Local Bandwidth Research Articles

Absolute error criteria for bandwidth selection in density estimation from censored data

In Kernel density estimation, a criticism of bandwidth selection techniques which minimize squared error expressions is that they perform poorly when estimating tails of probability density functions. Techniques minimizing absolute error expressions are thought to result in more uniform performance and be potentially superior. An asympotic mean absolute error expression for nonparametric kernel density estimators from right-censored data is developed here. This expression is used to obtain local and global bandwidths that are optimal in the sense that they minimize asymptotic mean absolute error and integrated asymptotic mean absolute error, respectively. These estimators are illustrated fro eight data sets from known distributions. Computer simulation results are discussed, comparing the estimation methods with squared-error-based bandwidth selection for right-censored data.

SimpleFit: a framework for analyzing design trade-offs in Raw architectures

The semiconductor industry roadmap projects that advances in VLSI technology will permit more than one billion transistors on a chip by the year 2010. The MIT Raw microprocessor is a proposed architecture that strives to exploit these chip-level resources by implementing thousands of tiles, each comprising a processing element and a small amount of memory, coupled by a static two-dimensional interconnect. A compiler partitions fine-grain instruction-level parallelism across the tiles and statically schedules intertile communication over the interconnect. Because Raw microprocessors fully expose their internal hardware structure to the software, they can be viewed as a gigantic FPGA with coarse-grained tiles in which software orchestrates communication over static interconnections. One open challenge in Raw architectures is to determine their optimal grain size and balance. The grain size is the area of each tile and the balance is the proportion of area in each tile devoted to memory, processing, communication, and off-chip global I/O. If the total chip area is fixed, higher processing power per tile requires large tiles and hence reduces the total number of tiles on the chip. This paper presents SimpleFit, a novel analytical framework that designers can use to reason about the design space of Raw microprocessors. Our model is also generalizable to multiprocessors on a chip. Based on an architectural model, an application model, and a VLSI cost analysis, the framework computes the performance of applications and uses an optimization process to identify designs that will execute these applications most cost-effectively, Although the optimal machine configurations obtained vary for different applications, problem sizes, and budgets, the general trends for various applications are similar. Accordingly, for the applications studied, assuming a one billion logic transistor equivalent area, we recommend building a Raw chip with approximately 1,000 tiles, 30 words/cycle global I/O, 20 Kbytes of local memory per tile, three to four words/cycle local communication bandwidth, and single-issue processors. This configuration will give performance near the global optimum for most applications.

Relative density estimation and local bandwidth selection for censored data

This paper is concerned with local bandwidth selection for a kernel-type estimator of the relative density (or grade density) when the samples come from two populations under a right random censorship mechanism. A two-stage smoothed plug-in local bandwidth selector with a beta reference is proposed. This estimation procedure is used to analyze the lifetime density in a two-sample problem related to breast cancer. A simulation study is carried out to examine the practical performance of the bandwidth selector.

Relative efficiency of local bandwidths in kernel density estimation*

The concept of relative efficiency can be used in kernel density estimation to analyze the global benefits of the optimal local bandwidth selector with respect to the optima* global one, in terms of MISE. This index, as well as its asymptotic counterpart -the asymptotic relative efficiency-, can be used to regard if the use of a local selector is worthwhile. This, of course, depends on the underlying density. Some bootstrap estimator of the relative efficiency index is presented in this paper and proved to be consistent at the rate n-i/7. The theory is illustrated with the computation of the relative efficiency of local bandwidths for a set of common densities and some simulations that show the good performance of its bootstrap estimator.

Service discovery in DEAPspace

The DEAPspace group at IBM Research, Zurich Lab is developing a system in which services can be shared between proximate devices. The target environment is single-hop short-range wireless systems, and the solution must offer prompt responsiveness for transient devices. In this article, the service discovery algorithm is presented. Through this new algorithm, a computing device can detect the presence of neighboring devices, share configuration and service information with those devices, and also notice when devices become unavailable. Targeted for wireless ad hoc single-hop networks, this solution improves the responsiveness of devices to changes in their environments. The effectiveness of this algorithm is demonstrated through comparison with others that achieve similar goals, considering the responsiveness to changes in the local environment, power consumption, and bandwidth requirement of devices.

Local Cross‐validation for Spectrum Bandwidth Choice

We investigate an automatic method of determining a local bandwidth for non‐parametric kernel spectral density estimates at a single frequency. This procedure is a modification of a cross‐validation technique for global bandwidth choices, avoiding the computation of any pilot estimate based on initial bandwidths or on approximate parametric models. Only local conditions on the spectral density around the frequency of interest are assumed. We illustrate with a Monte Carlo study the performance in finite samples of the bandwidth estimates proposed.

Dynamic viscoplastic approach for strain localization in a sandstone

In this paper, the viscoplastic regularization method is used to simulate strain localization in a compressed prismatical sample of Fontainebleau sandstone. First of all, the limits of Rice's formalism concerning the prediction of the localization band width and the post-bifurcation treatment are recalled. Then the theoretical aspects of the viscoplastic regularization method is reviewed. The non-objectivity or mesh-sensitivity of the solution is pointed out when a simple plastic model without viscosity is used. Finally, a fictitious viscosity parameter in the form of relaxation time is introduced in the model and regularizes the problem.

Local likelihood density estimation in line transect sampling

A novel semiparametric estimator for the probability density function of detected distances in line transect sampling is proposed. The estimator is obtained using a local likelihood density estimation approach, a technique recently proposed which affords the advantages of both parametric and nonparametric methods, i.e. accuracy and robustness. Moreover, a procedure for the selection of the local likelihood bandwidth is obtained. The performance of the proposed estimator with respect to some existing nonparametric and semiparametric estimators is assessed by means of a Monte Carlo study. Finally, a real data set is analyzed. Copyright © 2000 John Wiley & Sons, Ltd.

Sequential design for local bandwidth response curve estimator

Suppose we want to estimate the response curve using a kernel estimator. When we perform an experiment to estimate the response curve, we usually decide on the location of all the design points in advance of the experiment. Müller and Schmitt (1988) propose the optimal design, but in practice, we need some prior information about response curve to construct such a design. Park and Faraway (1998) present the sequential design for the global bandwidth estimator and show that their sequential design is asymptotically equivalent to the optimal design and is more efficient than the evenly spaced design. However, the sequential design points will be unequally spaced and so using a global bandwidth may not be appropriate. In this paper, we present the sequential design for the local bandwidth response curve estimator.

Block length selection in the bootstrap for time series

The blockwise bootstrap is a modification of Efron's bootstrap designed to give correct results for dependent stationary observations. One drawback of the method is that it depends critically on a block length which has to be chosen by the user. Here we propose a fully data-driven method to select this block length. It is based on the equivalence of the blockwise bootstrap variance to a lag weight estimator of a spectral density at the origin. The relevant spectral density is the one of the process given by the influence function of the statistic to be bootstrapped. In this equivalence the block length is the inverse of the bandwidth. We thus apply a recently developed local bandwidth selection procedure to the time series given by the estimated influence function. Simulations show that this procedure gives good results in a wide range of situations.

The representation of spectrally dynamic signals in primary auditory cortex

Natural sounds, including speech, are traditionally characterized by their patterns of pitch, timbre, loudness, and modulation, all of which are closely related to the instantaneous spectrum of these signals. The central auditory system has developed elegant mechanisms to encode this spectro-temporal information. Thus, at the level of the primary auditory cortex (AI), the dynamic spectrum is repeatedly represented over a wide range of spectral and temporal resolutions. This multi-scale representation is derived from response patterns of cells selective to such parameters as local bandwidth, asymmetry of spectral peaks, and onset/offset transition rates. Physiological experiments measured the spectral and dynamic properties of neurons in AI using linear system analysis. One set of experiments used broadband signals with sinusoidally modulated spectral envelopes (‘‘ripples’’). Varying the density (or frequency), amplitude, phase, and velocity of the ripple pattern in a systematic fashion was used to derive a ripple transfer function. From this function it is possible to derive a spectro-temporal (impulse) response function via an inverse Fourier transform. A separate technique computed the neuronal impulse response (via reverse correlation) using signals with random and dynamic spectral characteristics. This presentation summarizes the application of these two analytical techniques to characterizing AI neuronal responses and discusses the scientific and technical implications of these multi-scale, cortical representations.

An optimal local bandwidth selector for kernel density estimation

The problem of selecting the bandwidth for optimal kernel density estimation at a point is considered. A class of local bandwidth selectors which minimize smoothed bootstrap estimates of mean-squared error in density estimation is introduced. It is proved that the bandwidth selectors in the class achieve optimal relative rates of convergence, dependent upon the local smoothness of the target density. Practical implementation of the bandwidth selection methodology is discussed. The use of Gaussian-based kernels to facilitate computation of the smoothed bootstrap estimate of mean-squared error is proposed. The performance of the bandwidth selectors is investigated empirically.

Interactive local bandwidth choice

A tool for user choice of the local bandwidth function for kernel density and nonparametric regression estimates is developed using KDE, a graphical object-oriented package for interactive kernel density estimation written in LISP-STAT. The bandwidth function is a parameterized spline, whose knots are manipulated by the user in one window, while the resulting estimate appears in another window. A real data illustration of this method raises concerns, because an extremely large family of estimates is available. Suggestions are made to overcome this problem so that this tool can be used effectively for presenting final results of a data analysis.

Local Adaptivity of Kernel Estimates with Plug‐in Local Bandwidth Selectors

In non‐parametric function estimation selection of a smoothing parameter is one of the most important issues. The performance of smoothing techniques depends highly on the choice of this parameter. Preferably the bandwidth should be determined via a data‐driven procedure. In this paper we consider kernel estimators in a white noise model, and investigate whether locally adaptive plug‐in bandwidths can achieve optimal global rates of convergence. We consider various classes of functions: Sobolev classes, bounded variation function classes, classes of convex functions and classes of monotone functions. We study the situations of pilot estimation with oversmoothing and without oversmoothing. Our main finding is that simple local plug‐in bandwidth selectors can adapt to spatial inhomogeneity of the regression function as long as there are no local oscillations of high frequency. We establish the pointwise asymptotic distribution of the regression estimator with local plug‐in bandwidth.

Goodness‐of‐fit Test in Parametric Time Series Models

A goodness‐of‐fit test is proposed which uses nonparametric curve estimation methods to investigate the fit of parametric models for the spectral density. A test of the null hypothesis that the function has parametric form is considered with a test statistic which compares parametric estimates and nonparametric kernel estimates of the function and its derivatives at a preselected number of points. An important issue for the nonparametric estimator is bandwidth choice, and we propose a data‐adaptive method for local bandwidth choice. Under the null hypothesis, asymptotically the test statistic has a χ2 distribution. Some practical issues are discussed.

Kernel bandwidth selection for a first order nonparametric streamflow simulation model

A new approach for streamflow simulation using nonparametric methods was described in a recent publication (Sharma et al. 1997). Use of nonparametric methods has the advantage that they avoid the issue of selecting a probability distribution and can represent nonlinear features, such as asymmetry and bimodality that hitherto were difficult to represent, in the probability structure of hydrologic variables such as streamflow and precipitation. The nonparametric method used was kernel density estimation, which requires the selection of bandwidth (smoothing) parameters. This study documents some of the tests that were conduced to evaluate the performance of bandwidth estimation methods for kernel density estimation. Issues related to selection of optimal smoothing parameters for kernel density estimation with small samples (200 or fewer data points) are examined. Both reference to a Gaussian density and data based specifications are applied to estimate bandwidths for samples from bivariate normal mixture densities. The three data based methods studied are Maximum Likelihood Cross Validation (MLCV), Least Square Cross Validation (LSCV) and Biased Cross Validation (BCV2). Modifications for estimating optimal local bandwidths using MLCV and LSCV are also examined. We found that the use of local bandwidths does not necessarily improve the density estimate with small samples. Of the global bandwidth estimators compared, we found that MLCV and LSCV are better because they show lower variability and higher accuracy while Biased Cross Validation suffers from multiple optimal bandwidths for samples from strongly bimodal densities. These results, of particular interest in stochastic hydrology where small samples are common, may have importance in other applications of nonparametric density estimation methods with similar sample sizes and distribution shapes.

Local bandwidth selection for kernel density estimation from right-censored data based on asymptotic mean absolute error

Local bandwidth selection for kernel density estimation from right-censored data based on asymptotic mean absolute error

Locating Exotherms in Differential Thermal Analysis with Nonparametric Regression

Differential Thermal Analysis (DTA) is a technique used to quantify cold tolerance in plants. Plant tissue is cooled, and the ambient temperature and the temperature differential between tissue and ambient are measured. Freezing episodes, called exotherms, can be identified as changepoints, local minima, or selected inflection points in a plot of differential temperature against ambient temperature. The primary exotherm typically manifests itself as a discontinuity and can be identified using techniques similar to those described by Muller (1992) and Loader (1996). Modifications for locally weighted polynomial regression are proposed here to locate additional exotherms through estimation of first and second derivatives. These techniques involve an innovative combination of local and global bandwidth selection. Finally, the estimators are applied to crepe myrtle (Lagerstremia indica) and pecan (Carya illinoensis) DTA data.

Empirical-Bias Bandwidths for Local Polynomial Nonparametric Regression and Density Estimation

A data-based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials. The estimator, called the empirical-bias bandwidth selector (EBBS), is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function. EBBS minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term. The variance term used is exact, not asymptotic, though it involves the conditional variance of the response given the predictors that must be estimated. The bias term is estimated empirically, not from an asymptotic expression. Thus EBBS is similar to the “double smoothing” approach of Härdle, Hall, and Marron and a local bandwidth selector of Schucany, but is developed here for a far wider class of estimation problems than what those authors considered. EBBS is tested on simulated data, and its performance seems quite satisfactory. Local polynomial smoothing of a histogram is a highly effective technique for density estimation, and several of the examples involve density estimation by EBBS applied to binned data.

Methodology for nonparametric regression from independent sources

Three nonparametric curve estimators with information from different sources were introduced by Gerard and Schucany (1996). Here, automatic local bandwidth selection is adapted to local linear estimation with two independent data sets. Consistency results are provided for this bandwidth selection procedure. The asymptotic results are supported by two simulation trials with the proposed data-based local bandwidth selection.