Gaussian Kernel Density Estimation Research Articles

A distributed density estimation algorithm and its application to naive Bayes classification

We consider the problem of learning a density function from observations of an unknown underlying model in a distributed setting, where the observations are partitioned into different sites. Applying commonly used density estimation methods such as Gaussian Mixture Model (GMM) or Kernel Density Estimation (KDE) to distributed data leads to an extensive amount of communication. A familiar approach to address this issue is to sample a small subset of data and collect them into a central node to run the density estimation algorithms on them. In this paper, we follow an alternative to the sub-sampling approach by proposing the nested Log-Poly model. This model provides an accurate density estimation from a small sized statistic of the entire data. In distributed settings, it transfers the small sized statistics from the client nodes to a central node. The estimation process is then run in the central node. The proposed model can be used in different learning tasks such as classification in supervised learning and clustering in unsupervised learning. However, the properties of nested Log-Poly make it a suitable model for one-dimensional density estimations in the distributed settings. This makes Log-Poly a good choice for naive Bayes classifier, where one-dimensional density estimation is required for every feature conditioned on the class label. We provide a theoretical analysis of the efficiency of our model in estimating a wide range of probability density functions. Our experiments show that nested Log-Poly outperforms the state of the art density estimators on several synthetic datasets. We compare the accuracy and the communication load of naive Bayes classifier using nested Log-Poly and other related density estimators on several real datasets. The experimental outcomes depict that nested Log-Poly has less communication load, while maintaining a competitive classification accuracy compared to similar methods that use the entire data. Moreover, we present a comprehensive comparison between nested Log-Poly and validated KDE with sub-sampling, in terms of the number of communicated variables and the number of bytes transferred between the clients and the central node. Nested Log-Poly provides comparable accuracy with the validated KDE with sub-sampling, while communicating fewer variables. However, our method needs to compute and transmit the variables with a high precision in order to accurately capture the details of the underlying distributions.

A Fast Method for Estimating the Number of Clusters Based on Score and the Minimum Distance of the Center Point

Clustering is widely used as an unsupervised learning algorithm. However, it is often necessary to manually enter the number of clusters, and the number of clusters has a great impact on the clustering effect. At present, researchers propose some algorithms to determine the number of clusters, but the results are not very good for determining the number of clusters of data sets with complex and scattered shapes. To solve these problems, this paper proposes using the Gaussian Kernel density estimation function to determine the maximum number of clusters, use the change of center point score to get the candidate set of center points, and further use the change of the minimum distance between center points to get the number of clusters. The experiment shows the validity and practicability of the proposed algorithm.

Super-Resolution Doppler Velocity Estimation by Kernel-Based Range–$\tau$ Point Conversions for UWB Short-Range Radars

Lower band ultrawideband (UWB) Doppler radar is promising for through-wall imaging, e.g., human body detection in rescue scenarios. The inherent problem with pulse-Doppler radar is the tradeoff between the Doppler velocity resolution and the resulting temporal resolution that makes it difficult to conduct real-time target tracking, because the separation of micro-Doppler velocities of the human body requires a higher Doppler velocity resolution. This problem is particularly severe for lower band UWB radar systems, which are required to attain a sufficient penetration depth in concrete material in the through-the-wall imaging scenario. Because UWB signals generally have large fractional bandwidths, the reflected pulse is located over a range gate along the slow-time direction; this is well known as the range walk problem. As a promising solution to this problem, this article newly introduces a technique for a super-resolution Doppler velocity estimation algorithm based on Gaussian kernel density estimation, which converts observed range– $\tau $ points to Doppler-associated ranges. In addition, this approach makes an important contribution for super-resolution range extraction with a compressed sensing (CS) filter, which is combined with the range-point migration (RPM) method for human body imaging associated with micro-Doppler components. 2-D or 3-D numerical simulations, including human body imaging scenario, demonstrate that the proposed method allows both accurate Doppler velocity estimation and human body imaging, which can be updated at the pulse-repetition interval.

Estimating load spectra probability distributions of train bogie frames by the diffusion-based kernel density method

The load spectra of a train bogie frame are used for bogie frame design, simulated fatigue testing and reliability evaluation and fatigue life prediction, however, modelling the bogie load spectrum is one of the most challenging tasks in this field. In this paper, diffusion-based kernel density estimation (DKDE) is used to obtain the load spectra distributions of train bogie frames. A full-scale B-type subway motor bogie and a full-scale high-speed EMU trailer bogie are analysed. DKDE, Gaussian kernel density estimation (GKDE) and parametric estimation are used to fit the distributions of the loads. Goodness-of-fit tests suggest that the distributions obtained by DKDE are close to the actual distributions. Fatigue damage are calculated to evaluate the distributions estimated by DKDE and GKDE. From the distributions obtained by DKDE, reasonable maximum load values and reasonable extrapolated load spectra are deduced.

Dynamic driving and counterfactual decomposition of the influencing factors of household energy consumption among provinces in China

Based on the panel data of 30 provinces in China from 2006 to 2016, the Gaussian kernel density estimation was initially used to describe the characteristics of the inter-provincial differences in household energy consumption (HEC), a dynamic panel data model was established secondly to study the main factors affecting HEC, and finally the counterfactual decomposition method was used to calculate the contribution of each factor to the inter-provincial differences in HEC. The research results are shown as follows: First, the inter-provincial HEC in China shows a phenomenon of agglomeration and growth, and the HEC level of different regions presents difference and imbalance. Second, the logarithms of consumption habits, total population and residents' income have significant effects on the logarithm of HEC, the influence coefficients are 0.191, 0.073 and 0.745, respectively; HEC not only has significant inertia characteristics, but also has certain population size effect and wealth effect. Third, the logarithms of technological progress, urbanization and household energy price have significant effects on the logarithm of HEC, the influence coefficients are 0.864, 0.195 and −0.346, respectively; the HEC saved by technological transformation is not sufficient to offset the increase in HEC demand, urbanization can improve HEC significantly, but the household energy price significantly inhibits HEC. Finally, from the perspective of contribution, the total population, residents' income and urbanization contribute a lot to the inter-provincial differences, totaling 73.02%. The possible innovations of this paper are as follows: First, the contribution of the influencing factors to the difference of inter-provincial HEC is studied, which is rarely seen in the previous literature. Second, compared with traditional decomposition method, counterfactual decomposition method is more convenient for economic interpretation. Third, the divergent conclusions of scholars are pointed out in this paper, and the actual impact of influencing factors on HEC are clarified.

Adaptive DBSCAN Algorithm Based on Sample Density Gradient

DBSCAN is a classic and commonly applied density-based clustering algorithm, but its clustering accuracy depends on the choice of two input parameters. This paper presents a new algorithm for adaptative parameter determination in DBSCAN. The assumption of this new algorithm is that regions with larger sample density gradient usually corresponds to the edge areas of clusters. The main idea is to generate some pre-clusters and determine the values of parameters by their statistics information. We first randomly pick pairs of points in the sample space to form circles, which are called “disks”. Then we estimate the density information of a disk by random sampling, and define the criteria of disk quality to select disks with larger sample density gradient. Finally, we obtain the suitable parameters of DBSCAN in terms of the distributions of radius and points number of these extracted disks by Gaussian kernel density estimation. Experimental results show that the new algorithm improves the accuracy of DBSCAN and performs better than classic algorithms like k-means and Birch in some cases.

Spatiotemporal Statistical Channel Model for Indoor Corridor at 14 GHz, 18 GHz, and 22 GHz Bands

Several techniques have been proposed to overcome challenges of meeting demands for higher data rates in wireless communication. Space-time diversity method is proposed to exploit spatiotemporal nature of the channel; hence, a comprehensive knowledge of the spatiotemporal properties of a channel is required. In this paper, a measurement-based channel model that considers both delay and angular domains of an indoor corridor channel for 14 GHz, 18 GHz, and 22 GHz is proposed. A nonparametric Gaussian kernel density estimation method is applied for cluster identification for the three frequency bands. This work proposes a spatiotemporal model that conditions the model parameters on the azimuthal spatial domain. The clusters are modeled on the complete azimuth plane and a Gaussian estimation distribution is fitted onto the empirical data plot. Both clusters and multipath components are modeled and results are compared with Saleh-Valenzuela model parameter values. The results show that both clusters and multipath components can be estimated by probability density functions that follow Gaussian and Laplacian fits on the spatial domain for indoor corridor environment, respectively.

Study on probability distribution of electrified railway traction loads based on kernel density estimator via diffusion

The probabilistic modeling for traction load is one of the most basic and challenging work in the field of electrified railway. The improved diffusion-based kernel density estimator (DKDE) is used for the first time to establish the probability distribution of traction loads. Based on the diffusion partial differential equation of finite domain, DKDE can be obtained by discrete and inverse discrete cosine transform. The DKDE effectively accounts for both the optimal bandwidth selection and boundary correction. Based on the measured data (feeder currents and re/active power), four goodness-of-fit tests are applied to test the estimated probability distribution of traction loads. Compared with the parametric estimation models and Gaussian kernel density estimator (GKDE) respectively, the results show that this probability distribution of traction loads by DKDE is more accurate and suitable. Moreover, this DKDE has strong applicability and versatility for the random variation of different traction loads.

Saliency Preprocessing Locality-Constrained Linear Coding for Remote Sensing Scene Classification

Locality-constrained Linear Coding (LLC) shows superior image classification performance due to its underlying properties of local smooth sparsity and good construction. It encodes the visual features in remote sensing images and realizes the process of modeling human visual perception of an image through a computer. However, it ignores the consideration of saliency preprocessing in the human visual system. Saliency detection preprocessing can effectively enhance a computer’s perception of remote sensing images. To better implement the task of remote sensing image scene classification, this paper proposes a new approach by combining saliency detection preprocessing and LLC. This saliency detection preprocessing approach is realized using spatial pyramid Gaussian kernel density estimation. Experiments show that the proposed method achieved a better performance for remote sensing scene classification tasks.

An adaptive point-of-interest recommendation method for location-based social networks based on user activity and spatial features

Point-of-interest (POI) recommendations can help users effectively explore new locations according to their preferences, which is an important research aspect for location-based social networks (LBSNs). However, most existing POI recommendation methods lack adaptability when making recommendations for users with different preferences, which causes unsatisfactory recommendation results. To this end, in this paper, we propose an adaptive POI recommendation method by combining user activity and spatial features, which can operate adaptively according to user activity. First, we extract three-dimensional user activity, time-based POI popularity and distance features using a probabilistic statistical analysis method from historical check-in datasets on LBSNs. Second, we devise a user activity clustering algorithm that is based on fuzzy c-means and compute POI popularity by applying smoothing technology to adjacent continuous time slots. Finally, we propose an adaptive recommendation scheme, which includes a two-dimensional Gaussian kernel density estimation algorithm and a one-dimensional power-law function algorithm with POI popularity according to user activity. Extensive experiments on Foursquare and Gowalla datasets show that the proposed method outperforms the baseline methods in terms of both precision and recall.

New probabilistic price forecasting models: Application to the Iberian electricity market

This article presents original Probabilistic Price Forecasting Models, for day-ahead hourly price forecasts in electricity markets, based on a Nadaraya–Watson Kernel Density Estimator approach. A Gaussian Kernel Density Estimator function is used for each input variable, which allows to calculate the parameters of the probability density function (PDF) of a Beta distribution for the hourly price variable. Thus, valuable information is obtained from PDFs such as point forecasts, variance values, quantiles, probabilities of prices, and time series representations of forecast uncertainty. A Reliability Indicator is also introduced to give a measure of “reliability” of forecasts. The Probabilistic Price Forecasting Models were satisfactorily applied to the real-world case study of the Iberian Electricity Market. Input variables of these models include recent prices, power demands and power generations in the previous day, power demands in the previous week, forecasts of demand, wind power generation and weather for the day-ahead, and chronological data. The best model, corresponding to the best combination of input variables that achieves the lowest MAE, obtains one of the highest Reliability Indicator values. A systematic analysis of MAE values of the Probabilistic Price Forecasting Models for different combinations of input variables showed that as more types of input variables were considered in these models, MAE values improved and Reliability Indicator values usually increased.

Short-term power load probability density forecasting based on Yeo-Johnson transformation quantile regression and Gaussian kernel function

Penetration of renewable resources into power systems, such as wind and solar power, has significantly grown the complexity and level of uncertainty in both power generation and demand sides, which are highly desirable to exploit more advanced methods to address the uncertainties. The probability density forecasting method using quantile regression can describe probability distributions of future power load. However, existing quantile regression probability density forecasting methods may encounter embarrassing cross phenomenon, affecting the effectiveness of probability density forecasting. To avoid the crossing issue, this study proposes a probability density forecasting method based on Yeo-Johnson transformation quantile regression using Gaussian kernel function. Gaussian kernel density estimation using a rule of thumb bandwidth is innovatively hybridized with Yeo-Johnson transformation quantile regression for short-term power load probability density forecasting. The evaluation metrics for forecasting errors and prediction interval are adopted to carry out a comprehensive study on load uncertainty handling. One-hourly historical load data of August 2014 in summer and December 2014 in winter from Ottawa, Canada are used to evaluate the performance of proposed model. The results show that the proposed method not only efficiently avoids the quantile crossing problem but also obtains smooth probability density curves and more accurate forecasting results.

Geographical heterogeneity and inequality of access to improved drinking water supply and sanitation in Nepal

BackgroundPer United Nations’ Sustainable Development Goals, Nepal is aspiring to achieve universal and equitable access to safe and affordable drinking water and provide access to adequate and equitable sanitation for all by 2030. For these goals to be accomplished, it is important to understand the country’s geographical heterogeneity and inequality of access to its drinking-water supply and sanitation (WSS) so that resource allocation and disease control can be optimized. We aimed 1) to estimate spatial heterogeneity of access to improved WSS among the overall Nepalese population at a high resolution; 2) to explore inequality within and between relevant Nepalese administrative levels; and 3) to identify the specific administrative areas in greatest need of policy attention.MethodsWe extracted cluster-sample data on the use of the water supply and sanitation that included 10,826 surveyed households from the 2011 Nepal Demographic and Health Survey, then used a Gaussian kernel density estimation with adaptive bandwidths to estimate the distribution of access to improved WSS conditions over a grid at 1 × 1 km. The Gini coefficient was calculated for the measurement of inequality in the distribution of improved WSS; the Theil L measure and Theil T index were applied to account for the decomposition of inequality.Results57% of Nepalese had access to improved sanitation (range: 18.1% in Mahottari to 100% in Kathmandu) and 92% to drinking-water (range: 41.7% in Doti to 100% in Bara). The most unequal districts in Gini coefficient among improved sanitation were Saptari, Sindhuli, Banke, Bajura and Achham (range: 0.276 to 0.316); and Sankhuwasabha, Arghakhanchi, Gulmi, Bhojpur, Kathmandu (range: 0.110 to 0.137) among improved drinking-water. Both the Theil L and Theil T showed that within-province inequality was substantially greater than between-province inequality; while within-district inequality was less than between-district inequality. The inequality of several districts was higher than what is calculated by regression of the Gini coefficient and our estimates.ConclusionsThis study showed considerable geographical heterogeneity and inequality not evidenced in previous national statistics. Our findings may be useful in prioritizing resources to reduce inequality and expand the coverage of improved water supply and sanitation in Nepal.

Homogeneity quantification of nanoparticles dispersion in composite materials

As the quantitative evaluation of dispersion of nanoparticles are crucial for the research of nanocomposites, including ceramic, metal, and polymer nanocomposites, we proposed two quantifiable tools for the evaluation of dispersion in the nanocomposites by using machine‐learning algorithms, that is the coefficient of variation of K‐nearest neighboring distances (CVKD value) and the information entropy derived from the probability density function. In total 230 different type of dispersion morphologies of nanocomposites were investigated by using K‐nearest neighboring algorithms, Gaussian mixture model, expectation maximization, and Kernel density estimation algorithms. The smaller values of CVKD or information entropy represented much more homogeneous dispersion of nanoparticles. It might be expected the information exhibited in the present study could assist in the rational design and fabrication of superior nanocomposites. POLYM. COMPOS., 40:1000–1005, 2019. © 2018 Society of Plastics Engineers

Rigorous Analysis for Efficient Statistically Accurate Algorithms for Solving Fokker--Planck Equations in Large Dimensions

This article presents a rigorous analysis for efficient statistically accurate algorithms for solving the Fokker--Planck equations associated with high-dimensional nonlinear stochastic systems with conditional Gaussian structures. Despite the conditional Gaussianity, these nonlinear systems can contain strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy that requires only a small number of samples $L$ to capture both the transient and the equilibrium non-Gaussian PDFs with high accuracy. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious Gaussian kernel density estimation in the remaining low-dimensional subspace. Rigorous analysis shows that the mean integrated squared error in the recovered PDFs in the high-dimensional subspace is bounded by the inverse square root of the determinant of the conditional covariance, where th...

Beating the curse of dimension with accurate statistics for the Fokker–Planck equation in complex turbulent systems

Solving the Fokker-Planck equation for high-dimensional complex dynamical systems is an important issue. Recently, the authors developed efficient statistically accurate algorithms for solving the Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures, which contain many strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy with a small number of samples [Formula: see text], where a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious Gaussian kernel density estimation in the remaining low-dimensional subspace. In this article, two effective strategies are developed and incorporated into these algorithms. The first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions, which allows an extremely efficient parallel computation due to the small size of each individual block. The second strategy exploits statistical symmetry for a further reduction of [Formula: see text] The resulting algorithms can efficiently solve the Fokker-Planck equation with strongly non-Gaussian PDFs in much higher dimensions even with orders in the millions and thus beat the curse of dimension. The algorithms are applied to a [Formula: see text]-dimensional stochastic coupled FitzHugh-Nagumo model for excitable media. An accurate recovery of both the transient and equilibrium non-Gaussian PDFs requires only [Formula: see text] samples! In addition, the block decomposition facilitates the algorithms to efficiently capture the distinct non-Gaussian features at different locations in a [Formula: see text]-dimensional two-layer inhomogeneous Lorenz 96 model, using only [Formula: see text] samples.

Data-driven kernel representations for sampling with an unknown block dependence structure under correlation constraints

The multidimensional Gaussian kernel-density estimation (G-KDE) is a powerful tool to identify the distribution of random vectors when the maximal information is a set of independent realizations. For these methods, a key issue is the choice of the kernel and the optimization of the bandwidth matrix. To optimize these kernel representations, two adaptations of the classical G-KDE are presented. First, it is proposed to add constraints on the mean and the covariance matrix in the G-KDE formalism. Secondly, it is suggested to separate in different groups the components of the random vector of interest that could reasonably be considered as independent. This block by block decomposition is carried out by looking for the maximum of a cross-validation likelihood quantity that is associated with the block formation. This leads to a tensorized version of the classical G-KDE. Finally, it is shown on a series of examples how these two adaptations can improve the nonparametric representations of the densities of random vectors, especially when the number of available realizations is relatively low compared to their dimensions.

Efficient statistically accurate algorithms for the Fokker–Planck equation in large dimensions

Solving the Fokker–Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions (PDFs) of complex systems in turbulence, neuroscience and excitable media. In this article, efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of Fokker–Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. The algorithms involve a hybrid strategy that requires only a small number of ensembles. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace. Particularly, the parametric method provides closed analytical formulae for determining the conditional Gaussian distributions in the high-dimensional subspace and is therefore computationally efficient and accurate. The full non-Gaussian PDF of the system is then given by a Gaussian mixture. Different from traditional particle methods, each conditional Gaussian distribution here covers a significant portion of the high-dimensional PDF. Therefore a small number of ensembles is sufficient to recover the full PDF, which overcomes the curse of dimensionality. Notably, the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high-dimensional non-Gaussian PDFs, and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems. It is shown in a stringent set of test problems that the method only requires an order of O(100) ensembles to successfully recover the highly non-Gaussian transient PDFs in up to 6 dimensions with only small errors.

New approaches for delineating n‐dimensional hypervolumes

Abstract Hutchinson's n‐dimensional hypervolume concept underlies many applications in contemporary ecology and evolutionary biology. Estimating hypervolumes from sampled data has been an ongoing challenge due to conceptual and computational issues. We present new algorithms for delineating the boundaries and probability density within n‐dimensional hypervolumes. The methods produce smooth boundaries that can fit data either more loosely (Gaussian kernel density estimation) or more tightly (one‐classification via support vector machine). Further, the algorithms can accept abundance‐weighted data, and the resulting hypervolumes can be given a probabilistic interpretation and projected into geographic space. We demonstrate the properties of these methods on a large dataset that characterises the functional traits and geographic distribution of thousands of plants. The methods are available in version ≥2.0.7 of the hypervolume r package. These new algorithms provide: (i) a more robust approach for delineating the shape and density of n‐dimensional hypervolumes; (ii) more efficient performance on large and high‐dimensional datasets; and (iii) improved measures of functional diversity and environmental niche breadth.

RCMC

During the past few years, the analysis of data generated from Location-Based Social Networks (LBSNs) have aided in the identification of urban patterns, understanding activity behaviours in urban areas, as well as producing novel recommender systems that facilitate users’ choices. Recognizing crowd-mobility patterns in cities is very important for public safety, traffic managment, disaster management, and urban planning. In this article, we propose a framework for Recognizing the Crowd Mobility Patterns in Cities using LBSN data. Our proposed framework comprises four main components: data gathering, recurrent crowd-mobility patterns extraction, temporal functional regions detection, and visualization component. More specifically, we employ a novel approach based on Non-negative Matrix Factorization and Gaussian Kernel Density Estimation for extracting the recurrent crowd-mobility patterns in cities illustrating how crowd shifts from one area to another during each day across various time slots. Moreover, the framework employs a hierarchical clustering-based algorithm for identifying what we refer to as temporal functional regions by modeling functional areas taking into account temporal variation by means of check-ins’ categories. We build the framework using a spatial-temporal dataset crawled from Twitter for two entire years (2013 and 2014) for the area of Manhattan in New York City. We perform a detailed analysis of the extracted crowd patterns with an exploratory visualization showing that our proposed approach can identify clearly obvious mobility patterns that recur over time and location in the urban scenario. Using same time interval, we show that correlating the temporal functional regions with the recognized recurrent crowd-mobility patterns can yield to a deeper understanding of city dynamics and the motivation behind the crowd mobility. We are confident that our proposed framework not only can help in managing complex city environments and better allocation of resources based on the expected crowd mobility and temporal functional regions but also can have a direct implication on a variety of applications such as personalized recommender systems, anomalous event detection, disaster resilience management systems, and others.