Positive linear decomposition for classifying patterns in dynamic measurements: a case study

The purpose of this article is to show the effectiveness of a positive linear decomposition in the derivation of robust features of high-dimensional dynamic measurements, in order to achieve effective pattern recognition and classification. The method begins with the singular value decomposition, projecting a matrix of dynamic process measurements (taken at uniform intervals over some time-window) onto a low-dimensional subspace. A convex cone, defined by the non-negativity of measurements, is then created. For normalization purposes a polygon, whose corners specify the feature vectors of the data, is formed by intersecting the cone with a plane. This polygon is reduced to a triangle with only the three most representative corners. The net effect of these steps is that the original orthogonal basis of the subspace (consisting of the first three principal components) is replaced by a new, non-orthogonal basis, which offers the advantage of containing only positive measurements and requiring only positive superposition of basis vectors to span the physically meaningful portion of the subspace. One of the vectors in this basis is selected as the feature vector for pattern recognition; a spanning tree created from the feature vectors classifies the patterns. The feature vectors from the new basis are much more robust with respect to changes in the width of the time window, and classification was possible even with feature vectors of differing time windows.     

Improving problem solving ability in mathematics by using a mathematical model: A computerized approach

The present study investigates the improvement of students’ mathematical performance by using a mathematical model through a computerized approach. We had developed an intervention program and 11 years students worked independently on a mathematical model in order to improve their self-representation in mathematics, to self-regulate their performance and consequently to improve their problem solving ability. The emphasis of using the specific model was on dividing the problem solving procedure into stages, the concentration on the students’ cognitive processes at each stage and the self-regulation of those cognitive processes in order to overcome cognitive obstacles. The use of the computer offered the opportunity to give students general comments, hints and feedback without the involvement of their teachers. Students had to communicate with a cartoon animation presenting a human being who faced difficulties and cognitive obstacles during problem solving procedure. Three tools were constructed for pre- and post-test (self-representation, mathematical performance and self-regulation). There were administered to 255 students (11 years old), who constituted the experimental and the control group. Results confirmed that providing students with the opportunity to self-reflect on their learning behavior when they encounter obstacles in problem solving is one possible way to enhance students’ self-regulation and consequently their mathematical performance.     

A multi-scale approach for modeling fire occurrence probability using satellite data and classification trees: A case study in a mountainous Mediterranean region

Fires constitute one major ecological disturbance which influences the natural cycle of vegetation succession and the structure and function of ecosystems. There is no single natural scale at which ecological phenomena are completely understood and thus the capacity to handle scale is beneficial to methodological frameworks for analyzing and monitoring ecosystems. Although satellite imagery has been widely applied for the assessment of fire related topics, there are few studies that consider fire at several spatial scales simultaneously. This research explores the relationships between fire occurrence and several families of environmental factors at different spatial observation scales by means of classification and regression tree models. Predictors accounting for vegetation status (estimated by spectral indices derived from Landsat imagery), fire history, topography, accessibility and vegetation types were included in the models of fire occurrence probability. We defined four scales of analysis by identifying four meaningful thresholds related to fire sizes in the study site. Sampling methodology was based on random points and the power-law distribution describing the local fire regime. The observation scale drastically affected tree size, and therefore the achieved level of detail, and the most explanatory variables in the trees. As a general trend, trees considering all the variables showed a spectral index ruling the most explicative split. According to the comparison of the four pre-determined analysis scales, we propose the existence of three eventual organization levels: landscape patch or ecosystem level, local level and the basic level, the most heterogeneous and complex scale. Rules with three levels of complexity and applicability for management were defined in the tree models: (i) the repeated critical thresholds (predictor values across which fire characteristics change rapidly), (ii) the meaningful final probability classes and (iii) the trees themselves.