Particle Density Estimation with Grid-Projected Adaptive Kernels

The reconstruction of smooth density fields from scattered data points is a procedure that has multiple applications in a variety of disciplines, including Lagrangian (particle-based) models of solute transport in fluids. In random walk particle tracking (RWPT) simulations, particle density is directly linked to solute concentrations, which is normally the main variable of interest, not just for visualization and post-processing of the results, but also for the computation of non-linear processes, such as chemical reactions. Previous works have shown the superiority of kernel density estimation (KDE) over other methods such as binning, in terms of its ability to accurately estimate the "true" particle density relying on a limited amount of information. Here, we develop a grid-projected KDE methodology to determine particle densities by applying kernel smoothing on a pilot binning; this may be seen as a "hybrid" approach between binning and KDE. The kernel bandwidth is optimized locally. Through simple implementation examples, we elucidate several appealing aspects of the proposed approach, including its computational efficiency and the possibility to account for typical boundary conditions, which would otherwise be cumbersome in conventional KDE

Reducing variance in univariate smoothing

A variance reduction technique in nonparametric smoothing is proposed: at each point of estimation, form a linear combination of a preliminary estimator evaluated at nearby points with the coefficients specified so that the asymptotic bias remains unchanged. The nearby points are chosen to maximize the variance reduction. We study in detail the case of univariate local linear regression. While the new estimator retains many advantages of the local linear estimator, it has appealing asymptotic relative efficiencies. Bandwidth selection rules are available by a simple constant factor adjustment of those for local linear estimation. A simulation study indicates that the finite sample relative efficiency often matches the asymptotic relative efficiency for moderate sample sizes. This technique is very general and has a wide range of applications.Comment: Published at http://dx.doi.org/10.1214/009053606000001398 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Examining the influence of cell size and bandwidth size on kernel density estimation crime hotspot maps for predicting spatial patterns of crime

Hotspot mapping is a popular technique used for helping target police patrols and other crime reduction initiatives. There are a number of spatial analysis techniques that can be used for identifying hotspots, but the most popular in recent years is kernel density estimation (KDE). KDE is popular because of the visually appealing way it represents the spatial distribution of crime, and because it is considered to be the most accurate of the commonly used hotspot mapping techniques. To produce KDE outputs, the researcher is required to enter values for two main parameters: the cell size and bandwidth size. To date little research has been conducted on the influence these parameters have on KDE hotspot mapping output, and none has been conducted on the influence these parameter settings have on a hotspot map’s central purpose – to identify where crime may occur in the future. We fill this gap with this research by conducting a number of experiments using different cell size and bandwidth values with crime data on residential burglary and violent assaults. We show that cell size has little influence on KDE crime hotspot maps for predicting spatial patterns of crime, but bandwidth size does have an influence. We conclude by discussing how the findings from this research can help inform police practitioners and researchers make better use of KDE for targeting policing and crime prevention initiatives

Poverty Analysis Based on Kernel Density Estimates from Grouped Data

Kernel density estimation (KDE) has been prominently used to measure poverty from grouped data (representing mean incomes of a small number of population quantiles). In this paper I analyze the performance of this method. Using Monte Carlo simulations for plausible theoretical distributions and unit data from several household surveys, I compare KDE-based poverty estimates with their true and survey counterparts. It is shown that the technique gives rise to biases in poverty whose sign and magnitude vary with the smoothing parameter, the kernel, the number of data-points analyzed, and the poverty indicators used. I also demonstrate that KDE-based global poverty rates and headcounts are highly sensitive to the choice of smoothing parameter. Depending on the parameter, the estimated proportion of '$1/day poor' in 2000 varies by a factor of 1.8, while the estimated number of '$2/day poor' in 2000 varies by 287 million people. These findings give rise to concern about the validity and robustness of kernel density estimation in poverty analysis. However, they provide a framework for interpretation of existing results using this technique

Contingent Kernel Density Estimation

Kernel density estimation is a widely used method for estimating a distribution based on a sample of points drawn from that distribution. Generally, in practice some form of error contaminates the sample of observed points. Such error can be the result of imprecise measurements or observation bias. Often this error is negligible and may be disregarded in analysis. In cases where the error is non-negligible, estimation methods should be adjusted to reduce resulting bias. Several modifications of kernel density estimation have been developed to address specific forms of errors. One form of error that has not yet been addressed is the case where observations are nominally placed at the centers of areas from which the points are assumed to have been drawn, where these areas are of varying sizes. In this scenario, the bias arises because the size of the error can vary among points and some subset of points can be known to have smaller error than another subset or the form of the error may change among points. This paper proposes a “contingent kernel density estimation” technique to address this form of error. This new technique adjusts the standard kernel on a point-by-point basis in an adaptive response to changing structure and magnitude of error. In this paper, equations for our contingent kernel technique are derived, the technique is validated using numerical simulations, and an example using the geographic locations of social networking users is worked to demonstrate the utility of the method

Improving partial mutual information-based input variable selection by consideration of boundary issues associated with bandwidth estimation

Abstract not availableXuyuan Li, Aaron C. Zecchin, Holger R. Maie

Estimation of Dynamic Latent Variable Models Using Simulated Nonparametric Moments

Abstract. Given a model that can be simulated, conditional moments at a trial parameter value can be calculated with high accuracy by applying kernel smoothing methods to a long simulation. With such conditional moments in hand, standard method of moments techniques can be used to estimate the parameter. Because conditional moments are calculated using kernel smoothing rather than simple averaging, it is not necessary that the model be simulable subject to the conditioning information that is used to define the moment conditions. For this reason, the proposed estimator is applicable to general dynamic latent variable models. It is shown that as the number of simulations diverges, the estimator is consistent and a higher-order expansion reveals the stochastic difference between the infeasible GMM estimator based on the same moment conditions and the simulated version. In particular, we show how to adjust standard errors to account for the simulations. Monte Carlo results show how the estimator may be applied to a range of dynamic latent variable (DLV) models, and that it performs well in comparison to several other estimators that have been proposed for DLV models.dynamic latent variable models; simulation-based estimation; simulated moments; kernel regression; nonparametric estimation