Asymmetric dependence (AD) is defined as dependence that differs across opposing regions of the joint return distribution. Recent evidence of AD between equity returns suggests that dependence can be decomposed into a linear component, captured by the correlation matrix, and a higher order component. When these higher order terms are characterised by increased correlation in bear or bull markets, the effectiveness of diversification strategies is reduced. To the extent that an investor is unable to completely diversify these higher order terms of dependence, it follows that they should be reflected in asset prices and managed explicitly during the portfolio construction process. The aim of this thesis is to determine the extent of AD amongst asset returns, to investigate whether AD is priced and to develop a means of managing AD in the portfolio. I justify the existence of AD and the separation of AD from linear dependence via the bivariate Edgeworth expansion, finding that the joint return distribution may be described by an infinite number of higher order co-moments. Correlation (and hence β) describes one dimension of an infinite number of higher dimensions describing dependence. To determine the importance of AD in finance, I first develop measures that can detect AD independent of the level of linear dependence and idiosyncratic risk. These measures are used to determine the extent of AD amongst US stock returns and the market, to obtain an understanding of how AD changes through time and to re-examine the evidence of AD between equity portfolios. By measuring AD separate from linear dependence, I demonstrate several findings. First, I find evidence of non-stationary AD that can exists irrespective of the magnitude of linear dependence, measured by β. This time-varying AD consists of both significant upper tail dependence (UTD) and significant lower tail dependence (LTD), although LTD is found to occur more frequently than UTD, especially for small stocks and stocks displaying high idiosyncratic risk. Significant time-varying AD is also detected between domestic equity indices and international equity markets, implying that if a portfolio is weighted towards certain industries or countries, portfolio construction methods may need to be adjusted in order too meet risk and return targets, particularly if future AD cannot be adequately forecasted. Next, I investigate whether AD is priced in US equities using the Fama and MacBeth (1973) regression methodology in conjunction with my β invariant AD metrics. I find that AD is as important as linear dependence in explaining the variation in returns. In particular, a positive relationship between LTD and return is found. I document an AD risk premium of 2.7% pa, compared to a β risk premium of 6.18% pa. The AD risk premium increases to 6.9% pa for stocks with significant LTD. This result holds after controlling for size, book-to-market ratio, downside β and coskewness. I also find past AD is a significant variable in predicting the future returns of small firms, whilst neither AD nor linear dependence predict the future returns of large firms. I subsequently demonstrate a means of incorporating AD structures during the portfolio construction process using copula functions. I then investigate how asymmetric return dependencies affect the efficient frontier and subsequent portfolio performance under a dynamic rebalancing framework. By considering the problem of tactically allocating a small set of domestic equity indices, I demonstrate several findings. First, I show that a Mean-Variance efficient frontier differs from the efficient frontier constructed under AD. Constructing paper portfolios based upon these differences, I find that real economic value lies in correctly accounting for AD structures. The primary source of this economic value stems from the ability to better protect portfolio value and reduce the size of any erosion in return relative to the normal portfolio. Finally, I document the benefits of actively managing AD during the portfolio construction process and determine a number of portfolio management principles required to successfully manage AD. I illustrate that managing asymmetry risk in a portfolio of international equity indices results in increased return, decreased risk and decreased transaction costs. I show that in order to yield these benefits, investors must actively and dynamically manage their portfolio. Furthermore, I illustrate that the ability to short-sell assets provides most of the benefits described.

Asymmetric dependence (AD) is defined as dependence that differs across opposing regions of the joint return distribution. Recent evidence of AD between equity returns suggests that dependence can be decomposed into a linear component, captured by the correlation matrix, and a higher order component. When these higher order terms are characterised by increased correlation in bear or bull markets, the effectiveness of diversification strategies is reduced. To the extent that an investor is unable to completely diversify these higher order terms of dependence, it follows that they should be reflected in asset prices and managed explicitly during the portfolio construction process. The aim of this thesis is to determine the extent of AD amongst asset returns, to investigate whether AD is priced and to develop a means of managing AD in the portfolio. I justify the existence of AD and the separation of AD from linear dependence via the bivariate Edgeworth expansion, finding that the joint return distribution may be described by an infinite number of higher order co-moments. Correlation (and hence β) describes one dimension of an infinite number of higher dimensions describing dependence. To determine the importance of AD in finance, I first develop measures that can detect AD independent of the level of linear dependence and idiosyncratic risk. These measures are used to determine the extent of AD amongst US stock returns and the market, to obtain an understanding of how AD changes through time and to re-examine the evidence of AD between equity portfolios. By measuring AD separate from linear dependence, I demonstrate several findings. First, I find evidence of non-stationary AD that can exists irrespective of the magnitude of linear dependence, measured by β. This time-varying AD consists of both significant upper tail dependence (UTD) and significant lower tail dependence (LTD), although LTD is found to occur more frequently than UTD, especially for small stocks and stocks displaying high idiosyncratic risk. Significant time-varying AD is also detected between domestic equity indices and international equity markets, implying that if a portfolio is weighted towards certain industries or countries, portfolio construction methods may need to be adjusted in order too meet risk and return targets, particularly if future AD cannot be adequately forecasted. Next, I investigate whether AD is priced in US equities using the Fama and MacBeth (1973) regression methodology in conjunction with my β invariant AD metrics. I find that AD is as important as linear dependence in explaining the variation in returns. In particular, a positive relationship between LTD and return is found. I document an AD risk premium of 2.7% pa, compared to a β risk premium of 6.18% pa. The AD risk premium increases to 6.9% pa for stocks with significant LTD. This result holds after controlling for size, book-to-market ratio, downside β and coskewness. I also find past AD is a significant variable in predicting the future returns of small firms, whilst neither AD nor linear dependence predict the future returns of large firms. I subsequently demonstrate a means of incorporating AD structures during the portfolio construction process using copula functions. I then investigate how asymmetric return dependencies affect the efficient frontier and subsequent portfolio performance under a dynamic rebalancing framework. By considering the problem of tactically allocating a small set of domestic equity indices, I demonstrate several findings. First, I show that a Mean-Variance efficient frontier differs from the efficient frontier constructed under AD. Constructing paper portfolios based upon these differences, I find that real economic value lies in correctly accounting for AD structures. The primary source of this economic value stems from the ability to better protect portfolio value and reduce the size of any erosion in return relative to the normal portfolio. Finally, I document the benefits of actively managing AD during the portfolio construction process and determine a number of portfolio management principles required to successfully manage AD. I illustrate that managing asymmetry risk in a portfolio of international equity indices results in increased return, decreased risk and decreased transaction costs. I show that in order to yield these benefits, investors must actively and dynamically manage their portfolio. Furthermore, I illustrate that the ability to short-sell assets provides most of the benefits described.

No presente trabalho, exploramos dois temas de interesse em finanças, economia e antropologia social, através da aplicação de técnicas da teoria da informação e da mecânica estatística. No primeiro tópico, estudamos a conexão entre teoria de dependência estatística, teoria de informação e teoria da cópulas. O conceito de distribuição cópula é revisto e aplicado em reformulação das definições de medida de dependência dadas por Rényi 1. Em seguida, mostramos que a informação mútua satisfaz todos os requisitos para ser uma boa medida de dependência. Obtemos uma identidade entre a informação mútua e a entropia da distribuição cópula e uma decomposição mais específica da informação mútua de uma distribuição elíptica nas suas partes linear e não-linear. Avaliamos o risco de usar quantidades ingênuas como medidas de dependência estatística, mostrando que a correlação linear pode subestimar grosseiramente a dependência. Esses resultados são utilizados para desenvolver um método de detectação de desvios de dependência gaussiana em pares de variáveis aleatórias e aplicá-lo a séries temporais financeiras. Finalmente, discutimos um método para ajustar t-cópulas a dados empíricos 2 através da medida da informação mútua e do tau de Kendall. No segundo tópico, desenvolvemos um modelo para o surgimento de autoridade em sociedades humanas pré-agrícolas. Discutimos motivações empíricas com raízes em neurociência, primatologia e antropologia para um modelo matemático capaz de explicar a ampla variabilidade de formas de organização social humana no eixo igualitário hierárquico. O modelo resulta da aplicação de teoria da informação a uma hipótese sobre os custos evolutivos envolvidos na vida social. O modelo gera um diagrama de fase rico, com diferentes regimes que podem ser interpretados como diferentes tipos de organização social. Os parâmetros de controler do modelo estão ligados à capacidade cognitiva da espécie em questão, ao tamanho do grupo e a pressões ecológicas e sociais. / In the present work we explore two topics of interest in finance, economics and social anthropology through the application of techniques from information theory and statistical mechanics. In the first topic we study the connexion between statistical dependency theory, information theory and copula theory. The concept of copula distribution is reviewed and applied to the reformulation of the definition of dependency measures given by Rényi 3. It is then shown that the mutual information satisfy all the requirements to be a good dependency measure. We derive an identity between mutual information and the entropy of the copula distribution and a more specific decomposition of the mutual information of an elliptical distribution into its linear and non-linear parts. We evaluate the risk of using naive measures as statistical dependency measures by showing that linear correlation can grossly underestimate dependency. Those results are used to develop a method to detect deviation from gaussian dependence in pairs of random variables and apply it to financial time series. Finally, we discuss a method to adjust t-copulas to empirical data4 through the determination of the mutual information and Kendalls tau. In the second topic we develop a model for the emergence of authority in pre-agricultural human societies. We discuss empirical motivations with roots in neuroscience, primatology and anthropology for a mathematical model able to explain the ample variability of forms of human social organization in the egalitarian-hierarchical axis. The model results from the application of information theory on a hypothesis about the evolutive costs involved in social life. It generates a rich phase diagram, with different regimes which can be interpreted as different types of societal organization, from egalitarian to hierarchical. The control parameters of the model are connected to the cognitive capacity of the species in question, the size of the group and ecological and social pressures.

No presente trabalho, exploramos dois temas de interesse em finanças, economia e antropologia social, através da aplicação de técnicas da teoria da informação e da mecânica estatística. No primeiro tópico, estudamos a conexão entre teoria de dependência estatística, teoria de informação e teoria da cópulas. O conceito de distribuição cópula é revisto e aplicado em reformulação das definições de medida de dependência dadas por Rényi 1. Em seguida, mostramos que a informação mútua satisfaz todos os requisitos para ser uma boa medida de dependência. Obtemos uma identidade entre a informação mútua e a entropia da distribuição cópula e uma decomposição mais específica da informação mútua de uma distribuição elíptica nas suas partes linear e não-linear. Avaliamos o risco de usar quantidades ingênuas como medidas de dependência estatística, mostrando que a correlação linear pode subestimar grosseiramente a dependência. Esses resultados são utilizados para desenvolver um método de detectação de desvios de dependência gaussiana em pares de variáveis aleatórias e aplicá-lo a séries temporais financeiras. Finalmente, discutimos um método para ajustar t-cópulas a dados empíricos 2 através da medida da informação mútua e do tau de Kendall. No segundo tópico, desenvolvemos um modelo para o surgimento de autoridade em sociedades humanas pré-agrícolas. Discutimos motivações empíricas com raízes em neurociência, primatologia e antropologia para um modelo matemático capaz de explicar a ampla variabilidade de formas de organização social humana no eixo igualitário hierárquico. O modelo resulta da aplicação de teoria da informação a uma hipótese sobre os custos evolutivos envolvidos na vida social. O modelo gera um diagrama de fase rico, com diferentes regimes que podem ser interpretados como diferentes tipos de organização social. Os parâmetros de controler do modelo estão ligados à capacidade cognitiva da espécie em questão, ao tamanho do grupo e a pressões ecológicas e sociais. / In the present work we explore two topics of interest in finance, economics and social anthropology through the application of techniques from information theory and statistical mechanics. In the first topic we study the connexion between statistical dependency theory, information theory and copula theory. The concept of copula distribution is reviewed and applied to the reformulation of the definition of dependency measures given by Rényi 3. It is then shown that the mutual information satisfy all the requirements to be a good dependency measure. We derive an identity between mutual information and the entropy of the copula distribution and a more specific decomposition of the mutual information of an elliptical distribution into its linear and non-linear parts. We evaluate the risk of using naive measures as statistical dependency measures by showing that linear correlation can grossly underestimate dependency. Those results are used to develop a method to detect deviation from gaussian dependence in pairs of random variables and apply it to financial time series. Finally, we discuss a method to adjust t-copulas to empirical data4 through the determination of the mutual information and Kendalls tau. In the second topic we develop a model for the emergence of authority in pre-agricultural human societies. We discuss empirical motivations with roots in neuroscience, primatology and anthropology for a mathematical model able to explain the ample variability of forms of human social organization in the egalitarian-hierarchical axis. The model results from the application of information theory on a hypothesis about the evolutive costs involved in social life. It generates a rich phase diagram, with different regimes which can be interpreted as different types of societal organization, from egalitarian to hierarchical. The control parameters of the model are connected to the cognitive capacity of the species in question, the size of the group and ecological and social pressures.

[pt] A preocupação com o aquecimento global e a poluição tem aumentado significativamente o interesse no desenvolvimento de fontes renováveis de energia. Este estudo tem como eixo principal a energia eólica, o uso dessa energia elimina resíduos indesejados e prejudiciais à saúde e ao meio ambiente causados por outras fontes de energia, como carvão e usinas nucleares. Este trabalho objetiva analisar a relação de dependência entre a velocidade do vento e a geração de energia eólica, esta é uma relação bastante complexa, por isso busca-se entender a natureza estocástica de ambas as variáveis. Como ferramenta metodológica foi utilizada a teoria da cópula. O estudo baseia-se na análise e modelagem da dependência entre dados de velocidade do vento e geração de energia eólica, para um banco de dados horário de um parque eólico do estado da Bahia, no período de janeiro a dezembro de 2017, após encontrar a cópula correspondente a estrutura de dependência para o ano completo e para cada mês individualmente, foram geradas simulações e apresentadas as probabilidades de ocorrência dos cenários em intervalos pré-definidos, os resultados obtidos foram significativos, testes estatísticos adequados foram realizados, evidenciando a qualidade do ajuste. / [en] Concern about global warming and pollution has significantly increased interest in developing renewable energy sources. This study has wind energy as its main axis, the use of this energy eliminates unwanted and harmful waste to health and the environment caused by other energy sources, such as coal and nuclear power plants. This work aims to analyze the dependence relationship between wind speed and wind energy generation, this is a very complex relationship, so we seek to understand the stochastic nature of both variables. As a methodological tool, the copula theory was used. The study is based on the analysis and modeling of the dependence between wind speed data and wind energy generation, for an hourly database of a wind farm in the state of Bahia, from January to December 2017, after finding the copula corresponding to the dependency structure for the entire year and for each month individually, simulations were generated and the probabilities of occurrence of the scenarios were presented at pre-defined intervals, the results obtained were significant, adequate statistical tests were performed, evidencing the quality of the fit .

Cette thèse est une contribution à la modélisation imbriquée de systèmes complexes. Elle propose une méthodologie globale pour quantifier les incertitudes et leurs origines dans une chaîne de calcul formée par plusieurs modèles pouvant être reliés les uns aux autres de façon complexe. Ce travail est organisé selon trois axes. D’abord, la structure dedépendance des paramètres du modèle, induite par la modélisation imbriquée, est modélisée de façon rigoureuse grâce à la théorie des copules. Puis, deux méthodes d’analyse de sensibilité adaptées aux modèles à paramètres d’entrée corrélés sont présentées : l’une est basée sur l’analyse de la distribution de la réponse du modèle, l’autre sur la décomposition de la covariance. Enfin, un cadre de travail inspiré de la théorie des graphes est proposé pour la description de l’imbrication des modèles. La méthodologie proposée est appliquée à des exemples industriels d’envergure : un modèle multiéchelles de calcul des propriétés mécaniques du béton par une méthode d’homogénéisation et un modèle multiphysique de calcul de dommage sur la culasse d’un moteur diesel. Les résultats obtenus fournissent des indications importantes pour une amélioration significative de la performance d’une structure. / This thesis is a contribution to the nested modelling of complex systems. A global methodology to quantify uncertainties and their origins in a workflow composed of several models that can be intricately linked is proposed. This work is organized along three axes. First, the dependence structure of the model parameters induced by the nested modelling is rigorously described thanks to the copula theory. Then, two sensitivity analysis methods for models with correlated inputs are presented : one is based on the analysis of the model response distribution and the other one is based on the decomposition of the covariance. Finally, a framework inspired by the graph theory is proposed for the description of the imbrication of the models. The proposed methodology is applied to different industrial applications : a multiscale modelling of the mechanical properties of concrete by homogenization method and a multiphysics approach of the damage on the cylinder head of a diesel engine. The obtained results provide the practitioner with essential informations for a significant improvement of the performance of the structure.

As Europe’s power production becomes increasingly reliant on intermittent renewable energy sources, uncertainties are likely to arise in power generation plans. Similarly, with the growing prevalence of electric vehicles, electric demand is also becoming more uncertain. These uncertainties in both production and demand can lead to challenges for European power systems. This thesis proposes the use of Monte-Carlo simulations to translate uncertainties in power generation and demand into uncertainties in the power grid. To integrate stochasticity in the forecasts, this thesis separates the multivariate probabilistic forecasting problem by first forecasting the marginal loads individually and probabilistically. Copula theory is then used to integrate spatial correlations and create realistic scenarios. These scenarios serve as inputs for Monte-Carlo simulations to estimate uncertainties in the power system. The methodology is tested using power injection data and the power system model of Switzerland. The results demonstrate that integrating stochasticity in forecasts improves the reliability of the power system. The proposed approach effectively models the uncertainty in both production and demand and provides valuable information for decision-making. / I takt med att Europas elproduktion blir alltmer beroende av intermittenta förnybara energikällor kommer det sannolikt att uppstå osäkerheter i planerna för elproduktion. På samma sätt blir efterfrågan på elektricitet mer osäker i takt med att elfordon blir allt vanligare. Dessa osäkerheter i både produktion och efterfrågan kan leda till utmaningar för de europeiska kraftsystemen. I denna avhandling föreslås att Monte-Carlo-simuleringar används för att omvandla osäkerheter i elproduktion och efterfrågan till osäkerheter i elnätet. För att integrera stokasticitet i prognoserna separerar denna avhandling det multivariata probabilistiska prognosproblemet genom att först individuellt och probabilistiskt prognostisera belastningar. Kopulateori används sedan för att integrera rumsliga korrelationer och skapa realistiska scenarier. Dessa scenarier tjänar som indata för Monte-Carlo-simuleringar för att uppskatta osäkerheterna i kraftsystemet. Metodiken testas med hjälp av data om inmatning av el och med hjälp av Schweiz kraftsystem. Resultaten visar att integrering av stokasticitet i prognoser förbättrar kraftsystemets tillförlitlighet. Den föreslagna metoden modellerar effektivt osäkerheten i både produktion och efterfrågan och ger värdefull information för beslutsfattandet.

Avec l’avènement de nouvelles techniques d’acquisition d’image et l’émergence des systèmes satellitaires à haute résolution, les données de télédétection à exploiter sont devenues de plus en plus riches et variées. Leur combinaison est donc devenue essentielle pour améliorer le processus d’extraction des informations utiles liées à la nature physique des surfaces observées. Cependant, ces données sont généralement hétérogènes et imparfaites ce qui pose plusieurs problèmes au niveau de leur traitement conjoint et nécessite le développement de méthodes spécifiques. C’est dans ce contexte que s’inscrit cette thèse qui vise à élaborer une nouvelle méthode de fusion évidentielle dédiée au traitement des images de télédétection hétérogènes à haute résolution. Afin d’atteindre cet objectif, nous axons notre recherche, en premier lieu, sur le développement d’une nouvelle approche pour l’estimation des fonctions de croyance basée sur la carte de Kohonen pour simplifier l’opération d’affectation des masses des gros volumes de données occupées par ces images. La méthode proposée permet de modéliser non seulement l’ignorance et l’imprécision de nos sources d’information, mais aussi leur paradoxe. Ensuite, nous exploitons cette approche d’estimation pour proposer une technique de fusion originale qui permettra de remédier aux problèmes dus à la grande variété des connaissances apportées par ces capteurs hétérogènes. Finalement, nous étudions la manière dont la dépendance entre ces sources peut être considérée dans le processus de fusion moyennant la théorie des copules. Pour cette raison, une nouvelle technique pour choisir la copule la plus appropriée est introduite. La partie expérimentale de ce travail est dédiée à la cartographie de l’occupation des sols dans les zones agricoles en utilisant des images SPOT-5 et RADARSAT-2. L’étude expérimentale réalisée démontre la robustesse et l’efficacité des approches développées dans le cadre de cette thèse. / With the advent of new image acquisition techniques and the emergence of high-resolution satellite systems, remote sensing data to be exploited have become increasingly rich and varied. Their combination has thus become essential to improve the process of extracting useful information related to the physical nature of the observed surfaces. However, these data are generally heterogeneous and imperfect, which poses several problems in their joint treatment and requires the development of specific methods. It is in this context that falls this thesis that aimed at developing a new evidential fusion method dedicated to heterogeneous remote sensing images processing at high resolution. In order to achieve this objective, we first focus our research, firstly, on the development of a new approach for the belief functions estimation based on Kohonen’s map in order to simplify the masses assignment operation of the large volumes of data occupied by these images. The proposed method allows to model not only the ignorance and the imprecision of our sources of information, but also their paradox. After that, we exploit this estimation approach to propose an original fusion technique that will solve problems due to the wide variety of knowledge provided by these heterogeneous sensors. Finally, we study the way in which the dependence between these sources can be considered in the fusion process using the copula theory. For this reason, a new technique for choosing the most appropriate copula is introduced. The experimental part of this work isdevoted to land use mapping in case of agricultural areas using SPOT-5 and RADARSAT-2 images. The experimental study carried out demonstrates the robustness and effectiveness of the approaches developed in the framework of this thesis.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-27T13:57:54Z No. of bitstreams: 1 DissACM.pdf: 2683283 bytes, checksum: 013455f7d8c0d48a1566d18bcdd0fbe8 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-10-03T18:18:22Z (GMT) No. of bitstreams: 1 DissACM.pdf: 2683283 bytes, checksum: 013455f7d8c0d48a1566d18bcdd0fbe8 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-10-03T18:18:38Z (GMT) No. of bitstreams: 1 DissACM.pdf: 2683283 bytes, checksum: 013455f7d8c0d48a1566d18bcdd0fbe8 (MD5) / Made available in DSpace on 2016-10-03T18:29:41Z (GMT). No. of bitstreams: 1 DissACM.pdf: 2683283 bytes, checksum: 013455f7d8c0d48a1566d18bcdd0fbe8 (MD5) Previous issue date: 2014-10-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The main objective of this research is to provide statistical tools that allow the comparison of individuals in a speci ed sports category. Particularly, the present study is focused on the performance evaluation in football using univariate and multivariate methods. The univariate approach is given by Z-CELAFISCS methodology, which was developed with the purpose of identifying talents in the sport. The multivariate approaches are given by the construction of indicators, speci cally by means of principal component analysis, factor analysis and copulas. These indicators allows the reduction of the dimensionality of the data in studying, providing better interpretation of the results and improving comparability between the performance and assortment of individuals. To facilitate the use of the methodology studied here was built an online statistical system called i-Sports. / principal objetivo do trabalho é apresentar ferramentas estatísticas que permitam a comparação de indivíduos em uma determinada modalidade esportiva. Particularmente, o estudo exposto é voltado à avaliação de desempenho em futebol, utilizando métodos univariados e multivariados. A abordagem univariada é dada pela metodologia Z-CELAFISCS, desenvolvida com o propósito de identi car talentos no esporte. As abordagens multivariadas são dadas pela construção de indicadores, mais especi camente por meio da análise de componentes principais, análise fatorial e cópulas. A obtenção desses indicadores possibilita a redução da dimensionalidade do estudo, fornecendo melhor interpretação dos resultados e melhor comparabilidade entre o desempenho e rankeamento dos indivíduos. Para facilitar a utilização da metodologia aqui estudada foi construído um sistema estat ístico online chamado de i-Sports.

In my thesis I~ deal with the design, implementation and testing of the advanced parallel Estimation of Distribution Algorithm (EDA) utilizing copula theory to create a~ probabilistic model. A~new population is created by the process of sampling the joint distribution function, which models the current distribution of the subpopulation of promising individuals . The usage of copulas increases the efficiency of the learning process and sampling the probabilistic model. It can be separated into mutually independent marginal distributions and the copula , which represents the correlations between the variables of the solved problem. This concept initiated the usage of the parallel island architecture , in which the migration of probabilistic models belonging to individual islands ' subpopulations was used instead of the migration of individuals . The statistical tests used in the comparison of the proposed algorithm ( mCEDA = migrating Copula - based Estimation of Distribution Algorithm ) and the algorithms of other authors confirmed the effectiveness of the proposed concept .