Francesco Mola

Multivariate data analysis and modeling through classification and regression trees

Abstract This paper provides a multivariate approach to binary segmentation in order to deal with more response variables. Splitting criteria are proposed to grow decision trees with multivariate classification/prediction. These are derived as extensions of criteria used in two-stage binary segmentation. The proposed methodology can be fruitfully performed not only to define decision rules for new cases but also to explore dependency in multivariate data. The feasibility of the method and the interpretation of the final decision trees are discussed in a practical example using a survey of the Bank of Italy.

A fast splitting procedure for classification trees

This paper provides a faster method to find the best split at each node when using the CART methodology. The predictability index τ is proposed as a splitting rule for growing the same classification tree as CART does when using the Gini index of heterogeneity as an impurity measure. A theorem is introduced to show a new property of the index τ: the τ for a given predictor has a value not lower than the τ for any split generated by the predictor. This property is used to make a substantial saving in the time required to generate a classification tree. Three simulation studies are presented in order to show the computational gain in terms of both the number of splits analysed at each node and the CPU time. The proposed splitting algorithm can prove computational efficiency in real data sets as shown in an example.

A statistical approach to growing a reliable honest tree

The introduction of a third stage in tree growing approach is suggested. The aim is to find a honest tree, that is, a tree which is not only understandable and accurate, but also statistically reliable. Testing procedures for both classification and regression trees are introduced. These procedures guide the search for those parts in tree structures which are statistically significant.

A field experiment on the use of Pistacia lentiscus L. and Scrophularia canina L. subsp. bicolor (Sibth. et Sm.) Greuter for the phytoremediation of abandoned mining areas

Abstract A two-year study has been conducted in an abandoned Pb/Zn mining site, with the aim of investigating the feasibility of phytoremediation using two native Mediterranean plants (Pistacia lentiscus and Scrophularia bicolor) and of assessing the performance of amendments able to reduce the toxic effects of heavy metals. The amendments used were compost, chemical fertilizer, and zeolites, used singly or in combination. Depending on the amendments applied, the two species showed different mortality rates in the different plots, but all produced an increase in P. lentiscus survival, while S. bicolor survival improved only when amended with zeolite or zeolite + fertilizer. Scrophularia bicolor proved to be a more efficient accumulator than P. lentiscus, especially for Pb uptake. Pistacia lentiscus accumulated metals mostly in the roots. The effect of amendments was to generally reduce the bioavailable metal fraction, especially lead, in the plots amended with compost. Pistacia lentiscus proved to be the most suitable species for phytostabilization and environmental restoration, both for its resistance to metals and high phytomass production. The experiments demonstrate that the use of compost not only encourages this kind of revegetation in degraded areas, but is also an economical option that uses a by-product of solid municipal waste treatment.

Generalized additive multi-mixture model for data mining

The main idea of this paper is to make statistical modelling into a feasible and valuable approach to data mining. The class of generalized additive multi-models (GAM-M) is considered in the framework of non-linear regression methods and data mining. GAM-M are based on a combined model integration approach that aims to associate estimations derived from smoothing functions as well as by either parametric or non-parametric models. We extend this approach to provide a class of models based on a mixture model combination. Bootstrap averaging and model fit scoring are exploited in order to prevent overfitting as well as to improve the prediction accuracy of the GAM-M models. The benchmarking of the proposed methodology is shown using a simulated data set.

Logistic Classification Trees

This paper provides a methodology how to grow exploratory trees enabling to understand, through statistical modeling, which variables are the most significant for determination why an object is in one class rather than in another. Logistic regression is used for modeling the dependence of the response dichotomous variable on the set of given predictors. The application on real data allows to discuss main advantages of the proposed procedure, especially for the analysis of real data sets whose dimensionality requires some sort of variable selection.

Alternative strategies and CATANOVA testing in two-stage binary segmentation

In the framework of binary segmentation, we introduce alternative splitting criteria based on the predictability r index of Goodman and Kruskal. We use such splitting criteria in a two-stage predictive splitting procedure. Furthermore, we introduce as stopping rule a statistical test based on the CATANOVA statistic of Light and Margolin. We show an example on a real data set.

Ternary Classification Trees: A Factorial Approach

Publisher Summary This chapter presents a methodology for constructing ternary trees via an exploratory multidimensional method—the nonsymmetrical correspondence analysis (NSCA). The proposed approach is convenient when the sample is very large and many predictors are considered. The analysis is organized as a sequence of NSCAs that leads to the construction of a decision tree for classifying new cases with unknown responses, as well as to explore the dataset by constructing maps at each node of the classification tree where splitting occurs. When the sample is very large, splitting nodes can be considered according to the category combinations of a pair of variables in the style of the multiple NSCA. The chapter constructs exploratory trees for emphasizing the most significant predictor at each level of the tree. For this purpose, a factorial approach such as the NSCA is used to construct the classification trees. The chapter provides an insight into the graphic displays of the NSCA to define a partitioning criterion into three classes. Factorial coordinates and predictability measures are used to distinguish among categories with strong and weak predictive power.

An Alternative Pruning Method Based on the Impurity-Complexity Measure

This paper provides a new pruning method for classification trees based on the impurity-complexity measure. Advantages of the proposed approach compared to the error-complexity pruning method are outlined showing an example on a real data set.

Modelling for Recursive Partitioning and Variable Selection

We present a binary segmentation methodology in which it is possible to select simultaneously sub-groups of variables as well as sub-groups of cases in each node of the binary tree. To a recursive partition procedure which defines either a classification tree or a regression tree we add a hierarchy of models. The main advantages of this approach, especially for large samples, are: to abandon immediately unsignificant variables; to reduce rapidly the number of possible splits in each node; the amalgamation procedure becomes faster and is of higher interpretative value.