Objetivou-se, com este trabalho, estudar os métodos hierárquicos de análise de agrupamentos (ligação simples, ligação completa, ligação média, centróide e de Ward com base nas distâncias Euclidiana, Euclidiana ao quadrado, Manhattan e Mahalanobis), de modo a identificar qual é o mais adequado para uma base de dados arqueológicos. Utilizou-se uma base de dados fornecida pelo Grupo de Estudos Arqueométricos do IPEN CNEN/SP, na qual foram analisadas 146 amostras de fragmentos cerâmicos de três sítios arqueológicos por análise por ativação com nêutrons instrumental, sendo determinadas as frações de massa de 24 elementos químicos: As, Ba, Ce, Co, Cr, Cs, Eu, Fe, Hf, K, La, Lu, Na, Nd, Rb, Sb, Sc, Sm, Ta, Tb, Th, U, Yb e Zn. Para a determinação do melhor método, foram avaliados os dendrogramas conjuntamente com o valor dos coeficientes de correlação cofenética (CCC), obtidos para cada método. O método da ligação média mostrou-se mais coerente na formação dos agrupamentos, apresentando também os maiores valores do CCC. Por último, um script com funções do programa estatístico R foi desenvolvido para calcular o CCC, com o intuito de auxiliar os pesquisadores a encontrar o método de agrupamento mais apropriado para sua base de dados. / The objective of this work was to study the hierarchical methods of cluster analysis (single linkage, complete linkage, average linkage, centroid and Ward with base in the Euclidean distance, Squared Euclidean distance, Manhattan and Mahalanobis) to identify which is the most appropriate in archaeological data set. It was used a data set provided by the Archaeometry Studies Group from IPEN CNEN/SP in which 146 samples of ceramic fragments from three archaeological sites analyzed by instrumental neutron activation analysis, being determined the mass fractions of 24 chemical elements: As, Ba, Ce, Co, Cr, Cs, Eu, Fe, Hf, K, La, Lu, Na, Nd, Rb, Sb, Sc, Sm, Ta, Tb, Th, U, Yb and Zn. For the determination of the best method, it was evaluated the dendrograms together with value of the cophenetic correlation coefficient (CCC) obtained for each method. The average linkage method was more coherent in formation of the clusters, and with higher CCC values. Finally, a script with functions of the R statistical software was developed to calculate the CCC with aim to be a tool for researchers find the most appropriate method of cluster analysis for their data set.

Objetivou-se, com este trabalho, estudar os métodos hierárquicos de análise de agrupamentos (ligação simples, ligação completa, ligação média, centróide e de Ward com base nas distâncias Euclidiana, Euclidiana ao quadrado, Manhattan e Mahalanobis), de modo a identificar qual é o mais adequado para uma base de dados arqueológicos. Utilizou-se uma base de dados fornecida pelo Grupo de Estudos Arqueométricos do IPEN CNEN/SP, na qual foram analisadas 146 amostras de fragmentos cerâmicos de três sítios arqueológicos por análise por ativação com nêutrons instrumental, sendo determinadas as frações de massa de 24 elementos químicos: As, Ba, Ce, Co, Cr, Cs, Eu, Fe, Hf, K, La, Lu, Na, Nd, Rb, Sb, Sc, Sm, Ta, Tb, Th, U, Yb e Zn. Para a determinação do melhor método, foram avaliados os dendrogramas conjuntamente com o valor dos coeficientes de correlação cofenética (CCC), obtidos para cada método. O método da ligação média mostrou-se mais coerente na formação dos agrupamentos, apresentando também os maiores valores do CCC. Por último, um script com funções do programa estatístico R foi desenvolvido para calcular o CCC, com o intuito de auxiliar os pesquisadores a encontrar o método de agrupamento mais apropriado para sua base de dados. / The objective of this work was to study the hierarchical methods of cluster analysis (single linkage, complete linkage, average linkage, centroid and Ward with base in the Euclidean distance, Squared Euclidean distance, Manhattan and Mahalanobis) to identify which is the most appropriate in archaeological data set. It was used a data set provided by the Archaeometry Studies Group from IPEN CNEN/SP in which 146 samples of ceramic fragments from three archaeological sites analyzed by instrumental neutron activation analysis, being determined the mass fractions of 24 chemical elements: As, Ba, Ce, Co, Cr, Cs, Eu, Fe, Hf, K, La, Lu, Na, Nd, Rb, Sb, Sc, Sm, Ta, Tb, Th, U, Yb and Zn. For the determination of the best method, it was evaluated the dendrograms together with value of the cophenetic correlation coefficient (CCC) obtained for each method. The average linkage method was more coherent in formation of the clusters, and with higher CCC values. Finally, a script with functions of the R statistical software was developed to calculate the CCC with aim to be a tool for researchers find the most appropriate method of cluster analysis for their data set.

For a class of two-dimensional boundary value problems including diffusion and elasticity problems it is proved that the constants in the corresponding strengthened Cauchy-Buniakowski-Schwarz (C.B.S.) inequality in the cases of h -hierarchical and p -hierarchical finite element discretizations with triangular meshes differ by the factor 0.75. For plane linear elasticity problems and triangulations with right isosceles tri- angles formulas are presented that show the dependence of the constant in the C.B.S. inequality on the Poisson's ratio. Furthermore, numerically determined bounds of the constant in the C.B.S. inequality are given for three-dimensional elasticity problems discretized by means of tetrahedral elements. Finally, the robustness of iterative solvers for elasticity problems is discussed briefly.

Cluster analysis is one of the classification methods of multivariate statistical analysis. The task of this analysis is to classify the objects into clusters so that objects inside these clusters are as similar as possible. The aim of this study is to evaluate the success of the classification of objects using six hierarchical cluster analysis methods. To reflect the distance between the objects, are used squared Euclidean and Mahalanobis distances. The success methods are evaluated through the information, which cluster the object belongs to, and this information is already contained in the data files. This thesis pointed out that the Ward's method is one of the most successful hierarchical method in a classification of objects into clusters. This method has been more successful in sorting objects than the other hierarchical methods, both in the case of leaving the correlated variables in the data file as well as removing them. The results of this work show that the highest success of classification objects into clusters is when the data set is cleaned of correlated variables. If the data file is not cleaned, the methods reach better results when the distance between objects is measured by Euclidean metric.

For a class of two-dimensional boundary value problems including diffusion and elasticity problems it is proved that the constants in the corresponding strengthened Cauchy-Buniakowski-Schwarz (C.B.S.) inequality in the cases of h -hierarchical and p -hierarchical finite element discretizations with triangular meshes differ by the factor 0.75. For plane linear elasticity problems and triangulations with right isosceles tri- angles formulas are presented that show the dependence of the constant in the C.B.S. inequality on the Poisson's ratio. Furthermore, numerically determined bounds of the constant in the C.B.S. inequality are given for three-dimensional elasticity problems discretized by means of tetrahedral elements. Finally, the robustness of iterative solvers for elasticity problems is discussed briefly.