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Robust Knowledge Discovery from High-Dimensional Data
Kalina, Jan
The paper is devoted to advanced robust methods for information extraction from highdimensional data. The concept of knowledge discovery is discussed together with its two important aspects: high dimensionality of the data and sensitivity to the presence of outlying data values. We propose new robust methods for knowledge discovery suitable for highdimensional data. They are based on the idea of implicit weighting, which is inspired by the least weighted squares regression estimator. We propose a highly robust method for a dimension reduction, which can be described as a robust alternative of the principal component analysis based on implicit down-weighting of less reliable data values. Further, we propose a novel robust approach to cluster analysis, which is a popular knowledge discovery method of unsupervised learning. A two-stage cluster analysis method tailor-made for highdimensional data is obtained by combining the robust principal component analysis with the robust cluster analysis. The procedure can be interpreted as a robust knowledge discovery method tailor made for high-dimensional data.
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Nonlinear Trend Modeling in the Analysis of Categorical Data
Kalina, Jan
This paper studies various approaches to testing trend in the context of categorical data. While the linear trend is far more popular in econometric applications, a nonlinear modeling of the trend allows a more subtle information extraction from real data, especially if the linearity of the trend cannot be expected and verified by hypothesis testing. We exploit the exact unconditional approach to propose alternative versions of some trend tests. One of them is the test of relaxed trend (Liu, 1998), who proposed a generalization of the classical Cochran- Armitage test of linear trend. A numerical example on real data reveals the advantages of the test of relaxed trend compared to the classical test of linear trend. Further, we propose an exact unconditional test also for modeling association between an ordinal response and nominal regressor. Further, we propose a robust estimator of parameters in the logistic regression model, which is based on implicit weighting of individual observations. We assess the breakdown point of the newly proposed robust estimator.
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Logistic and Poisson regression
Kalina, Jan
The work describes methods of logistic and Poisson regression for the analysis of contingency tables. It explains general principles of hypothesis testing about the significance of their parameters. Particular methods can be derived also within a general theory of generalized linear models. For this general context also principles of variable selection are investigated. It is shown how properties of particular methods correspond to properties, which can be derived in the framework of generalized linear models.
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Analysis of contingency tables
Kalina, Jan
The paper compares asymptotic, exact conditional and exact unconditional tests for the evaluation of contingency tables. Relationship of asymptotic methods with generalized linear models is studied. The work focuses on exact methods, which yield promising results in the analysis of data with small samples. Some of the methods are alternatively derived in the context of logistic and Poisson regression.
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On Heteroscedasticity in Robust Regression
Kalina, Jan
This work studies the phenomenon of heteroscedasticity and its consequences for various methods of linear regression, including the least squares, least weighted squares and regression quantiles. We focus on hypothesis tests for these regression methods. The new approach consists in deriving asymptotic heteroscedasticity tests for robust regression, which are asymptotically equivalent to standard tests computed for the least squares regression. One approach to modeling heteroscedasticity assumes a prior knowledge or specific model for the variability of random regression errors. Another (and more general) approach does not assume a specific form of heteroscedasticity. The paper also describes heteroscedastic regression, which is a tool to incorporate heteroscedasticity to the model. This allows us to define the heteroscedastic least weighted squares regression.
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Some models for categorical data
Kalina, Jan
The work investigates connections between log-linear and logit models, which are two special cases of generalized linear models, suitable for statistical modeling of categorical data. Further the Markov-McNemar test is described, which is a test of homogeneity of limit marginal probabilities originally proposed for evaluating DNA sequences.
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Classification analysis
Kalina, Jan
The paper compares properties of methods, which are commonly used for the task of classification analysis in both statistics and informatics. Some relationships are derived concerning the usage of classification methods for such data, which do not fit fulfill usual assumptions.
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