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.

中文翻译：

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随时间变化的流数据上基于Parzen核的概率密度函数学习程序及其在模式分类中的应用

在本文中，我们提出了Parzen核密度估计器（KDE）的递归变量，以跟踪非平稳环境中数据流上动态密度的变化。在固定环境中，完善的传统KDE技术具有良好的渐近特性。它们用于处理流数据的现有扩展主要基于各种启发式概念（失去收敛性）。在本文中，我们研究了称为递归概念漂移跟踪KDE的递归KDE，并证明了它们的弱（概率）和强（概率为1）收敛，随着样本大小接近无穷大，可以实现完美的跟踪特性。在三个定理和随后的示例中，我们展示了如何选择递归KDE的带宽和学习率，以确保弱收敛和强收敛。