L∞ Error and Bandwidth Selection for Kernel Density Estimates of Large Data

Numerous facets of scientific research implicitly or explicitly call for the estimation of probability densities. Histograms and kernel density estimates (KDEs) are two commonly used techniques for estimating such information, with the KDE generally providing a higher fidelity representation of the probability density function (PDF). Both methods require specification of either a bin width or a kernel bandwidth. While techniques exist for choosing the kernel bandwidth optimally and objectively, they are computationally intensive, since they require repeated calculation of the KDE. A solution for objectively and optimally choosing both the kernel shape and width has recently been developed by Bernacchia and Pigolotti (2011). While this solution theoretically applies to multidimensional KDEs, it has not been clear how to practically do so.A method for practically extending the Bernacchia–Pigolotti KDE to multidimensions is introduced. This multidimensional extension is combined with a recently-developed computational improvement to their method that makes it computationally efficient: a 2D KDE on 105 samples only takes 1 s on a modern workstation. This fast and objective KDE method, called the fastKDE method, retains the excellent statistical convergence properties that have been demonstrated for univariate samples. The fastKDE method exhibits statistical accuracy that is comparable to state-of-the-science KDE methods publicly available in R, and it produces kernel density estimates several orders of magnitude faster. The fastKDE method does an excellent job of encoding covariance information for bivariate samples. This property allows for direct calculation of conditional PDFs with fastKDE. It is demonstrated how this capability might be leveraged for detecting non-trivial relationships between quantities in physical systems, such as transitional behavior.

Kernel Density Estimators (KDEs) are a non-parametric density estimation technique that has recently been applied to Monte Carlo radiation transport simulations. Kernel density estimators are an alternative to histogram tallies for obtaining global solutions in Monte Carlo tallies. With KDEs, a single event, either a collision or particle track, can contribute to the score at multiple tally points with the uncertainty at those points being independent of the desired resolution of the solution. Thus, KDEs show potential for obtaining estimates of a global solution with reduced variance when compared to a histogram. Previously, KDEs have been applied to neutronics for one-group reactor physics problems and fixed source shielding applications. However, little work was done to obtain reaction rates using KDEs. This paper introduces a new form of the MFP KDE that is capable of handling general geometries. Furthermore, extending the MFP KDE to 2-D problems in continuous energy introduces inaccuracies to the solution. An ad-hoc solution to these inaccuracies is introduced that produces errors smaller than 4% at material interfaces.

Abstract: Home‐range estimators are commonly tested with simulated animal locational data in the laboratory before the estimators are used in practice. Although kernel density estimation (KDE) has performed well as a home‐range estimator for simulated data, several recent studies have reported its poor performance when used with data collected in the field. This difference may be because KDE and other home‐range estimators are generally tested with simulated point locations that follow known statistical distributions, such as bivariate normal mixtures, which may not represent well the space‐use patterns of all wildlife species. We used simulated animal locational data of 5 point pattern shapes that represent a range of wildlife utilization distributions to test 4 methods of home‐range estimation: 1) KDE with reference bandwidths, 2) KDE with least‐squares cross‐validation, 3) KDE with plug‐in bandwidths, and 4) minimum convex polygon (MCP). For the point patterns we simulated, MCP tended to produce more accurate area estimates than KDE methods. However, MCP estimates were markedly unstable, with bias varying widely with both sample size and point pattern shape. The KDE methods performed best for concave distributions, which are similar to bivariate normal mixtures, but still overestimated home ranges by about 40–50% even in the best cases. For convex, linear, perforated, and disjoint point patterns, KDE methods overestimated home‐range sizes by 50–300%, depending on sample size and method of bandwidth selection. These results indicate that KDE does not produce home‐range estimates that are as accurate as the literature suggests, and we recommend exploring other techniques of home‐range estimation.

Multivariate online kernel density estimation with Gaussian kernels

Recently Hazelton and Turlach (2009) proposed a weighted kernel density estimatorfor the deconvolution problem. In the case of Gaussian kernels and measurement er-ror, they argued that the weighted kernel density estimator is a competitive estimatorover the classical deconvolution kernel estimator. In this paper we consider weightedkernel density estimators when sample observations are contaminated by double expo-nentially distributed errors. The performance of the weighted kernel density estimatorsis compared over the classical deconvolution kernel estimator and the kernel density es-timator based on the support vector regression method by means of a simulation study.The weighted density estimator with the Gaussian kernel shows numerical instabilityin practical implementation of optimization function. However the weighted densityestimates with the double exponential kernel has very similar patterns to the classicalkernel density estimates in the simulations, but the shape is less satisfactory than theclassical kernel density estimator with the Gaussian kernel.Keywords: Deconvolution, kernel density estimator, support vector regression, weightedkernel density estimator.

An urban, commercial central district is often regarded as the heart of a city. Therefore, quantitative research on commercial central districts plays an important role when studying the development and evaluation of urban spatial layouts. However, conventional planar kernel density estimation (KDE) and network kernel density estimation (network KDE) do not reflect the fact that the road network density is high in urban, commercial central districts. To solve this problem, this paper proposes a new method (commercial-intersection KDE), which combines road intersections with KDE to identify commercial central districts based on point of interest (POI) data. First, we extracted commercial POIs from Amap (a Chinese commercial, navigation electronic map) based on existing classification standards for urban development land. Second, we calculated the commercial kernel density in the road intersection neighborhoods and used those values as parameters to build a commercial intersection density surface. Finally, we used the three standard deviations method and the commercial center area indicator to differentiate commercial central districts from areas with only commercial intersection density. Testing the method using Nanjing City as a case study, we show that our new method can identify seven municipal, commercial central districts and 26 nonmunicipal, commercial central districts. Furthermore, we compare the results of the traditional planar KDE with those of our commercial-intersection KDE to demonstrate our method’s higher accuracy and practicability for identifying urban commercial central districts and evaluating urban planning.

Radio source counts provide a fundamental census of cosmic radio emission, yet their estimation is usually based on coarse histograms that suffer from bin-choice bias, boundary effects, and survey incompleteness. We apply and rigorously evaluate kernel density estimation (KDE) as a nonparametric alternative to the conventional binned method for estimating differential radio source counts. Using simulated flux-limited samples derived from an input luminosity function model, we compare the performance of standard KDE, adaptive KDE, and traditional binning methods. Our results show that KDE-based approaches yield more accurate and stable estimates, particularly in the high-flux regime where data are sparse and conventional methods struggle. We also apply the adaptive KDE method to real observational data from the LOFAR Two-Metre Sky Survey Deep Fields. Our analysis robustly confirms the pronounced “drop and bump” feature at sub-mJy flux densities, but also reveals that a secondary, modest bump seen in the binned data at ∼10 mJy is likely a binning artifact. We also demonstrate the flexibility of KDE in addressing observational incompleteness through weighted estimation, which applies weights continuously at the level of individual sources rather than averaging them in discrete bins. These strengths make KDE a powerful tool for source-count analyses in current and future radio surveys and, more broadly, in analogous studies at other wavelengths. All computations in this study are implemented with AstroKDE , a Python package we have developed for astronomical applications.

Particle density estimation with grid-projected and boundary-corrected adaptive kernels

This paper proposes a wind power probabilistic model (WPPM) using the reflection method and multi-kernel function kernel density estimation (KDE). With the increasing penetration of renewable energy sources (RESs) into power systems, several probabilistic approaches have been introduced to assess the impact of RESs on the power system. A probabilistic approach requires a wind power scenario (WPS), and the WPS is generated from the WPPM. Previously, WPPM was generated using a parametric density estimation, and it had limitations in reflecting the characteristics of wind power data (WPD) due to a boundary bias problem. The paper proposes a WPPM generated using the KDE, which is a non-parametric method. Additionally, the paper proposes a reflection method correcting for the boundary bias problem caused by the double-bounded characteristic of the WPD and the multi-kernel function KDE minimizing the effect of tied values. Six bandwidth selectors are used to calculate the bandwidth for the KDE, and one is selected by analyzing the correlation between the normalized WPD and the calculated bandwidth. The results were validated by generating WPPMs with WPDs in six regions of the Republic of Korea, and it was confirmed that the accuracy and goodness-of-fit are improved when the proposed method is used.

In this paper, we present a Markov chain Monte Carlo (MCMC) simulation algorithm for estimating parameters in the kernel density estimation of bivariate insurance claim data via transformations. Our data set consists of two types of auto insurance claim costs and exhibits a high-level of skewness in the marginal empirical distributions. Therefore, the kernel density estimator based on original data does not perform well. However, the density of the original data can be estimated through estimating the density of the transformed data using kernels. It is well known that the performance of a kernel density estimator is mainly determined by the bandwidth, and only in a minor way by the kernel. In the current literature, there have been some developments in the area of estimating densities based on transformed data, where bandwidth selection usually depends on pre-determined transformation parameters. Moreover, in the bivariate situation, the transformation parameters were estimated for each dimension individually. We use a Bayesian sampling algorithm and present a Metropolis-Hastings sampling procedure to sample the bandwidth and transformation parameters from their posterior density. Our contribution is to estimate the bandwidths and transformation parameters simultaneously within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is better captured through the bivariate density estimator based on transformed data.

Estimating the kernel density function of a random vector taking values on Riemannian manifolds are considered. More Precisely, we consider the problem of the estimation of the probability density of n i.i.d. random objects on the product of two compact Riemannian manifolds without boundary. The proposed methodology adapts the Pelletier’s approach which is the kernel density estimation on non Euclidean setting. Under sufficient regularity assumptions on the underlying density, L2 convergence rates are obtained. Riemannian products are the most natural and fruitful generalization of Cartesian products. These manifold models are of interest to many fields of applied mathematics, natural sciences and technology. Let us recall here as examples, the use of these products in the domain of computer vision and in biomedical sciences. In this note, we construct an estimator of the density on a product manifold from one of the estimators existing in the literature. Among the estimation methods, we mainly cite a method based on Fourier developments proposed in (Hendriks, 1990). Another method based on geodesic distance is proposed later by B. Pelletier in (Pelletier, 2005). This method used in the present work, is a generalization of the kernel method in the affine case. Another method recently developed by Kim et al. in (Kim, 2013) use the concept of exponential map to define a kernel estimator of the density on a Riemannian manifold. The aim in the present work, is to show that Pelletier's method is the best suited to the case of the Riemannian product.

In many data stream mining applications, traditional density estimation methods such as kernel density estimation, reduced set density estimation can not be applied to the density estimation of data streams because of their high computational burden, processing time and intensive memory allocation requirement. In order to reduce the time and space complexity, a novel density estimation method Dm-KDE over data streams based on the proposed algorithm m-KDE which can be used to design a KDE estimator with the fixed number of kernel components for a dataset is proposed. In this method, Dm-KDE sequence entries are created by algorithm m-KDE instead of all kernels obtained from other density estimation methods. In order to further reduce the storage space, Dm-KDE sequence entries can be merged by calculating their KL divergences. Finally, the probability density functions over arbitrary time or entire time can be estimated through the obtained estimation model. In contrast to the state-of-the-art algorithm SOMKE, the distinctive advantage of the proposed algorithm Dm-KDE exists in that it can achieve the same accuracy with much less fixed number of kernel components such that it is suitable for the scenarios where higher on-line computation about the kernel density estimation over data streams is required.We compare Dm-KDE with SOMKE and M-kernel in terms of density estimation accuracy and running time for various stationary datasets. We also apply Dm-KDE to evolving data streams. Experimental results illustrate the effectiveness of the proposed method.

A century ago, when Student’s t-statistic was introduced, no one ever imagined its increasing applicability in the modern era. It finds applications in highly multiple hypothesis testing, feature selection and ranking, high dimensional signal detection, etc. Student’s t-statistic is constructed based on the empirical distribution function (EDF). An alternative choice to the EDF is the kernel density estimate (KDE), which is a smoothed version of the EDF. The novelty of the work consists of an alternative to Student’s t-test that uses the KDE technique and exploration of the usefulness of KDE based t-test in the context of its application to large-scale simultaneous hypothesis testing. An optimal bandwidth parameter for the KDE approach is derived by minimizing the asymptotic error between the true p-value and its asymptotic estimate based on normal approximation. If the KDE-based approach is used for large-scale simultaneous testing, then it is interesting to consider, when does the method fail to manage the error rate? We show that the suggested KDE-based method can control false discovery rate (FDR) if total number tests diverge at a smaller order of magnitude than N3/2, where N is the total sample size. We compare our method to several possible alternatives with respect to FDR. We show in simulations that our method produces a lower proportion of false discoveries than its competitors. That is, our method better controls the false discovery rate than its competitors. Through these empirical studies, it is shown that the proposed method can be successfully applied in practice. The usefulness of the proposed methods is further illustrated through a gene expression data example.

The aim of this article is to construct and carry out an integrated evaluation of a system of indicators for regional logistic systems using an improved entropy weight method. Promoting the development of regional logistic systems is crucial for reducing logistic costs in the real economy and enhancing the endogenous component of economic development. This article establishes an evaluation system consisting of 17 indicators organized into four dimensions: result scale, operational quality, social contribution, and green development. The system reflects both the scale and efficiency of the logistic sector and its environmental impact during development. Additionally, it captures changes within the logistic sector itself as well as its contributions to other industries and to people’s livelihoods. An improved entropy-weight method and kernel density estimation were applied to assess the development level of regional logistic systems, in order to determine their spatial and temporal distribution, dynamic evolution, and regional differences. The key advantage of the improved entropy-weight system is that it utilizes indicator information in both spatial and temporal dimensions, using a global reference area, which allows for both vertical and horizontal comparisons of evaluation results. The comprehensive analysis of the characteristics of regional logistic systems in terms of spatial and temporal evolution showed that the level of regional logistic systems changed with an average annual growth rate of -0.04% between 2002 and 2024, though with significant regional and inter-provincial differences. From 2002 to 2024, based on the average annual growth rates of regional logistic systems, only the central, western, and northern regions achieved effective growth. In other words, regional logistic systems across the country and in the five main regions were bipolar or multipolar, with characteristic gradient features. To analyze the dynamics of the distribution of regional logistic systems and to clearly depict the evolutionary model of regional logistic system levels, the kernel density estimation – a nonparametric method capable of describing the distribution of a random variable in terms of a continuous density curve was used. The study revealed that, due to the full-scale military invasion, the gap between logistic systems in different regions gradually widened, showing a gradual reduction in the differentiation feature of «east–center–west».

Kernel density estimation (KDE) is an important method in nonparametric learning. While KDE has been studied extensively in the context of accuracy of distribution estimation, it has not been studied extensively in the context of classification. This paper studies nine bandwidth selection schemes for kernel density estimation in Naive Bayesian classification context, using 52 machine learning benchmark datasets. The contributions of this paper are threefold. First, it shows that some commonly used and very sophisticated bandwidth selection schemes do not give good performance in Naive Bayes. Surprisingly, some very simple bandwidth selection schemes give statistically significantly better performance. Second, it shows that kernel density estimation can achieve statistically significantly better classification performance than a commonly used discretization method in Naive Bayes, but only when appropriate bandwidth selection schemes are applied. Third, this study gives bandwidth distribution patterns for the investigated bandwidth selection schemes.