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.

Data-driven clustering : new methods and applications

Abstract : This work is concerned with the development and application of novel unsupervised learning methods, having in mind two target applications: the analysis of forensic case data and the classification of remote sensing images. First, a method based on a symbolic optimization of the inter-sample distance measure is proposed to improve the flexibility of spectral clustering algorithms, and applied to the problem of forensic case data. This distance is optimized using a loss function related to the preservation of neighborhood structure between the input space and the space of principal components, and solutions are found using genetic programming. Results are compared to a variety of state-of--the-art clustering algorithms. Subsequently, a new large-scale clustering method based on a joint optimization of feature extraction and classification is proposed and applied to various databases, including two hyperspectral remote sensing images. The algorithm makes uses of a functional model (e.g., a neural network) for clustering which is trained by stochastic gradient descent. Results indicate that such a technique can easily scale to huge databases, can avoid the so-called out-of-sample problem, and can compete with or even outperform existing clustering algorithms on both artificial data and real remote sensing images. This is verified on small databases as well as very large problems. Résumé : Ce travail de recherche porte sur le développement et l'application de méthodes d'apprentissage dites non supervisées. Les applications visées par ces méthodes sont l'analyse de données forensiques et la classification d'images hyperspectrales en télédétection. Dans un premier temps, une méthodologie de classification non supervisée fondée sur l'optimisation symbolique d'une mesure de distance inter-échantillons est proposée. Cette mesure est obtenue en optimisant une fonction de coût reliée à la préservation de la structure de voisinage d'un point entre l'espace des variables initiales et l'espace des composantes principales. Cette méthode est appliquée à l'analyse de données forensiques et comparée à un éventail de méthodes déjà existantes. En second lieu, une méthode fondée sur une optimisation conjointe des tâches de sélection de variables et de classification est implémentée dans un réseau de neurones et appliquée à diverses bases de données, dont deux images hyperspectrales. Le réseau de neurones est entraîné à l'aide d'un algorithme de gradient stochastique, ce qui rend cette technique applicable à des images de très haute résolution. Les résultats de l'application de cette dernière montrent que l'utilisation d'une telle technique permet de classifier de très grandes bases de données sans difficulté et donne des résultats avantageusement comparables aux méthodes existantes.

Data visualisation and exploration with prior knowledge

Visualising data for exploratory analysis is a major challenge in many applications. Visualisation allows scientists to gain insight into the structure and distribution of the data, for example finding common patterns and relationships between samples as well as variables. Typically, visualisation methods like principal component analysis and multi-dimensional scaling are employed. These methods are favoured because of their simplicity, but they cannot cope with missing data and it is difficult to incorporate prior knowledge about properties of the variable space into the analysis; this is particularly important in the high-dimensional, sparse datasets typical in geochemistry. In this paper we show how to utilise a block-structured correlation matrix using a modification of a well known non-linear probabilistic visualisation model, the Generative Topographic Mapping (GTM), which can cope with missing data. The block structure supports direct modelling of strongly correlated variables. We show that including prior structural information it is possible to improve both the data visualisation and the model fit. These benefits are demonstrated on artificial data as well as a real geochemical dataset used for oil exploration, where the proposed modifications improved the missing data imputation results by 3 to 13%.

Making better biogeographical predictions of species' distributions

1. Biogeographical models of species' distributions are essential tools for assessing impacts of changing environmental conditions on natural communities and ecosystems. Practitioners need more reliable predictions to integrate into conservation planning (e.g. reserve design and management). 2. Most models still largely ignore or inappropriately take into account important features of species' distributions, such as spatial autocorrelation, dispersal and migration, biotic and environmental interactions. Whether distributions of natural communities or ecosystems are better modelled by assembling individual species' predictions in a bottom-up approach or modelled as collective entities is another important issue. An international workshop was organized to address these issues. 3. We discuss more specifically six issues in a methodological framework for generalized regression: (i) links with ecological theory; (ii) optimal use of existing data and artificially generated data; (iii) incorporating spatial context; (iv) integrating ecological and environmental interactions; (v) assessing prediction errors and uncertainties; and (vi) predicting distributions of communities or collective properties of biodiversity. 4. Synthesis and applications. Better predictions of the effects of impacts on biological communities and ecosystems can emerge only from more robust species' distribution models and better documentation of the uncertainty associated with these models. An improved understanding of causes of species' distributions, especially at their range limits, as well as of ecological assembly rules and ecosystem functioning, is necessary if further progress is to be made. A better collaborative effort between theoretical and functional ecologists, ecological modellers and statisticians is required to reach these goals.

The Effect of Public Spending on Consumption: Reconciling Theory and Evidence

Recent empirical evidence from vector autoregressions (VARs) suggests that public spending shocks increase (crowd in) private consumption. Standard general equilibrium models predict the opposite. We show that a standard real business cycle (RBC) model in which public spending is chosen optimally can rationalize the crowding-in effect documented in the VAR literature. When such a model is used as a data-generating process, a VAR estimated using the artificial data yields a positive consumption response to an increase in public spending, consistent with the empirical findings. This result holds regardless of whether private and public purchases are complements or substitutes.

Factor Analysis of a Large DSGE Model

We study the workings of the factor analysis of high-dimensional data using artificial series generated from a large, multi-sector dynamic stochastic general equilibrium (DSGE) model. The objective is to use the DSGE model as a laboratory that allow us to shed some light on the practical benefits and limitations of using factor analysis techniques on economic data. We explain in what sense the artificial data can be thought of having a factor structure, study the theoretical and finite sample properties of the principal components estimates of the factor space, investigate the substantive reason(s) for the good performance of di¤usion index forecasts, and assess the quality of the factor analysis of highly dissagregated data. In all our exercises, we explain the precise relationship between the factors and the basic macroeconomic shocks postulated by the model.

Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation

Many modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data with summary statistics of the observed data. Here we show how to construct appropriate summary statistics for ABC in a semi-automatic manner. We aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that optimal summary statistics are the posterior means of the parameters. Although these cannot be calculated analytically, we use an extra stage of simulation to estimate how the posterior means vary as a function of the data; and we then use these estimates of our summary statistics within ABC. Empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate ABC analyses than the ad hoc choices of summary statistics that have been proposed in the literature. We also demonstrate advantages over two alternative methods of simulation-based inference.

A semi-supervised classification technique based on interacting forces

Semi-supervised learning is a classification paradigm in which just a few labeled instances are available for the training process. To overcome this small amount of initial label information, the information provided by the unlabeled instances is also considered. In this paper, we propose a nature-inspired semi-supervised learning technique based on attraction forces. Instances are represented as points in a k-dimensional space, and the movement of data points is modeled as a dynamical system. As the system runs, data items with the same label cooperate with each other, and data items with different labels compete among them to attract unlabeled points by applying a specific force function. In this way, all unlabeled data items can be classified when the system reaches its stable state. Stability analysis for the proposed dynamical system is performed and some heuristics are proposed for parameter setting. Simulation results show that the proposed technique achieves good classification results on artificial data sets and is comparable to well-known semi-supervised techniques using benchmark data sets.

Topographic time-frequency decomposition of the EEG

Frequency-transformed EEG resting data has been widely used to describe normal and abnormal brain functional states as function of the spectral power in different frequency bands. This has yielded a series of clinically relevant findings. However, by transforming the EEG into the frequency domain, the initially excellent time resolution of time-domain EEG is lost. The topographic time-frequency decomposition is a novel computerized EEG analysis method that combines previously available techniques from time-domain spatial EEG analysis and time-frequency decomposition of single-channel time series. It yields a new, physiologically and statistically plausible topographic time-frequency representation of human multichannel EEG. The original EEG is accounted by the coefficients of a large set of user defined EEG like time-series, which are optimized for maximal spatial smoothness and minimal norm. These coefficients are then reduced to a small number of model scalp field configurations, which vary in intensity as a function of time and frequency. The result is thus a small number of EEG field configurations, each with a corresponding time-frequency (Wigner) plot. The method has several advantages: It does not assume that the data is composed of orthogonal elements, it does not assume stationarity, it produces topographical maps and it allows to include user-defined, specific EEG elements, such as spike and wave patterns. After a formal introduction of the method, several examples are given, which include artificial data and multichannel EEG during different physiological and pathological conditions.

Training a Linear Neural Network with a Stable LSP Solution for Jamming Cancellation

Two jamming cancellation algorithms are developed based on a stable solution of least squares problem (LSP) provided by regularization. They are based on filtered singular value decomposition (SVD) and modifications of the Greville formula. Both algorithms allow an efficient hardware implementation. Testing results on artificial data modeling difficult real-world situations are also provided.

Bayesian Dependence Tests for Continuous, Binary and Mixed Continuous-Binary Variables

Tests for dependence of continuous, discrete and mixed continuous-discrete variables are ubiquitous in science. The goal of this paper is to derive Bayesian alternatives to frequentist null hypothesis significance tests for dependence. In particular, we will present three Bayesian tests for dependence of binary, continuous and mixed variables. These tests are nonparametric and based on the Dirichlet Process, which allows us to use the same prior model for all of them. Therefore, the tests are “consistent” among each other, in the sense that the probabilities that variables are dependent computed with these tests are commensurable across the different types of variables being tested. By means of simulations with artificial data, we show the effectiveness of the new tests.

Evaluating tests for convergence of economic series using Monte Carlo methods with an application to real GDP's per head.

Doutoramento em Economia.

Artificial neural networks analysis of Sao Paulo subway tunnel settlement data

Artificial neural networks have been used to analyze a number of engineering problems, including settlement caused by different tunneling methods in various types of ground mass. This paper focuses on settlement over shotcrete- supported tunnels on Sao Paulo subway line 2 (West Extension) that were excavated in Tertiary sediments using the sequential excavation method. The adjusted network is a good tool for predicting settlement above new tunnels to be excavated in similar conditions. The influence of network training parameters on the quality of results is also discussed. (C) 2007 Elsevier Ltd. All rights reserved.

Stratigraphic interpretation of Well-Log data of the Athabasca Oil Sands of Alberta Canada through Pattern recognition and Artificial Intelligence

Dissertation submitted in partial fulfillment of the requirements for the Degree of Master of Science in Geospatial Technologies.