Abstract
Using problem-specific background knowledge, computer programs developed within the framework of Inductive Logic Programming (ILP) have been used to construct restricted first-order logic solutions to scientific problems. However, their approach to the analysis of data with substantial numerical content has been largely limited to constructing clauses that: (a) provide qualitative descriptions ("high", "low" etc.) of the values of response variables; and (b) contain simple inequalities restricting the ranges of predictor variables. This has precluded the application of such techniques to scientific and engineering problems requiring a more sophisticated approach. A number of specialised methods have been suggested to remedy this. In contrast, we have chosen to take advantage of the fact that the existing theoretical framework for ILP places very few restrictions of the nature of the background knowledge. We describe two issues of implementation that make it possible to use background predicates that implement well-established statistical and numerical analysis procedures. Any improvements in analytical sophistication that result are evaluated empirically using artificial and real-life data. Experiments utilising artificial data are concerned with extracting constraints for response variables in the text-book problem of balancing a pole on a cart. They illustrate the use of clausal definitions of arithmetic and trigonometric functions, inequalities, multiple linear regression, and numerical derivatives. A non-trivial problem concerning the prediction of mutagenic activity of nitroaromatic molecules is also examined. In this case, expert chemists have been unable to devise a model for explaining the data. The result demonstrates the combined use by an ILP program of logical and numerical capabilities to achieve an analysis that includes linear modelling, clustering and classification. In all experiments, the predictions obtained compare favourably against benchmarks set by more traditional methods of quantitative methods, namely, regression and neural-network.

Abstract
Given C-2 infinitely renormalizable unimodal maps f and g with a quadratic critical point and the same bounded combinatorial type, we prove that they are C1+alpha conjugate along the closure of the corresponding forward orbits of the critical points, for some alpha > 0.

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Abstract
Discretization of continuous attributes is an important task for certain types of machine learning algorithms. Bayesian approaches, for instance, require assumptions about data distributions. Decision Trees on the other hand, require sorting operations to deal with continuous attributes, which largely increase learning times. This paper presents a new method of discretization, whose main characteristic is that it takes into account interdependencies between attributes. Detecting interdependencies can be seen as discovering redundant attributes. This means that our method performs attribute selection as a side effect of the discretization. Empirical evaluation on five benchmark datasets from UCI repository, using C4.5 and a naive Bayes, shows a consistent reduction of the features without loss of generalization accuracy.

Abstract
Using multiple classifiers for increasing learning accuracy is an active research area. In this paper we present a new general method for merging classifiers. The basic idea of Cascade Generalization is to sequentially run the set of classifiers, at each step performing an extension of the original data set by adding new attributes. The new attributes are derived from the probability class distribution given by a base classifier. This constructive step extends the representational language for the high level classifiers, relaxing their bias. Cascade Generalization produces a single but structured model for the data that combines the model class representation of the base classifiers. We have performed an empirical evaluation of Cascade composition of three well known classifiers: Naive Bayes, Linear Discriminant, and C4.5. Composite models show an increase of performance, sometimes impressive, when compared with the corresponding single models, with significant statistical confidence levels. © Springer-Veriag Berlin Heidelberg 1998.

Abstract