Orthogonal Series Density Estimation Research Articles

Nonparametric density estimation over its unknown support for right censored data

The long-standing problem of nonparametric density estimation of the lifetime of interest over its unknown support, when only right-censored observations are available, is considered. Two cases, when the support of the lifetime of interest is a proper subset of the support of censoring variable and when the supports are the same, are explored. An orthogonal series density estimation, over a random interval defined by a specially chosen sequence of order statistics, is proposed and rates of the mean integrated squared error convergence are evaluated.

A machine-vision method for automatic classification of stellar halo substructure

ABSTRACT Tidal debris structures formed from disrupted satellites contain important clues about the assembly histories of galaxies. To date, studies of these structures have been hampered by reliance on by-eye identification and morphological classification which leaves their interpretation significantly uncertain. In this work, we present a new machine-vision technique based on the Subspace-Constrained Mean Shift (SCMS) algorithm which can perform these tasks automatically. SCMS finds the location of the high-density ‘ridges’ that define substructure morphology. After identification, the coefficients of an orthogonal series density estimator are used to classify points on the ridges as part of a continuum between shell-like or stream-like debris, from which a global morphological classification can be determined. We dub this procedure Subspace-Constrained Unsupervised Detection of Structure (SCUDS). By applying this tool to controlled N-body simulations of minor mergers, we demonstrate that the extracted classifications correspond to the well-understood underlying physics of phase mixing. The application of SCUDS to resolved stellar population data from near-future surveys will inform our understanding of the buildup of galaxies’ stellar haloes.

Orthogonal series density estimation for complex surveys

ABSTRACTWe propose an orthogonal series density estimator for complex surveys, where samples are neither independent nor identically distributed. The proposed estimator is proved to be design-unbiased and asymptotically design-consistent. The asymptotic normality is proved under both design and combined spaces. Two data driven estimators are proposed based on the proposed oracle estimator. We show the efficiency of the proposed estimators in simulation studies. A real survey data example is provided for an illustration.

Local orthogonal polynomial expansion for density estimation

ABSTRACTA local orthogonal polynomial expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localised expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalises straightforwardly to multivariate settings. By manner of construction, it is similar to local likelihood density estimation (LLDE). In the limit of small bandwidths, LOrPE functions as kernel density estimation (KDE) with high-order (effective) kernels inherently free of boundary bias, a natural consequence of kernel reshaping to accommodate endpoints. Consistency and faster asymptotic convergence rates follow. In the limit of large bandwidths LOrPE is equivalent to orthogonal series density estimation (OSDE) with Legendre polynomials, thereby inheriting its consistency. We compare the performance of LOrPE to KDE, LLDE, and OSDE, in a number of simulation studies. In terms of mean integrated squared error, the results suggest that with a proper balance of the two tuning parameters, bandwidth and degree, LOrPE generally outperforms these competitors when estimating densities with sharply truncated supports.

Deconvolving the detector in Fourier space

We discuss algorithms for the analysis of hadronic final states, with application to single top t-channel production, heavy Higgs decaying as H → W+W- in the lepton plus jets mode, and others. In our examples, nature has arranged for the triple differential decay rates in angles we call θ1, θ2, ϕ to be a short finite series in orthogonal functions, √2πamk,lYmk (θ1, ϕ)Yml (θ2, 0) (summation implied), where the amk,l are complex constants. This observation can be exploited in two ways; first, a technique called orthogonal series density estimation may be employed to extract coefficients amk,l of the decay and physics parameters related to these coefficients; second, an angular analog of the convolution theorem may be employed to analytically deconvolve detector resolution effects from an observed signal.

Non parametric sequential estimation of the probability density function by orthogonal series

ABSTRACTWe present in this work an orthogonal series density estimator based on random number of observations Nt. We give its statistical properties (bias, variance, mean square error, and mean square integrated error) and some asymptotic properties. We consider that Nt is independent of the observations and as t → ∞. We show that Kronmal–Tarter method for choosing the smoothing parameter is still valid in the case where the sample size is random. A detailed study with cosine basis is presented. The estimator obtained is a probability density, asymptotically unbiased and consistent. A simulation is used in order to study the behavior of the density estimator and shows that the estimator is performant.

Orthogonal series density estimation

AbstractOrthogonal series density estimation is a powerful nonparametric estimation methodology that allows one to analyze and present data at hand without any prior opinion about shape of an underlying density. The idea of construction of an adaptive orthogonal series density estimator is explained on the classical example of a direct sample from a univariate density. Data‐driven estimators, which have been used for years, as well as recently proposed procedures, are reviewed. Orthogonal series estimation is also known for its sharp minimax properties which are explained. Furthermore, applications of the orthogonal series methodology to more complicated settings, including censored and biased data as well as estimation of the density of regression errors and the conditional density, are also presented. Copyright © 2010 John Wiley & Sons, Inc.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Density Estimation

Asymptotic normality of the mixture density estimator in a disaggregation scheme

The paper concerns the asymptotic distribution of the mixture density estimator, proposed by Leipus et al. [Leipus, R., Oppenheim, G., Philippe, A., and Viano, M.-C. (2006), ‘Orthogonal Series Density Estimation in a Disaggregation Scheme’, Journal of Statistical Planning and Inference, 136, 2547–2571], in the aggregation/disaggregation problem of random parameter AR(1) process. We prove that, under mild conditions on the (semiparametric) form of the mixture density, the estimator is asymptotically normal. The proof is based on the limit theory for the quadratic form in linear random variables developed by Bhansali et al. [Bhansali, R.J., Giraitis, L., and Kokoszka, P.S. (2007), Approximations and Limit Theory for Quadratic Forms of Linear Processes’, Stochastic Processes and their Applications, 117, 71–95]. The moving average representation of the aggregated process is investigated. A simulation study illustrates the result.

Adaptive density estimation using an orthogonal series for global illumination

In Monte-Carlo photon-tracing methods energy-carrying particles are traced in an environment to generate hit points on object surfaces for simulating global illumination. The surface illumination can be reconstructed from particle hit points by solving a density estimation problem using an orthogonal series. The appropriate number of terms of an orthogonal series used for approximating surface illumination depends on the numbers of hit points (i.e. the number of samples) as well as illumination discontinuity (i.e. shadow boundaries) on a surface. Existing photon-tracing methods based on orthogonal series density estimation use a pre-specified or fixed number m of terms of an orthogonal series; this results in undesirable visual artifacts, i.e. either near-constant shading across a surface which conceals the true illumination variation when m is very small or excessive illumination oscillation when m is very large. On the other hand, interactive user specification of the number of terms for different surface patches is inefficient and inaccurate, and thus is not a practical solution. In this paper an algorithm is presented for automatically determining on the fly the optimal number of terms to be used in an orthogonal series in order to reconstruct surface illumination from surface hit points. When the optimal number of terms required is too high due to illumination discontinuity of a surface, a heuristic scheme is used to subdivide the surface along the discontinuity boundary into some smaller patches, called sub-patches, so as to allow a smaller number of terms in the orthogonal series to optimally represent illumination on these sub-patches. Experimental results are presented to show that the new method improves upon other existing orthogonal series-based density estimation methods used for global illumination in both running time and memory requirements.

Orthogonal series density estimation in a disaggregation scheme

In this paper we consider long-memory processes obtained by aggregation of independent random parameter AR(1) processes. We propose an estimator of the density of the underlying random parameter. This estimator is based on the expansion of the density function on the basis of Gegenbauer polynomials. Rate of convergence to zero of the mean integrated square error (MISE) and of the uniform error are obtained. The results are illustrated by Monte-Carlo simulations.

Orthogonal series density estimation and the kernel eigenvalue problem.

Kernel principal component analysis has been introduced as a method of extracting a set of orthonormal nonlinear features from multivariate data, and many impressive applications are being reported within the literature. This article presents the view that the eigenvalue decomposition of a kernel matrix can also provide the discrete expansion coefficients required for a nonparametric orthogonal series density estimator. In addition to providing novel insights into nonparametric density estimation, this article provides an intuitively appealing interpretation for the nonlinear features extracted from data using kernel principal component analysis.

Adaptive orthogonal series density estimation for small samples

Adaptive nonnegative orthogonal series estimators are investigated for the case of small sample sizes. The estimators are analyzed via intensive Monte Carlo study and oracle inequalities when an estimator is compared with optimal pseudo-estimators (“oracles”) based on an underlying (estimated) density. The results are favorable to a modified asymptotically efficient estimator. The paper also sheds light on that how to employ results of asymptotic theory for the practically important case of small sample sizes.

An orthogonal series density estimation approach to reconstructing positron emission tomography images

Positron emission tomography (PET) is an important medical imaging technique. Statistically, the PET image reconstruction problem comprises estimating the intensity function of a non-homogeneous Poisson process from a set of indirectly observed data (an integral transform is involved). In this paper, we investigate a new reconstruction method consisting in the adaptation of orthogonal series density estimation techniques to use with an idealised form of the PET problem. The method provides reasonable reconstructions quickly; its computational speed is its major advantage. It has further advantages (e.g. no pixellation required) and various disadvantages (e.g. difficulties with object boundaries, non-negativity not guaranteed) which are discussed. Its major disadvantage, however, is the difficulty associated with generalising the approach to cope with more realistic versions of the PET model.

The Selection of Terms in an Orthogonal Series Density Estimator

Abstract We show that Kronmal and Tarter's well-known rule for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases. These difficulties arise when the underlying density has a nonmonotone sequence of Fourier coefficients, as is likely to be the case with sharply peaked or multimodal distributions. We suggest a way of overcoming these shortcomings.

The Selection of Terms in an Orthogonal Series Density Estimator

The Selection of Terms in an Orthogonal Series Density Estimator