A data driven equivariant approach to constrained Gaussian mixture modeling

Maximum likelihood estimation of Gaussian mixture models with different class-specific covariance matrices is known to be problematic. This is due to the unboundedness of the likelihood, together with the presence of spurious maximizers. Existing methods to bypass this obstacle are based on the fact that unboundedness is avoided if the eigenvalues of the covariance matrices are bounded away from zero. This can be done imposing some constraints on the covariance matrices, i.e. by incorporating a priori information on the covariance structure of the mixture components. The present work introduces a constrained equivariant approach, where the class conditional covariance matrices are shrunk towards a pre-specified matrix Psi. Data-driven choices of the matrix Psi, when a priori information is not available, and the optimal amount of shrinkage are investigated. The effectiveness of the proposal is evaluated on the basis of a simulation study and an empirical example

Integrated smoothed location model and data reduction approaches for multi variables classification

Smoothed Location Model is a classification rule that deals with mixture of continuous variables and binary variables simultaneously. This rule discriminates groups in a parametric form using conditional distribution of the continuous variables given each pattern of the binary variables. To conduct a practical classification analysis, the objects must first be sorted into the cells of a multinomial table generated from the binary variables. Then, the parameters in each cell will be estimated using the sorted objects. However, in many situations, the estimated parameters are poor if the number of binary is large relative to the size of sample. Large binary variables will create too many multinomial cells which are empty, leading to high sparsity problem and finally give exceedingly poor performance for the constructed rule. In the worst case scenario, the rule cannot be constructed. To overcome such shortcomings, this study proposes new strategies to extract adequate variables that contribute to optimum performance of the rule. Combinations of two extraction techniques are introduced, namely 2PCA and PCA+MCA with new cutpoints of eigenvalue and total variance explained, to determine adequate extracted variables which lead to minimum misclassification rate. The outcomes from these extraction techniques are used to construct the smoothed location models, which then produce two new approaches of classification called 2PCALM and 2DLM. Numerical evidence from simulation studies demonstrates that the computed misclassification rate indicates no significant difference between the extraction techniques in normal and non-normal data. Nevertheless, both proposed approaches are slightly affected for non-normal data and severely affected for highly overlapping groups. Investigations on some real data sets show that the two approaches are competitive with, and better than other existing classification methods. The overall findings reveal that both proposed approaches can be considered as improvement to the location model, and alternatives to other classification methods particularly in handling mixed variables with large binary size

A Bound on the Error of Cross Validation Using the Approximation and Estimation Rates, with Consequences for the Training-Test Split

: We give an analysis of the generalization error of cross validation in terms of two natural measures of the difficulty of the problem under consideration: the approximation rate (the accuracy to which the target function can be ideally approximated as a function of the number of hypothesis parameters), and the estimation rate (the deviation between the training and generalization errors as a function of the number of hypothesis parameters). The approximation rate captures the complexity of the target function with respect to the hypothesis model, and the estimation rate captures the extent to which the hypothesis model suffers from overfitting. Using these two measures, we give a rigorous and general bound on the error of cross validation. The bound clearly shows the tradeoffs involved with making fl --- the fraction of data saved for testing --- too large or too small. By optimizing the bound with respect to fl, we then argue (through a combination of formal analysis, plotting, and ..

Data Science na modelação e previsão de séries económico-financeiras: das metodologias clássicas ao Deep Learning

A articulação de técnicas/ferramentas estatísticas, matemáticas e computacionais, no processo de análise, modelação e previsão de séries temporais, manifesta-se um claro suporte de apoio à tomada de decisão. O constante desafio na procura de previsões acuradas tem levado os investigadores à melhoraria das técnicas já existentes e a investir na procura de metodologias alternativas. Especificamente para séries económico-financeiras, a aplicação de metodologias baseadas em Inteligência Artificial, em particular de "Deep Learning", tem sido apontada com uma opção promissora. Neste estudo faz-se uma comparação crítica dos resultados obtidos por aplicação de metodologias clássicas de previsão (nomeadamente modelos autorregressivos e de alisamento exponencial) e de "Deep Learning" (mediante a implementação de algumas arquiteturas redes neuronais). O estudo empírico foi sustentando em quatro séries económico-financeiras distintas: "Consumer Price Index for All Urban Consumers: All Items in U.S. City Average" (CPIAUCSL); "Vehicle-Miles Travelled" (VMT); "Portuguese Stock Index 20" (PSI 20) e "Standard & Poor's 500 Exchange-Traded Fund" (SPY). A análise comparativa é feita tendo por base a qualidade preditiva e o custo computacional associado a cada um dos modelos de previsão. Reconhecidas vantagens na aplicação das metodologias de "Deep Learning", são discutidas possíveis alterações procurando melhorar a qualidade preditiva e reduzir o tempo de execução computacional. As alterações introduzidas em modelos de redes neuronais revelaram-se promissoras na redução do tempo computacional e nos valores da métrica de erro de previsão usada. Este sucesso é sobretudo evidente em séries que apresentam dinâmicas 'irregulares', como são exemplo as séries financeiras.The articulation of statistical, mathematical and computational techniques/tools, in the process of analysis, modelling and forecasting time series, manifests clear support for decision making. The constant challenge in the quest for the most accurate results possible has led researchers not only to improve the existing techniques, but also to invest in the search for alternative methodologies. Specifically, for economic and financial series, the application of methodologies based on Artificial Intelligence, in particular Deep Learning, has been pointed out as a promising option. This study makes a critical comparison of the results obtained by applying classical forecasting methodologies (namely autoregressive models and exponential smoothing) and Deep Learning (through the implementation of some neural network architectures). The empirical study focused on four economic-financial series with different characteristics: Consumer Price Index for All Urban Consumers: All Items in U.S. City Average (CPIAUCSL); Vehicle-Miles Travelled (VMT); Portuguese Stock Index 20 (PSI 20) and Standard & Poor's 500 Exchange-Traded Fund (SPY). The comparative analysis is made based on both predictive quality and computational cost associated with each of the forecasting models. Recognized the advantages in the application of Deep Learning methodologies, we discuss some changes to introduce in the existing models to improve their predictive quality while reducing computational execution time. The changes introduced in neural network models proved to be promising in reducing the associated computational time but and the values of the error metric used. This success is especially evident in series with ‘irregular’ dynamics, as is the case with financial series

Machine learning for network based intrusion detection: an investigation into discrepancies in findings with the KDD cup '99 data set and multi-objective evolution of neural network classifier ensembles from imbalanced data.

For the last decade it has become commonplace to evaluate machine learning techniques for network based intrusion detection on the KDD Cup '99 data set. This data set has served well to demonstrate that machine learning can be useful in intrusion detection. However, it has undergone some criticism in the literature, and it is out of date. Therefore, some researchers question the validity of the findings reported based on this data set. Furthermore, as identified in this thesis, there are also discrepancies in the findings reported in the literature. In some cases the results are contradictory. Consequently, it is difficult to analyse the current body of research to determine the value in the findings. This thesis reports on an empirical investigation to determine the underlying causes of the discrepancies. Several methodological factors, such as choice of data subset, validation method and data preprocessing, are identified and are found to affect the results significantly. These findings have also enabled a better interpretation of the current body of research. Furthermore, the criticisms in the literature are addressed and future use of the data set is discussed, which is important since researchers continue to use it due to a lack of better publicly available alternatives. Due to the nature of the intrusion detection domain, there is an extreme imbalance among the classes in the KDD Cup '99 data set, which poses a significant challenge to machine learning. In other domains, researchers have demonstrated that well known techniques such as Artificial Neural Networks (ANNs) and Decision Trees (DTs) often fail to learn the minor class(es) due to class imbalance. However, this has not been recognized as an issue in intrusion detection previously. This thesis reports on an empirical investigation that demonstrates that it is the class imbalance that causes the poor detection of some classes of intrusion reported in the literature. An alternative approach to training ANNs is proposed in this thesis, using Genetic Algorithms (GAs) to evolve the weights of the ANNs, referred to as an Evolutionary Neural Network (ENN). When employing evaluation functions that calculate the fitness proportionally to the instances of each class, thereby avoiding a bias towards the major class(es) in the data set, significantly improved true positive rates are obtained whilst maintaining a low false positive rate. These findings demonstrate that the issues of learning from imbalanced data are not due to limitations of the ANNs; rather the training algorithm. Moreover, the ENN is capable of detecting a class of intrusion that has been reported in the literature to be undetectable by ANNs. One limitation of the ENN is a lack of control of the classification trade-off the ANNs obtain. This is identified as a general issue with current approaches to creating classifiers. Striving to create a single best classifier that obtains the highest accuracy may give an unfruitful classification trade-off, which is demonstrated clearly in this thesis. Therefore, an extension of the ENN is proposed, using a Multi-Objective GA (MOGA), which treats the classification rate on each class as a separate objective. This approach produces a Pareto front of non-dominated solutions that exhibit different classification trade-offs, from which the user can select one with the desired properties. The multi-objective approach is also utilised to evolve classifier ensembles, which yields an improved Pareto front of solutions. Furthermore, the selection of classifier members for the ensembles is investigated, demonstrating how this affects the performance of the resultant ensembles. This is a key to explaining why some classifier combinations fail to give fruitful solutions

Machine learning for network based intrusion detection : an investigation into discrepancies in findings with the KDD cup '99 data set and multi-objective evolution of neural network classifier ensembles from imbalanced data

For the last decade it has become commonplace to evaluate machine learning techniques for network based intrusion detection on the KDD Cup '99 data set. This data set has served well to demonstrate that machine learning can be useful in intrusion detection. However, it has undergone some criticism in the literature, and it is out of date. Therefore, some researchers question the validity of the findings reported based on this data set. Furthermore, as identified in this thesis, there are also discrepancies in the findings reported in the literature. In some cases the results are contradictory. Consequently, it is difficult to analyse the current body of research to determine the value in the findings. This thesis reports on an empirical investigation to determine the underlying causes of the discrepancies. Several methodological factors, such as choice of data subset, validation method and data preprocessing, are identified and are found to affect the results significantly. These findings have also enabled a better interpretation of the current body of research. Furthermore, the criticisms in the literature are addressed and future use of the data set is discussed, which is important since researchers continue to use it due to a lack of better publicly available alternatives. Due to the nature of the intrusion detection domain, there is an extreme imbalance among the classes in the KDD Cup '99 data set, which poses a significant challenge to machine learning. In other domains, researchers have demonstrated that well known techniques such as Artificial Neural Networks (ANNs) and Decision Trees (DTs) often fail to learn the minor class(es) due to class imbalance. However, this has not been recognized as an issue in intrusion detection previously. This thesis reports on an empirical investigation that demonstrates that it is the class imbalance that causes the poor detection of some classes of intrusion reported in the literature. An alternative approach to training ANNs is proposed in this thesis, using Genetic Algorithms (GAs) to evolve the weights of the ANNs, referred to as an Evolutionary Neural Network (ENN). When employing evaluation functions that calculate the fitness proportionally to the instances of each class, thereby avoiding a bias towards the major class(es) in the data set, significantly improved true positive rates are obtained whilst maintaining a low false positive rate. These findings demonstrate that the issues of learning from imbalanced data are not due to limitations of the ANNs; rather the training algorithm. Moreover, the ENN is capable of detecting a class of intrusion that has been reported in the literature to be undetectable by ANNs. One limitation of the ENN is a lack of control of the classification trade-off the ANNs obtain. This is identified as a general issue with current approaches to creating classifiers. Striving to create a single best classifier that obtains the highest accuracy may give an unfruitful classification trade-off, which is demonstrated clearly in this thesis. Therefore, an extension of the ENN is proposed, using a Multi-Objective GA (MOGA), which treats the classification rate on each class as a separate objective. This approach produces a Pareto front of non-dominated solutions that exhibit different classification trade-offs, from which the user can select one with the desired properties. The multi-objective approach is also utilised to evolve classifier ensembles, which yields an improved Pareto front of solutions. Furthermore, the selection of classifier members for the ensembles is investigated, demonstrating how this affects the performance of the resultant ensembles. This is a key to explaining why some classifier combinations fail to give fruitful solutions.EThOS - Electronic Theses Online ServiceGBUnited Kingdo