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<p dir="ltr">It has been shown that formative assessments can lead to improvements in the learning process. Cognitive Diagnostic Models (CDMs) are a powerful formative assessment tool that can be used to provide individuals with valuable information regarding skill mastery in educational settings. These models provide each student with a ``skill mastery profile'' that shows the level of mastery they have obtained with regard to a specific set of skills. These profiles can be used to help both students and educators make more informed decisions regarding the educational process, which can in turn accelerate learning for students. However, despite their utility, these models are rarely used with small sample sizes. One reason for this is that these models are often complex, containing many parameters that can be difficult to estimate accurately when working with a small number of observations. This work aims to contribute to and expand upon previous work to make CDMs more accessible for a wider range of educators and students.</p><p dir="ltr">There are three main small sample statistical problems that we address in this work: 1) accurate estimation of the population distribution of skill mastery profiles, 2) accurate estimation of additional model parameters for CDMs as well as improved classification of individual skill mastery profiles, and 3) improved selection of an appropriate CDM for each item on the assessment. Each of these problems deals with a different aspect of educational measurement and the solutions provided to these problems can ultimately lead to improvements in the educational process for both students and teachers. By finding solutions to these problems that work well when using small sample sizes, we make it possible to improve learning in everyday classroom settings and not just in large scale assessment settings.</p><p dir="ltr">In the first part of this work, we propose novel algorithms for estimating the population distribution of skill mastery profiles for a popular CDM, the Deterministic Inputs Noisy ``and'' Gate (DINA) model. These algorithms borrow inspiration from the concepts behind popular machine learning algorithms. However, in contrast to these methods, which are often used solely for prediction, we illustrate how the ideas behind these methods can be adapted to obtain estimates of specific model parameters. Through studies involving simulated and real-life data, we illustrate how the proposed algorithms can be used to gain a better picture of the distribution of skill mastery profiles for an entire population students, but can do so by only using a small sample of students from that population. </p><p dir="ltr">In the second part of this work, we introduce a new method for regularizing high-dimensional CDMs using a class of Bayesian shrinkage priors known as catalytic priors. We show how a simpler model can first be fit to the observed data and then be used to generate additional pseudo-observations that, when combined with the original observations, make it easier to more accurately estimate the parameters in a complex model of interest. We propose an alternative, simpler model that can be used instead of the DINA model and show how the information from this model can be used to formulate an intuitive shrinkage prior that effectively regularizes model parameters. This makes it possible to improve the accuracy of parameter estimates for the more complex model, which in turn leads to better classification of skill mastery. We demonstrate the utility of this method in studies involving simulated and real-life data and show how the proposed approach is superior to other common approaches for small sample estimation of CDMs.</p><p dir="ltr">Finally, we discuss the important problem of selecting the most appropriate model for each item on assessment. Often, it is not uncommon in practice to use the same CDM for each item on an assessment. However, this can lead to suboptimal results in terms of parameter estimation and overall model fit. Current methods for item-level model selection rely on large sample asymptotic theory and are thus inappropriate when the sample size is small. We propose a Bayesian approach for performing item-level model selection using Reversible Jump Markov chain Monte Carlo. This approach allows for the simultaneous estimation of posterior probabilities and model parameters for each candidate model and does not require a large sample size to be valid. We again demonstrate through studies involving simulated and real-life data that the proposed approach leads to a much higher chance of selecting the best model for each item. This in turn leads to better estimates of item and other model parameters, which ultimately leads to more accurate information regarding skill mastery. </p>

Thesis (MComm)--University of Stellenbosch, 2003. / Please refer to full text for abstract.

The mechanisms that control growth processes in biology tissues have attracted continuous research interest despite their complexity. With the emergence of big data experimental approaches there is an urgent need to develop statistical and computational models to fit the experimental data and that can be used to make predictions to guide future research. In this work we apply statistical methods on growth process of different biological tissues, focusing on development of neuron dendrites and tumor cells. We first examine the neuron cell growth process, which has implications in neural tissue regenerations, by using a computational model with uniform branching probability and a maximum overall length constraint. One crucial outcome is that we can relate the parameter fits from our model to real data from our experimental collaborators, in order to examine the usefulness of our model under different biological conditions. Our methods can now directly compare branching probabilities of different experimental conditions and provide confidence intervals for these population-level measures. In addition, we have obtained analytical results that show that the underlying probability distribution for this process follows a geometrical progression increase at nearby distances and an approximately geometrical series decrease for far away regions, which can be used to estimate the spatial location of the maximum of the probability distribution. This result is important, since we would expect maximum number of dendrites in this region; this estimate is related to the probability of success for finding a neural target at that distance during a blind search. We then examined tumor growth processes which have similar evolutional evolution in the sense that they have an initial rapid growth that eventually becomes limited by the resource constraint. For the tumor cells evolution, we found an exponential growth model best describes the experimental data, based on the accuracy and robustness of models. Furthermore, we incorporated this growth rate model into logistic regression models that predict the growth rate of each patient with biomarkers; this formulation can be very useful for clinical trials. Overall, this study aimed to assess the molecular and clinic pathological determinants of breast cancer (BC) growth rate in vivo.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

L'objet principal de cette thèse est de développer des estimateurs adaptatifs en statistique pour données fonctionnelles. Dans une première partie, nous nous intéressons au modèle linéaire fonctionnel et nous définissons un critère de sélection de la dimension pour des estimateurs par projection définis sur des bases fixe ou aléatoire. Les estimateurs obtenus vérifient une inégalité de type oracle et atteignent la vitesse de convergence minimax pour le risque lié à l'erreur de prédiction. Pour les estimateurs définis sur une collection de modèles aléatoires, des outils de théorie de la perturbation ont été utilisés pour contrôler les projecteurs aléatoires de manière non-asymptotique. D'un point de vue numérique, cette méthode de sélection de la dimension est plus rapide et plus stable que les méthodes usuelles de validation croisée. Dans une seconde partie, nous proposons un critère de sélection de fenêtre inspiré des travaux de Goldenshluger et Lepski, pour des estimateurs à noyau de la fonction de répartition conditionnelle lorsque la covariable est fonctionnelle. Le risque de l'estimateur obtenu est majoré de manière non-asymptotique. Des bornes inférieures sont prouvées ce qui nous permet d'établir que notre estimateur atteint la vitesse de convergence minimax, à une perte logarithmique près. Dans une dernière partie, nous proposons une extension au cadre fonctionnel de la méthodologie des surfaces de réponse, très utilisée dans l'industrie. Ce travail est motivé par une application à la sûreté nucléaire. / The main purpose of this thesis is to develop adaptive estimators for functional data.In the first part, we focus on the functional linear model and we propose a dimension selection device for projection estimators defined on both fixed and data-driven bases. The prediction error of the resulting estimators satisfies an oracle-type inequality and reaches the minimax rate of convergence. For the estimator defined on a data-driven approximation space, tools of perturbation theory are used to solve the problems related to the random nature of the collection of models. From a numerical point of view, this method of dimension selection is faster and more stable than the usual methods of cross validation.In a second part, we consider the problem of bandwidth selection for kernel estimators of the conditional cumulative distribution function when the covariate is functional. The method is inspired by the work of Goldenshluger and Lepski. The risk of the estimator is non-asymptotically upper-bounded. We also prove lower-bounds and establish that our estimator reaches the minimax convergence rate, up to an extra logarithmic term.In the last part, we propose an extension to a functional context of the response surface methodology, widely used in the industry. This work is motivated by an application to nuclear safety.

Ce manuscrit se concentre sur deux problèmes d'estimation de fonction. Pour chacun, une garantie non asymptotique des performances de l'estimateur proposé est fournie par une inégalité d'oracle. Pour l'estimation de densité conditionnelle, des mélanges de régressions gaussiennes à poids exponentiels dépendant de la covariable sont utilisés. Le principe de sélection de modèle par maximum de vraisemblance pénalisé est appliqué et une condition sur la pénalité est établie. Celle-ci est satisfaite pour une pénalité proportionnelle à la dimension du modèle. Cette procédure s'accompagne d'un algorithme mêlant EM et algorithme de Newton, éprouvé sur données synthétiques et réelles. Dans le cadre de la régression à bruit sous-gaussien, l'agrégation à poids exponentiels d'estimateurs linéaires permet d'obtenir une inégalité d'oracle en déviation, au moyen de techniques PAC-bayésiennes. Le principal avantage de l'estimateur proposé est d'être aisément calculable. De plus, la prise en compte de la norme infinie de la fonction de régression permet d'établir un continuum entre inégalité exacte et inexacte. / This manuscript focuses on two functional estimation problems. A non asymptotic guarantee of the proposed estimator’s performances is provided for each problem through an oracle inequality.In the conditional density estimation setting, mixtures of Gaussian regressions with exponential weights depending on the covariate are used. Model selection principle through penalized maximum likelihood estimation is applied and a condition on the penalty is derived. If the chosen penalty is proportional to the model dimension, then the condition is satisfied. This procedure is accompanied by an algorithm mixing EM and Newton algorithm, tested on synthetic and real data sets. In the regression with sub-Gaussian noise framework, aggregating linear estimators using exponential weights allows to obtain an oracle inequality in deviation,thanks to pac-bayesian technics. The main advantage of the proposed estimator is to be easily calculable. Furthermore, taking the infinity norm of the regression function into account allows to establish a continuum between sharp and weak oracle inequalities.

As distribuições de abundância relativa das espécies tiveram um papel importante no desenvolvimento da ecologia de comunidades, revelando um dos padrões mais bem estabelecidos da ecologia, que é a alta dominância de algumas espécies nas comunidades biológicas. Este padrão provocou a criação de dezenas de modelos teóricos na tentativa de explicar quais mecanismos ecológicos poderiam gerá-lo. Os modelos teóricos de abundância relativa das espécies podem ser vistos como descritores das comunidades, e seus parâmetros, medidas sintéticas de dimensões da diversidade. Esses parâmetros podem ser utilizados não só como descritores biologicamente interpretáveis das comunidades, mas também como variáveis respostas a possíveis fatores ambientais que afetam as comunidades. Adotando então esta aplicação descritiva dos modelos, nosso objetivo foi comparar as comunidades de aves de áreas em um gradiente de fragmentação, utilizando como variável resposta os valores estimados do parâmetro do modelo série logarítmica, o &#945; de Fisher. Como todos os modelos teóricos de abundância relativa propostos têm como premissa, a igualdade de probabilidade de captura entre as espécies, o que para comunidades de espécies de organismos móveis, como aves, parece pouco realista, neste trabalho investigamos também o grau de sensibilidade dos modelos quanto à quebra dessa premissa. Assim, por meio de simulações de comunidades, analisamos o viés de seleção e estimação, e revelamos que o aumento do grau de heterogeneidade entre as probabilidades de captura das espécies acarreta no incremento do viés de seleção do modelo real e também de estimação dos parâmetros. Porém, como o objetivo do estudo era identificar os fatores que influenciam a diversidade das comunidades, mesmo com o viés de estimação, talvez ainda fosse possível revelar o grau de influência sobre os valores dos parâmetros, quando ele existir. Assim, prosseguimos com mais uma etapa de simulações, em que geramos comunidades cujos valores de parâmetros tinham uma relação linear com a área dos fragmentos. O que encontramos é que independente da igualdade ou desigualdade de capturabilidade das espécies, quando o efeito existe, ele é sempre detectado, porém dependendo do grau de diferença de probabilidade de captura das espécies, o efeito pode ser subestimado. E, na ausência de efeito, ele pode ser falsamente detectado, dependendo do grau de heterogeneidade de probabilidades de captura entre as espécies, mas sempre com estimativas bem baixas para o efeito inexistente. Com esses resultados então, pudemos quantificar os tipos de efeitos da heterogeneidade de probabilidades de captura e prosseguir com as análises dos efeitos de fragmentação. O que nossos resultados mostraram é que na paisagem com 10% de cobertura vegetal, a área parece influenciar a diversidade dos fragmentos mais que o isolamento, e que na paisagem de 50% de cobertura vegetal, a variável de isolamento se torna mais importante que a área para explicar os dados. Porém, em uma interpretação mais parcimoniosa, consideramos as estimativas dos efeitos muito baixas para considerar que ele de fato existia. Com isso, concluímos que o processo de fragmentação provavelmente não tem efeito sobre a hierarquia de abundância das espécies, e é independente da porcentagem de cobertura vegetal da paisagem. Contudo, em uma descrição do número de capturas de cada espécie nos fragmentos, ponderada pelo número de capturas amostrado em áreas contínuas adjacentes, revelaram que o tamanho do fragmento pode ser importante na determinação de quais espécies serão extintas ou beneficiadas e que talvez a qualidade da matriz seja decisiva para a manutenção de espécies altamente sensíveis em fragmentos pequenos. Assim, demonstramos que, embora as SADs sejam pouco afetadas pela fragmentação, a posição das espécies na hierarquia de abundâncias pode mudar muito, o que reflete as diferenças de sensibilidade das espécies a área e isolamento dos fragmentos. / Species abundance distribution (SADs) had an important role in community ecology, revealing one of the most well established pattern in ecology, which is the high dominance by just a few species. This pattern stimulated the proposal of innumerous theoretical models in an attempt to explain the ecological mechanism which could generate it. However these models can also be a descriptor of the communities and their parameters synthetic measures of diversity. Such parameters can be used as response variables to environmental impact affecting communities. Adopting this approach our objective was to compare bird communities through areas of different levels of fragmentation, using as response variable the estimates of &#945;, the parameter of Fishers logseries. Considering the implicit assumption of equal capture probabilities among species in SAD models we also investigated the degree of sensibility of the models when this assumption is disrespected, once it seems so unrealistic. Thus simulating communities in which species had equal and different capture probabilities among them we found that increases in the degrees of heterogeneity in species catchability lead to a gain in biases on the model selection and parameters estimations. Additionally, since our goal in this study was identify some factors that may influence the diversity in communities, even with the biases, if they were constant, maybe it was still possible to test the relation. In this context we proceed to another stage of simulations, where we generate communities whose parameter values had a linear relationship with remnant area. What we find is that regardless of equal or unequal in catchability of species, when the effect exists, it is always detected, but depending on the degree of difference in probability of catching the species, the effect may be underestimated. Further, in the absence of effect, it can be falsely detected, depending on the degree of heterogeneity of capture probabilities among species, but always with very low estimates for the effect non-existent. With these results, we could quantify the types of effects of heterogeneity on capture probabilities and proceed with the analysis of the effects of fragmentation. What we showed is that the landscape with 10% vegetation cover, the fragment area appears to influence the diversity of the fragments rather than isolation, and landscape in 50% of plant cover, the isolation variable becomes more important than area to explain the data. But in a more parsimonious interpretation, we consider the estimated of the effects too low to consider that they actually exist. Therefore, we conclude that the fragmentation process probably has no effect on the hierarchy of species abundance. However, in a description of the number of captures of each species in the fragments, weighted by the number of catches sampled in continuous adjacent areas revealed that the fragment size may be important in determining which species will be extinct or benefit and that perhaps the quality of matrix is decisive for the maintenance of highly sensitive species in small fragments. Thus, we demonstrated that while the SAD are not significantly affected by fragmentation, the position in the hierarchy of species abundances can change a lot, which reflects the different sensitivity of species to area and isolation in the fragments.

In dieser Arbeit wird die Smallest-Accepted Methode als neue Lepski-Typ Methode für Modellwahl im geordneten Fall eingeführt. In einem ersten Schritt wird die Methode vorgestellt und im Fall von Schätzproblemen mit bekannter Fehlervarianz untersucht. Die Hauptkomponenten der Methode sind ein Akzeptanzkriterium, basierend auf Modellvergleichen für die eine Familie von kritischen Werten mit einem Monte-Carlo-Ansatz kalibriert wird, und die Wahl des kleinsten (in Komplexität) akzeptierten Modells. Die Methode kann auf ein breites Spektrum von Schätzproblemen angewandt werden, wie zum Beispiel Funktionsschätzung, Schätzung eines linearen Funktionals oder Schätzung in inversen Problemen. Es werden allgemeine Orakelungleichungen für die Methode im Fall von probabilistischem Verlust und einer polynomialen Verlustfunktion gezeigt und Anwendungen der Methode in spezifischen Schätzproblemen werden untersucht. In einem zweiten Schritt wird die Methode erweitert auf den Fall einer unbekannten, möglicherweise heteroskedastischen Fehlerstruktur. Die Monte-Carlo-Kalibrierung wird durch eine Bootstrap-basierte Kalibrierung ersetzt. Eine neue Familie kritischer Werte wird eingeführt, die von den (zufälligen) Beobachtungen abhängt. In Folge werden die theoretischen Eigenschaften dieser Bootstrap-basierten Smallest-Accepted Methode untersucht. Es wird gezeigt, dass unter typischen Annahmen unter normalverteilten Fehlern für ein zugrundeliegendes Signal mit Hölder-Stetigkeits-Index s > 1/4 und log(n) (p^2/n) klein, wobei n hier die Anzahl der Beobachtungen und p die maximale Modelldimension bezeichnet, die Anwendung der Bootstrap-Kalibrierung anstelle der Monte-Carlo-Kalibrierung theoretisch gerechtfertigt ist. / In this thesis, the Smallest-Accepted method is presented as a new Lepski-type method for ordered model selection. In a first step, the method is introduced and studied in the case of estimation problems with known noise variance. The main building blocks of the method are a comparison-based acceptance criterion relying on Monte-Carlo calibration of a set of critical values and the choice of the model as the smallest (in complexity) accepted model. The method can be used on a broad range of estimation problems like function estimation, estimation of linear functionals and inverse problems. General oracle results are presented for the method in the case of probabilistic loss and for a polynomial loss function. Applications of the method to specific estimation problems are studied. In a next step, the method is extended to the case of an unknown possibly heteroscedastic noise structure. The Monte-Carlo calibration step is now replaced by a bootstrap-based calibration. A new set of critical values is introduced, which depends on the (random) observations. Theoretical properties of this bootstrap-based Smallest-Accepted method are then studied. It is shown for normal errors under typical assumptions, that the replacement of the Monte-Carlo step by bootstrapping in the Smallest-Accepted method is valid, if the underlying signal is Hölder-continuous with index s > 1/4 and log(n) (p^2/n) is small for a sample size n and a maximal model dimension p.

Pela necessidade de reverter o atual quadro de degradação da Mata Atlântica, ações de restauração se fazem urgentes e devem ser pensadas a fim de restabelecer a biodiversidade nessas áreas, envolvendo as diversas formas de vida vegetal, animal e suas interações. O restabelecimento da relação planta-frugívoro e consequente dispersão de sementes certamente são essenciais não só para a conservação de uma floresta existente, mas também na aceleração do processo de restauração florestal. Portanto, a atração dos agentes dispersores de sementes deve fazer parte dos esforços empregados em ações restauradoras. Uma forma de avaliar a contribuição destes animais em áreas restauradas é através do estudo da chuva de sementes, mais especificamente aquela que é resultado dos eventos de dispersão pela fauna (zoocoria). Este estudo buscou caracterizar e comparar a composição da chuva de sementes zoocóricas em uma área em processo de restauração florestal na Mata Atlântica, submetidas a duas técnicas de manejo distintas, uma por meio de plantio de restauração em área total e outra, através da indução e condução da regeneração natural, originando uma área de capoeira. Ainda, utilizando a ferramenta de seleção de modelos pelo critério de Akaike, foram avaliadas se variáveis de estrutura e composição da vegetação arbustivo-arbórea influenciaram a riqueza e abundância da chuva de sementes zoocóricas total e imigrantes. O estudo foi conduzido em uma área em processo de restauração florestal, com seis anos de idade, que abrange 28,86 ha da Fazenda Intermontes (24°11\'17\" S, 42°24\'49\" O; 24°12\'47\" S, 42° 26\'15\" O), próximo ao município de Ribeirão Grande, SP. Propágulos depositados nos coletores foram retirados mensalmente pelo período de 1 ano. Utilizou-se um total de 100 coletores de sementes, de 1 m x 1 m. Para a caracterização da vegetação presente, foi realizado um levantamento dos indivíduos arbustivo-arbóreos, num raio de 5 metros no entorno de cada coletor de semente. Os resultados mostraram que a composição da comunidade da chuva de sementes zoocóricas diferiu entre os ambientes de capoeira e plantio, porém a riqueza e abundância médias das sementes não diferiram significativamente entre os ambientes. Apesar da composição da comunidade ter sido diferente, as categorias funcionais das sementes presentes na chuva, em ambas as áreas, foram semelhantes entre si. Avaliando se houve influência de variáveis relacionadas à estrutura e composição da vegetação arbustivo-arbórea na chuva de sementes zoocóricas, os modelos gerados e selecionados indicaram que as variáveis estudadas não influenciaram a riqueza e abundância da chuva de sementes zoocóricas total e imigrantes. O estudo concluiu também que o processo de chegada de propágulos alóctones a área já se iniciou, demonstrando um grande potencial de incremento de novas espécies, pertencentes a outras formas de vida e a diferentes funções ecológicas. Isto possibilita a aceleração do processo de restauração florestal, aumento da complexidade estrutural da vegetação e uma contribuição para a heterogeneidade da floresta implantada, fator este importante para o processo de retorno e incremento da fauna dispersora. / The need to reverse the current degradation of the Atlantic Forest requires urgent restoration actions aimed at reestablishing biodiversity in these areas, involving various plant and animal life forms and their interactions. The reestablishment of plant-frugivore interactions and subsequent seed dispersal are essential not only for the conservation of an existing forest, but also for the acceleration of forest restoration processes. Therefore, seed dispersal agents should be employed in restoration actions. One way to assess animals\' contribution in seed dispersion is through the study of seed rain, more specifically through results of dispersal events by fauna (zoochory). This study aimed to characterize and compare the composition of animal-dispersed seed rain in an area of the Atlantic Forest undergoing a restoration process using two different management techniques. One area comprised of tree planting and another comprising a \"capoeira\" through assisted natural regeneration. We also used an Akaike information criterion of model selection tool to evaluate whether structure and composition variables of arbustive-arboreal vegetation influenced the richness and abundance of total and immigrant animal-dispersed seed rain. The study was conducted in an area undergoing a forest restoration process with six years of age, covering 28.86 ha of the Intermontes Farm (24°11\'17\"S, 42°24\'49\"W; 24°12\'47\"S, 42°26\'15\"W), near Ribeirão Grande city, São Paulo State, Brazil. Propagules deposited in traps were removed monthly for a period of one year. We used 100 seed collectors 1 m x 1 m. To characterize the vegetation in the region, we surveyed the arbustive-arboreal species in a 5-meter radius around each seed collector. The results showed that the community composition of the animal-dispersed seed rain differed between tree planting and \"capoeira\" environments; however, the richness and abundance averages of seeds did not differ significantly between the environments. Although the community composition was different, functional categories of seeds in the rain in both areas were similar. Assessing whether there was influence of variables related to structure and composition of arbustive-arboreal species on animal-dispersed seed rain, generated and selected models indicated that these variables did not influence the richness and abundance of total and immigrant animal-dispersed seed rain. The results also showed the presence of alien propagules in the region, demonstrating great potential for the growth of new species belonging to other life forms with different ecological functions. This allows the acceleration of forest restoration processes, increased structural complexity of vegetation and contribution to heterogeneity of deployed forest, which is important for the return and increase of animal dispersers.