On the Schoenberg transformations in data analysis: theory and illustrations

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and visualized on artificial data sets by classical multidimensional scaling. A distance-based discriminant algorithm and a robust multidimensional centroid estimate illustrate the theory, closely connected to the Gaussian kernels of Machine Learning.

Advanced Kernel Methods For Remote Sensing Image Classification

Résumé Suite aux recentes avancées technologiques, les archives d'images digitales ont connu une croissance qualitative et quantitative sans précédent. Malgré les énormes possibilités qu'elles offrent, ces avancées posent de nouvelles questions quant au traitement des masses de données saisies. Cette question est à la base de cette Thèse: les problèmes de traitement d'information digitale à très haute résolution spatiale et/ou spectrale y sont considérés en recourant à des approches d'apprentissage statistique, les méthodes à noyau. Cette Thèse étudie des problèmes de classification d'images, c'est à dire de catégorisation de pixels en un nombre réduit de classes refletant les propriétés spectrales et contextuelles des objets qu'elles représentent. L'accent est mis sur l'efficience des algorithmes, ainsi que sur leur simplicité, de manière à augmenter leur potentiel d'implementation pour les utilisateurs. De plus, le défi de cette Thèse est de rester proche des problèmes concrets des utilisateurs d'images satellite sans pour autant perdre de vue l'intéret des méthodes proposées pour le milieu du machine learning dont elles sont issues. En ce sens, ce travail joue la carte de la transdisciplinarité en maintenant un lien fort entre les deux sciences dans tous les développements proposés. Quatre modèles sont proposés: le premier répond au problème de la haute dimensionalité et de la redondance des données par un modèle optimisant les performances en classification en s'adaptant aux particularités de l'image. Ceci est rendu possible par un système de ranking des variables (les bandes) qui est optimisé en même temps que le modèle de base: ce faisant, seules les variables importantes pour résoudre le problème sont utilisées par le classifieur. Le manque d'information étiquétée et l'incertitude quant à sa pertinence pour le problème sont à la source des deux modèles suivants, basés respectivement sur l'apprentissage actif et les méthodes semi-supervisées: le premier permet d'améliorer la qualité d'un ensemble d'entraînement par interaction directe entre l'utilisateur et la machine, alors que le deuxième utilise les pixels non étiquetés pour améliorer la description des données disponibles et la robustesse du modèle. Enfin, le dernier modèle proposé considère la question plus théorique de la structure entre les outputs: l'intègration de cette source d'information, jusqu'à présent jamais considérée en télédétection, ouvre des nouveaux défis de recherche. Advanced kernel methods for remote sensing image classification Devis Tuia Institut de Géomatique et d'Analyse du Risque September 2009 Abstract The technical developments in recent years have brought the quantity and quality of digital information to an unprecedented level, as enormous archives of satellite images are available to the users. However, even if these advances open more and more possibilities in the use of digital imagery, they also rise several problems of storage and treatment. The latter is considered in this Thesis: the processing of very high spatial and spectral resolution images is treated with approaches based on data-driven algorithms relying on kernel methods. In particular, the problem of image classification, i.e. the categorization of the image's pixels into a reduced number of classes reflecting spectral and contextual properties, is studied through the different models presented. The accent is put on algorithmic efficiency and the simplicity of the approaches proposed, to avoid too complex models that would not be used by users. The major challenge of the Thesis is to remain close to concrete remote sensing problems, without losing the methodological interest from the machine learning viewpoint: in this sense, this work aims at building a bridge between the machine learning and remote sensing communities and all the models proposed have been developed keeping in mind the need for such a synergy. Four models are proposed: first, an adaptive model learning the relevant image features has been proposed to solve the problem of high dimensionality and collinearity of the image features. This model provides automatically an accurate classifier and a ranking of the relevance of the single features. The scarcity and unreliability of labeled. information were the common root of the second and third models proposed: when confronted to such problems, the user can either construct the labeled set iteratively by direct interaction with the machine or use the unlabeled data to increase robustness and quality of the description of data. Both solutions have been explored resulting into two methodological contributions, based respectively on active learning and semisupervised learning. Finally, the more theoretical issue of structured outputs has been considered in the last model, which, by integrating outputs similarity into a model, opens new challenges and opportunities for remote sensing image processing.

Recent advances in remote sensing image processing

Remote sensing image processing is nowadays a mature research area. The techniques developed in the field allow many real-life applications with great societal value. For instance, urban monitoring, fire detection or flood prediction can have a great impact on economical and environmental issues. To attain such objectives, the remote sensing community has turned into a multidisciplinary field of science that embraces physics, signal theory, computer science, electronics, and communications. From a machine learning and signal/image processing point of view, all the applications are tackled under specific formalisms, such as classification and clustering, regression and function approximation, image coding, restoration and enhancement, source unmixing, data fusion or feature selection and extraction. This paper serves as a survey of methods and applications, and reviews the last methodological advances in remote sensing image processing.

Interest rates mapping

The present study deals with the analysis and mapping of Swiss franc interest rates. Interest rates depend on time and maturity, defining term structure of the interest rate curves (IRC). In the present study IRC are considered in a two-dimensional feature space - time and maturity. Exploratory data analysis includes a variety of tools widely used in econophysics and geostatistics. Geostatistical models and machine learning algorithms (multilayer perceptron and Support Vector Machines) were applied to produce interest rate maps. IR maps can be used for the visualisation and pattern perception purposes, to develop and to explore economical hypotheses, to produce dynamic asset-liability simulations and for financial risk assessments. The feasibility of an application of interest rates mapping approach for the IRC forecasting is considered as well. (C) 2008 Elsevier B.V. All rights reserved.

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The quality of environmental data analysis and propagation of errors are heavily affected by the representativity of the initial sampling design [CRE 93, DEU 97, KAN 04a, LEN 06, MUL07]. Geostatistical methods such as kriging are related to field samples, whose spatial distribution is crucial for the correct detection of the phenomena. Literature about the design of environmental monitoring networks (MN) is widespread and several interesting books have recently been published [GRU 06, LEN 06, MUL 07] in order to clarify the basic principles of spatial sampling design (monitoring networks optimization) based on Support Vector Machines was proposed. Nonetheless, modelers often receive real data coming from environmental monitoring networks that suffer from problems of non-homogenity (clustering). Clustering can be related to the preferential sampling or to the impossibility of reaching certain regions.