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Gradient Descent Parameter Learning of Bayesian Networks under Monotonicity Restrictions
Plajner, Martin; Vomlel, Jiří
2018 - English
Learning parameters of a probabilistic model is a necessary step in most machine learning modeling tasks. When the model is complex and data volume is small the learning process may fail to provide good results. In this paper we present a method to improve learning results for small data sets by using additional information about the modelled system. This additional information is represented by monotonicity conditions which are restrictions on parameters of the model. Monotonicity simplifies the learning process and also these conditions are often required by the user of the system to hold. \n\nIn this paper we present a generalization of the previously used algorithm for parameter learning of Bayesian Networks under monotonicity conditions. This generalization allows both parents and children in the network to have multiple states. The algorithm is described in detail as well as monotonicity conditions are.\n\nThe presented algorithm is tested on two different data sets. Models are trained on differently sized data subsamples with the proposed method and the general EM algorithm. Learned models are then compared by their ability to fit data. We present empirical results showing the benefit of monotonicity conditions. The difference is especially significant when working with small data samples. The proposed method outperforms the EM algorithm for small sets and provides comparable results for larger sets.
Keywords:
Bayesian networks; Learning model parameters; monotonicity condition
Fulltext is available at external website.
Gradient Descent Parameter Learning of Bayesian Networks under Monotonicity Restrictions
Learning parameters of a probabilistic model is a necessary step in most machine learning modeling tasks. When the model is complex and data volume is small the learning process may fail to provide ...
Nonparametric Bootstrap Techniques for Implicitly Weighted Robust Estimators
Kalina, Jan
2018 - English
The paper is devoted to highly robust statistical estimators based on implicit weighting, which have a potential to find econometric applications. Two particular methods include a robust correlation coefficient based on the least weighted squares regression and the minimum weighted covariance determinant estimator, where the latter allows to estimate the mean and covariance matrix of multivariate data. New tools are proposed allowing to test hypotheses about these robust estimators or to estimate their variance. The techniques considered in the paper include resampling approaches with or without replacement, i.e. permutation tests, bootstrap variance estimation, and bootstrap confidence intervals. The performance of the newly described tools is illustrated on numerical examples. They reveal the suitability of the robust procedures also for non-contaminated data, as their confidence intervals are not much wider compared to those for standard maximum likelihood estimators. While resampling without replacement turns out to be more suitable for hypothesis testing, bootstrapping with replacement yields reliable confidence intervals but not corresponding hypothesis tests.
Keywords:
robust statistics; multivariate data; correlation coefficient; econometrics
Fulltext is available at external website.
Nonparametric Bootstrap Techniques for Implicitly Weighted Robust Estimators
The paper is devoted to highly robust statistical estimators based on implicit weighting, which have a potential to find econometric applications. Two particular methods include a robust correlation ...
SYNTHESIZED ENRICHMENT FUNCTIONS FOR EXTENDED FINITE ELEMENT ANALYSES WITH FULLY RESOLVED MICROSTRUCTURE
Doskar, M.; Novák, J.; Zeman, Jan
2017 - English
Inspired by the first order numerical homogenization, we present a method for extracting continuous fluctuation fields from the Wang tile based compression of a material microstructure. The fluctuation fields are then used as enrichment basis in Extended Finite Element Method (XFEM) to reduce number of unknowns in problems with fully resolved microstructural geometry synthesized by means of the tiling concept. In addition, the XFEM basis functions are taken as reduced modes of a detailed discretization in order to circumvent the need for non-standard numerical quadratures. The methodology is illustrated with a scalar steady-state problem.
Keywords:
Wang tiling; microstructure synthesis; microstructure-informed enrichment functions; extended finite element method
Fulltext is available at external website.
SYNTHESIZED ENRICHMENT FUNCTIONS FOR EXTENDED FINITE ELEMENT ANALYSES WITH FULLY RESOLVED MICROSTRUCTURE
Inspired by the first order numerical homogenization, we present a method for extracting continuous fluctuation fields from the Wang tile based compression of a material microstructure. The ...
Question Selection Methods for Adaptive Testing with Bayesian Networks
Plajner, Martin; Magauina, A.; Vomlel, Jiří
2017 - English
The performance of Computerized Adaptive Testing systems, which are used for testing of human knowledge, relies heavily on methods selecting correct questions for tested students. In this article we propose three different methods selecting questions with Bayesian networks as students’ models. We present the motivation to use these methods and their mathematical description. Two empirical datasets, paper tests of specific topics in mathematics and Czech language for foreigners, were collected for the purpose of methods’ testing. All three methods were tested using simulated testing procedure and results are compared for individual methods. The comparison is done also with the sequential selection of questions to provide a relation to the classical way of testing. The proposed methods are behaving much better than the sequential selection which verifies the need to use a better selection method. Individually, our methods behave differently, i.e., select different questions but the success rate of model’s predictions is very similar for all of them. This motivates further research in this topic to find an ordering between methods and to find the best method which would provide the best possible selections in computerized adaptive tests.
Keywords:
Computerized Adaptive Testing; Question Selection Methods; Bayesian Networks
Fulltext is available at external website.
Question Selection Methods for Adaptive Testing with Bayesian Networks
The performance of Computerized Adaptive Testing systems, which are used for testing of human knowledge, relies heavily on methods selecting correct questions for tested students. In this article we ...
Analysis of truncated data with application to the operational risk estimation
Volf, Petr
2017 - English
Analysis of operational risk often faces problems arising from the structure of available data, namely of left truncation and occurrence of heavy-tailed loss values. We deal with model given by lognormal dostribution contaminated by the Pareto one and to use of the Cramér-von Mises, Anderson-Darling, and Kolmogorov-Smirnov minimum distance estimators. Analysis is based on MC studies. The main objective is to propose a method of statistical analysis and modeling for the distribution of sum of\nlosses over a given period, particularly of its right quantiles.
Keywords:
operational risk; statistical analysis; truncated data
Fulltext is available at external website.
Analysis of truncated data with application to the operational risk estimation
Analysis of operational risk often faces problems arising from the structure of available data, namely of left truncation and occurrence of heavy-tailed loss values. We deal with model given by ...
Risk-Sensitive Optimality in Markov Games
Sladký, Karel; Martínez Cortés, V. M.
2017 - English
The article is devoted to risk-sensitive optimality in Markov games. Attention is focused on Markov games evolving on communicating Markov chains with two-players with opposite aims. Considering risk-sensitive optimality criteria means that total reward generated by the game is evaluated by exponential utility function with a given risk-sensitive coefficient. In particular, the first player (resp. the secondplayer) tries to maximize (resp. minimize) the long-run risk sensitive average reward. Observe that if the second player is dummy, the problem is reduced to finding optimal policy of the Markov decision chain with the risk-sensitive optimality. Recall that for the risk sensitivity coefficient equal to zero we arrive at traditional optimality criteria. In this article, connections between risk-sensitive and risk-neutral Markov decisionchains and Markov games models are studied using discrepancy functions. Explicit formulae for bounds on the risk-sensitive average long-run reward are reported. Policy iteration algorithm for finding suboptimal policies of both players is suggested. The obtained results are illustrated on numerical example.
Keywords:
two-person Markov games; communicating Markov chains; risk-sensitive optimality; dynamic programming
Fulltext is available at external website.
Risk-Sensitive Optimality in Markov Games
The article is devoted to risk-sensitive optimality in Markov games. Attention is focused on Markov games evolving on communicating Markov chains with two-players with opposite aims. Considering ...
Various Approaches to Szroeter’s Test for Regression Quantiles
Kalina, Jan; Peštová, B.
2017 - English
Regression quantiles represent an important tool for regression analysis popular in econometric applications, for example for the task of detecting heteroscedasticity in the data. Nevertheless, they need to be accompanied by diagnostic tools for verifying their assumptions. The paper is devoted to heteroscedasticity testing for regression quantiles, while their most important special case is commonly denoted as the regression median. Szroeter’s test, which is one of available heteroscedasticity tests for the least squares, is modified here for the regression median in three different ways: (1) asymptotic test based on the asymptotic representation for regression quantiles, (2) permutation test based on residuals, and (3) exact approximate test, which has a permutation character and represents an approximation to an exact test. All three approaches can be computed in a straightforward way and their principles can be extended also to other heteroscedasticity tests. The theoretical results are expected to be extended to other regression quantiles and mainly to multivariate quantiles.
Keywords:
Heteroscedasticity; Regression median; Diagnostic tools; Asymptotics
Fulltext is available at external website.
Various Approaches to Szroeter’s Test for Regression Quantiles
Regression quantiles represent an important tool for regression analysis popular in econometric applications, for example for the task of detecting heteroscedasticity in the data. Nevertheless, they ...
Flexible Moment Invariant Bases for 2D Scalar and Vector Fields
Bujack, R.; Flusser, Jan
2017 - English
Complex moments have been successfully applied to pattern detection tasks in two-dimensional real, complex, and vector valued functions. In this paper, we review the different bases of rotational moment invariants based on the generator approach with complex monomials. We analyze their properties with respect to independence, completeness, and existence and\npresent superior bases that are optimal with respect to all three criteria for both scalar and vector fields.
Keywords:
Pattern detection; moment invariants; scalar fields; vector fields; flow fields; generator; basis; complex; monomial
Fulltext is available at external website.
Flexible Moment Invariant Bases for 2D Scalar and Vector Fields
Complex moments have been successfully applied to pattern detection tasks in two-dimensional real, complex, and vector valued functions. In this paper, we review the different bases of rotational ...
Avoiding overfitting of models: an application to research data on the Internet videos
Jiroušek, Radim; Krejčová, I.
2017 - English
The problem of overfitting is studied from the perspective of information theory. In this context, data-based model learning can be viewed as a transformation process, a process transforming the information contained in data into the information represented by a model. The overfitting of a model often occurs when one considers an unnecessarily complex model, which usually means that the considered model contains more information than the original data. Thus, using one of the basic laws of information theory saying that any transformation cannot increase the amount of information, we get the basic restriction laid on models constructed from data: A model is acceptable if it does not contain more information than the input data file.
Keywords:
data-based learning; probabilistic models; information theory; MDL principle; lossless encoding
Fulltext is available at external website.
Avoiding overfitting of models: an application to research data on the Internet videos
The problem of overfitting is studied from the perspective of information theory. In this context, data-based model learning can be viewed as a transformation process, a process transforming the ...
Causality and Intervention in Business Process Management
Bína, V.; Jiroušek, Radim
2017 - English
The paper presents an algebraic approach to the modeling of causality in systems of stochastic variables. The methodology is based on an operator of a composition that provides the possibility of composing a multidimensional distribution from low-dimensional building blocks taking advantage of the dependence structure of the problem variables. The authors formally define and demonstrate on a hypothetical example a surprisingly elegant unifying approach to conditioning by a single variable and the evaluation of the effect of an intervention. Both operations are realized by the composition with a degenerated distribution and differ only in the sequence in which the operator of the composition is performed.
Keywords:
Compositional model; Operator of composition,; Causality; Conditioning; Intervention
Fulltext is available at external website.
Causality and Intervention in Business Process Management
The paper presents an algebraic approach to the modeling of causality in systems of stochastic variables. The methodology is based on an operator of a composition that provides the possibility of ...
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