Kernel Density Estimation Technique Research Articles

Bayesian approach in the nonparametric conditional density estimation

In the literature, it is well known that in kernel density estimation, plug-in and cross-validation techniques, for the selection of the smoothing parameter, tend to provide under-or over-smoothed estimators when the sample size is small or medium, or the function to be estimated is complex. To overcome this latest problem, recently, the Bayesian approach has been proposed as an alternative to these classical methods. In this paper, we restricted attention to extending the idea of the Bayes rule to estimate the smoothing parameter of the conditional density kernel estimation. We are interested in the impact of the use of the global Bayesian approach for the smoothing parameter selection on the performance (the average of the Integrated Squared Error (ISE) and the calculation time required for their implementation) of the kernel conditional density estimator compared to the classical method trough a Monté Carlo simulation. Moreover, based on simulated samples of different sizes from two different conditional models, we adopted the Bayesian approach compared with the classical smoothing parameter selection procedure (the cross-validation method), using the Gaussian kernel to construct the estimators in question.

Data driven multi-objective economic-environmental robust optimization for refinery planning with multiple modes under uncertainty

In the context of the global energy transition, the refining industry is facing a major challenge between economic profitability and environmental protection. Planning refineries using traditional methods ignores the complex interactions between market dynamics and environmental factors, which causes unsatisfactory decisions. This study proposes a novel model-driven multi-objective robust optimization framework designed to tackle both economic and environmental challenges faced by the refining industry during its transformation process. The framework aims to maximize refinery profitability while reducing price uncertainty and achieving low carbon emissions and university energy use. Firstly, we introduce a nonlinear mechanism model of key production units to construct a multi-objective planning model that aims to simultaneously maximize profits and minimize carbon emissions and energy consumption. Secondly, to capture market dynamics uncertainty, we construct an uncertainty set based on the principal component analysis-robust kernel density estimation technique. Thirdly, using robust optimization theory, we transform this uncertainty set into a solvable binary problem. Finally, a case study is presented to demonstrate the effectiveness of this optimization framework in real-world applications, where we can keep CO2 emissions constant and still make the refinery profitable up to 95% in the worst-case uncertainty optimization. Additionally, uncertainty optimization’s profitability increases as conservatism decreases, illustrating the framework’s flexibility in responding to changing markets. Using this framework for decision support can assist the refining industry in protecting the environment while safeguarding economic profitability.

The Kernel Density Estimation Technique for Spatio-Temporal Distribution and Mapping of Rain Heights over South Africa: The Effects on Rain-Induced Attenuation

The devastating effects of rain-induced attenuation on communication links operating above 10 GHz during rainy events can significantly degrade signal quality, leading to interruptions in service and reduced data throughput. Understanding the spatial and seasonal distribution of rain heights is crucial for predicting these attenuation effects and for network performance optimization. This study utilized ten years of atmospheric temperature and geopotential height data at seven pressure levels (1000, 850, 700, 500, 300, 200, and 100 hPa) obtained from the Copernicus Climate Data Store (CDS) to deduce rain heights across nine stations in South Africa. The kernel density estimation (KDE) method was applied to estimate the temporal variation of rain height. A comparison of the measured and estimated rain heights shows a correlation coefficient of 0.997 with a maximum percentage difference of 5.3%. The results show that rain height ranges from a minimum of 3.5 km during winter in Cape Town to a maximum of about 5.27 km during the summer in Polokwane. The spatial variation shows a location-dependent seasonal trend, with peak rain heights prevailing at the low-latitude stations. The seasonal variability indicates that higher rain heights dominate in the regions (Polokwane, Pretoria, Nelspruit, Mahikeng) where there is frequent occurrence of rainfall during the winter season and vice versa. Contour maps of rain heights over the four seasons (autumn, spring, winter, and summer) were also developed for South Africa. The estimated seasonal rain heights show that rain-induced attenuations were grossly underestimated by the International Telecommunication Union (ITU) recommended rain heights at most of the stations during autumn, spring, and summer but fairly overestimated during winter. Durban had a peak attenuation of 15.9 dB during the summer, while Upington recorded the smallest attenuation of about 7.7 dB during winter at a 0.01% time exceedance. Future system planning and adjustments of existing infrastructure in the study stations could be improved by integrating these localized, seasonal radio propagation data in link budget design.

Fruit Distribution Density Estimation in YOLO-Detected Strawberry Images: A Kernel Density and Nearest Neighbor Analysis Approach

Precise information on strawberry fruit distribution is of significant importance for optimizing planting density and formulating harvesting strategies. This study applied a combined analysis of kernel density estimation and nearest neighbor techniques to estimate fruit distribution density from YOLOdetected strawberry images. Initially, an improved yolov8n strawberry object detection model was employed to obtain the coordinates of the fruit centers in the images. The results indicated that the improved model achieved an accuracy of 94.7% with an mAP@0.5~0.95 of 87.3%. The relative error between the predicted and annotated coordinates ranged from 0.002 to 0.02, demonstrating high consistency between the model predictions and the annotated results. Subsequently, based on the strawberry center coordinates, the kernel density estimation algorithm was used to estimate the distribution density in the strawberry images. The results showed that with a bandwidth of 200, the kernel density estimation accurately reflected the actual strawberry density distribution, ensuring that all center points in high-density regions were consistently identified and delineated. Finally, to refine the strawberry distribution information, a comprehensive method based on nearest neighbor analysis was adopted, achieving target area segmentation and regional density estimation in the strawberry images. Experimental results demonstrated that when the distance threshold ϵ was set to 600 pixels, the correct grouping rate exceeded 94%, and the regional density estimation results indicated a significant positive correlation between the number of fruits and regional density. This study provides scientific evidence for optimizing strawberry planting density and formulating harvesting sequences, contributing to improved yield, harvesting efficiency, and reduced fruit damage. In future research, this study will further explore dynamic models that link fruit distribution density, planting density, and fruit growth status.

Data-driven Approximation of Cumulative Distribution Function Using Particle Swarm Optimization based Finite Mixtures of Logistic Distribution

This paper proposes a data-driven approximation of the Cumulative Distribution Function using the Finite Mixtures of the Cumulative Distribution Function of Logistic distribution. Since it is not possible to solve the logistic mixture model using the Maximum likelihood method, the mixture model is modeled to approximate the empirical cumulative distribution function using the computational intelligence algorithms. The Probability Density Function is obtained by differentiating the estimate of the Cumulative Distribution Function. The proposed technique estimates the Cumulative Distribution Function of different benchmark distributions. Also, the performance of the proposed technique is compared with the state-of-the-art kernel density estimator and the Gaussian Mixture Model. Experimental results on κ−μ distribution show that the proposed technique performs equally well in estimating the probability density function. In contrast, the proposed technique outperforms in estimating the cumulative distribution function. Also, it is evident from the experimental results that the proposed technique outperforms the state-of-the-art Gaussian Mixture model and kernel density estimation techniques with less training data.

STAR‐ESDM: A Generalizable Approach to Generating High‐Resolution Climate Projections Through Signal Decomposition

AbstractHigh‐resolution climate projections are critical to assessing climate risk and developing climate resilience strategies. However, they remain limited in quality, availability, and/or geographic coverage. The Seasonal Trends and Analysis of Residuals empirical statistical downscaling model (STAR‐ESDM) is a computationally‐efficient, flexible approach to generating such projections that can be applied globally using predictands and predictors sourced from weather stations, gridded data sets, satellites, reanalysis, and global or regional climate models. It uses signal processing combined with Fourier filtering and kernel density estimation techniques to decompose and smooth any quasi‐Gaussian time series, gridded or point‐based, into multi‐decadal long‐term means and/or trends; static and dynamic annual cycles; and probability distributions of daily variability. Long‐term predictor trends are bias‐corrected and predictor components used to map predictand components to future conditions. Components are then recombined for each station or grid cell to produce a continuous, high‐resolution bias‐corrected and downscaled time series at the spatial and temporal scale of the predictand time series. Comparing STAR‐ESDM output driven by coarse global climate model simulations with daily temperature and precipitation projections generated by a high‐resolution version of the same global model demonstrates it is capable of accurately reproducing projected changes for all but the most extreme temperature and precipitation values. For most continental areas, biases in 1‐in‐1000 hottest and coldest temperatures are <0.5°C and biases in the 1‐in‐1000 wet day precipitation amounts are <5 mm/day. As climate impacts intensify, STAR‐ESDM represents a significant advance in generating consistent high‐resolution projections to comprehensively assess climate risk and optimize resilience globally.

Evolving demographics: a dynamic clustering approach to analyze residential segregation in Berlin

This paper examines the phenomenon of residential segregation in Berlin over time using a dynamic clustering analysis approach. Previous research has examined the phenomenon of residential segregation in Berlin at a high spatial and temporal aggregation and statically, i.e. not over time. We propose a methodology to investigate the existence of clusters of residential areas according to migration background, age group, gender, and socio-economic dimension over time. To this end, we have developed a sequential mixed methods approach that includes a multivariate kernel density estimation technique to estimate the density of subpopulations and a dynamic cluster analysis to discover spatial patterns of residential segregation over time (2009-2020). The dynamic analysis shows the emergence of clusters on the dimensions of migration background, age group, gender and socio-economic variables. We also identified a structural change in 2015, resulting in a new cluster in Berlin that reflects the changing distribution of subpopulations with a particular migratory background. Finally, we discuss the findings of this study with previous research and suggest possibilities for policy applications and future research using a dynamic clustering approach for analyzing changes in residential segregation at the city level.

Photovoltaic Power Prediction Based on Hybrid Deep Learning Networks and Meteorological Data.

Conventional point prediction methods encounter challenges in accurately capturing the inherent uncertainty associated with photovoltaic power due to its stochastic and volatile nature. To address this challenge, we developed a robust prediction model called QRKDDN (quantile regression and kernel density estimation deep learning network) by leveraging historical meteorological data in conjunction with photovoltaic power data. Our aim is to enhance the accuracy of deterministic predictions, interval predictions, and probabilistic predictions by incorporating quantile regression (QR) and kernel density estimation (KDE) techniques. The proposed method utilizes the Pearson correlation coefficient for selecting relevant meteorological factors, employs a Gaussian Mixture Model (GMM) for clustering similar days, and constructs a deep learning prediction model based on a convolutional neural network (CNN) combined with a bidirectional gated recurrent unit (BiGRU) and attention mechanism. The experimental results obtained using the dataset from the Australian DKASC Research Centre unequivocally demonstrate the exceptional performance of QRKDDN in deterministic, interval, and probabilistic predictions for photovoltaic (PV) power generation. The effectiveness of QRKDDN was further validated through ablation experiments and comparisons with classical machine learning models.

Nonparametric Copula Density Estimation Methodologies

This paper proposes several methodologies whose objective consists of securing copula density estimates. More specifically, this aim will be achieved by differentiating bivariate least-squares polynomials fitted to Deheuvels’ empirical copulas, by making use of Bernstein’s approximating polynomials of appropriately selected orders; by differentiating linearized distribution functions evaluated at optimally spaced grid points; and by implementing the kernel density estimation technique in conjunction with a repositioning of the pseudo-observations and a certain criterion for determining suitable bandwidths. Smoother representations of such density estimates can further be secured by approximating them by means of moment-based bivariate polynomials. The various copula density estimation techniques being advocated herein are successfully applied to an actual dataset as well as a random sample generated from a known distribution.

DKPE-GraphSYN: a drug synergy prediction model based on joint dual kernel density estimation and positional encoding for graph representation.

Introduction: Synergistic medication, a crucial therapeutic strategy in cancer treatment, involves combining multiple drugs to enhance therapeutic effectiveness and mitigate side effects. Current research predominantly employs deep learning models for extracting features from cell line and cancer drug structure data. However, these methods often overlook the intricate nonlinear relationships within the data, neglecting the distribution characteristics and weighted probability densities of gene expression data in multi-dimensional space. It also fails to fully exploit the structural information of cancer drugs and the potential interactions between drug molecules. Methods: To overcome these challenges, we introduce an innovative end-to-end learning model specifically tailored for cancer drugs, named Dual Kernel Density and Positional Encoding (DKPE) for Graph Synergy Representation Network (DKPEGraphSYN). This model is engineered to refine the prediction of drug combination synergy effects in cancer. DKPE-GraphSYN utilizes Dual Kernel Density Estimation and Positional Encoding techniques to effectively capture the weighted probability density and spatial distribution information of gene expression, while exploring the interactions and potential relationships between cancer drug molecules via a graph neural network. Results: Experimental results show that our prediction model achieves significant performance enhancements in forecasting drug synergy effects on a comprehensive cancer drug and cell line synergy dataset, achieving an AUPR of 0.969 and an AUC of 0.976. Discussion: These results confirm our model's superior accuracy in predicting cancer drug combinations, providing a supportive method for clinical medication strategy in cancer.

Effect of exposure conditions on the chloride concentration profiles in a marine concrete structure, studied by means of the kernel density estimation technique and a Gaussian–Lorentzian model

The corrosion of the embedded steel due to chloride action is one of the main degradation problems limiting the service life of reinforced concrete structures in a marine environment. Modelling chloride profiles into concrete is a key factor to predict durability of structures. Experimental profiles from the harbour of a Mediterranean city are here studied fitting them to a Gaussian–Lorentzian peak model. This model is suited for the observed peak-shaped profiles, yielding descriptive transport parameters for both the convective and the diffusive zones. The parameters of the diffusive zone are found to be in agreement with the solution of the Fick’s second law of diffusion. The obtained parameters describing the profiles are statistically analysed by means of the kernel density estimation technique in order to determine how exposure conditions affect these parameters. A criterion of probability density distributions overlapping is used to draw conclusions about how exposure time, orientation respect to sea, wind, and splash affect the height and location of the chloride peak concentration, and the transport coefficients in the convective and diffusive zones of the profiles.

Ultrasound Entropy Imaging Based on the Kernel Density Estimation: A New Approach to Hepatic Steatosis Characterization.

In this paper, we present the kernel density estimation (KDE)-based parallelized ultrasound entropy imaging and apply it for hepatic steatosis characterization. A KDE technique was used to estimate the probability density function (PDF) of ultrasound backscattered signals. The estimated PDF was utilized to estimate the Shannon entropy to construct parametric images. In addition, the parallel computation technique was incorporated. Clinical experiments of hepatic steatosis were conducted to validate the feasibility of the proposed method. Seventy-two participants and 204 patients with different grades of hepatic steatosis were included. The experimental results show that the KDE-based entropy parameter correlates with log10 (hepatic fat fractions) measured by magnetic resonance spectroscopy in the 72 participants (Pearson's r = 0.52, p < 0.0001), and its areas under the receiver operating characteristic curves for diagnosing hepatic steatosis grades ≥ mild, ≥moderate, and ≥severe are 0.65, 0.73, and 0.80, respectively, for the 204 patients. The proposed method overcomes the drawbacks of conventional histogram-based ultrasound entropy imaging, including limited dynamic ranges and histogram settings dependence, although the diagnostic performance is slightly worse than conventional histogram-based entropy imaging. The proposed KDE-based parallelized ultrasound entropy imaging technique may be used as a new ultrasound entropy imaging method for hepatic steatosis characterization.

An integrated system to significant wave height prediction: Combining feature engineering, multi-criteria decision making, and hybrid kernel density estimation

Accurate prediction of significant wave height is paramount for the effective design, operation, and maintenance of wave energy converters. However, current research falls short in achieving precise and stable point predictions, along with comprehensive uncertainty analysis of significant wave height. To address this gap, this study presents a comprehensive significant wave height combined prediction system. This integrated system encompasses outlier detection utilizing Autoencoders, sophisticated feature engineering, a multi-criteria decision-based model selection methodology, a multi-objective homogeneous nuclear molecular optimization algorithm, and a hybrid kernel density estimation technique. To tackle the critical issue of model selection within ensemble prediction, we introduce a multi-criteria compromise solution ranking algorithm known as VIKOR for the selection of sub-models. Additionally, a novel multi-objective homogeneous nuclear molecular optimization algorithm is proposed, which incorporates joint opposing selection and an elite retention strategy to effectively manage multiple objectives simultaneously, yielding Pareto optimal solutions for combining weights. Furthermore, a hybrid kernel density estimation approach is developed, surpassing previous methods reliant on a single kernel function and fixed bandwidth, thereby achieving a more precise fit to the distribution of wave height data. The effectiveness of the proposed model is rigorously evaluated using wave height datasets from three distinct locations. The experimental results convincingly demonstrate that the significant wave height combined prediction system outperforms existing solutions, excelling in both point and interval predictions of significant wave height.

Inverse Distance Weighting as an alternative interpolation method to create radiometric maps of natural radionuclide concentrations using QGIS

Abstract. A previous study by the authors illustrated the distribution of naturally occurring radionuclides, i.e., potassium (K40), thorium (Th232), and uranium (U238), in the sediment of the Berg River estuary. The study also described the Delta Underwater Gamma detection System (DUGS) and its usage to measure the concentrations of these natural radionuclides. It also proposed a novel radiometric mapping technique with QGIS and, more importantly, highlighted the geospatial process through Kernel Density Estimation (KDE) to create radiometric maps. The present study used the same data and a similar design but proposed the use of the Inverse Distance Weighted (IDW) interpolation method in QGIS to display natural radionuclide concentrations. The radiometric maps created in the previous study using the KDE technique created smooth and visually attractive maps. However, the IDW method is an exact interpolation method that predicts a value at a sampling location that is identical to the observed value. This is a requirement for the analysis of natural radionuclide concentrations in sediments. However, the effectiveness of the interpolation methods was evaluated using SPSS statistics software. First, probability-probability (P-P) plots were produced for each interpolation method. Secondly, descriptive regression statistics, including 'goodness-of-fit', Analysis-of-Variance (ANOVA), coefficients, and residuals were evaluated for both the IDW and KDE interpolation methods. This was done to assess which method was more effective for calculating the radionuclide concentrations (actual vs predicted values) in the Berg River area. The results showed that both methods experienced problems to predict unknown values. However, IDW consistently performed better than KDE across most of the interpolation tests. Natural radionuclides are useful predictors to track sedimentation and the results of this paper can serve as a benchmark for future work in tidal and non-tidal coastal environments.

A novel system for multivariate analysis of discontinuities in fractured rock masses based on manifold learning and fractal models

The engineering-geological characteristics of fractured rock masses depend on the discontinuity properties, which can be analyzed by the stereonet-based system. However, this system faces challenges in describing orientations, which are spherical data, and it cannot incorporate other scalar parameters (e.g., trace length, aperture, etc.) that affect engineering practice. Therefore, a novel system based on manifold learning is proposed to analyze the distribution patterns of orientation data and characterize multiple discontinuity properties. The proposed system uses manifold learning to explore and visualize the manifold structure of orientation data by a two-dimensional kernel density estimation (2D-KDE) technique. It also integrates other parameters into the manifold structure to capture multiple properties. To quantify the manifold structure and reveal the real fractal behaviors of orientation data, a detailed analysis process including monofractal and multifractal models is proposed. The effectiveness of the proposed system is verified by comparison with the stereonet-based system in several aspects. The equivalent manifold structure is also proposed to enhance the engineering applicability of the system, and one of its applications is demonstrated. Finally, the proposed system is applied to a significant road project and further developed by unmanned aerial vehicle (UAV) photogrammetry. The proposed system provides new insights for research on fractured rock masses and has great potential for solving multi-parameter problems in engineering geology.

Informed expansion for informative path planning via online distribution learning

Mobile robots are essential tools for gathering knowledge of the environment and monitoring areas of interest as well as industrial assets. Informative Path Planning methodologies have been successfully applied making robots able to autonomously acquire information and explore unknown surroundings. Rapidly-exploring Information Gathering approaches have been validated in real-world applications, proving they are the way to go when aiming for Information Gathering tasks. In fact, RIG can plan paths for robots with several degrees of freedom and rapidly explore complex workspaces by using the state-of-the-art Voronoi-biased expansion. Nevertheless, it is an efficient solution when most of the area is unknown but its effectiveness decreases as the exploration/gathering evolves. This paper introduces an innovative informed expansion for IG tasks that combines the Kernel Density Estimation technique and a rejection sampling algorithm. By learning online the distribution of the acquired information (i.e., the discovered map), the proposed methodology generates samples in the unexplored regions of the workspace, and thus steers the tree toward the most promising areas. Realistic simulations and an experimental campaign, conducted in the underwater robotics domain, provide a proof-of-concept validation for the developed informed expansion methodology and demonstrate that it enhances the performance of the RIG algorithm.

Stochastic inversion combining seismic data, facies properties, and advanced multiple-point geostatistics

We proposed a novel seismic inversion approach that integrates the physical properties of litho-facies, and geophysical data, within a multiple-point geostatistical framework to reduce the uncertainty in predictions of litho-facies spatial arrangement away from wells or control points. The litho-facies groups (rock-type) in the well locations are defined and conditioned to the distribution of elastic properties, including the values of P-wave velocity (Vp) and facies density (ρ) within the borehole. A conceptual geological model (training image) is utilized within a wavelet-based multiple-point geostatistical simulation (WAVESIM) algorithm to generate litho-facies realizations. In our inversion algorithm, the forward model is created by implementing the bivariate Kernel density estimation (KDE) technique of the litho-facies properties (Vp and ρ) that are distributed within the well locations. We utilize an iterative inversion framework, where a particular number of elastic properties (Vp and ρ) for each WAVESIM realization are drawn. For each generated set of the WAVESIM realizations, we simulate reflectivity series by KDE that are convolved with the seismic wavelet to create synthetic seismograms. Then, using a normalized, cross-correlation function we select the realization that provides the best-match between synthetic seismogram and the input seismic data. Our inversion technique is successfully applied to synthetic and field datasets. Our results demonstrate the efficiency of our inversion to characterize highly heterogeneous reservoirs in a reasonable computational time and control the connectivity between the channels.

A Design and Comparative Analysis of a Home Energy Disaggregation System Based on a Multi-Target Learning Framework

Insightful information on energy use encourages home residents to conduct home energy conservation. This paper proposes an experimental design for an energy disaggregation system based on the low-computational-cost approaches of multi-target classification and multi-target regression, which are under the multi-target learning framework. The experiments are set up to determine the optimal learning algorithm and model parameters. In addition, the designated system can provide inference of the appliance power state and the estimated power consumption from both approaches. The kernel density estimation technique is utilized to formulate the appliance power state as a finite-state machine for the multi-target classification approach. Multi-target regression can directly provide the estimation of appliance power demand from the aggregate data, and this work unifies the system’s design together with multi-target classification. The predictive performances obtained through the F-score (micro-averaged) and power estimation accuracy index for the power state inference and the estimated power demand, respectively, are shown to outperform a deep-learning-based denoising autoencoder network under the same data settings from both approaches. The results lead to a recommendation to apply the approach in home energy monitoring, which is mainly based on the characteristics of appliance power and the information that the residents wish to perceive.

Data-driven two-stage distributionally robust optimization for refinery planning under uncertainty

This work investigates the refinery planning problem under uncertainty in product prices. A novel data-driven Wasserstein distributionally robust optimization framework is proposed for handling uncertainties in the refinery-wide planning operations. A data-driven ambiguity set is constructed based on the Wasserstein metric to model the distributional uncertainty. The robust kernel density estimation technique is adopted to establish the support set to reduce the effect of the potential outliers. Based on the derived support set and ambiguity set, a data-driven two-stage distributionally robust optimization model for refinery planning is developed. Then, the robust counterpart of the proposed model is formulated to make the problem computationally tractable. Finally, a real-world case study on a petroleum refinery is presented to illustrate the effectiveness and applicability of the proposed framework.

A mixed-copula-based integral method for reliability analysis of a novel multi-functional rescue end-effector

To improve the performance of a conventional rescue equipment, we designed a novel multi-functional rescue end-effector comprising a separation mechanism and a grasping mechanism. The failure causes of the separation mechanism are analyzed, and three main failure modes are determined. Meanwhile, their performance functions are established as well. To solve the complex correlations among multiple failure modes, a novel mixed copula function that can capture the upper and lower tail dependencies simultaneously is proposed, which comprises the Gumbel copula and the Clayton copula. Further, a novel procedure that adopts the non-parametric kernel density estimation (NKDE) technique to estimate the probability density function (PDF) and the cumulative distribution function (CDF) for each performance function of the separation mechanism is proposed. The parameters of the mixed copula function are estimated using the method of maximum likelihood estimation. Subsequently, to further study the separation mechanism’s reliability during working, a time-variant parameter estimation method is introduced to build the time-variant mixed copula function. Finally, both the reliability and time-variant reliability of the separation mechanism are analyzed via the proposed integral methods. Comparisons with other analysis schemes show that the proposed integral method is not only effective but also efficient in analyzing the reliability of our target mechanism.