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Alan Julian Izenman - One of the best experts on this subject based on the ideXlab platform.

  • Recent Developments in Nonparametric Density Estimation
    Journal of the American Statistical Association, 1991
    Co-Authors: Alan Julian Izenman
    Abstract:

    Advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research, and especially the development of nonparametric data analysis procedures. In particular, theoretical and applied research on nonparametric Density Estimation has had a noticeable influence on related topics, such as nonparametric regression, nonparametric discrimination, and nonparametric pattern recognition. This article reviews recent developments in nonparametric Density Estimation and includes topics that have been omitted from review articles and books on the subject. The early Density Estimation methods, such as the histogram, kernel estimators, and orthogonal series estimators are still very popular, and recent research on them is described. Different types of restricted maximum likelihood Density estimators, including order-restricted estimators, maximum penalized likelihood estimators, and sieve estimators, are discussed, where restrictions are imposed upon the class of densities or on the form of the likelihood function. Nonparametric Density estimators that are data-adaptive and lead to locally smoothed estimators are also discussed; these include variable partition histograms, estimators based on statistically equivalent blocks, nearest-neighbor estimators, variable kernel estimators, and adaptive kernel estimators. For the multivariate case, extensions of methods of univariate Density Estimation are usually straightforward but can be computationally expensive. A method of multivariate Density Estimation that did not spring from a univariate generalization is described, namely, projection pursuit Density Estimation, in which both dimensionality reduction and Density Estimation can be pursued at the same time. Finally, some areas of related research are mentioned, such as nonparametric Estimation of functionals of a Density, robust parametric Estimation, semiparametric models, and Density Estimation for censored and incomplete data, directional and spherical data, and Density Estimation for dependent sequences of observations.

Clayton Scott - One of the best experts on this subject based on the ideXlab platform.

  • robust kernel Density Estimation
    arXiv: Machine Learning, 2011
    Co-Authors: Clayton Scott
    Abstract:

    We propose a method for nonparametric Density Estimation that exhibits robustness to contamination of the training sample. This method achieves robustness by combining a traditional kernel Density estimator (KDE) with ideas from classical $M$-Estimation. We interpret the KDE based on a radial, positive semi-definite kernel as a sample mean in the associated reproducing kernel Hilbert space. Since the sample mean is sensitive to outliers, we estimate it robustly via $M$-Estimation, yielding a robust kernel Density estimator (RKDE). An RKDE can be computed efficiently via a kernelized iteratively re-weighted least squares (IRWLS) algorithm. Necessary and sufficient conditions are given for kernelized IRWLS to converge to the global minimizer of the $M$-estimator objective function. The robustness of the RKDE is demonstrated with a representer theorem, the influence function, and experimental results for Density Estimation and anomaly detection.

Robert D. Nowak - One of the best experts on this subject based on the ideXlab platform.

  • Multiresolution nonparametric intensity and Density Estimation
    2002 IEEE International Conference on Acoustics Speech and Signal Processing, 2002
    Co-Authors: Rebecca M. Willett, Robert D. Nowak
    Abstract:

    This paper introduces a new multiscale method for nonparametric piecewise polynomial intensity and Density Estimation of point processes. Fast, piecewise polynomial, maximum penalized likelihood methods for intensity and Density Estimation are developed. The recursive partitioning scheme underlying these methods is based on multiscale likelihood factorizations which, unlike conventional wavelet decompositions, are very well suited to applications with point process data. Experimental results demonstrate that multiscale methods can outperform wavelet and kernel based Density Estimation methods.

Jim E. Griffin - One of the best experts on this subject based on the ideXlab platform.

  • On the Bayesian analysis of species sampling mixture models for Density Estimation
    2020
    Co-Authors: Jim E. Griffin
    Abstract:

    The mixture of normals model has been extensively applied to Density Estimation problems. This paper proposes an alternative parameterisation that naturally leads to new forms of prior distribution. The parameters can be interpreted as the location, scale and smoothness of the Density. Priors on these parameters are often easier to specify. Alternatively, improper and default choices lead to automatic Bayesian Density Estimation. The ideas are extended to multivariate Density Estimation.

  • Bayesian multivariate Density Estimation for observables and random effects
    2020
    Co-Authors: Jim E. Griffin
    Abstract:

    Multivariate Density Estimation is approached using Bayesian nonparametric mixture of normals models. Two models are developed which are both centred over a multivariate normal distribution but make different prior assumptions about how the unknown distribution departs from a normal distribution. The priors are applied to Density Estimation of both observables and random effects (or other unobservable random quantities). Markov chain Monte Carlo methods are described for Estimation of all models. The models are applied to Density Estimation for observables and the application of a nonparametric linear mixed model to repeated cholesterol measurements from the Framingham study.

  • Default priors for Density Estimation with mixture models
    Bayesian Analysis, 2010
    Co-Authors: Jim E. Griffin
    Abstract:

    The infinite mixture of normals model has become a popular method for Density Estimation problems. This paper proposes an alternative hierarchical model that leads to hyperparameters that can be interpreted as the location, scale and smoothness of the Density. The priors on other parts of the model have little effect on the Density estimates and can be given default choices. Automatic Bayesian Density Estimation can be implemented by using uninformative priors for location and scale and default priors for the smoothness. The performance of these methods for Density Estimation are compared to previously proposed default priors for four data sets.

Adrian M. Peter - One of the best experts on this subject based on the ideXlab platform.

  • Wavelet based Density Estimation for multidimensional streaming data
    2020
    Co-Authors: Daniel Weinand, Gedeon Nyengele, Mark Moyou, Adrian M. Peter
    Abstract:

    In the present information economy, the colossal amount of data generated daily has spawned the need for realtime data driven algorithms that extract actionable intelligence with minimal user inuence. Density Estimation is one path to extract this intelligence from the data and the incorporation of wavelets serves to boost the accuracy of the Density Estimation framework. The goal of this project was to develop a multidimensional computational implementation of this wavelet Density Estimation framework and showcase its utility on a relevant application. The report contains background information on wavelets, Density Estimation and all the details about the Matlab and Java implementations of the multidimensional wavelet Density estimator. Finally, we demonstrate the utility of the approach by applying it to nancial market analysis.

  • Multiwavelet Density Estimation
    Applied Mathematics and Computation, 2013
    Co-Authors: J. B. Locke, Adrian M. Peter
    Abstract:

    Accurate Density Estimation methodologies play an integral role in a variety of scientific disciplines with applications including simulation models, decision support tools, and exploratory data analysis. In the past, histograms and kernel Density estimators have been the predominant tools of choice, primarily due to their ease of use and mathematical simplicity. More recently, the use of wavelets for Density Estimation has gained in popularity due to their ability to approximate a large class of functions, including those with localized, abrupt variations. However, a well-known attribute of wavelet bases is that they cannot be simultaneously symmetric, orthogonal, and compactly supported. Multiwavelets-more general, vector-valued constructions of wavelets-overcome this disadvantage, making them natural choices for estimating Density functions, many of which exhibit local symmetries around features such as a mode. We extend the methodology of wavelet Density Estimation to use multiwavelet bases and illustrate several empirical examples of multiwavelet Density Estimation.

  • Multiwavelet Density Estimation
    arXiv: Statistics Theory, 2012
    Co-Authors: J. B. Locke, Adrian M. Peter
    Abstract:

    Accurate Density Estimation methodologies play an integral role in a variety of scientific disciplines, with applications including simulation models, decision support tools, and exploratory data analysis. In the past, histograms and kernel Density estimators have been the predominant tools of choice, primarily due to their ease of use and mathematical simplicity. More recently, the use of wavelets for Density Estimation has gained in popularity due to their ability to approximate a large class of functions, including those with localized, abrupt variations. However, a well-known attribute of wavelet bases is that they can not be simultaneously symmetric, orthogonal, and compactly supported. Multiwavelets-a more general, vector-valued, construction of wavelets-overcome this disadvantage, making them natural choices for estimating Density functions, many of which exhibit local symmetries around features such as a mode. We extend the methodology of wavelet Density Estimation to use multiwavelet bases and illustrate several empirical results where multiwavelet estimators outperform their wavelet counterparts at coarser resolution levels.