Abstract
In this paper we present system Ltree for propositional supervised learning. Ltree is able to define decision surfaces both orthogonal and oblique to the axes defined by the attributes of the input space. This is done combining a decision tree with a linear discriminant by means of constructive induction. At each decision node Ltree defines a new instance space by insertion of new attributes that are projections of the examples that fall at this node over the hyper-planes given by a linear discriminant function. This new instance space is propagated down through the tree. Tests based on those new attributes are oblique with respect to the original input space. Ltree is a probabilistic tree in the sense that it outputs a class probability distribution for each query example. The class probability distribution is computed at learning time, taking into account the different class distributions on the path from the root to the actual node. We have carried out experiments on twenty one benchmark datasets and compared our system with other well known decision tree systems (orthogonal and oblique) like C4.5, OC1, LMDT, and CART. On these datasets we have observed that our system has advantages in what concerns accuracy and learning times at statistically significant confidence levels.

Abstract
Naive Bayes is a well known and studied algorithm both in statistics and machine learning. Bayesian learning algorithms represent each concept with a single probabilistic summary. In this paper we present an iterative approach to naive Bayes. The iterative Bayes begins with the distribution tables built by the naive Bayes. Those tables are iteratively updated in order to improve the probability class distribution associated with each training example. Experimental evaluation of Iterative Bayes on 25 benchmark datasets shows consistent gains in accuracy. An interesting side effect of our algorithm is that it shows to be robust to attribute dependencies.

Abstract

Abstract
In a previous work, we presented system Ltree, a multivariate tree that combines a decision tree with a linear discriminant by means of constructive induction. We have shown that it performs quite well, in terms of accuracy and learning times, in comparison with other multivariate systems like LMDT, OC1, and CART. In this work, we extend the previous work by using two new discriminant functions: a quadratic discriminant and a logistic discriminant. Using the same architecture as Ltree we obtain two new multivariate trees Qtree and LgTree. The three systems have been evaluate on 17 UCI datasets. From the empirical study, we argue that these systems can be shown as a composition of classifiers with low correlation error. From a bias-variance analysis of the error rate, the error reduction of all the systems in comparison to a univariate tree, is due to a reduction on both components.

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.