Multiwavelet density estimation

2000

In this paper we describe theoretical properties of wavelet-based random densities and give algorithms for their generation. We exhibit random densities subject to some standard constraints: smoothness, modality, and skewness. We also give a relevant example of use of random densities.

Statistical Methodology, 2014

We consider the estimation of a two dimensional continuous-discrete density function with applications to competing risks. We construct two new wavelet estimators (non-adaptive and adaptive) for the joint density function taking into account this special continuous-discrete structure. The rates of convergence of the proposed estimators are established under the L 2 risk over Besov balls. Our main result proves that our adaptive wavelet estimator (based on hard thresholding) attains a sharp rate of convergence. A simulation study illustrates the usefulness of the proposed estimators.

Journal of the Royal Statistical Society: Series D (The Statistician), 2000

In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this paper gives a relatively accessible introduction to standard wavelet analysis and provides a review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be effective, rather than to researchers who are already familiar with the ®eld. Given that objective, we do not emphasize mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in greater detail and generality if required. The paper ®rst establishes some necessary basic mathematical background and terminology relating to wavelets. It then reviews the more well-established applications of wavelets in statistics including their use in nonparametric regression, density estimation, inverse problems, changepoint problems and in some specialized aspects of time series analysis. Possible extensions to the uses of wavelets in statistics are then considered. The paper concludes with a brief reference to readily available software packages for wavelet analysis.

2012

The density estimation problem under bias and multiplicative censoring is considered. Adopting the wavelet approach, we construct a linear nonadaptive estimator and a nonlinear adaptive estimator. The adaptive one belongs to the family of the hard thresholding estimators. We evaluate their performances by determining upper bounds of the mean integrated squared error over a wide range of functions. Sharp upper bounds are obtained.

Quantitative evidence is presented in order to study the performance of a wavelet-based, second order estimator of relevance to mul-tiscaling phenomena such as telecommunications traffic. Special attention is given to the behavior of the confidence intervals of the scaling exponent under Gaussian assumptions with synthetic long-range dependent trajectories, allowing for a straightforward contrast with the theoretical results presented in the bibliography. The bias-variance tradeoff decision behind the choice of the onset of the scaling region is also re-visited, along with its influence on the confidence of the intervals of the exponent.

We propose a method of estimation of the derivatives of probability density based wavelets methods for a sequence of negatively associated random variables with a common one-dimensional probability density function and obtain an upper bound on L p -losses for the such estimators.

Proceedings of WSCG, 2006

Silvia Castro Dpto. de Cs e Ing Comp. Univ. Nac. del Sur Avda. Alem 1253 Argentina (8000) Bahía Blanca smc@cs.uns.edu.ar ... Liliana Castro Dpto. de Matemática Univ. Nac. del Sur Avda. Alem 1253 Argentina (8000) Bahía Blanca lcastro@uns.edu.ar

Journal of Economic Surveys, 2013

Economists are already familiar with the Discrete Wavelet Transform. However, a body of work using the Continuous Wavelet Transform has also been growing. We provide a selfcontained summary on continuous wavelet tools, such as the Continuous Wavelet Transform, the Cross-Wavelet, the Wavelet Coherency and the Phase-Di¤erence. Furthermore, we generalize the concept of simple coherency to Partial Wavelet Coherency and Multiple Wavelet Coherency, akin to partial and multiple correlations, allowing the researcher to move beyond bivariate analysis. Finally, we describe the Generalized Morse Wavelets, a class of analytic wavelets recently proposed. A user-friendly toolbox, with examples, is attached to this paper.

We consider the estimation of a density function on the basis of a random sample from a weighted distribution. We propose linear and nonlinear wavelet density estimators, and provide their asymptotic formulae for mean integrated squared error. In particular, we derive an analogue of the asymptotic formula of the mean integrated square error in the context of kernel density estimators for weighted data, admitting an expansion with distinct squared bias and variance components. For nonlinear wavelet density estimators, unlike the analogous situation for kernel or linear wavelet density estimators, this asymptotic formula of the mean integrated square error is relatively unaffected by assumptions of continuity, and it is available for densities which are smooth only in a piecewise sense. We illustrate the behavior of the proposed linear and nonlinear wavelet density estimators in finite sample situations both in simulations and on a real-life dataset. Comparisons with a kernel density estimator are also given.

IEEE Computational Science and Engineering, 1995

Wavelets are mathematical functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Wavelets were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, and earthquake prediction. This paper introduces wavelets to the interested technical person outside of the digital signal processing field. I describe the history of wavelets beginning with Fourier, compare wavelet transforms with Fourier transforms, state properties and other special aspects of wavelets, and finish with some interesting applications such as image compression, musical tones, and de-noising noisy data. 1. WAVELETS OVERVIEW The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers in the wavelet field feel that, by using wavelets, one is adopting a whole new mindset or perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. This idea is not new. Approximation using superposition of functions has existed since the early 1800's, when Joseph Fourier discovered that he could superpose sines and cosines to represent other functions. However, in wavelet analysis, the scale that we use to look at data plays a special role. Wavelet algorithms process data at different scales or resolutions. If we look at a signal with a large "window," we would notice gross features. Similarly, if we look at a signal with a small "window," we would notice small features. The result in wavelet analysis is to see both the forest and the trees, so to speak. This makes wavelets interesting and useful. For many decades, scientists have wanted more appropriate functions than the sines and cosines which comprise the bases of Fourier analysis, to approximate choppy signals (1). By their definition, these functions are non-local (and stretch out to infinity). They therefore do a very poor job in approximating sharp spikes. But with wavelet analysis, we can use approximating functions that are contained neatly in finite domains. Wavelets are well-suited for approximating data with sharp discontinuities. The wavelet analysis procedure is to adopt a wavelet prototype function, called an analyzing wavelet or mother wavelet. Temporal analysis is performed with a contracted, high-frequency version of the prototype wavelet, while frequency analysis is performed with a dilated, low-frequency version of the same wavelet. Because the original signal or function can be represented in terms of a wavelet 1