Introduction Methods Experiments Robust cartogram visualization of outliers in manifold leaning Alessandra Tosi and Alfredo Vellido atosi@lsi.upc.edu - avellido@lsi.upc.edu LSI Department - UPC, Barcelona Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction Methods Experiments Introduction 1 Goals Methods 2 NLDR methods: Generative Topographic Mapping Distortion measures in NLDR: Magnification Factor Cartogram-based representation Experiments 3 Cartograms representations for GTM and its variants Results Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction Methods Goals Experiments Table of Contents Introduction 1 Goals Methods 2 Experiments 3 Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction Methods Goals Experiments PROBLEM: Increasing amount of available high-dimensional data sets, with different levels of complexity and growing diversity of characteristics. Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction Methods Goals Experiments PROBLEM: Increasing amount of available high-dimensional data sets, with different levels of complexity and growing diversity of characteristics. CHALLENGE: Translation of raw data into useful information that can be acted upon in practical terms. Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction Methods Goals Experiments PROBLEM: Increasing amount of available high-dimensional data sets, with different levels of complexity and growing diversity of characteristics. CHALLENGE: Translation of raw data into useful information that can be acted upon in practical terms. Nonlinear Dimensionality Reduction : Nonlinear techniques are applied to reduce dimensionality of data in order to explore multivariate data. It is almost impossible to completely avoid geometrical distortions while reducing dimensionality Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction Methods Goals Experiments PROBLEM: Increasing amount of available high-dimensional data sets, with different levels of complexity and growing diversity of characteristics. CHALLENGE: Translation of raw data into useful information that can be acted upon in practical terms. Nonlinear Dimensionality Reduction : Nonlinear techniques are applied to reduce dimensionality of data in order to explore multivariate data. It is almost impossible to completely avoid geometrical distortions while reducing dimensionality Distortion Measures : Quantify and visualize this distortion itself in order to interpret data in a more faithful way. Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction Methods Goals Experiments PROBLEM: Increasing amount of available high-dimensional data sets, with different levels of complexity and growing diversity of characteristics. CHALLENGE: Translation of raw data into useful information that can be acted upon in practical terms. Nonlinear Dimensionality Reduction : Nonlinear techniques are applied to reduce dimensionality of data in order to explore multivariate data. It is almost impossible to completely avoid geometrical distortions while reducing dimensionality Distortion Measures : Quantify and visualize this distortion itself in order to interpret data in a more faithful way. Visualization : Explicitly reintroducing the local distortion created by NLDR models into the low-dimensional representation of the MVD for visualization that they produce. Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation Table of Contents Introduction 1 Methods 2 NLDR methods: Generative Topographic Mapping Distortion measures in NLDR: Magnification Factor Cartogram-based representation 3 Experiments Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation NLDR methods for MVD visualization To successfully analyse real data, more complex models are often required: Nonlinear Dimensionality Reduction models ( NLDR ). Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation NLDR methods for MVD visualization To successfully analyse real data, more complex models are often required: Nonlinear Dimensionality Reduction models ( NLDR ). Manifold learning attempts to describe MVD through nonlinear low-dimensional manifolds embedded in the observed data space. The aim is to discover the underlying geometry of data, while preserving the topology rather than pairwise distances and generating a low- dimensionality model. Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation NLDR methods for MVD visualization To successfully analyse real data, more complex models are often required: Nonlinear Dimensionality Reduction models ( NLDR ). Latent Variables Models attempt to provide an additional set of variables (latent or hidden variables) in addition to the observed ones. Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation NLDR methods for MVD visualization To successfully analyse real data, more complex models are often required: Nonlinear Dimensionality Reduction models ( NLDR ). Vector quantization reduces the number of observation by replacing original data with a smaller set of vectors of the same dimension, called prototypes (units, neurons, centroids, weight vectors) 1 0.8 0.6 0.4 0.2 0 1 0.8 1 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0 0 Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation Generative Topographic Mapping (GTM) The Generative Topographic Mapping (GTM) is a nonlinear Latent Variable Model developed by Bishop, Svens´ en and Williams in the late nineties. Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation Generative Topographic Mapping (GTM) The Generative Topographic Mapping (GTM) is a nonlinear Latent Variable Model developed by Bishop, Svens´ en and Williams in the late nineties. Basic GTM defines a Gaussian probability distribution in the latent space, in order to induce the corresponding probability distribution in the observed data space, using concepts of Bayesian inference. Images of sampled data points, or prototypes , are defined according to the following rule: y k = W Φ( u k ) Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation Generative Topographic Mapping (GTM) The Generative Topographic Mapping (GTM) is a nonlinear Latent Variable Model developed by Bishop, Svens´ en and Williams in the late nineties. Basic GTM defines a Gaussian probability distribution in the latent space, in order to induce the corresponding probability distribution in the observed data space, using concepts of Bayesian inference. Images of sampled data points, or prototypes , are defined according to the following rule: y k = W Φ( u k ) The basic GTM model has some limitations when dealing with atypical data or outliers , as they are likely to bias the estimation of its parameters. More robust formulations of GTM have been proposed using a mixture of Student’s t-distributions ( t-GTM ). Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation Magnification Factor Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
Introduction NLDR methods: Generative Topographic Mapping Methods Distortion measures in NLDR: Magnification Factor Experiments Cartogram-based representation Magnification Factor � det ( JJ T ) dA’ / dA = J is the Jacobian (of dimension 2 × d ) of the mapping transformation. Robust cartogram visualization of outliers in manifold leaning A. Tosi and A. Vellido
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