Choice Of Bandwidth Research Articles

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Confidence bands for Horvitz–Thompson estimators using sampled noisy functional data

When collections of functional data are too large to be exhaustively observed, survey sampling techniques provide an effective way to estimate global quantities such as the population mean function. Assuming functional data are collected from a finite population according to a probabilistic sampling scheme, with the measurements being discrete in time and noisy, we propose to first smooth the sampled trajectories with local polynomials and then estimate the mean function with a Horvitz-Thompson estimator. Under mild conditions on the population size, observation times, regularity of the trajectories, sampling scheme, and smoothing bandwidth, we prove a Central Limit theorem in the space of continuous functions. We also establish the uniform consistency of a covariance function estimator and apply the former results to build confidence bands for the mean function. The bands attain nominal coverage and are obtained through Gaussian process simulations conditional on the estimated covariance function. To select the bandwidth, we propose a cross-validation method that accounts for the sampling weights. A simulation study assesses the performance of our approach and highlights the influence of the sampling scheme and bandwidth choice.

Learning mixtures of truncated basis functions from data

In this paper we investigate methods for learning hybrid Bayesian networks from data. First we utilize a kernel density estimate of the data in order to translate the data into a mixture of truncated basis functions (MoTBF) representation using a convex optimization technique. When utilizing a kernel density representation of the data, the estimation method relies on the specification of a kernel bandwidth. We show that in most cases the method is robust wrt. the choice of bandwidth, but for certain data sets the bandwidth has a strong impact on the result. Based on this observation, we propose an alternative learning method that relies on the cumulative distribution function of the data.Empirical results demonstrate the usefulness of the approaches: Even though the methods produce estimators that are slightly poorer than the state of the art (in terms of log-likelihood), they are significantly faster, and therefore indicate that the MoTBF framework can be used for inference and learning in reasonably sized domains. Furthermore, we show how a particular sub-class of MoTBF potentials (learnable by the proposed methods) can be exploited to significantly reduce complexity during inference.

The estimation of misspecified long memory models

We consider time series that, possibly after integer differencing or integrating or other detrending, are covariance stationary with spectral density that is regularly varying near zero frequency, and unspecified elsewhere. This semiparametric framework includes series with short, long and negative memory. We consider the consistency of the popular log-periodogram memory estimate that, conventionally but wrongly, assumes the spectral density obeys a pure power law. The local-to zero misspecification leads to increased bias, such that the usual central limit theorem may only hold for bandwidths entailing considerable imprecision. The order of the bias is calculated for several slowly-varying factors, and some discussion of mean squared error and bandwidth choice is included.

Representing Data Distributions with a Nonparametric Kernel Density: The Way to Estimate the Optimal Oil Contents of Palm Mesocarp at Various Periods

The most popular nonparametric density estimates is kernel density estimate. This estimate depends on the bandwidth choice which was given the optimization to kernel optimality process. We proposed Epanechnikov kernel which is the most optimal kernel in the AMISE. The resample data as replicate samples has been obtained by using bootstrap mechanism to provide the information about the sampling distribution. Then the resample data was used in Epanechnikov kernel simulation to estimate the optimal solution. This study was simulated using oil contents (%) data at various periods after pollination. The oil contents (%) were obtained by extraction of oil palm mesocarp. The result show that, Epanechnikov kernel using resamples data from bootstrap could be used for nonparametric optimization cases such as oil content (%) of oil palm mesocarp.

Recognizing and visualizing departures from independence in bivariate data using local Gaussian correlation

It is well known that the traditional Pearson correlation in many cases fails to capture non-linear dependence structures in bivariate data. Other scalar measures capable of capturing non-linear dependence exist. A common disadvantage of such measures, however, is that they cannot distinguish between negative and positive dependence, and typically the alternative hypothesis of the accompanying test of independence is simply "dependence". This paper discusses how a newly developed local dependence measure, the local Gaussian correlation, can be used to construct local and global tests of independence. A global measure of dependence is constructed by aggregating local Gaussian correlation on subsets of $\mathbb{R}^{2}$ , and an accompanying test of independence is proposed. Choice of bandwidth is based on likelihood cross-validation. Properties of this measure and asymptotics of the corresponding estimate are discussed. A bootstrap version of the test is implemented and tried out on both real and simulated data. The performance of the proposed test is compared to the Brownian distance covariance test. Finally, when the hypothesis of independence is rejected, local independence tests are used to investigate the cause of the rejection.

Bayesian nonparametric estimation of pair correlation function for inhomogeneous spatial point processes

The pair correlation function (PCF) is a useful tool for studying spatial point patterns. It is often estimated by some nonparametric approach such as kernel smoothing. However, the statistical properties of the kernel estimator are highly dependent on the choice of bandwidth. An inappropriate value of the bandwidth may lead to an estimator with a large bias or variance or both. In this work, we present an alternative PCF estimator based on Bayesian nonparametric regression. The method provides data-driven smoothing and intuitive uncertainty measures, together with efficient computation. The merits of our method are demonstrated via a simulation study and a couple of applications involving astronomy data and data on restaurant locations.

Noise effect on the interpolation equation for near infrared thermography

This paper investigates the performance of interpolation equations for a near infrared thermal imager operating over wavelengths from 0.9 µm to 1.7 µm with various filter bandwidths and a broad temperature range from 300 °C to 1000 °C. The equations are based on a general formulation of the effective wavelength as a function of the temperature. The quality of the interpolation is assessed in relation to the order of the effective wavelength. However, the noise induced by the imperfections of the thermal imager significantly disturbs the signal, and this phenomenon is enhanced as the bandwidth of the filter increases (i.e. for low-temperature applications). The main purpose of this paper is to establish the right choice of the filter bandwidth and the expression and order of the interpolation equation in relation to the noise level on the thermal imager and the desired accuracy. This paper first outlines the background on interpolation equations and then tests them on synthetic data from signals delivered first by an ideal thermal imager (i.e. free from noise) and then from noisy signals. This simulation study provides a framework for users to select an interpolation equation with an adequate order for near infrared thermal imagers. The performances of the selected interpolation equations are finally demonstrated on real images performed by a near infrared thermal imager.

Jointly optimal bandwidth selection for the planar kernel-smoothed density-ratio

The kernel-smoothed density-ratio or ‘relative risk’ function for planar point data is a useful tool for examining disease rates over a certain geographical region. Instrumental to the quality of the resulting risk surface estimate is the choice of bandwidth for computation of the required numerator and denominator densities. The challenge associated with finding some ‘optimal’ smoothing parameter for standalone implementation of the kernel estimator given observed data is compounded when we deal with the density-ratio per se. To date, only one method specifically designed for calculation of density-ratio optimal bandwidths has received any notable attention in the applied literature. However, this method exhibits significant variability in the estimated smoothing parameters. In this work, the first practical comparison of this selector with a little-known alternative technique is provided. The possibility of exploiting an asymptotic MISE formulation in an effort to control excess variability is also examined, and numerical results seem promising.

Preaveraging-Based Estimation of Quadratic Variation in the Presence of Noise and Jumps: Theory, Implementation, and Empirical Evidence

This article contributes to the theory for preaveraging estimators of the daily quadratic variation of asset prices and provides novel empirical evidence. We develop asymptotic theory for preaveraging estimators in the case of autocorrelated microstructure noise and propose an explicit test for serial dependence. Moreover, we extend the theory on preaveraging estimators for processes involving jumps. We discuss several jump-robust measures and derive feasible central limit theorems for the general quadratic variation. Using transaction data of different stocks traded at the New York Stock Exchange, we analyze the estimators’ sensitivity to the choice of the preaveraging bandwidth. Moreover, we investigate the dependence of preaveraging-based inference on the sampling scheme, the sampling frequency, microstructure noise properties, and the occurrence of jumps. As a result of a thorough empirical study, we provide guidance for optimal implementation of preaveraging estimators and discuss potential pitfalls in practice.

Discrete triangular associated kernel and bandwidth choices in semiparametric estimation for count data

This work deals with semiparametric kernel estimator of probability mass functions which are assumed to be modified Poisson distributions. This semiparametric approach is based on discrete associated kernel method appropriated for modelling count data; in particular, the famous discrete symmetric triangular kernels are used. Two data-driven bandwidth selection procedures are investigated and an explicit expression of optimal bandwidth not available until now is provided. Moreover, some asymptotic properties of the cross-validation criterion adapted for discrete semiparametric kernel estimation are studied. Finally, to measure the performance of semiparametric estimator according to each type of bandwidth parameter, some applications are realized on three real count data-sets from sociology and biology.

Bandwidth selection for the estimation of transition probabilities in the location-scale progressive three-state model

Times between consecutive events are often of interest in medical studies. Usually the events represent different states of the disease process and are modeled using multi-state models. This paper introduces and studies a feasible estimation method for the transition probabilities in a progressive three-state model. We assume that the vector of gap times $$(T_1,T_2)$$ satisfies a nonparametric location-scale regression model $$T_2=m(T_1)+\sigma (T_1)\epsilon $$ , where the functions $$m$$ and $$\sigma $$ are `smooth', and $$\epsilon $$ is independent of $$T_1$$ . Under this model, Van Keilegom et al. (J Stat Plan Inference 141:1118---1131, 2011) proposed estimators of the transition probabilities. However, the important issue of automatic bandwidth choice in this setting has not been examined, making the analysis of real datasets rather difficult. In this paper, we study the performance of their estimator in practice, we propose some modifications and study practical issues related to the implementation of the estimator, which involves the choice of an appropriate bandwidth. In an extensive simulation study the good performance of the method is shown. Simulations also demonstrate that the proposed estimator compares favorably with alternative estimators. Furthermore, the proposed methodology is illustrated with a real database on breast cancer.

A rule-based ad hoc method for selecting a bandwidth in kernel home-range analyses

Background: An important issue in conducting kernel home-range analyses is the choice of bandwidth or smoothing parameter. To examine the effects of this choice, telemetry data were collected at high sampling rates (843 to 5,069 locations) on 20 North American elk, Cervus elaphus, in northeastern Oregon, USA, during 2000, 2002, and 2003. The elk had their collars replaced annually, hence none were monitored for more than a single year. True home ranges were defined by buffering the actual paths of individuals. Fixed-kernel and adaptive-kernel estimates were then determined with reference bandwidths (href), least-squares cross-validation bandwidths (hlscv), and rule-based ad hoc bandwidths designed to prevent under-smoothing (had hoc). Both raw data and sub-sampled sparse datasets (1, 2, 4, 6, 12, and 24 locations/elk/day) were used. Results: With fixed-kernel and adaptive-kernel analyses, reference bandwidths were positively biased (including areas not part of an animal’s home range) but performed better (lower bias, closer match between estimated and true home ranges) with increasing sample size. Least-squares cross-validation bandwidths were positively biased with very small sample sizes, but quickly became negatively biased with increasing sample size, as home-range estimates broke up into disjoint polygons. Ad hoc bandwidths outperformed reference and least-squares crossvalidation bandwidths, exhibited only moderate positive bias, were relatively unaffected by sample size, and were characterized by lower Type I errors (falsely including areas not part of the true home range). Ad hoc bandwidths also exhibited lower Type II errors (failure to include portions of the true home range) than did least-squares crossvalidation bandwidths, although reference bandwidths resulted in lowest Type II error rates. Auto-correlation indices increased to about 150 to 200 locations per elk, and then stabilized. Bias of fixed-kernel analyses with ad hoc bandwidths was not affected by auto-correlation, but did increase with irregularly shaped home ranges with high fractal dimensions. Conclusions: The rule-based ad hoc bandwidths, specifically designed to prevent fragmentation of estimated home ranges, outperformed both href and hlscv, and gave the smallest value for h consistent with a contiguous home-range estimate. The protocol for choosing the ad hoc bandwidth was shown to be consistent and repeatable.

Joint Probability Mass Function Estimation From Asynchronous Samples

A common approach to study the relationship between different signals is to model them as random processes and estimate their joint probability distribution from the observed data. When synchronous samples of the random processes are available, then the empirical distribution gives a reliable estimate. However, in several situations, such as sensors spread over a vast region or a software probing smart phone sensors for readings, synchronous samples are either not available, or are difficult/costly to obtain. In such cases, we have to depend on non-periodic, asynchronous samples to obtain good estimates of the joint distribution. In this paper, we consider independent Poisson sampling of the individual random processes and we propose a kernel based estimate of the joint probability mass function. We prove that our estimate is consistent (in the mean-square sense) for strong mixing processes, which is a wide class of random processes including Markov processes. We also provide expressions for the asymptotic mean-square error (MSE), study the bias-variance tradeoff, and discuss the choice of the kernel bandwidth. By appropriately choosing the kernel, we show that we can obtain an asymptotic rate of T-4/5 for the MSE, where T is the interval of observation. We also present several numerical results to discuss the accuracy of our asymptotic approximations for finite T.

Data-driven bandwidth choice for gamma kernel estimates of density derivatives on the positive semi-axis

In some applications it is necessary to estimate derivatives of probability densities defined on the positive semi-axis. The quality of nonparametric estimates of the probability densities and their derivatives are strongly influenced by smoothing parameters (bandwidths). In this paper an expression for the optimal smoothing parameter of the gamma kernel estimate of the density derivative is obtained. For this parameter data-driven estimates based on methods called “rule of thumb” and “cross-validation” are constructed. The quality of the estimates is verified and demonstrated on examples of density derivatives generated by Maxwell and Weibull distributions.

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Transformations In Hazard Rate Estimation For Heavy-Tailed Data

A new estimate of the hazard rate function is proposed, specifically designed for situations when the underlying data are heavy tailed. The estimate is nonparametric in nature and is based on the concept that estimation bias is reduced both in body and in the tail through an appropriate transformation of the sample. With the use of a practical bandwidth choice rule the estimate is illustrated graphically on distributional and real world data.

A comparison of estimators of the geographical relative risk function

The geographical relative risk function is a useful tool for investigating the spatial distribution of disease based on case and control data. The most common way of estimating this function is using the ratio of bivariate kernel density estimates constructed from the locations of cases and controls, respectively. An alternative is to use a local-linear (LL) estimator of the log-relative risk function. In both cases, the choice of bandwidth is critical. In this article, we examine the relative performance of the two estimation techniques using a variety of data-driven bandwidth selection methods, including likelihood cross-validation (CV), least-squares CV, rule-of-thumb reference methods, and a new approximate plug-in (PI) bandwidth for the LL estimator. Our analysis includes the comparison of asymptotic results; a simulation study; and application of the estimators on two real data sets. Our findings suggest that the density ratio method implemented with the least-squares CV bandwidth selector is generally best, with the LL estimator with PI bandwidth being competitive in applications with strong large-scale trends but much worse in situations with elliptical clusters.

A new time domain estimation of k-factors GARMA processes

We address the problem of parameter estimation of long memory time series. We consider k-factors Gegenbauer Autoregressive Moving Average (k-GARMA) processes and we estimate their parameters by the minimum Hellinger distance estimator. We establish the consistency of the estimator and the asymptotic normality for some bandwidth choice.

Evaluating temporally weighted kernel density methods for predicting the next event location in a series

One aspect of tactical crime or terrorism analysis is predicting the location of the next event in a series. The objective of this article is to present a methodology to identify the optimal parameters and to test the performance of temporally weighted kernel density estimation models for predicting the next event in a criminal or terrorist event series. By placing event series in a space–time point pattern framework, the next event prediction models are shown to be based on estimating a conditional spatial density function. We use temporal weights that indicate how much influence past events have toward predicting future event locations, which can also incorporate uncertainty in the event timing. Results of applying this methodology to crime series in Baltimore County, MD, indicate that performance can vary greatly by crime type and little by series length and is fairly robust to choice of bandwidth.