Optimal [formula omitted] bandwidth selection for variable kernel density estimates

We present several multivariate histogram density estimates that are universallyL 1-optimal to within a constant factor and an additive term\(O\left( {\sqrt {\log {n \mathord{\left/ {\vphantom {n n}} \right. \kern-\nulldelimiterspace} n}} } \right)\). The bin widths are chosen by the combinatorial method developed by the authors inCombinatorial Methods in Density Estimation (Springer-Verlag, 2001). The present paper solves a problem left open in that book.

Methods of density estimation on the Grassmann manifold

Density estimation by the penalized combinatorial method

For eight years (2007–2014), Greeneridge Sciences deployed, collected, and analyzed an immense passive acoustic dataset to study the response of bowhead whales (Balaena mysticetus) to oil exploration activities along a 280 km swath of the Alaskan Arctic continental shelf. Each year, up to 40 Directional Autonomous Seafloor Acoustic Recorders (DASARs) detected bowhead calls during the whales’ annual fall migration, resulting in over 2.4 million localized calls over the study period. These enormous sample sizes, combined with the unique localization capability of the DASARs, have enabled the investigation and cross-validation of several acoustic density estimation (DE) methods. Here, we compared results among three density estimation methods: (i) direct census, (ii) distance sampling, and (iii) spatially explicit capture recapture (SECR). In the course of our investigation, we encountered and addressed two challenges: (i) how to apply point transect theory to distributed array systems, and (ii) how to implement SECR in situations where detections between sensors are not statistically independent or where false positives exist on single sensors. These density estimates were then used to estimate relative abundance of the Western Arctic bowhead whale population over multiple years. [Work sponsored by the Joint Industry Programme.]

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

Kernel density estimation is a widely used statistical tool and bandwidth selection is critically important. The Sheather and Jones’ (SJ) selector [A reliable data-based bandwidth selection method for kernel density estimation, J. R. Stat. Soc. Ser. B 53 (1991), pp. 683–690] remains the best available data-driven bandwidth selector. It can, however, perform poorly if the true density deviates too much in shape from normal. This paper first develops an alternative selector by following ideas in Park and Marron [On the use of pilot estimators in bandwidth selection, Nonparametr. Stat. 1 (1992), pp. 231–240] to reduce the impact of the normal reference density. The selector can bring drastic improvement to less smooth densities that the SJ selector has difficulty with, but may be slightly worse off otherwise. We then propose to combine the alternative selector and SJ selector by using the smaller of the two bandwidths, which has the effect of automatically picking the better one for a particular density. In our extensive simulation, study using the 15 benchmark densities in Marron and Wand [Exact mean integrated squared error, Ann. Statist. 20 (1992), pp. 712–736], the combined selector significantly improves the SJ selector for 5 difficult densities and retains the superior performance of the SJ selector for the other 10. A ready-to-use R function is provided.

Nonparametric density estimation is a fundamental problem of statistics and data mining. Even though kernel density estimation is the most widely used method, its performance highly depends on the choice of the kernel bandwidth, and it can become computationally expensive for large data sets. We present an adaptive sparse-grid-based density estimation method which discretizes the estimated density function on basis functions centered at grid points rather than on kernels centered at the data points. Thus, the costs of evaluating the estimated density function are independent from the number of data points. We give details on how to estimate density functions on sparse grids and develop a cross validation technique for the parameter selection. We show numerical results to confirm that our sparse-grid-based method is well-suited for large data sets, and, finally, employ our method for the classification of astronomical objects to demonstrate that it is competitive to current kernel-based density estimation approaches with respect to classification accuracy and runtime.

This study compared the results from three freeway density estimation methods based on point detector data with the results obtained from the density estimation procedure of the Highway Capacity Manual 2010 (HCM 2010). The three methods were the cumulative volume–based method, the occupancy-based method, and the fundamental relationship–based method. The study also developed and tested a new method that integrated data from point traffic detectors and automatic vehicle identification readers to estimate the density of freeway segments for offline and real-time applications. The four density estimation methods were compared with each other and the HCM 2010 method by using two case studies based on simulation modeling and real-world data. Results showed that the density estimates based on the proposed segmentation method, cumulative volume method, and HCM 2010 method were generally closer to each other compared with the estimates based on the other two tested methods. The simulation case study showed that the density estimates from these three methods were also closer to density measurements obtained on the basis of vehicle trajectories from simulation. The two case study results indicate that the selection of the density estimation method affects mainly the level-of-service value during intermediate congested conditions.

Kernel density estimation (KDE) is one of the most widely used nonparametric density estimation methods. The fact that it is a memory-based method, i.e., it uses the entire training data set for prediction, makes it unsuitable for most current big data applications. Several strategies, such as tree-based or hashing-based estimators, have been proposed to improve the efficiency of the kernel density estimation method. The novel density kernel density estimation method (DMKDE) uses density matrices, a quantum mechanical formalism, and random Fourier features, an explicit kernel approximation, to produce density estimates. This method has its roots in the KDE and can be considered as an approximation method, without its memory-based restriction. In this paper, we systematically evaluate the novel DMKDE algorithm and compare it with other state-of-the-art fast procedures for approximating the kernel density estimation method on different synthetic data sets. Our experimental results show that DMKDE is on par with its competitors for computing density estimates and advantages are shown when performed on high-dimensional data. We have made all the code available as an open source software repository.

The elements that constitute the literary space can mainly be described by categories such as settings, zones where actions take place and routes along which characters move. Apart from the presentation of the specific locations and spatial distribution of literary places, we are also interested in the spatial pattern such places form, the boundaries that separate the literarily populated from the void regions and the varying density of the literary space. From a GIS point of view, the elements of the literary space correspond to point, line and area data types. However, the established method used to calculate the spatially varying density – the method of density estimation – is restricted to point data only. In this paper, we will present an improved method that is able to estimate the density regardless of the underlying data type. Our approach aims at adopting the typically radially symmetric kernel function to approximate also linear and areal features. We claim that this method treats point, line and area data in a consistent way by taking equivalent density contributions into account. The different steps of the improved method can visually be examined by the accompanying map examples.

We introduce a recursive method for estimating a probability density subject to constraints of unimodality or monotonicity. It uses an empirical estimate of the probability transform to construct a sequence of maps of a known template, which satisfies the constraints. The algorithm may be employed without a smoothing step, in which case it produces step-function approximations to the sampling density. More satisfactorily, a certain amount of smoothing may be interleaved between each recursion, in which case the estimate is smooth. The amount of smoothing may be chosen using a standard cross-validation algorithm. Unlike other methods for density estimation, however, the recursive approach is robust against variation of the amount of smoothing, and so choice of bandwidth is not critical.

We propose a method for nonparametric density estimation that exhibits robustness to contamination of the training sample. This method achieves robustness by combining a traditional kernel density estimator (KDE) with ideas from classical M-estimation. We interpret the KDE based on a positive semi-definite kernel as a sample mean in the associated reproducing kernel Hilbert space. Since the sample mean is sensitive to outliers, we estimate it robustly via M-estimation, yielding a robust kernel density estimator (RKDE). An RKDE can be computed efficiently via a kernelized iteratively re-weighted least squares (IRWLS) algorithm. Necessary and sufficient conditions are given for kernelized IRWLS to converge to the global minimizer of the M-estimator objective function. The robustness of the RKDE is demonstrated with a representer theorem, the influence function, and experimental results for density estimation and anomaly detection.

This work reviews the literature on spline local basis methods for non-parametric density estimation. Particular attention is paid to B-spline density estimators which have experienced recent advances in both theory and methodology. These estimators occupy a very interesting space in statistics, which lies aptly at the cross-section of numerous statistical frameworks. New insights, experiments, and analyses are presented to cast the various estimation concepts in a unified context, while parallels and contrasts are drawn to the more familiar contexts of kernel density estimation. Unlike kernel density estimation, the study of local basis estimation is not yet fully mature, and this work also aims to highlight the gaps in existing literature which merit further investigation.

In the linear model y = x′β + u where x is a regressor vector and E(u | x)= 0, we estimate β in E(y | x)= x′β However, the assumption of the linear model, or any nonlinear model for that matter, is a strong one. In nonparametric regression, we try to estimate E(y | x) without specifying the functional form. Since $$E(y{\kern 1pt} |{\kern 1pt} x) = \smallint y \cdot f(y|x)dy = \smallint y \cdot \{ f(y,x)/f(x)\} dy $$ (1.1) if we can estimate f(y, x) and f(x), we can also estimate E(y | x). In this chapter, we study nonparametric density estimation for x as a prelude to nonparametric regression in the next chapter. We will assume that x has a continuous density function f (x) If x is discrete, one can estimate P(x = x 0) either with the same estimation method used for the continuous case or with the number of observations with x i = x 0. There are several nonparametric density estimation methods available. The most popular is “kernel density estimation method” which we explore mainly. Other methods will be examined briefly in the last section. See Prakasa Rao (1983), Silverman (1986), Izenman (1991), Rosenblatt (1991) and Scott (1992) for more on nonparametric density estimation in general.

Bandwidth selection is the main problem of kernel density estimation, the most popular method of density estimation. The classical normal reference bandwidth usually oversmoothes the density estimate. The existing hi‐tech bandwidths have computational problems (even may not exist) and are not robust against outliers in the sample. A highly robust normal reference bandwidth is proposed, which adapts to different types of densities.