Density Estimator Research Articles

Kernel Density Estimators for Axisymmetric Particle Beams

Bright beams are commonly represented by sampled data in the numerical algorithms used to simulate their properties. However, in these calculations and the analyses of their outputs, the beam’s density is sometimes required and must be calculated from the samples. Axisymmetric beams, which possess a rotational symmetry and are naturally expressed in polar coordinates, pose a particular challenge to density estimators. The area element in polar coordinates shrinks as the radius becomes small, and weighting the samples to account for their reduced frequency may cause unwelcome artifacts. In this work, we derive analytical expressions for two kernel density estimators, which solve these problems in the spatial coordinates and in the transverse phase space. We show how the kernels can be found by averaging the Gaussian kernel in Cartesian coordinates over the polar angle and demonstrate their use on test problems. These results show that particle beam symmetries can be taken advantage of in density estimation while avoiding artifacts.

Estimation of plant density based on presence/absence data using hybrid inference

Monitoring of plant populations has become more and more important, especially in the current context of environmental change. In this paper, we propose methods to estimate plant density from presence/absence surveys, wherein the presence or absence of each species is recorded on sample plots. Presence/absence sampling is a useful and relatively simple method for monitoring state and change of plant communities. Moreover, it has advantages compared to traditional plant cover assessment, the latter being more prone to observer bias. We present a hybrid estimation framework, that combines model- and design-based inference features, in which a generalised linear model (for binary presence/absence data) and an inhomogeneous Poisson model (for plant locations) are used to estimate plant density in a region of interest. We look at two different cases, the first one with a known area and the second one where the area is unknown and must be estimated. Our methods are applied to real data on Vaccinium vitis-idaea from the Swedish National Forest Inventory as well as simulated data to assess the performance of our estimators of plant density and corresponding variance estimators. The results obtained are promising and indicate that this method has a potential to add considerable analytic strength to monitoring programmes that collect presence/absence data.

A new wavelet-based estimation of conditional density via block threshold method

A new wavelet-based estimator of the conditional density is investigated. The estimator is constructed by combining a special ratio technique and applying a non negative estimator to the density function in the denominator. We used a wavelet shrinkage technique to find an adaptive estimator for this problem. In particular, a block thresholding estimator is proposed, and we prove that it enjoys powerful mean integrated squared error properties over Besov balls. Moreover, it is shown that convergence rates for the mean integrated squared error (MISE) of the adaptive estimator are optimal under some mild assumptions. Finally, a numerical example has been considered to illustrate the performance of the estimator.

A diffusion-based kernel density estimator (diffKDE, version 1) with optimal bandwidth approximation for the analysis of data in geoscience and ecological research

Abstract. Probability density functions (PDFs) provide information about the probability of a random variable taking on a specific value. In geoscience, data distributions are often expressed by a parametric estimation of their PDF, such as, for example, a Gaussian distribution. At present there is growing attention towards the analysis of non-parametric estimation of PDFs, where no prior assumptions about the type of PDF are required. A common tool for such non-parametric estimation is a kernel density estimator (KDE). Existing KDEs are valuable but problematic because of the difficulty of objectively specifying optimal bandwidths for the individual kernels. In this study, we designed and developed a new implementation of a diffusion-based KDE as an open source Python tool to make diffusion-based KDE accessible for general use. Our new diffusion-based KDE provides (1) consistency at the boundaries, (2) better resolution of multimodal data, and (3) a family of KDEs with different smoothing intensities. We demonstrate our tool on artificial data with multiple and boundary-close modes and on real marine biogeochemical data, and compare our results against other popular KDE methods. We also provide an example for how our approach can be efficiently utilized for the derivation of plankton size spectra in ecological research. Our estimator is able to detect relevant multiple modes and it resolves modes that are located closely to a boundary of the observed data interval. Furthermore, our approach produces a smooth graph that is robust to noise and outliers. The convergence rate is comparable to that of the Gaussian estimator, but with a generally smaller error. This is most notable for small data sets with up to around 5000 data points. We discuss the general applicability and advantages of such KDEs for data–model comparison in geoscience.

NON-QUALITY RISK EVALUATION FROM TOOL WEAR ASSESSMENT

The development of part quality virtual sensors requires knowledge and observability of cutting conditions and in particular tool wear as tool are consumables. This paper presents an unsupervised anomalies detection approach to assess tool wear from standard machine load sensors in order to evaluate a non-quality risk metric. The developed methodology combines physics and business rules with density estimators to analyse the behaviour of axes and spindle loads. Industrial data from an automotive production line are used to illustrate the methodology application.

Development and prediction of a robust multivariate trophic state index for the classification of lentic water bodies

This study presents a novel parametric-non parametric composite approach for developing Multivarite Trophic State Index (MTSI) for the classification of lentic water bodies (LWB). The newly developed framework involves the use of water quality parameters (WQPs), Principal Component Analysis (PCA) and Non-Parametric Gaussian Kernal Density (NPGKD) estimator. The generic framework is demonstrated for developing MTSI, based on the datasets of 18 LWBs in Kerala, India. The method could capture the hypereutrophic status and Akkulam lake was found to be in this state and three other lakes are in eutrophic state for the 2012–2018 database used in this study. The method of developing MTSI is generic and with discrete threshold values which enable its application to any hydroclimatic region. Subsequently, a novel Machine Learning based framework is proposed for prediction of MTSI considering WQPs as inputs. Four machine learning methods and the linear regression are applied for the prediction of MTSI of LWBs of Kerala, considering six combinations of input datasets of WQPs. The rigorous performance evaluation based on multitude of error, coefficient and graphical measures confirmed the superiority of Random Forest as the best method for the prediction of MTSI, for the chosen database. The combination of nine WQPs involving both physical and chemical parameters is found to be the best for MTSI prediction and machine learning offers flexibility to get comparable performance even with lesser number of input variables.

Class-relevant feature density estimator for open set fault classification of industrial equipment using vibration signals

Recently, the application of vibration signals to the fault diagnosis of industrial equipment has attracted increased attention. Open set recognition allows deep networks to detect unknown faults while maintaining a high-classification accuracy for known faults, which is necessary for practical industrial applications. Existing distance- and probability-based open set recognition methods show significant potential in handling this challenging task. In this study, we combine a deep backbone network and probabilistic model to construct a class-relevant feature density estimator (CRFDE) that integrates the advantages of distance- and probability-based methods. Specifically, a loss function is developed to enable the backbone network to extract the class-relevant features and probabilistic model to estimate them, where the employed probabilistic model is improved from variational auto-encoder (VAE) to class-conditional VAE (CCVAE) that can estimate the class-conditional data densities. The output class-relevant feature densities of CCVAE are treated as special distances to determine the class belongingness of the input samples. We validated the proposed CRFDE on two public motor bearing datasets and one laboratory gas-insulated switchgear vibration signal dataset. The results revealed that the proposed CRFDE achieved higher F1 scores than the existing methods.

Smoothing level selection for density estimators based on the moments

This paper introduces an approach to select the bandwidth or smoothing parameter in multiresolution (MR) density estimation and nonparametric density estimation. It is based on the evolution of the second, third and fourth central moments and the shape of the estimated densities for different bandwidths and resolution levels. The proposed method has been applied to density estimation by means of multiresolution densities as well as kernel density estimation (MRDE and KDE respectively). The results of the simulations and the empirical application demonstrate that the level of resolution resulting from the moments method performs better with multimodal densities than the Bayesian Information Criterion (BIC) for multiresolution densities estimation and the plug-in for kernel densities estimation.

Nonparametric density estimation with nonuniform B-spline bases

Recently, Kirkby et al. introduced a general and efficient nonparametric density estimation procedure for local bases that is based on the statistical Galerkin method. However, the resulting density estimators using the statistical Galerkin method are not guaranteed to be proper, i.e., the density functions are not ensured to be nonnegative and normalized. To address this issue, we present a constrained Galerkin method with B-spline functions for nonparametric density estimation. Furthermore, we also provide a procedure to generate the optimal knot vectors for density estimations with nonuniform B-spline bases, which are based on the cumulative distribution of the feature function. Extensive numerical experiments are conducted to demonstrate the efficiency and accuracy of our framework.

TAKDE: Temporal Adaptive Kernel Density Estimator for Real-Time Dynamic Density Estimation.

Real-time density estimation is ubiquitous in many applications, including computer vision and signal processing. Kernel density estimation is arguably one of the most commonly used density estimation techniques, and the use of "sliding window" mechanism adapts kernel density estimators to dynamic processes. In this article, we derive the asymptotic mean integrated squared error (AMISE) upper bound for the "sliding window" kernel density estimator. This upper bound provides a principled guide to devise a novel estimator, which we name the temporal adaptive kernel density estimator (TAKDE). Compared to heuristic approaches for "sliding window" kernel density estimator, TAKDE is theoretically optimal in terms of the worst-case AMISE. We provide numerical experiments using synthetic and real-world datasets, showing that TAKDE outperforms other state-of-the-art dynamic density estimators (including those outside of kernel family). In particular, TAKDE achieves a superior test log-likelihood with a smaller run-time.

Characterizing residential mobility among people with HIV in Tennessee and its impact on HIV care outcomes.

Our objective was to assess the prevalence and patterns of mobility among people with HIV (PWH) in Tennessee and its impact on HIV care outcomes. Retrospective cohort study. We combined residential address and HIV surveillance data from PWH in Tennessee from 2016 to 2018. Using Poisson regression, we estimated associations between in-state mobility (change in address or total miles moved) in 1 year and outcomes in the subsequent year; retention: having two CD4 + /HIV RNA values (labs) in a calendar year at least 3 months apart, loss to follow-up (LTFU): having labs at baseline but not the subsequent year, and viral suppression: HIV RNA less than 200 copies/ml. We applied a kernel density estimator to origin-destination address lines to visualize mobility patterns across demographic subgroups. Among 17 428 PWH [median age 45 years (interquartile range; IQR 34-53)], 6564 (38%) had at least one move. Median miles moved was 8.9 (IQR 2.6-143.4)). We observed in-state movement between major cities (Chattanooga, Knoxville, Memphis and Nashville) and out-of-state movement to and from Georgia and Florida. Having at least one in-state move was associated with a decreased likelihood of retention [adjusted relative risk (aRR) = 0.91; 95% confidence interval (CI) 0.88-0.95], and an increased risk of LTFU (aRR = 1.17; 95% CI 1.04-1.31, two to three moves vs. none). Greater distance moved in-state was associated with decreased retention and increased LTFU (aRR = 0.53; 95% CI 0.49-0.58, aRR = 2.52; 95% CI 2.25-2.83, respectively for 1000 vs. 0 miles). There was no association between mobility and viral suppression. Mobility is common among PWH in Tennessee and is associated with initial poor engagement in HIV care.

Integrated methodology to link geochemical and geophysical-lab data in a geophysical investigation of a slag heap for resource quantification

The increasing need to find alternative stocks of critical raw materials drives to revisit the residues generated during the former production of mineral and metallic raw materials. Geophysical methods contribute to the sustainable characterization of metallurgical residues inferring on their composition, zonation and volume(s) estimation. Nevertheless, more quantitative approaches are needed to link geochemical or mineralogical analyses with the geophysical data. In this contribution, we describe a methodology that integrates geochemical and geophysical laboratory measurements to interpret geophysical field data solving a classification problem. The final aim is to estimate volume(s) of different types of materials to assess the potential resource recovery. We illustrate this methodology with a slag heap composed of residues from a former iron and steel factory. First, we carried out a 3D field acquisition using electrical resistivity tomography (ERT) and induced polarization (IP), based on which, a sampling survey was designed. We conducted laboratory measurements of ERT, IP, spectral induced polarization (SIP), and X-ray fluorescence analysis, based on which, 4 groups of different chemical composition were identified. Then we carried out a 3D probabilistic classification of the field data, based on 2D kernel density estimators (for each group) fitted to the inverted data collocated with the samples. The estimated volumes based on the classification model were: 4.17 × 103 m3 ± 12 %, 1.888 × 105 m3 ± 12 %, 59.4 × 103 m3 ± 19 %, and 2.30 × 104 m3 ± 21% for the groups ordered with an increasing metallic content. The uncertainty ranges were derived from comparing the volumes with and without considering the probabilities associated to the classification. We found that a representative sampling and the definition of the KDE bandwidths are defining elements in the classification and ultimately the estimation of volumes. This methodology is suitable to quantitatively interpret geophysical data in terms of the geochemical composition of the materials, integrating uncertainties both in the classification and the estimation of volumes. Furthermore, several crucial elements in the investigation of metallurgical residues could be applied in a real case study, e.g., geophysical field acquisition, sampling and lab measurements.

Penalized sieve estimation of zero‐inefficiency stochastic frontiers

SummaryStochastic frontier models for cross‐sectional data typically assume that the one‐sided distribution of firm‐level inefficiency is continuous. However, it may be reasonable to hypothesize that inefficiency is continuous except for a discrete mass at zero capturing fully efficient firms (zero‐inefficiency). We propose a sieve‐type density estimator for such a mixture distribution in a nonparametric stochastic frontier setting under a unimodality‐at‐zero assumption. Consistency, rates of convergence and asymptotic normality of the estimators are established, as well as a test of the zero‐inefficiency hypothesis. Simulations and two applications are provided to demonstrate the practicality of the method.

Livin' la vida local: philopatry results in consistent patterns of annual space use in a long‐lived lizard

AbstractFor animals exhibiting range residency, the home range is a useful framework to quantify space use. Some reptiles can live decades in the wild and experience extreme environmental variation that influences patterns of habitat use. Individuals may modify their use of space over time, reducing the utility of single‐year home range estimates. Very high frequency (VHF) telemetry data were collected for Gila monsters (Heloderma suspectum) at three Mojave Desert sites in Clark County, Nevada, and home range utilization distributions were calculated using an autocorrelated kernel density estimator. Home range size was consistent within individuals and populations, and home range size did not vary across years at any site. To measure home range fidelity (year‐to‐year reuse), we calculated Bhattacharyya's coefficient (BC) for each combination of years in which an individual was tracked and averaged estimates across individuals and populations. The average BC score was 0.86 (scale from 0 to 1; 0 = no overlap and 1 = complete overlap) and did not vary among populations. We modeled home range area accumulation to estimate the minimum sample size needed for asymptotic stability and found home range accumulation to be dynamic and variable within and across years and individuals. Analysis of the frequency of movement by individuals, average distance traveled per movement, and cumulative distance traveled per active season revealed that movement patterns vary considerably by year. Heterogeneity of space use among populations and individuals suggests that individual and local environmental variation, rather than annual variation in resource availability, may drive home range size and movement patterns of Gila monsters in southern Nevada. Annual variability in movement patterns did not translate to variability in home range size or location, and the species exhibits extremely high philopatry, using the same areas for periods of at least 3–5 years.

Density estimation for toroidal data using semiparametric mixtures

Toroidal data is an extension of circular data on a torus and plays a critical part in various scientific fields. This article studies the density estimation of multivariate toroidal data based on semiparametric mixtures. One of the major challenges of semiparametric mixture modelling in a multi-dimensional space is that one can not directly maximize the likelihood over the unrestricted component density as it will result in a degenerate estimate with an unbounded likelihood. To overcome this problem, we propose to fix the maximum of the component density, which subsequently bounds the maximum of the mixture and its likelihood function, hence providing a satisfactory density estimate. The product of univariate circular distributions are utilized to form multivariate toroidal densities as candidates for mixture components. Numerical studies show that the mixture-based density estimator is superior in general to the kernel density estimator.

Distance sampling and spatial capture-recapture for estimating density of Northern Bobwhite

Obtaining accurate density estimates is critical for managers making decisions regarding wildlife populations. A variety of methods have been used to estimate population density, with two of the most common being distance sampling (DS) and spatial capture recapture (SCR). We evaluated precision and bias of density estimators through a simulation study of DS using data collected from either human observer point counts or automated recording units (ARUs), and SCR using trapping data or trapping data augmented with telemetry data. We then fit each of the four models to empirical data collected during autumn 2016–2018 for a population of northern bobwhite Colinus virginianus on Di-Lane Wildlife Management Area in Georgia. Density estimates in our simulation study were relatively unbiased using all four methods but were most accurate for the two SCR methods. Empirical density estimates were similar across methods and years, with annual averages of 0.41 birds per ha in 2016, 0.44 birds per ha in 2017 and 0.31 birds per ha in 2018. In general, both SCR methods were also the most precise using the empirical data, although ARU distance sampling did also produce comparable density estimates and precision values. Although SCR methods performed the best overall, cost and increased labor of these monitoring programs should be evaluated in relation to the relative ease of ARU deployment or point count surveys, which also provided adequate, and often similar, density estimates. Integrating these constraints into a structured decision-making framework will aid managers weighing decisions regarding how to monitor, and ultimately make management decisions.

A Parzen–Rosenblatt type density estimator for circular data: exact and asymptotic optimal bandwidths

. For the Parzen–Rosenblatt type density estimator for circular data, we prove the existence of a minimizer, h MISE ( f ; K , n ) , of its exact mean integrated squared error (MISE) and show that it is asymptotically equivalent to the bandwidth h AMISE ( f ; K , n ) that minimizes the leading terms of the MISE, together with the order of convergence of the relative error h AMISE ( f ; K , n ) / h MISE ( f ; K , n ) − 1 . Some small and moderate sample size comparisons between the two bandwidths are also presented when the underlying density is a mixture of von Mises densities.

Marginal post-processing of Bayesian inference products with normalizing flows and kernel density estimators

ABSTRACT Bayesian analysis has become an indispensable tool across many different cosmological fields, including the study of gravitational waves, the cosmic microwave background, and the 21-cm signal from the Cosmic Dawn, among other phenomena. The method provides a way to fit complex models to data describing key cosmological and astrophysical signals and a whole host of contaminating signals and instrumental effects modelled with ‘nuisance parameters’. In this paper, we summarize a method that uses masked autoregressive flows and kernel density estimators to learn marginal posterior densities corresponding to core science parameters. We find that the marginal or ‘nuisance-free’ posteriors and the associated likelihoods have an abundance of applications, including the calculation of previously intractable marginal Kullback–Leibler divergences and marginal Bayesian model dimensionalities, likelihood emulation, and prior emulation. We demonstrate each application using toy examples, examples from the field of 21-cm cosmology, and samples from the Dark Energy Survey. We discuss how marginal summary statistics like the Kullback–Leibler divergences and Bayesian model dimensionalities can be used to examine the constraining power of different experiments and how we can perform efficient joint analysis by taking advantage of marginal prior and likelihood emulators. We package our multipurpose code up in the pip-installable code margarine for use in the wider scientific community.

Geotechnologies applied to geographic information system (GIS) of Fish farming in Rondônia state, Western Amazon

This research demonstrated a Geographic Information System (GIS) of licensed fish farms in Rondônia state, Brazil. Based on structuring of the GIS, spatial analyzes of location and distribution of fish farms were carried out in relation to highway network; to drainage; to microregions of Rondônia and the verification of the density. Methodological procedure consisted of modeling the Database (DB), whose information was obtained from Secretaria do Estado de Rondônia para Desenvolvimento Ambiental (SEDAM/RO), which holds the references of licensed fish farms processed in SPRING and ARCGIS 9 Arcmap 9.3 software. For spatial statistics, the Kernel density estimator was applied. The main result is the fact that GIS made it quick and easy to search for data and information about the fish farms studied. The highest density was 4937.64 fish farms per unit area in Ji-Paraná microregion, which is located in the Central region of Rondônia state. In thematic mapping, the fish farms showed some spatial dependencies, as follows: I – They depend on main access, highway BR 364. II – The cluster of fish farms is arranged where there is greater availability of water, that is, they depend on water courses. Therefore, positioning and distribution of fish farms take place in the three main microregions, Ji-Paraná 40.30% of licensed fish farms, followed by microregions of Cacoal 16.02% and Ariquemes 15.87%.

New Feature Selection Approach for Photovoltaïc Power Forecasting Using KCDE

Feature selection helps improve the accuracy and computational time of solar forecasting. However, FS is often passed by or conducted with methods that do not suit the solar forecasting issue, such as filter or linear methods. In this study, we propose a wrapper method termed Sequential Forward Selection (SFS), with a Kernel Conditional Density Estimator (KCDE) named SFS-KCDE, as FS to forecast day-ahead regional PV power production in French Guiana. This method was compared to three other FS methods used in earlier studies: the Pearson correlation method, the RReliefF (RRF) method, and SFS using a linear regression. It has been shown that SFS-KCDE outperforms other FS methods, particularly for overcast sky conditions. Moreover, Wrapper methods show better forecasting performance than filter methods and should be used.