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Gérard Biau

  1. An Affine Invariant $k$-Nearest Neighbor Regression Estimate.

    Mon, 05/21/2012 - 15:01 — SomNambul
    Authors: Gérard Biau, Luc Devroye, Vida Dujmovic, Adam Krzyzak
    Subjects: Statistics
    Abstract

    We design a data-dependent metric in $\mathbb R^d$ and use it to define the
    $k$-nearest neighbors of a given point. Our metric is invariant under all
    affine transformations. We show that, with this metric, the standard
    $k$-nearest neighbor regression estimate is asymptotically consistent under the
    usual conditions on $k$, and minimal requirements on the input data.

  2. Sparse single-index model.

    Tue, 01/18/2011 - 16:01 — SomNambul
    Authors: Gérard Biau, Pierre Alquier
    Subjects: Statistics
    Abstract

    The single-index model is known to offer a flexible way to model a variety of
    high-dimensional real-world phenomena. However, despite its relative implicity,
    this dimension reduction scheme is faced with severe complications as soon as
    the underlying dimension becomes larger than the number of observations ("p
    larger than n" paradigm). To circumvent this difficulty, we consider the
    single-index model estimation problem from a sparsity perspective using a
    PAC-Bayesian approach.

  3. Statistical analysis of $k$-nearest neighbor collaborative recommendation.

    Tue, 10/05/2010 - 15:01 — SomNambul
    Authors: Gérard Biau, Laurent Rouvière, Benoît Cadre
    Subjects: Statistics
    Abstract

    Collaborative recommendation is an information-filtering technique that
    attempts to present information items that are likely of interest to an
    Internet user. Traditionally, collaborative systems deal with situations with
    two types of variables, users and items. In its most common form, the problem
    is framed as trying to estimate ratings for items that have not yet been
    consumed by a user. Despite wide-ranging literature, little is known about the
    statistical properties of recommendation systems.

  4. Analysis of a Random Forests Model.

    Tue, 05/04/2010 - 15:01 — SomNambul
    Authors: Gérard Biau
    Subjects: Machine Learning
    Abstract

    Random forests are a scheme proposed by Leo Breiman in the 00's for building
    a predictor ensemble with a set of decision trees that grow in randomly
    selected subspaces of data. Despite growing interest and practical use, there
    has been little exploration of the statistical properties of random forests,
    and little is known about the mathematical forces driving the algorithm. In
    this paper, we offer an in-depth analysis of a random forests model suggested
    by Breiman in 2004, which is very close to the original algorithm.

  5. PCA-Kernel Estimation.

    Mon, 03/29/2010 - 15:01 — SomNambul
    Authors: Gérard Biau, André Mas
    Subjects: Statistics
    Abstract

    Many statistical estimation techniques for high-dimensional or functional
    data are based on a preliminary dimension reduction step, which consists in
    projecting the sample $\bX_1, \hdots, \bX_n$ onto the first $D$ eigenvectors of
    the Principal Component Analysis (PCA) associated with the empirical projector
    $\hat \Pi_D$. Classical nonparametric inference methods such as kernel density
    estimation or kernel regression analysis are then performed in the (usually
    small) $D$-dimensional space.

  6. A Stochastic Model for Collaborative Recommendation.

    Wed, 10/14/2009 - 11:03 — SomNambul
    Authors: Gérard Biau, Benoit Cadre, Laurent Rouvière
    Subjects: Machine Learning
    Abstract

    Collaborative recommendation is an information-filtering technique that
    attempts to present information items (movies, music, books, news, images, Web
    pages, etc.) that are likely of interest to the Internet user. Traditionally,
    collaborative systems deal with situations with two types of variables, users
    and items. In its most common form, the problem is framed as trying to estimate
    ratings for items that have not yet been consumed by a user. Despite
    wide-ranging literature, little is known about the statistical properties of
    recommendation systems.

  7. A Stochastic Model for Collaborative Recommendation.

    Wed, 10/14/2009 - 11:03 — SomNambul
    Authors: Gérard Biau, Benoit Cadre, Laurent Rouvière
    Subjects: Machine Learning
    Abstract

    Collaborative recommendation is an information-filtering technique that
    attempts to present information items (movies, music, books, news, images, Web
    pages, etc.) that are likely of interest to the Internet user. Traditionally,
    collaborative systems deal with situations with two types of variables, users
    and items. In its most common form, the problem is framed as trying to estimate
    ratings for items that have not yet been consumed by a user. Despite
    wide-ranging literature, little is known about the statistical properties of
    recommendation systems.

  8. Sequential Quantile Prediction of Time Series.

    Wed, 08/19/2009 - 19:30 — SomNambul
    Authors: Gérard Biau, Benoît Patra
    Subjects: Methodology
    Abstract

    Motivated by a broad range of potential applications, we address the quantile
    prediction problem of real-valued time series. We present a sequential quantile
    forecasting model based on the combination of a set of elementary nearest
    neighbor-type predictors called "experts" and show its consistency under a
    minimum of conditions. Our approach builds on the methodology developed in
    recent years for prediction of individual sequences and exploits the quantile
    structure as a minimizer of the so-called pinball loss function.

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