lima bean seeds

Lima bean cultivation has not received proper national and regional attention by research agencies, impairing the knowledge of the agronomic characteristics and its potential, information that is essential for the selection and launching of superior genotypes. The objective of this work was to characterize the seeds of 17 lima bean genotypes produced by family farmers in Paraíba State. The characterization was based on the descriptors for Phaseolus lunatus. A completely randomized design was used with 15 replicates. Data were submitted to analysis of variance with a subsequent grouping of means by the Scott-Knott test at 1% of probability. For analysis of the genetic diversity, the Tocher method was used based on the generalized Mahalanobis distance. There is variability among the genotypes. According to the means test, genotype 10 has a higher seed length, width, and weight. Considering the diversity among the groups and aiming to select light-colored and higher means of the evaluated traits, the selection of genotypes 8, 13, 4, and 10 is recommended to establish a breeding program that satisfies both the consumer and the producer.

Phenotypical characterization of Portulaca umbraticola: A non-conventional edible ornamental crop

Purslanes are frequently used in Brazil as potted plants and for landscaping because of their beautiful colored flowers. Despite their commercial importance, the genetic diversity within ornamental purslane remains largely unknown. Thus, a complete characterization of ornamental purslane resources is essential for its utilization since genetic variability is indispensable for the efficient development of new cultivars. This study aimed to evaluate genetic variability and recommend accessions to start a genetic breeding program for purslane (Portulaca umbraticola). Twenty purslane accessions were collected in public places in Areia City and Santa Rita City, Paraiba State, Brazil. The experiment was conducted in a greenhouse. An entirely randomized experimental design with 20 accessions and 5 replicates was utilized. Plants were evaluated for quantitative and qualitative traits. Data were submitted to analysis of variance and grouped by Scott–Knott’s criteria (p ≤ 0.05). The accessions were grouped using Tocher’s method based on the generalized Mahalanobis distance. Principal component analysis was also used to analyze the genetic divergence. There were significant differences (p ≤ 0.05) for all evaluated traits except for internode distance, plant height, and leaf length. The Scott–Knott criteria clustered the accessions into two groups for all traits except days to flowering (five groups). Tocher’s grouping based on Mahalanobis distance allowed the accessions to be separated into eight clusters. The genetic diversity observed in this study was supported by the consonance between the Tocher cluster method and the PC analyses. The traits that most contributed to genetic divergence among accessions were days to flowering (70.37%), flower length (6.10%), leaf width (3.89%), branch number (3.57%), and stem length (3.36%). Considering the criteria for use in outdoor spaces, wider plants with several large colored flowers, thicker stems, greener leaves, and earliness were interesting. Accessions 5, 7, 13, 14, 16, 18, and 20 should be selected as parents for further breeding programs. However, if the objective is to produce potted plants, dwarf plants with bigger leaves are desired, and accession 1 should be selected. This will create greater variability in segregating populations, which will increase the overall quality of ornamental use of this species. For those accessions with no reproductive verticils but with other excellent attributes, for example, high chlorophyll content breeding strategies, such as induced mutations or protoplast fusion, can be used.

Genetic trends and multivariate interrelationships for grain quality of irrigated rice genotypes

For genetic improvement programs, researches with multivariate approaches in rice are fundamental, to define genetic trends, clusters and correlations of agronomic characters that together help selection procedures. This work aimed to reveal the agronomic performance, trends and genetic interrelationships of grain quality based on multivariate models applied to elite lines of irrigated rice. The experiment took place in the 2017/2018 harvest, held at Estação Experimental Terras Baixas (ETB), of Embrapa Clima Temperado. The study used randomized blocks design, with three replications. There were fifteen F6 lines and four control cultivars. Evaluation of intrinsic physical quality attributes with the aid of S21 grain statistical analyzer, as well as grain yield and mill yield (whole and broken grains). Performance of analysis of variance, genetic parameters and Scott Knott cluster test, linear correlation, canonical correlations, cluster analysis via generalized Mahalanobis distance, using the Toucher method, in addition to BIPLOT principal component analysis. The results showed that PH18502 and PH18701 strains presented better agronomic performance for the studied characters, by univariate analysis. The linear and canonical correlations presented demonstrate potential in the direction of selection of multiple characters and point to the possibility of indirect selection among the relevant agronomic characters for the production chain of irrigated rice.

Psychiatric Inpatient Healthcare Resource Utilization and Treatment Patterns Among Patients With Predominant Negative Symptoms in Schizophrenia

AbstractIntroductionCurrently approved treatments for schizophrenia (antipsychotics) have demonstrated effectiveness for treating positive symptoms; however, these agents are largely ineffective in treating other domains. Negative symptoms, including avolition, alogia, blunted affect, and asociality, are difficult to treat, and often persist despite adequate control of positive symptoms. Additionally, some patients experience “predominant” (moderate-to-severe negative symptoms that have greater relative severity than co-occurring positive symptoms) or “prominent” (severity of negative symptoms [moderate-to-severe] without any reference to positive symptoms) negative symptoms. These symptoms are known to have great impact on patient social functioning and quality of life, and are associated with poorer clinical course and outcomes for patients. Here, we examined inpatient healthcare resource utilization in patients with schizophrenia experiencing predominantly negative symptoms (PNS).MethodsDe-identified data were extracted from electronic health records in the NeuroBlu Database across 25 US mental healthcare providers. Positive and negative symptom data were derived from free-text records using natural language processing. PNS was defined as the presence of three or more negative symptoms and three or fewer positive symptoms at first clinical contact following schizophrenia diagnosis. Groups were balanced for baseline demographic and clinical characteristics by minimizing the generalized Mahalanobis distance and compared using chi-square and t-tests. Treatment patterns were visualized using Sankey diagrams.ResultsA total of 4444 patients with schizophrenia were identified and 8% were classified as PNS. A balanced cohort of 720 patients (50% PNS) was generated. Patients with PNS were more likely to be hospitalized in the 12 months following diagnosis (PNS: 76%, non-PNS: 60%, χ2: 22.5, p < 0.001) and were switched to a second-line antipsychotic after a shorter first-line treatment duration. The most frequently prescribed antipsychotics differed between groups (PNS: risperidone, aripiprazole, haloperidol; non-PNS: risperidone, olanzapine, other atypical).DiscussionThis study demonstrates that negative symptoms in schizophrenia may be associated with worse illness course and higher healthcare resource utilization. There remains a need for new treatment options for patients with persistent, prominent, or predominant negative symptoms which specifically improve this historically hard-to-treat and assess symptom domain.FundingSunovion Pharmaceuticals, Inc.

Generalized Mahalanobis distance and its application in detecting matrix outliers

In this paper, a new distance for matrix observations called generalized Mahalanobis distance is introduced, some of its properties are studied, and its distribution is obtained for the observations of the matrix variate elliptically contoured distributions. Also, as a significant application, the introduced distance is used in detecting matrix outliers, and its method is described. Finally, some examples are provided for illustrative purposes, and the performance of the presented approach of detecting outliers is investigated by a simulation study.

Genetic Divergence and Selection of Genitors Aiming to Generate Variability in Common Bean Lines

The objective of this work was to evaluate the genetic divergence between common bean strains from the breeding program of the Agricultural Research Institute of Mozambique (IIAM)-Zonal Centre (CZC) and to identify potential genitors to generate segregating populations with high average and wide genetic variability. Five common bean lines were evaluated in a randomized block design with three repetitions. The quantitative traits evaluated were leaf area index, days to 50% flowering, days to maturity, several pods per plant, 100 seed weight, and yield, while the qualitative traits were growth habit, epicotyl colour, leaf colour intensity, leaf roughness, fresh pod colour, grain colour, grain size, and flower colour. The data obtained were submitted to variance analysis and grouping of means by the Scott-Knott test. The relative importance of characters was evaluated by the Singh. Clustering was performed by the Tocher optimization method and distance projection in the 2D plane, using the generalized Mahalanobis distance as a measure of dissimilarity. The degree of preservation of genetic distances in the 2D plane was assessed by the Cohenetic Correlation Coefficient, distortion, and stress. The LPA54 and Bonus lines were highlighted by direct selection based on the performance per se of their characteristics. Genetic variability was checked for the trait’s days to 50% flowering several of pods per plant, and 100-seed weight and the strains were separated into two groups. The crosses between (G19833, LPA54, Manteiga, and Bonus) with the strain DOR364 there is the potential for obtaining hybrids with higher heterotic effect and segregating populations with high variability. The traits of 100-seed weight, pods per plant, epicotyl colour, and grain size were the most important in the discrimination of the strains

Genetic diversity in creole genotypes of Amazon chicory

ABSTRACT Amazon chicory is still a little-known vegetable despite its great agronomic potential. The characterization of chicory genotypes concerning genetic divergence is a key step for breeding programs, as it allows the selection of superior individuals and to explore the variability and complementarity of characteristics via interbreeding between newly generated genotypes. In this context, this study aimed to evaluate the genetic divergence among Amazon chicory creole genotypes from the northern Brazilian states of Pará and Rondônia based on morpho-agronomic traits. We conducted an experiment in a randomized block design with eight chicory genotypes (treatments) and four replications. Both quantitative and qualitative characteristics were evaluated. Genetic divergence was estimated via squared generalized Mahalanobis distance (D2), considering only quantitative characters, and the genotypes were subsequently clustered via the UPGMA method. Analysis of variance showed significant differences among genotypes for all studied characteristics, except shoot fresh weight. The UPGMA grouped the genotypes into three clusters, which demonstrated that the genotypes from Colares and Santarém Novo (Pará) (Chic-02 and Chic-04) were the most divergent as compared to the genotypes from Castanhal and Santa Isabel do Pará (Pará). Qualitative characteristics showed a monomorphic behavior and, therefore, were not used to assess genetic divergences. To obtain segregating populations with complementary characteristics, crossbreeding between the two most divergent clusters is recommended.

Phenotypic divergence for morphological and yield-related traits in black oat (Avena strigosa)

This study aims to evaluate the phenotypic divergence among 46 accessions of A. strigosa from Mesoregions Northwest and Northeast of Rio Grande do Sul. The experiment was carried out in the field, in a randomized block design, with three replications. The germplasm was characterized using 40 morphoagronomic, 21 of which were either binary or multicategoric, while 19 were quantitative traits as cycle (days to heading), plant length, peduncle, glume and lemma, and production of biomass and grains. The data were submitted to descriptive, univariate, and multivariate statistical analysis. The accessions were monomorphic in 17 out of 21 among qualitative characters. Vegetative habit, diameter, length, and wall thickness of culm were the main quantitative traits in the discrimination of the accessions. The maximum and minimum dissimilarity, expressed by the generalized Mahalanobis distance, ranged from 2.98 up to 55.71. Based on the magnitude of divergence and performance towards agronomic traits in attributes, some accessions may be incorporated into breeding programs of the species

Genetic Divergence and Selection of Common Bean Genotypes with High Agronomic Performance in Mozambique

The deep knowledge of genetic diversity in access, in the breeding program, and the contribution of genetic factors as well as the phenotypic expression is of extreme importance for the development of new cultivars and maintenance of the program. This research aimed to study the genetic diversity and selection of the best common bean genotypes with high agronomic performance in the Agrarian Station of Sussundenga-Mozambique. Twenty-two common bean genotypes were evaluated in a randomized block design with three repetitions in the 2021/2022 agricultural season. The agronomic characteristics evaluated were: the number of days to flowering, plant height, plant architecture, number of pods per plant, the weight of 100 seeds, and grain yield. The data obtained were submitted to individual variance analysis, followed by the Scott-Knott grouping test. For the effect of genetic divergence between the genotypes, multivariate analysis was used based on the generalized Mahalanobis distance and Tocher optimization method, UPGMA, and canonical variables for the projection of distance in the 2D plane. It was also verified by the Cophenetic Correlation Coefficient. The Singh criterion (1981) was used, indicating the characteristics Number of pods per plant (42.81%), the weight of 100 seeds (19.59%), and Plant height (16.08%) as the most important for the genetic divergence of bean genotypes.

Morphological divergence of seedlings Calophyllum brasiliense cambes collected in São Paulo and Tocantins

Objective: The objective was to quantify the morphological diversity of Calophyllum brasiliense Cambes from four municipalities of the State of Tocantins employing morphological characteristics, to strengthen information on conservation and future breeding of the species. Methodology: Seeds were collected from four municipalities in the State of Tocantins and one in the State of São Paulo. The following were evaluated: plant height, stem diameter, root length, number of leaves, leaf area, root dry mass, shoot dry mass, total dry mass, and Dickson´s quality index. The data were subjected to univariate analysis of variance, Tocher grouping method, and UPGMA, obtaining a dendrogram through the generalized Mahalanobis distance. Results: The results showed a statistical difference of 1 and 5% probability. Dueré stood out in stem diameter(5.52 mm), Sandolandia in height (34.84 cm) and root length (42.13 cm). Formoso stood out in the number of leaves (34 leaves). Lagoa da Confusão in leaf area (856.28 cm 2 ) and São Paulo in root dry mass (16.20 g), shoot dry mass (12.38 g), total dry mass (16.20 g), and Dickson´s quality index (1.57). Implications: Variations in morphological characteristics can be used as a tool for genetic studies of guanandi progeny accordingto their similarity and/or differences. Conclusions: The morphological divergence evidenced that among the five studied areas it is possible to direct the collection of seeds to subsidize conservation strategies and future breeding of the species.

Caracterização morfoagronômica de genótipos de feijoeiro comum quanto à precocidade

O feijoeiro comum é uma cultura de grande importância alimentar, econômica e social, em termos de produção e consumo, é uma leguminosa herbácea, com ciclo em torno de 95 dias, a maioria das cultivares utilizadas para o cultivo no mercado brasileiro expressa grande variabilidade genéticas quanto ao início de florescimento e a duração total do ciclo. Diante disso, objetivou-se com este trabalho avaliar a caracterização morfoagronômica de genótipos de feijoeiro comum quanto à precocidade de cultivo. O experimento foi conduzido na EMAPER, no município de Cáceres-MT, adotando o delineamento experimental em blocos ao acaso com três repetições, avaliando 58 genótipos de feijão comum. As parcelas foram compostas por quatro linhas de 4 m, com espaçamento de 0,5 m entre linhas. As características avaliadas foram o número de dias para o florescimento e ciclo de cultivo, os dados foram submetidos à análise de variância e a comparação das medias pelo teste de agrupamento Scott-Knott a 1% de probabilidade. A divergência genética foi estimada método de otimização de Tocher e hierárquico UPGMA, baseada na Distância Generalizada de Mahalanobis utilizando o programa computacional GENES. Pode-se concluir que existe variabilidade genética entre os genótipos de feijão comum avaliados, e os genótipos 3 (BG-UNEMAT-11) e 57 (TB 02-23) apresentaram menor número de dias para florescimento (31 dias) e menor ciclo da cultura (73 a 82 dias), sendo os mais indicados para programas de melhoramento genético visando precocidade.

Use of computational intelligence in the genetic divergence of colored cotton plants

The objective of this work was to analyze the genetic diversity using conventional methods and artificial neural networks among 12 colored fiber cotton genotypes, using technological characteristics of the fiber and productivity in terms of cottonseed and cotton fiber yield. The experiment was conducted in an experimental area located at Fazenda Capim Branco, belonging to the Federal University of Uberlândia, in the city of Uberlândia, Minas Gerais. Twelve genotypes of colored fiber cotton were evaluated, 10 from the Cotton Genetic Improvement Program (PROMALG): UFUJP - 01, UFUJP - 02, UFUJP - 05, UFUJP - 08, UFUJP - 09, UFUJP - 10, UFUJP - 11, UFUJP - 13, UFUJP - 16, UFUJP - 17 and two commercial cultivars: BRS Rubi (RC) and BRS Topázio (TC). The experimental design used was complete randomized block (CRB) with three replications. The following evaluations were carried out at full maturation: yield of cottonseed (kg ha-1) and the technological characteristics, which include, fiber length, micronaire, maturation, length uniformity, short fiber index, elongation and strength, using the HVI (High volume instrument) device. Genetic dissimilarity was measured using the generalized Mahalanobis distance and after obtaining the dissimilarity matrix, the genotypes were grouped using a hierarchical clustering method (UPGMA). A discriminant analysis and the Kohonen Self-Organizing Map (SOM) by Artificial Neural Networks (ANN’s) were performed through computational intelligence. SOM was able to detect differences and organize the similarities between accesses in a more coherent way, forming a larger number of groups, when compared to the method that uses the Mahalanobis matrix. It was also more accurate than the discriminant analysis, since it made it possible to differentiate groups more coherently when comparing their phenotypic behavior. The methods that use computational intelligence proved to be more efficient in detecting similarity, with Kohonen's Self-Organizing Map being the most adequate to classify and group cotton genotypes.

Genetic divergence in maize regarding grain yield and tassel traits

Maize tassels with a longer length and higher number of branches prevent the passage of solar radiation into the upper plant canopy and act as a drain for photoassimilates that could be used for grain production. This study aimed to verify if there is genetic divergence regarding grain yield and tassel traits among maize cultivars in three agricultural years. Twenty maize cultivars were evaluated and 11 tassel traits and grain yield were measured. Individual and joint analyses of variance were performed for each trait. Principal component analysis was performed, the generalized Mahalanobis distance matrix between cultivars was determined, and the genetic divergence analysis was performed, using the unweighted pair group method with arithmetic mean (UPGMA), which allowed constructing the dendrogram. The cophenetic correlation coeffi cient was calculated to validate the cluster. There is genetic divergence among maize cultivars and six groups of cultivars were formed. Tassel length, total branches number, central spike dry matter, and grain yield are the traits that most contribute to the genetic divergence among maize cultivars.

GENETIC DISSIMILARITY FOR RESISTANCE TO FOLIAR DISEASES ASSOCIATED WITH THE AGRONOMIC POTENTIAL IN MAIZE

ABSTRACT In the present study the objective was to evaluate the genetic diversity among families of maize siblings for resistance to foliar diseases associated with their agronomic potential, identifying groups of families that can be used as sources of resistance in maize crop. The experiments were conducted in the experimental area of the Federal University of Goiás at the Jataí Regional Unit, in Jataí, GO, Brazil, constituted by 182 half-sibling families of maize and two commercial hybrids as a control. The 182 half-sibling families were divided into three experiments with 60, 60 and 62 families, respectively. The experimental design used was randomized blocks, with three replicates. Eight quantitative characters and 4 foliar diseases were evaluated. The multivariate analysis technique was used to measure the genetic divergence for the four foliar diseases represented by the generalized Mahalanobis distance. Based on the genetic dissimilarity matrix, the dendrogram was constructed using the clustering method of the average distance between groups (Unweighted Pair Group Method with Arithmetic Mean - UPGMA). After defining the groups, univariate analysis of variance was performed in order to evaluate the effects of the groups on each character studied. Comparisons were made between the means of the groups, using the Tukey test (p <0.05). White spot (32.53%) was the disease that most contributed to the total divergence between families. Group 10 stood out among the others as a source of resistance to the disease complex associated with yield. The genetic variability of families for foliar disease complex reveals potential for future studies facing pyramiding genes.

Genetic divergence in biomass sorghum genotypes through agronomic and physical-chemical characters

The present research aimed to evaluate the genetic divergence in 34 sorghum biomass genotypes via agronomic and physicochemical characters. The design used was randomized blocks with three replications. The agronomic and physical-chemical characteristics evaluated were: days for flowering, number of stems, plant height, number of leaves, green mass production, dry mass production, determination of total ash, determination of volatile content, insoluble lignin and determination of fixed carbon content. The data were submitted to analysis of variance and then, to estimate divergence, the generalized Mahalanobis distance was used as a measure of dissimilarity. Based on this matrix, the methods of Tocher's optimization clusters and the Hierarchical method of Average Grouping Between Groups (UPGMA) were used, and analysis of canonical variables, and the projection based on the first two canonical variables arranged in two-dimensional space. Singh criterion was also used to quantify the relative contribution of these characteristics to genetic divergence. The evaluated genotypes showed significant differences for all the evaluated characteristics. The combination between the 201429B001 and 201429B028 (394.98) genotype pairs was the most divergent and the combination between the 201429B015 and 201429B031 (6.31) genotypes was the most similar. The grouping generated by the Tocher Optimization method, hierarchical UPGMA and graphical dispersion showed similarity in the grouping of genotypes. The first two canonical variables were sufficient to explain about 81.78% of the total variation observed. The results showed a wide genetic diversity among the 34 genotypes of sorghum biomass.

Diversidade genética entre acessos de feijão-guandu por meio de caracteres morfoagronômicos em região do cariri paraibano

Com objetivo de quantificar a divergência genética entre 22 acessos de feijão-guandu por meio da análise de agrupamentos com base em caracteres morfoagronômicos, além de indicar cruzamentos mais promissores para dar continuidade ao programa de melhoramento da espécie de feijão- guandu foi realizado a presente pesquisa. O delineamento experimental empregado foi o delineamento inteiramente casualizado - DIC. Foi realizado o teste de agrupamento de médias, segundo Scott e Knott a 5% de probabilidade. Os dados de médias dos caracteres desses acessos foram submetidos à análise de divergência genética. Foi aplicado o método de agrupamento pela ligação média entre grupos (UPGMA), utilizando a distância generalizada de Mahalanobis (D2) e método de agrupamento de Tocher. A importância relativa dos caracteres, também foram avaliadas. Existe ampla variabilidade genética entre os acessos de feijão-guandu, presentes no banco de germoplasma, e facilmente identificada por meio de caracteres morfoagronômicos. Quatro grupos distintos de acessos de feijão-guandu foram detectados, evidenciando a existência de diversidade entre os acessos. Os cruzamentos mais promissores são aqueles envolvendo como parentais os acessos dos grupos distintos e mais divergentes: Grupos IV e I, IV e II, IV e III, III e II, e II e I. O número de dias para o florescimento foi o caractere que mais contribuiu para divergência genética.

Genetic divergence among eggplant genotypes under high temperatures

ABSTRACT The aim of this study was to estimate the genetic divergence among eggplant genotypes for agronomic traits in order to gather information for the selection of genotypes in eggplant breeding programs for tolerance to high temperatures. Ten traits recommended by the International Board for Plant Genetic Resources were analyzed in 24 genotypes, arranged in a randomized complete block design with four replicates. Data were submitted to analysis of variance (P<0.01) and later to the UPGMA and Tocher grouping methods, using the generalized Mahalanobis distance (D2) as dissimilarity measure. Three and six groups of similarity were obtained, respectively, for the multivariate techniques used, UPGMA and Tocher, with concordance in the grouping of 87.50% of the genotypes. The characters fruit length (34.71%), fruit width (35.96%) and fruit length/width ratio (14.08%) were the main contributors to genetic divergence, explaining 90.72% of total genetic dissimilarity. The genotypes presented considerable genetic variability for all agronomic traits analyzed and can be used in eggplant genetic breeding programs for high temperatures.

A k-means procedure based on a Mahalanobis type distance for clustering multivariate functional data

This paper proposes a clustering procedure for samples of multivariate functions in $$(L^2(I))^{J}$$ , with $$J\ge 1$$ . This method is based on a k-means algorithm in which the distance between the curves is measured with a metric that generalizes the Mahalanobis distance in Hilbert spaces, considering the correlation and the variability along all the components of the functional data. The proposed procedure has been studied in simulation and compared with the k-means based on other distances typically adopted for clustering multivariate functional data. In these simulations, it is shown that the k-means algorithm with the generalized Mahalanobis distance provides the best clustering performances, both in terms of mean and standard deviation of the number of misclassified curves. Finally, the proposed method has been applied to two case studies, concerning ECG signals and growth curves, where the results obtained in simulation are confirmed and strengthened.

Implicações da análise univariada e multivariada na dissimilaridade de acessos de feijão comum

Univariate and multivariate analyses have been routinely used to discriminate genotypes in breeding programs, but their relationship is not always considered. The objective of this study was to relate univariate and multivariate analyses on the dissimilarity among common bean accessions based on characteristics such as days until emergence, plant height, first pod insertion height, harvest days, final stand, number of pods per plant, pod length, 100-grain weight, grain yield and seed length, width and thickness. Data were submitted to univariate analysis of variance, with a comparison of means by the Scott-Knott test, and multivariate analysis to estimate the genetic divergence based on the generalized Mahalanobis distance, grouping the accesses through Tocher’s method. The relative importance of the characters was also estimated by Singh's method, performing the multicollinearity diagnosis. There was a significant difference for all the evaluated characteristics, showing the existence of variability between the accesses. The greater variability found was related to seed characteristics. It was possible to relate univariate and multivariate analyses, since there was a relationship between the groups generated by Scott-Knott’s test and the percentage of the relative contribution of the evaluated characters. However, the clustering obtained in the univariate analysis for the characteristic that contributed the most to the divergence between the accessions was not compatible with the cluster generated by Tocher’s method. The coupled use of these analyses can help breeders to decide, choosing superior genotypes and that present wide genetic divergence, mainly for the characteristics of greater interest.

Simplicial variances, potentials and Mahalanobis distances

The average squared volume of simplices formed by k independent copies from the same probability measure μ on Rd defines an integral measure of dispersion ψk(μ), which is a concave functional of μ after suitable normalization. When k=1 it corresponds to tr(Σμ) and when k=d we obtain the usual generalized variance det(Σμ), with Σμ the covariance matrix of μ. The dispersion ψk(μ) generates a notion of simplicial potential at any x∈Rd, dependent on μ. We show that this simplicial potential is a quadratic convex function of x, with minimum value at the mean aμ for μ, and that the potential at aμ defines a central measure of scatter similar to ψk(μ), thereby generalizing results by Wilks (1960) and van der Vaart (1965) for the generalized variance. Simplicial potentials define generalized Mahalanobis distances, expressed as weighted sums of such distances in every k-margin, and we show that the matrix involved in the generalized distance is a particular generalized inverse of Σμ, constructed from its characteristic polynomial, when k=rank(Σμ). Finally, we show how simplicial potentials can be used to define simplicial distances between two distributions, depending on their means and covariances, with interesting features when the distributions are close to singularity.