Medidas de dependência entre séries temporais são estudadas com a perspectiva de evidenciar como diferentes regiões do cérebro interagem, por meio da aplicação a sinais eletrofisiológicos. Baseado na representação auto-regressiva e espectral de séries temporais, diferentes medidas são comparadas entre si, incluindo coerência espectral e a coerência parcial direcionada, e introduz-se uma nova medida, denominada transferência parcial direcionada. As medidas são analisadas pelas propriedades de parcialização, relações diretas ou indiretas e direcionalidade temporal, e são mostradas suas relações com a correlação quadrática. Conclui-se que, entre as medidas analisadas, a coerência parcial direcionada e a transferência parcial direcionada possuem o maior número de características desejáveis, fundamentadas no conceito de causalidade de Granger. A estatística assintótica é desenvolvida para todas as medidas, incluindo intervalo de confiança e teste de hipótese nula, assim como sua implementação computacional. A aplicação a séries simuladas e a análise de dados eletrofisiológicos reais ilustram o estudo comparativo e a aplicabilidade das novas estatísticas apresentadas. / Measures of dependence between temporal series are studied in the context of revealing how different brain regions interact, through their application to electrophysiology. Based on the spectral and autoregressive model of time series, different measures are compared, including coherence and partial directed coherence, and a new measure is introduced, named partial directed transfer. The measures are analyzed through the properties of partialization, direct or indirect relations and temporal directionality, and their relation to quadratic correlation is shown. It results that among the presented measures, partial directed coherence and partial directed transfer reveal the highest number of desirable properties, being grounded on the concept of Granger causality. The asymptotic statistics for all measures are developed, including confidence intervals and null hypothesis testing, as well as their computational implementation. The application to simulated series and the analysis of electrophysiological data illustrate the comparative study and the applicability of the newly presented statistics.

Les modèles de croissance de plantes peuvent être utilisés afin de prédire des quantités d’intérêt ou évaluer la variabilité génotypique au sein d’une population de plantes ; ce double usage est mis en évidence au sein de ce travail. Trois modèles de plantes sont ainsi considérés (LNAS pour la betterave et le blé, GreenLab pour Arabidopsis thaliana) au sein du cadre mathématique des modèles à espace d’états généraux.Une nouvelle plate-forme de calcul générique pour la modélisation et l’inférence statistique (ADJUSTIN’) a été développée en Julia, permettant la simulation des modèles de croissance de plantes considérés ainsi que l’utilisation de techniques d’estimation de pointe telles que les méthodes de Monte Carlo par chaînes de Markov ou de Monte Carlo séquentielles.L’inférence statistique au sein des modèles de croissance de plantes étant de première importance pour des applications concrètes telles que la prédiction de rendement, les méthodes d’estimation de paramètres et d’états au sein de modèles à espaces d’états et dans un cadre bayésien furent tout d’abord étudiées, et plusieurs cas d’étude pour les plantes considérées sont analysés pour le cas d’une plante individuelle.La caractérisation de la variabilité au sein d’une population de plantes est envisagée à travers les distributions des paramètres de population au sein de modèles hiérarchiques bayésiens. Cette approche requérant l’acquisition de nombreuses données pour chaque individu, un algorithme de segmentation-suivi pour l’analyse d’images d’Arabidopsis thaliana, obtenues grâce au Phénoscope, une plate-forme de phénotypage à haut rendement de l’INRA Versailles, est proposé.Finalement, l’intérêt de l’utilisation des modèles hiérarchiques bayésiens pour la mise en évidence de la variabilité au sein d’une population de plantes est discutée. D’abord par l’étude de différents scénarios sur des données simulées, et enfin en utilisant les données expérimentales obtenues à partir de l’analyse d’images pour une population d’Arabidopsis thaliana comprenant 48 individus. / Plant growth models can be used in order to predict quantities of interest or assess the genotypic variability of a population of plants; this dual use is emphasized throughout this work.Three plant growth models are therefore considered (LNAS for sugar beet and wheat, GreenLab for Arabidopsis thaliana) within the mathematical framework of general state space models.A new generic computing platform for modelling and statistical inference (ADJUSTIN’) has been developed in Julia, allowing to simulate the plant growth models considered as well as the use of state-of-the-art estimation techniques such as Markov chain Monte Carlo and sequential Monte Carlo methods.Statistical inference within plant growth models is of primary importance for concrete applications such as yield prediction, parameter and state estimation methods within general state-space models in a Bayesian framework were first studied and several case studies for the plants considered are then investigated in the case of an individual plant.The characterization of the variability of a population of plants is envisioned through the distributions of parameters using Bayesian hierarchical models. This approach requiring the acquisition of numerous data for each individual, a segmentation-tracking algorithm for the analysis of images of Arabidopsis thaliana, obtained thanks to the Phenoscope, a high-throughput phenotyping platform of INRA Versailles, is proposed.Finally, the interest of using Bayesian hierarchical models to evidence the variability of a population of plants is discussed. First through the study of different scenarios on simulated data, and then by using the experimental data acquired via image analysis for the population of Arabidopsis thaliana comprising 48 individuals.

<p>In this thesis, we present algorithms to allow agents to estimate certain properties in a robust, online, and distributed manner. Each agent receives a sequence of observations, and through communication, collectively infers properties of the data gathered by all agents by communicating.</p> <p><br></p> <p>In the first part of the thesis, we provide algorithms to infer the correlations between interacting entities from these large datasets. Gaussian graphical models have been well studied to represent the relationships between the various random variables which generate data, and numerous algorithms have been proposed to learn the dependencies in such models. However, existing algorithms typically process data in a batch at a central location, limiting their applications in scenarios where data arrive in real-time and are gathered by different agents.  </p> <p><br></p> <p>To address these challenges, first, we propose an online sparse inverse covariance algorithm to infer the static network structure (i.e., dependencies between nodes) in real-time from time-series data, in a centralized location. Subsequently, we propose a distributed algorithm to cooperatively learn the network structure in real-time from data collected by distributed agents. We characterize the theoretical convergence properties and provide simulations using synthetic datasets and real-world hurricane Twitter datasets in disaster management applications.    </p> <p><br></p> <p>The second part of this thesis addresses the robustness of online and distributed learning under arbitrary data corruption. We propose online and distributed algorithms for robust mean, covariance, and sparse inverse covariance estimation. These algorithms are capable of operating effectively even in the presence of adversarial data attacks. We provide theoretical bounds on the error and rate of convergence of these methods and evaluate their performance under various settings.</p> <p><br></p> <p>Finally, we consider the problem of classification with a network of heterogeneous and partially informative agents, each receiving local data from an underlying true class, and equipped with a classifier that only distinguishes between a subset of the entire set of classes. We propose an iterative algorithm that uses the posterior probabilities of any classifier and recursively updates each agent's local belief based on its local signals and belief information from its neighbors. We then adopt a novel distributed min-rule to update each agent’s global belief and enable learning of the true class for all agents. We analyze the convergence properties of our proposed algorithm, and subsequently, demonstrate and compare its performance with local averaging and global average consensus through simulations and with a visual image dataset.</p>

This thesis investigates the possibility of using Sequential Monte Carlo methods (SMC) to create an online algorithm to infer properties from a dataset, such as unknown model parameters. Statistical inference from data streams tends to be difficult, and this is particularly the case for parametric models, which will be the focus of this paper. We develop a sequential Monte Carlo algorithm sampling sequentially from the model's posterior distributions. As a key ingredient of this approach, unknown static parameters are jittered towards the shrinking support of the posterior on the basis of an artificial Markovian dynamic allowing for correct pseudo-marginalisation of the target distributions. We then test the algorithm on a simple Gaussian model, a Gausian Mixture Model (GMM), as well as a variable dimension GMM. All tests and coding were done using Matlab. The outcome of the simulation is promising, but more extensive comparisons to other online algorithms for static parameter models are needed to really gauge the computational efficiency of the developed algorithm. / Detta examensarbete undersöker möjligheten att använda Sekventiella Monte Carlo metoder (SMC) för att utveckla en algoritm med syfte att utvinna parametrar i realtid givet en okänd modell. Då statistisk slutledning från dataströmmar medför svårigheter, särskilt i parameter-modeller, kommer arbetets fokus ligga i utvecklandet av en Monte Carlo algoritm vars uppgift är att sekvensiellt nyttja modellens posteriori fördelningar. Resultatet är att okända, statistiska parametrar kommer att förflyttas mot det krympande stödet av posterioren med hjälp utav en artificiell Markov dynamik, vilket tillåter en korrekt pseudo-marginalisering utav mål-distributionen. Algoritmen kommer sedan att testas på en enkel Gaussisk-modell, en Gaussisk mixturmodell (GMM) och till sist en GMM vars dimension är okänd. Kodningen i detta projekt har utförts i Matlab.

No presente trabalho, conceitos de inferência estatística são utilizados para aplicação e comparação de diferentes métodos de análise de sinais de ressonância magnética funcional. A idéia central baseia-se na obtenção da distribuição de probabilidade da variável aleatória de interesse, para cada método estudado e sob diferentes valores da relação sinal-ruído (SNR). Este objetivo é atingido através de simulações numéricas da função resposta hemodinâmica (HRF) acrescida de ruído gaussiano. Tal procedimento nos permite avaliar a sensibilidade e a especificidade dos métodos empregados através da construção das curvas ROC (receiver operating characteristic) para diferentes valores de SNR. Sob específicas condições experimentais, aplicamos métodos clássicos de análise (teste t de Student e correlação), medidas de informação (distância de Kullback-Leibler e sua forma generalizada) e um método Bayesiano (método do pixel independente). Em especial, mostramos que a distância de Kullback-Leibler (D) (ou entropia relativa) e sua forma generalizada são medidas úteis para análise de sinais dentro do cenário de teoria da informação. Estas entropias são usadas como medidas da \"distância\"entre as funções de probabilidade p1 e p2 dos níveis do sinal relacionados a estímulo e repouso. Para prevenir a ocorrência de valores divergentes de D, introduzimos um pequeno parâmetro d nas definições de p1 e p2. Estendemos a análise, apresentando um estudo original da distância de Kullback-Leibler generalizada Dq (q é o parâmetro de Tsallis). Neste caso, a escolha apropriada do intervalo 0 < q < 1 permite assegurar que Dq seja finito. Obtemos as densidades de probabilidade f (D) e f (Dq) das médias amostrais das variáveis D e Dq , respectivamente, calculadas ao longo das N épocas de todo o experimento. Para pequenos valores de N (N < 30), mostramos que f (D) e f (Dq) são muito bem aproximadas por distribuições Gamma (qui^2 < 0,0009). Em seguida, estudamos o método (Bayesiano) do pixel independente, considerando a probabilidade a posteriori como variável aleatória e obtendo sua distribuição para várias SNR\'s e probabilidades a priori. Os resultados das simulações apontam para o fato de que a correlação e o método do pixel independente apresentam melhor desempenho do que os demais métodos empregados (para SNR > -20 dB). Contudo, deve-se ponderar que o teste t e os métodos entrópicos compartilham da vantagem de não se utilizarem de um modelo para HRF na análise de dados reais. Finalmente, para os diferentes métodos, obtemos os mapas funcionais correspondentes a séries de dados reais de um voluntário assintomático submetido a estímulo motor de evento relacionado, os quais demonstram ativação nas áreas cerebrais motoras primária e secundária. Enfatizamos que o procedimento adotado no presente estudo pode, em princípio, ser utilizado em outros métodos e sob diferentes condições experimentais. / In the present work, concepts of statistical inference are used for application and comparison of different methods of signal analysis in functional magnetic resonance imaging. The central idea is based on obtaining the probability distribution of the random variable of interest, for each method studied under different values of signal-to-noise ratio (SNR). This purpose is achieved by means of numerical simulations of the hemodynamic response function (HRF) with gaussian noise. This procedure allows us to assess the sensitivity and specificity of the methods employed by the construction of the ROC curves (receiver operating characteristic) for different values of SNR. Under specific experimental conditions, we apply classical methods of analysis (Student\'s t test and correlation), information measures (distance of Kullback-Leibler and its generalized form) and a Bayesian method (independent pixel method). In particular, we show that the distance of Kullback-Leibler D (or relative entropy) and its generalized form are useful measures for analysis of signals within the information theory scenario. These entropies are used as measures of the \"distance\"between the probability functions p1 and p2 of the signal levels related to stimulus and non-stimulus. In order to avoid undesirable divergences of D, we introduced a small parameter d in the definitions of p1 and p2. We extend such analysis, by presenting an original study of the generalized Kullback-Leibler distance Dq (q is Tsallis parameter). In this case, the appropriate choice of range 0 < q < 1 ensures that Dq is finite. We obtain the probability densities f (D) and f (Dq) of the sample averages of the variables D and Dq, respectively, calculated over the N epochs of the entire experiment. For small values of N (N < 30), we show that f (D) and f (Dq) are well approximated by Gamma distributions (qui^2 < 0.0009). Afterward, we studied the independent pixel bayesian method, considering the probability a posteriori as a random variable, and obtaining its distribution for various SNR\'s and probabilities a priori. The results of simulations point to the fact that the correlation and the independent pixel method have better performance than the other methods used (for SNR> -20 dB). However, one should consider that the Student\'s t test and the entropic methods share the advantage of not using a model for HRF in real data analysis. Finally, we obtain the maps corresponding to real data series from an asymptomatic volunteer submitted to an event-related motor stimulus, which shows brain activation in the primary and secondary motor brain areas. We emphasize that the procedure adopted in this study may, in principle, be used in other methods and under different experimental conditions.

Il est attendu que les radios flexibles constituent un tournant technologique majeur dans le domaine des communications sans fil. Le point de vue adopté en radios flexibles est de considérer les canaux de communication comme un ensemble de ressources qui peuvent être accédées sur demande par un réseau primaire sous licence ou de manière opportuniste par un réseau secondaire à plus faible priorité. Du point de vue de la couche physique, le réseau primaire n’a aucune information sur l’existence de réseaux secondaires, de sorte que ces derniers doivent explorer l’environnement aérien de manière autonome à la recherche d’opportunités spectrales et exploiter ces ressources de manière optimale. Les phases d’exploration et d’exploitation, qui impliquent la gestion de nombreux agents, doivent être très fiables, rapides et efficaces. L’objectif de cette thèse est de modéliser, d’analyser et de proposer des solutions efficaces et quasi optimales pour ces dernières opérations.En ce qui concerne la phase d’exploration, nous calculons le test optimal de Neyman-Pearson de détection de plusieurs sources primaires via un réseau de capteurs. Cette procédure permet à un réseau secondaire d’établir la présence de ressources spectrales disponibles. La complexité calculatoire de l’approche optimale appelle cependant la mise en place de méthodes moins onéreuses, que nous rappelons et discutons. Nous étendons alors le test de détection en l’estimation aveugle de la position de sources multiples, qui permet l’acquisition d’informations détaillées sur les ressources spectrales disponibles.Le second volet de cette thèse est consacré à la phase d’exploitation optimale des ressources au niveau du réseau secondaire. Pour ce faire, nous obtenons une approximation fine du débit ergodique d’un canal multi-antennes à accès multiples et proposons des solutions peu coûteuses en termes de feedback afin que les réseaux secondaires s’adaptent rapidement aux évolutions rapides du réseau primaire. / Future cognitive radio networks are expected to come as a disruptive technological advance in the currently saturated field of wireless communications. The idea behind cognitive radios is to think of the wireless channels as a pool of communication resources, which can be accessed on-demand by a primary licensed network or opportunistically preempted (or overlaid) by a secondary network with lower access priority. From a physical layer point of view, the primary network is ideally oblivious of the existence of a co-localized secondary networks. The latter are therefore required to autonomously explore the air in search for resource left-overs, and then to optimally exploit the available resource. The exploration and exploitation procedures, which involve multiple interacting agents, are requested to be highly reliable, fast and efficient. The objective of the thesis is to model, analyse and propose computationally efficient and close-to-optimal solutions to the above operations.Regarding the exploration phase, we first resort to the maximum entropy principle to derive communication models with many unknowns, from which we derive the optimal multi-source multi-sensor Neyman-Pearson signal sensing procedure. The latter allows for a secondary network to detect the presence of spectral left-overs. The computational complexity of the optimal approach however calls for simpler techniques, which are recollected and discussed. We then proceed to the extension of the signal sensing approach to the more advanced blind user localization, which provides further valuable information to overlay occupied spectral resources.The second part of the thesis is dedicaded to the exploitation phase, that is, the optimal sharing of available resources. To this end, we derive an (asymptotically accurate) approximated expression for the uplink ergodic sum rate of a multi-antenna multiple-access channel and propose solutions for cognitive radios to adapt rapidly to the evolution of the primary network at a minimum feedback cost for the secondary networks.

Essa dissertação apresenta o desenvolvimento métodos para caracterização da conectividade entre séries temporais neurofisiológicas. Utilizam-se metodologias provenientes da Teoria da Informação Entropias Aproximada e Amostral para representar a complexidade da série no tempo, o que permite inferir como sua variabilidade se transfere a outras sequências, através do uso da coerência parcial direcionada. Para cada sistema analisado: (1) Faz-se uma transformação em outro, relacionando-o às medidas de entropia, (2) Estima-se a conectividade pela coerência parcial direcionada e (3) Avalia-se a robustez do procedimento via simulações de Monte Carlo e análise de sensibilidade. Para os exemplos simulados, a técnica proposta é capaz de oferecer resultados plausíveis, através da correta inferência da direção de conectividade em casos de acoplamento não-linear (quadrático), com número reduzido de amostras temporais dos sinais, em que outras abordagens falham. Embora de simples implementação, conclui-se que o processo mostra-se como uma extensão da causalidade de Granger para o caso não-linear. / The purpose of this work is to present the development of methods for characterizing the connectivity between nonlinear neurophysiological time series. Methodologies from Information Theory Approximate and Sample Entropies are used to represent the complexity of the series in a period of time, which allows inferring on how its variability is transferred to other sequences, using partial directed coherence. Methods: For each system under consideration, (1) It is done a transformation in another, relating it to measures of entropy, (2) The connectivity is estimated by the use of partial directed coherence and (3) The robustness of the procedure is analyzed via Monte Carlo simulations and sensitivity analysis. Results: For the simulated examples, the proposed technique is able to offer plausible results, through the correct inference of the connectivity direction, in cases of nonlinear coupling (quadratic), with a reduced number of signals samples, where other approaches fail. Conclusion: The process proves to be an extension of the Granger causality to the nonlinear case.

Essa dissertação apresenta o desenvolvimento métodos para caracterização da conectividade entre séries temporais neurofisiológicas. Utilizam-se metodologias provenientes da Teoria da Informação Entropias Aproximada e Amostral para representar a complexidade da série no tempo, o que permite inferir como sua variabilidade se transfere a outras sequências, através do uso da coerência parcial direcionada. Para cada sistema analisado: (1) Faz-se uma transformação em outro, relacionando-o às medidas de entropia, (2) Estima-se a conectividade pela coerência parcial direcionada e (3) Avalia-se a robustez do procedimento via simulações de Monte Carlo e análise de sensibilidade. Para os exemplos simulados, a técnica proposta é capaz de oferecer resultados plausíveis, através da correta inferência da direção de conectividade em casos de acoplamento não-linear (quadrático), com número reduzido de amostras temporais dos sinais, em que outras abordagens falham. Embora de simples implementação, conclui-se que o processo mostra-se como uma extensão da causalidade de Granger para o caso não-linear. / The purpose of this work is to present the development of methods for characterizing the connectivity between nonlinear neurophysiological time series. Methodologies from Information Theory Approximate and Sample Entropies are used to represent the complexity of the series in a period of time, which allows inferring on how its variability is transferred to other sequences, using partial directed coherence. Methods: For each system under consideration, (1) It is done a transformation in another, relating it to measures of entropy, (2) The connectivity is estimated by the use of partial directed coherence and (3) The robustness of the procedure is analyzed via Monte Carlo simulations and sensitivity analysis. Results: For the simulated examples, the proposed technique is able to offer plausible results, through the correct inference of the connectivity direction, in cases of nonlinear coupling (quadratic), with a reduced number of signals samples, where other approaches fail. Conclusion: The process proves to be an extension of the Granger causality to the nonlinear case.

Recent experimental techniques in biology made possible the acquisition of overwhelming amounts of data concerning complex biological networks, such as neural networks, gene regulation networks and protein-protein interaction networks. These techniques are able to record states of individual components of such networks (neurons, genes, proteins) for a large number of configurations. However, the most biologically relevantinformation lies in their connectivity and in the way their components interact, information that these techniques aren't able to record directly. The aim of this thesis is to study statistical methods for inferring information about the connectivity of complex networks starting from experimental data. The subject is approached from a statistical physics point of view drawing from the arsenal of methods developed in the study of spin glasses. Spin-glasses are prototypes of networks of discrete variables interacting in a complex way and are widely used to model biological networks. After an introduction of the models used and a discussion on the biological motivation of the thesis, all known methods of network inference are introduced and analysed from the point of view of their performance. Then, in the third part of the thesis, a new method is proposed which relies in the remark that the interactions in biology are not necessarily symmetric (i.e. the interaction from node A to node B is not the same as the one from B to A). It is shown that this assumption leads to methods that are both exact and efficient. This means that the interactions can be computed exactly, given a sufficient amount of data, and in a reasonable amount of time. This is an important original contribution since no other method is known to be both exact and efficient.

No presente trabalho, conceitos de inferência estatística são utilizados para aplicação e comparação de diferentes métodos de análise de sinais de ressonância magnética funcional. A idéia central baseia-se na obtenção da distribuição de probabilidade da variável aleatória de interesse, para cada método estudado e sob diferentes valores da relação sinal-ruído (SNR). Este objetivo é atingido através de simulações numéricas da função resposta hemodinâmica (HRF) acrescida de ruído gaussiano. Tal procedimento nos permite avaliar a sensibilidade e a especificidade dos métodos empregados através da construção das curvas ROC (receiver operating characteristic) para diferentes valores de SNR. Sob específicas condições experimentais, aplicamos métodos clássicos de análise (teste t de Student e correlação), medidas de informação (distância de Kullback-Leibler e sua forma generalizada) e um método Bayesiano (método do pixel independente). Em especial, mostramos que a distância de Kullback-Leibler (D) (ou entropia relativa) e sua forma generalizada são medidas úteis para análise de sinais dentro do cenário de teoria da informação. Estas entropias são usadas como medidas da \"distância\"entre as funções de probabilidade p1 e p2 dos níveis do sinal relacionados a estímulo e repouso. Para prevenir a ocorrência de valores divergentes de D, introduzimos um pequeno parâmetro d nas definições de p1 e p2. Estendemos a análise, apresentando um estudo original da distância de Kullback-Leibler generalizada Dq (q é o parâmetro de Tsallis). Neste caso, a escolha apropriada do intervalo 0 < q < 1 permite assegurar que Dq seja finito. Obtemos as densidades de probabilidade f (D) e f (Dq) das médias amostrais das variáveis D e Dq , respectivamente, calculadas ao longo das N épocas de todo o experimento. Para pequenos valores de N (N < 30), mostramos que f (D) e f (Dq) são muito bem aproximadas por distribuições Gamma (qui^2 < 0,0009). Em seguida, estudamos o método (Bayesiano) do pixel independente, considerando a probabilidade a posteriori como variável aleatória e obtendo sua distribuição para várias SNR\'s e probabilidades a priori. Os resultados das simulações apontam para o fato de que a correlação e o método do pixel independente apresentam melhor desempenho do que os demais métodos empregados (para SNR > -20 dB). Contudo, deve-se ponderar que o teste t e os métodos entrópicos compartilham da vantagem de não se utilizarem de um modelo para HRF na análise de dados reais. Finalmente, para os diferentes métodos, obtemos os mapas funcionais correspondentes a séries de dados reais de um voluntário assintomático submetido a estímulo motor de evento relacionado, os quais demonstram ativação nas áreas cerebrais motoras primária e secundária. Enfatizamos que o procedimento adotado no presente estudo pode, em princípio, ser utilizado em outros métodos e sob diferentes condições experimentais. / In the present work, concepts of statistical inference are used for application and comparison of different methods of signal analysis in functional magnetic resonance imaging. The central idea is based on obtaining the probability distribution of the random variable of interest, for each method studied under different values of signal-to-noise ratio (SNR). This purpose is achieved by means of numerical simulations of the hemodynamic response function (HRF) with gaussian noise. This procedure allows us to assess the sensitivity and specificity of the methods employed by the construction of the ROC curves (receiver operating characteristic) for different values of SNR. Under specific experimental conditions, we apply classical methods of analysis (Student\'s t test and correlation), information measures (distance of Kullback-Leibler and its generalized form) and a Bayesian method (independent pixel method). In particular, we show that the distance of Kullback-Leibler D (or relative entropy) and its generalized form are useful measures for analysis of signals within the information theory scenario. These entropies are used as measures of the \"distance\"between the probability functions p1 and p2 of the signal levels related to stimulus and non-stimulus. In order to avoid undesirable divergences of D, we introduced a small parameter d in the definitions of p1 and p2. We extend such analysis, by presenting an original study of the generalized Kullback-Leibler distance Dq (q is Tsallis parameter). In this case, the appropriate choice of range 0 < q < 1 ensures that Dq is finite. We obtain the probability densities f (D) and f (Dq) of the sample averages of the variables D and Dq, respectively, calculated over the N epochs of the entire experiment. For small values of N (N < 30), we show that f (D) and f (Dq) are well approximated by Gamma distributions (qui^2 < 0.0009). Afterward, we studied the independent pixel bayesian method, considering the probability a posteriori as a random variable, and obtaining its distribution for various SNR\'s and probabilities a priori. The results of simulations point to the fact that the correlation and the independent pixel method have better performance than the other methods used (for SNR> -20 dB). However, one should consider that the Student\'s t test and the entropic methods share the advantage of not using a model for HRF in real data analysis. Finally, we obtain the maps corresponding to real data series from an asymptomatic volunteer submitted to an event-related motor stimulus, which shows brain activation in the primary and secondary motor brain areas. We emphasize that the procedure adopted in this study may, in principle, be used in other methods and under different experimental conditions.