SOMKE: kernel density estimation over data streams by sequences of self-organizing maps

Multivariate Density Estimation by Neural Networks

We propose nonparametric methods to obtain the Probability Density Function (PDF) to assess the properties of the underlying data generating process (DGP) without imposing any assumptions on the DGP, using neural networks (NNs). The proposed NN has advantages compared to well-known parametric and nonparametric density estimators. Our approach builds on literature on cumulative distribution function (CDF) estimation using NN. We extend this literature by providing analytical derivatives of this obtained CDF. Our approach hence removes the numerical approximation error in differentiating the CDF output, leading to more accurate PDF estimates. The proposed solution applies to any NN model, i.e., for any number of hidden layers or hidden neurons in the multilayer perceptron (MLP) structure. The proposed solution applies the PDF estimation by NN to continuous distributions as well as discrete distributions. We also show that the proposed solution to obtain the PDF leads to good approximations when applied to correlated variables in a multivariate setting. We test the performance of our method in a large Monte Carlo simulation using various complex distributions. Subsequently, we apply our method to estimate the density of the number of vehicle counts per minute measured with road sensors for a time window of 24 h.

Granular Ball Sampling for Noisy Label Classification or Imbalanced Classification

This article presents a general sampling method, called granular-ball sampling (GBS), for classification problems by introducing the idea of granular computing. The GBS method uses some adaptively generated hyperballs to cover the data space, and the points on the hyperballs constitute the sampled data. GBS is the first sampling method that not only reduces the data size but also improves the data quality in noisy label classification. In addition, because the GBS method can be used to exactly describe the boundary, it can obtain almost the same classification accuracy as the results on the original datasets, and it can obtain an obviously higher classification accuracy than random sampling. Therefore, for the data reduction classification task, GBS is a general method that is not especially restricted by any specific classifier or dataset. Moreover, the GBS can be effectively used as an undersampling method for imbalanced classification. It has a time complexity that is close to O( N ), so it can accelerate most classifiers. These advantages make GBS powerful for improving the performance of classifiers. All codes have been released in the open source GBS library at http://www.cquptshuyinxia.com/GBS.html.

Adaptive Robust Local Online Density Estimation for Streaming Data

Accurate online density estimation is crucial to numerous applications that are prevalent with streaming data. Existing online approaches for density estimation somewhat lack prompt adaptability and robustness when facing concept-drifting and noisy streaming data, resulting in delayed or even deteriorated approximations. To alleviate this issue, in this work, we first propose an adaptive local online kernel density estimator (ALoKDE) for real-time density estimation on data streams. ALoKDE consists of two tightly integrated strategies: (1) a statistical test for concept drift detection and (2) an adaptive weighted local online density estimation when a drift does occur. Specifically, using a weighted form, ALoKDE seeks to provide an unbiased estimation by factoring in the statistical hallmarks of the latest learned distribution and any potential distributional changes that could be introduced by each incoming instance. A robust variant of ALoKDE, i.e., R-ALoKDE, is further developed to effectively handle data streams with varied types/levels of noise. Moreover, we analyze the asymptotic properties of ALoKDE and R-ALoKDE, and also derive their theoretical error bounds regarding bias, variance, MSE and MISE. Extensive comparative studies on various artificial and real-world (noisy) streaming data demonstrate the efficacies of ALoKDE and R-ALoKDE in online density estimation and real-time classification (with noise).

On the Parzen Kernel-Based Probability Density Function Learning Procedures Over Time-Varying Streaming Data With Applications to Pattern Classification

In this paper, we propose a recursive variant of the Parzen kernel density estimator (KDE) to track changes of dynamic density over data streams in a nonstationary environment. In stationary environments, well-established traditional KDE techniques have nice asymptotic properties. Their existing extensions to deal with stream data are mostly based on various heuristic concepts (losing convergence properties). In this paper, we study recursive KDEs, called recursive concept drift tracking KDEs, and prove their weak (in probability) and strong (with probability one) convergence, resulting in perfect tracking properties as the sample size approaches infinity. In three theorems and subsequent examples, we show how to choose the bandwidth and learning rate of a recursive KDE in order to ensure weak and strong convergence. The simulation results illustrate the effectiveness of our algorithm both for density estimation and classification over time-varying stream data.

Online Density Estimation of Nonstationary Sources Using Exponential Family of Distributions

We investigate online probability density estimation (or learning) of nonstationary (and memoryless) sources using exponential family of distributions. To this end, we introduce a truly sequential algorithm that achieves Hannan-consistent log-loss regret performance against true probability distribution without requiring any information about the observation sequence (e.g., the time horizon $T$ and the drift of the underlying distribution $C$ ) to optimize its parameters. Our results are guaranteed to hold in an individual sequence manner. Our log-loss performance with respect to the true probability density has regret bounds of $O(({CT})^{1/2})$ , where $C$ is the total change (drift) in the natural parameters of the underlying distribution. To achieve this, we design a variety of probability density estimators with exponentially quantized learning rates and merge them with a mixture-of-experts notion. Hence, we achieve this square-root regret with computational complexity only logarithmic in the time horizon. Thus, our algorithm can be efficiently used in big data applications. Apart from the regret bounds, through synthetic and real-life experiments, we demonstrate substantial performance gains with respect to the state-of-the-art probability density estimation algorithms in the literature.

Topology-based clustering using polar self-organizing map

Cluster analysis of unlabeled data sets has been recognized as a key research topic in varieties of fields. In many practical cases, no a priori knowledge is specified, for example, the number of clusters is unknown. In this paper, grid clustering based on the polar self-organizing map (PolSOM) is developed to automatically identify the optimal number of partitions. The data topology consisting of both the distance and density is exploited in the grid clustering. The proposed clustering method also provides a visual representation as PolSOM allows the characteristics of clusters to be presented as a 2-D polar map in terms of the data feature and value. Experimental studies on synthetic and real data sets demonstrate that the proposed algorithm provides higher clustering accuracy and lower computational cost compared with six conventional methods.

Confabulation-inspired association rule mining for rare and frequent itemsets

A new confabulation-inspired association rule mining (CARM) algorithm is proposed using an interestingness measure inspired by cogency. Cogency is only computed based on pairwise item conditional probability, so the proposed algorithm mines association rules by only one pass through the file. The proposed algorithm is also more efficient for dealing with infrequent items due to its cogency-inspired approach. The problem of associative classification is used here for evaluating the proposed algorithm. We evaluate CARM over both synthetic and real benchmark data sets obtained from the UC Irvine machine learning repository. Experiments show that the proposed algorithm is consistently faster due to its one time file access and consumes less memory space than the Conditional Frequent Patterns growth algorithm. In addition, statistical analysis reveals the superiority of the approach for classifying minority classes in unbalanced data sets.

Reacting to different types of concept drift: the Accuracy Updated Ensemble algorithm

Data stream mining has been receiving increased attention due to its presence in a wide range of applications, such as sensor networks, banking, and telecommunication. One of the most important challenges in learning from data streams is reacting to concept drift, i.e., unforeseen changes of the stream's underlying data distribution. Several classification algorithms that cope with concept drift have been put forward, however, most of them specialize in one type of change. In this paper, we propose a new data stream classifier, called the Accuracy Updated Ensemble (AUE2), which aims at reacting equally well to different types of drift. AUE2 combines accuracy-based weighting mechanisms known from block-based ensembles with the incremental nature of Hoeffding Trees. The proposed algorithm is experimentally compared with 11 state-of-the-art stream methods, including single classifiers, block-based and online ensembles, and hybrid approaches in different drift scenarios. Out of all the compared algorithms, AUE2 provided best average classification accuracy while proving to be less memory consuming than other ensemble approaches. Experimental results show that AUE2 can be considered suitable for scenarios, involving many types of drift as well as static environments.