Adaptive Kernel Density Estimation Research Articles

A Flexible Method for Estimating Luminosity Functions via Kernel Density Estimation

Abstract We propose a flexible method for estimating luminosity functions (LFs) based on kernel density estimation (KDE), the most popular nonparametric density estimation approach developed in modern statistics, to overcome issues surrounding the binning of LFs. One challenge in applying KDE to LFs is how to treat the boundary bias problem, as astronomical surveys usually obtain truncated samples predominantly due to the flux-density limits of surveys. We use two solutions, the transformation KDE method ( ) and the transformation–reflection KDE method ( ) to reduce the boundary bias. We develop a new likelihood cross-validation criterion for selecting optimal bandwidths, based on which the posterior probability distribution of the bandwidth and transformation parameters for and are derived within a Markov Chain Monte Carlo sampling procedure. The simulation result shows that and perform better than the traditional binning method, especially in the sparse data regime around the flux limit of a survey or at the bright end of the LF. To further improve the performance of our KDE methods, we develop the transformation–reflection adaptive KDE approach ( ). Monte Carlo simulations suggest that it has good stability and reliability in performance, and is around an order of magnitude more accurate than using the binning method. By applying our adaptive KDE method to a quasar sample, we find that it achieves estimates comparable to the rigorous determination in a previous work, while making far fewer assumptions about the LF. The KDE method we develop has the advantages of both parametric and nonparametric methods.

Method to estimate sag frequency in doubly fed induction generator integrated power system based on adaptive kernel density estimation

Voltage sag frequency estimation is necessary for understanding the voltage sag severity in power system and offering full information for the interested parties to mitigate voltage sag. The high penetration of wind power in the power system and the uncertainty of the fault distribution raise new challenges to accurate voltage sag frequency estimation. This study presents a systematic voltage sag frequency estimation method, considering the fault distribution density, fault ride-through (FRT) process of wind turbines during voltage sag and the interval characteristic of voltage sag frequency. First, this study proposes a fault distribution estimation model based on adaptive kernel density. Second, this study proposes a method for calculating the residual voltage and duration of voltage sag during FRT and combines the common distance protection action to analyse the effect on voltage sag by FRT process of wind turbines. Lastly, this study proposes an interval-valued voltage sag estimation method considering the interval characteristics of fault rate in the power system. IEEE 30-bus test system is used to verify the proposed method, the estimation results show better performance of the proposed method compared with the typical estimation methods.

A Multivariate Statistics-Based Approach for Detecting Diesel Engine Faults with Weak Signatures

The problem of timely detecting the engine faults that make engine operating parameters exceed their control limits has been well-solved. However, in practice, a fault of a diesel engine can be present with weak signatures, with the parameters fluctuating within their control limits when the fault occurs. The weak signatures of engine faults bring considerable difficulties to the effective condition monitoring of diesel engines. In this paper, a multivariate statistics-based fault detection approach is proposed to monitor engine faults with weak signatures by taking the correlation of various parameters into consideration. This approach firstly uses principal component analysis (PCA) to project the engine observations into a principal component subspace (PCS) and a residual subspace (RS). Two statistics, i.e., Hotelling’s T 2 and Q statistics, are then introduced to detect deviations in the PCS and the RS, respectively. The Hotelling’s T 2 and Q statistics are constructed by taking the correlation of various parameters into consideration, so that faults with weak signatures can be effectively detected via these two statistics. In order to reasonably determine the control limits of the statistics, adaptive kernel density estimation (KDE) is utilized to estimate the probability density functions (PDFs) of Hotelling’s T 2 and Q statistics. The control limits are accordingly derived from the PDFs by giving a desired confidence level. The proposed approach is demonstrated by using a marine diesel engine. Experimental results show that the proposed approach can effectively detect engine faults with weak signatures.

A novel reliability analysis method based on Gaussian process classification for structures with discontinuous response

Reliability analysis techniques combining with various surrogate models have attracted increasing attention because of their accuracy and great efficiency. However, they primarily focus on the structures with continuous response, while very rare researches on the reliability analysis for structures with discontinuous response are carried out. Furthermore, existing adaptive reliability analysis methods based on importance sampling (IS) still have some intractable defects when dealing with small failure probability, and there is no related research on reliability analysis for structures involving discontinuous response and small failure probability. Therefore, this paper proposes a novel reliability analysis method called AGPC-IS for such structures, which combines adaptive Gaussian process classification (GPC) and adaptive-kernel-density-estimation-based IS. In AGPC-IS, an efficient adaptive strategy for design of experiments (DoE), taking into consideration the classification uncertainty, the sampling uniformity and the regional classification accuracy improvement, is developed with the purpose of improving the accuracy of Gaussian process classifier. The adaptive kernel density estimation is introduced for constructing the quasi-optimal density function of IS. In addition, a novel and more precise stopping criterion is also developed from the perspective of the stability of failure probability estimation.The efficiency, superiority and practicability of AGPC-IS are verified by three examples

A Multilevel and Spatial Analysis of the Infant and Young Child Feeding Practices and Associated Factors among the Under-2 Aged Children in Bangladesh

ABSTRACT Infant and young child feeding (IYCF) is crucial for the growth, development, and survival of a child. This study aims to examine the factors associated with IYCF practices among the under-2 aged children in Bangladesh and spatial variability. A total of 2336 children aged 6–23 months were selected from the 2014 Bangladesh Demographic and Health Survey (BDHS). A multi-level logistic regression was used to determine the association between socio-demographic and health-related factors accounting for the cluster effects. An adaptive bandwidth kernel density estimator (KDE) was used to investigate the spatial variability of IYCF practices in Bangladesh. The prevalence of minimum dietary diversity (MDD), minimum meal frequency (MMF), and minimum acceptable diet (MAD) were 29%, 47%, and 19% respectively, where the suboptimal IYCF practice was 96% among them. The study findings show that higher maternal age at childbirth and maternal education level are positively associated with MDD, MMF, and MAD. The likelihood of MMF and MAD was higher for children whose mothers had the decision-making abilities (adjusted odds ratio, [AOR] 1.23 and 1.26, respectively) and had exposure to media (AOR 1.43 and 1.38, respectively). Four or more antenatal care (ANC) visits positively associated with the MDD (AOR 1.47; 95% confidence interval [CI] 1.07–2.02) and ICFI (AOR 3.06; 95% CI 1.24–7.54). The heatmap shows Northern parts of Bangladesh performed better than South-West parts in child feeding practices. Overall, age-appropriate IYCF practices are not satisfactory, and many maternal characteristics contribute significantly to this. Geospatial inequalities are also notable across Bangladesh. The current scenario reinforces the need for programmes and interventions that address the spectrum of IYCF practices and help reduce spatial disparities.

Probabilistic Optimal Power Flow Calculation Method Based on Adaptive Diffusion Kernel Density Estimation

In order to accurately evaluate the influence of the uncertainty and correlation of photovoltaic (PV) output and load on the running state of power system, a probabilistic optimal power flow calculation method based on adaptive diffusion kernel density estimation is proposed in paper. Firstly, based on the distribution characteristics of PV output, the adaptive diffusion kernel density estimation model of PV output is constructed, which can fit the distribution of arbitrary distribution of PV power. This model can improve the local adaptability of PV output model and reflect the uncertainty and volatility of PV output more accurately. Secondly, The Kendall rank correlation coefficient and the least Euclidean distance are used as correlation measure and index of fitting to select the optimal Copula function, and the joint probability distribution model of PV output and load is constructed. After extracting the correlated PV output and load samples, a probabilistic optimal power flow calculation method considering the correlation of PV output and load is proposed. Finally, simulation studies are conducted with the measured data of a PV power plant of China and the IEEE 30-bus power system. The results show that considering the correlation between PV output and load can improve the accuracy of probabilistic optimal power flow calculation and effectively reduce the power generation cost of power system.

Improved trajectory similarity-based approach for turbofan engine prognostics

Trajectory similarity-based prediction (TSBP) is an emerging real-time remaining useful life (RUL) prediction method that has drawn considerable attention in the field of data-driven prognostics. TSBP is fast, and the corresponding model is easy to train. However, TBSP only provides a point estimation of RUL, which is insufficient for some specific prognostic applications. Hence, this study introduces an improved TSBP method to handle the issue of prognostic uncertainty. On the basis of an adaptive kernel density estimation technique and β-criterion, the improved TSBP method not only provides an accurate and precise point prediction of RUL but also specifies the confidence interval of RUL prediction. The capability of obtaining the confidence interval of RUL can enhance the TSBP method for uncertainty management. The effectiveness of the proposed method is validated through two cases studies, which are related to turbofan engine prognostics.

Closed-Form Nonparametric GLRT Detector for Subpixel Targets in Hyperspectral Images

The generalized likelihood ratio test (GLRT) is here combined with the nonparametric approach to derive a new adaptive detector for subpixel targets in hyperspectral images. Specifically, a variable bandwidth kernel density estimator (KDE) is employed for estimating the conditional probability density functions composing the GLRT. Although KDE has generally a low mathematical tractability, an approximated closed-form solution is here derived, thanks to an innovative and uncommon choice for the kernel function. Experimental results in subpixel target detection scenarios show that the proposed detector represents not only the natural evolution of but also a successful alternative to both very widely employed and very recently proposed GLRT-based detectors.

Data-Driven Nonparametric Chance-Constrained Optimization for Microgrid Energy Management

In this article, we present a data-driven nonparametric chance-constrained optimization for microgrid energy management. The proposed approach imposes no assumption on probability density and distribution functions of solar generation and load. Adaptive kernel density estimator is utilized to construct a confidence set for each random parameter based on the historical data. The constructed confidence sets encompass the ambiguous true distribution and density functions. The concept of $\phi $ -divergence tolerance is applied to compute the distance between the estimated and true probability distribution functions (PDF)s. The estimated distributions are used to formulate a set of data-driven nonparametric chance constraints and model system/component restrictions. To account for the impact of errors in the forecast distributions on system economics and security, confidence levels of the chance constraints are adjusted with respect to pointwise errors of the estimated PDFs. This adjustment ensures that the microgrid chance constraints are satisfied with a predetermined confidence level even if the true realizations of solar generation and load do not exactly fit on the estimated PDFs. The chance constraints are converted into algebraic constraints. Numerical results show the effectiveness of the proposed approach for microgrid management.

Probabilistic deep neural network price forecasting based on residential load and wind speed predictions

Precise price forecasting can lessen the risk of participation in the deregulated electricity market. On account of a large amount of historical data, deep learning based methods can be a promising solution to achieve an accurate forecast. This study presents a deep neural network algorithm to estimate the probability density function of price, incorporating the prediction of wind speed and residential load as two other high volatile parameters. To this end, first, a combination of convolution neural network (CNN) and gated recurrent unit (GRU) is utilised to predict wind speed and residential load. Then, the results are integrated into historical price information to form the input dataset for price forecasting. The proposed price forecast procedure consists of CNN, GRU, and adaptive kernel density estimator (AKDE). AKDE is used as a numerical algorithm to capture probabilistic characterisation of real-time and day-ahead prices. Several deep and shallow networks and the proposed algorithm are implemented, and the results are compared. Furthermore, the effectiveness of AKDE in providing complete statistical information is verified through comparison with conventional and fixed smooth KDEs. In addition, the gradient boosting tree method is incorporated to verify the dependence of the price to the wind and the residential loads.

Nonparametric Probabilistic Unbalanced Power Flow With Adaptive Kernel Density Estimator

This paper presents a nonparametric algorithm to estimate probability density functions of power flow outputs in unbalanced distribution systems. The proposed algorithm is based on an adaptive kernel density estimation. As the main advantage, the proposed algorithm is: 1) applicable for power flow problems with unknown classes of probability distributions for output random variables; 2) highly accurate; 3) non-time-consuming; 4) capable of modeling uncertainties of distributed generators; 5) able to provide complete statistical information on the probabilistic power flow problem; and 6) applicable to unbalanced power distribution systems. The proposed nonparametric estimator is applied to the IEEE 13-bus and IEEE 37-bus test cases, which consist of wind farms and photovoltaic units. We use several statistical parameters to compare outcomes of the proposed nonparametric estimator with results obtained by the 2n + 1 point estimation, and unscented transforms methods. In addition, probability density functions of power flow output variables are plotted using the proposed algorithm, Monte-Carlo simulation, and diffusion methods, and the results are compared.

A high-performance approach for predicting donor splice sites based on short window size and imbalanced large samples

BackgroundSplice sites prediction has been a long-standing problem in bioinformatics. Although many computational approaches developed for splice site prediction have achieved satisfactory accuracy, further improvement in predictive accuracy is significant, for it is contributing to predict gene structure more accurately. Determining a proper window size before prediction is necessary. Overly long window size may introduce some irrelevant features, which would reduce predictive accuracy, while the use of short window size with maximum information may performs better in terms of predictive accuracy and time cost. Furthermore, the number of false splice sites following the GT–AG rule far exceeds that of true splice sites, accurate and rapid prediction of splice sites using imbalanced large samples has always been a challenge. Therefore, based on the short window size and imbalanced large samples, we developed a new computational method named chi-square decision table (χ2-DT) for donor splice site prediction.ResultsUsing a short window size of 11 bp, χ2-DT extracts the improved positional features and compositional features based on chi-square test, then introduces features one by one based on information gain, and constructs a balanced decision table aimed at implementing imbalanced pattern classification. With a 2000:271,132 (true sites:false sites) training set, χ2-DT achieves the highest independent test accuracy (93.34%) when compared with three classifiers (random forest, artificial neural network, and relaxed variable kernel density estimator) and takes a short computation time (89 s). χ2-DT also exhibits good independent test accuracy (92.40%), when validated with BG-570 mutated sequences with frameshift errors (nucleotide insertions and deletions). Moreover, χ2-DT is compared with the long-window size-based methods and the short-window size-based methods, and is found to perform better than all of them in terms of predictive accuracy.ConclusionsBased on short window size and imbalanced large samples, the proposed method not only achieves higher predictive accuracy than some existing methods, but also has high computational speed and good robustness against nucleotide insertions and deletions.ReviewersThis article was reviewed by Ryan McGinty, Ph.D. and Dirk Walther.

Adaptive Nonparametric Kernel Density Estimation Approach for Joint Probability Density Function Modeling of Multiple Wind Farms

The uncertainty of wind power brings many challenges to the operation and control of power systems, especially for the joint operation of multiple wind farms. Therefore, the study of the joint probability density function (JPDF) of multiple wind farms plays a significant role in the operation and control of power systems with multiple wind farms. This research was innovative in two ways. One, an adaptive bandwidth improvement strategy was proposed. It replaced the traditional fixed bandwidth of multivariate nonparametric kernel density estimation (MNKDE) with an adaptive bandwidth. Two, based on the above strategy, an adaptive multi-variable non-parametric kernel density estimation (AMNKDE) approach was proposed and applied to the JPDF modeling for multiple wind farms. The specific steps of AMNKDE were as follows: First, the model of AMNKDE was constructed using the optimal bandwidth. Second, an optimal model of bandwidth based on Euclidean distance and maximum distance was constructed, and the comprehensive minimum of these distances was used as a measure of optimal bandwidth. Finally, the ordinal optimization (OO) algorithm was used to solve this model. The scenario results indicated that the overall fitness error of the AMNKDE method was 8.81% and 11.6% lower than that of the traditional MNKDE method and the Copula-based parameter estimation method, respectively. After replacing the modeling object the overall fitness error of the comprehensive Copula method increased by as much as 1.94 times that of AMNKDE. In summary, the proposed approach not only possesses higher accuracy and better applicability but also solved the local adaptability problem of the traditional MNKDE.

Space–time inhomogeneous background intensity estimators for semi-parametric space–time self-exciting point process models

Histogram maximum likelihood estimators of semi-parametric space–time self-exciting point process models via expectation–maximization algorithm can be biased when the background process is inhomogeneous. We explore an alternative estimation method based on the variable bandwidth kernel density estimation (KDE) and EM algorithm. The proposed estimation method involves expanding the semi-parametric models by incorporating an inhomogeneous background process in space and time and applying the variable bandwidth KDE to estimate the background intensity function. Using an example, we show how the variable bandwidth KDE can be estimated this way. Two simulation examples based on residual analysis are designed to evaluate and validate the ability of our methods to recover the background intensity function and parametric triggering intensity function.

A quad-tree-based fast and adaptive Kernel Density Estimation algorithm for heat-map generation

ABSTRACTKernel Density Estimation (KDE) is a classic algorithm for analyzing the spatial distribution of point data, and widely applied in spatial humanities analysis. A heat-map permits intuitive visualization of spatial point patterns estimated by KDE without any overlapping. To achieve a suitable heat-map, KDE bandwidth parameter selection is critical. However, most generally applicable bandwidth selectors of KDE with relatively high accuracy encounter intensive computation issues that impede or limit the applications of KDE in big data era. To solve the complex computation problems, as well as make the bandwidths adaptively suitable for spatially heterogenous distributions, we propose a new Quad-tree-based Fast and Adaptive KDE (QFA-KDE) algorithm for heat-map generation. QFA-KDE captures the aggregation patterns of input point data through a quad-tree-based spatial segmentation function. Different bandwidths are adaptively calculated for locations in different grids calculated by the segmentation function; and density is estimated using the calculated adaptive bandwidths. In experiments, through comparisons with three mostly used KDE methods, we quantitatively evaluate the performance of the proposed method in terms of correctness, computation efficiency and visual effects. Experimental results demonstrate the power of the proposed method in computation efficiency and heat-map visual effects while guaranteeing a relatively high accuracy.

An Energy Aware Adaptive Kernel Density Estimation Approach to Unequal Clustering in Wireless Sensor Networks

Energy conservation is one of the most important challenges in wireless sensor networks (WSNs). Therefore, compared with the traditional networks, the WSNs not only need high-quality services with high throughput or low transmission delay, but also pay greater attention to energy utilization to extend network lifetime. The clustering routing algorithm is considered to be among the effective ways to collect and transmit data in WSNs. Cluster head (CH) plays a vital role in the cluster which is in charge of data aggregation and data transmission, so their energy consumption is higher than non-CH nodes. The traditional clustering algorithm tends to have the same size in each cluster. However, due to the randomness of the node distribution, the equal clustering mechanism obviously cannot reduce energy consumption. In order to solve this problem, this paper contributes a new unequal clustering algorithm, an energy-aware adaptive kernel density estimation algorithm (EAKDE), which aims to balance the energy dissipation among the CHs. EAKDE utilizes fuzzy logic to determine the priority of nodes competing for CH. In order to adapt the dynamic change of node conditions, adaptive kernel density estimation algorithm is utilized to assign the appropriate unequal cluster radius to sensor nodes. The simulation results demonstrate that, in different scenarios, EAKDE outperforms the other well-known algorithms in terms of network stability, network lifetime, and energy efficiency.

Adaptive Kernel Density Estimation with Generalized Least Square Cross-Validation

Adaptive kernel density estimator is an efficient estimator when the density to be estimated has long tail or multi-mode. They use varying bandwidths at each observation point by adapting a fixed bandwidth for data. It is well-known that bandwidth selection is too important for performance of kernel estimators. An efficient recent method is the generalized least square cross-validation which improves the least squares cross-validation. In this paper, performances of the adaptive kernel estimators obtained based on the generalized least square cross-validation are investigated. We performed a simulation study to inform about performances of the modified adaptive kernel estimators. For the simulation, we use also the bandwidth selection methods of normal reference, least squares cross-validation, biased cross-validation, and plug-in methods. Simulation study shows that the adaptive kernel estimators improve the performances of the kernel estimators with fixed bandwidth selected based on generalized least square cross-validation.

Photovoltaic Output Power Probability Model Based on Adaptive Beta Kernel Density Estimation

Due to the current non-parametric kernel density estimation method for photovoltaic(PV) output power probability modeling using fixed bandwidth, lacking of local adaptability, existing boundary bias, etc, the probability model of PV output power based on adaptive Beta kernel density estimation theory is proposed. The historical data of the PV power in Ashland region of the US was used for simulation analysis. The simulation results verified the accuracy and effectiveness of the proposed model.

Evaluation of threshold selection methods for adaptive kernel density estimation in disease mapping

BackgroundMaps of disease rates produced without careful consideration of the underlying population distribution may be unreliable due to the well-known small numbers problem. Smoothing methods such as Kernel Density Estimation (KDE) are employed to control the population basis of spatial support used to calculate each disease rate. The degree of smoothing is controlled by a user-defined parameter (bandwidth or threshold) which influences the resolution of the disease map and the reliability of the computed rates. Methods for automatically selecting a smoothing parameter such as normal scale, plug-in, and smoothed cross validation bandwidth selectors have been proposed for use with non-spatial data, but their relative utilities remain unknown. This study assesses the relative performance of these methods in terms of resolution and reliability for disease mapping.ResultsUsing a simulated dataset of heart disease mortality among males aged 35 years and older in Texas, we assess methods for automatically selecting a smoothing parameter. Our results show that while all parameter choices accurately estimate the overall state rates, they vary in terms of the degree of spatial resolution. Further, parameter choices resulting in desirable characteristics for one sub group of the population (e.g., a specific age-group) may not necessarily be appropriate for other groups.ConclusionWe show that the appropriate threshold value depends on the characteristics of the data, and that bandwidth selector algorithms can be used to guide such decisions about mapping parameters. An unguided choice may produce maps that distort the balance of resolution and statistical reliability.

On the utility of asymptotic bandwidth selectors for spatially adaptive kernel density estimation

Implementation of the spatially adaptive kernel estimator relies on choice of a ‘global bandwidth’. We derive the closed-form asymptotic bias for this estimator with the aim of developing relevant selectors, and note non-uniform convergence hinders its usability for this purpose.