Gradual drifts in data streams are usually hard to detect and often do not necessarily trigger the evolution of new fuzzy rules during model adaptation steps in order to represent the new, drifted data distribution(s) appropriately in the fuzzy model. Over time, they thus lead to oversized rules with untypically large local errors (typically also worsening the global model error), as representing joint local data distributions before and after a drift happened likewise. We therefore propose an incremental rule splitting concept for generalized fuzzy rules in order to autonomously compensate these negative effects of gradual drifts. Our splitting condition is based on the local error of rules measured in terms of a weighted contribution to the whole model error and on the size of the rules measured in terms of the volume of the associated clusters. We use the concept of statistical process control in order to omit an extra threshold parameter in our splitting condition. The splitting technique relies on the eigendecomposition of the rule covariance matrix to adequately manipulate the largest eigenvector and eigenvalues in order to retrieve the new centers and contours of the two split rules. Furthermore, we guarantee sufficient flexibility in adapting the shapes and consequents of the split rules to the new drifted situation in the stream by integrating a specific dynamic and smooth forgetting concept of older samples, which formed the original (nonsplit) rules. Robustness against outliers is guaranteed by the realization of a two-layer model building process, where one layer represents the cluster partition and the other layer the rule partition: Only clusters becoming significant over time are accepted as rules in the fuzzy model. The splitting concepts are integrated in the generalized smart evolving learning engine for fuzzy systems (termed as Gen-Smart-EFS) and successfully tested on two real-world application scenarios, engine test benches and rolling mills, the latter including a real-occurring gradual drift (whose position in the data is known). Results show clearly improved error trend lines over time when splitting is applied, compared to the case when it is not applied: reduction of the mean absolute model error by about one third (rolling mills) and about one half (engine test benches).

中文翻译：

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用于自主漂移补偿的广义演化模糊系统中的增量规则分裂

数据流中的逐渐漂移通常难以检测，并且通常不一定会在模型适应步骤期间触发新模糊规则的演变，以便在模糊模型中适当地表示新的漂移数据分布。随着时间的推移，它们因此导致具有异常大的局部错误（通常也会使全局模型错误恶化）的过大规则，因为表示同样发生漂移前后的联合局部数据分布。因此，我们提出了广义模糊规则的增量规则分裂概念，以便自主补偿逐渐漂移的这些负面影响。我们的分裂条件是基于规则的局部误差（以对整个模型误差的加权贡献来衡量）和规则的大小（以相关集群的体积衡量）。我们使用统计过程控制的概念来在我们的分裂条件中省略额外的阈值参数。分裂技术依赖于规则协方差矩阵的特征分解来充分操纵最大的特征向量和特征值，以便检索两个分裂规则的新中心和轮廓。此外，我们通过整合旧样本的特定动态和平滑遗忘概念来保证在使拆分规则的形状和结果适应流中新的漂移情况方面具有足够的灵活性，这些概念形成了原始（非拆分）规则。通过实现两层模型构建过程来保证对异常值的鲁棒性，其中一层表示集群分区，另一层表示规则分区：只有随着时间的推移变得重要的集群被接受为模糊模型中的规则。分裂概念被集成到模糊系统的广义智能进化学习引擎（称为 Gen-Smart-EFS）中，并在两个实际应用场景中成功测试，发动机测试台和轧机，后者包括一个真实发生的渐进式漂移（其在数据中的位置已知）。结果显示，与不应用拆分的情况相比，应用拆分时随时间推移的误差趋势线明显改善：平均绝对模型误差减少约三分之一（轧机）和约二分之一（发动机测试台）。分裂概念被集成到模糊系统的广义智能进化学习引擎（称为 Gen-Smart-EFS）中，并在两个实际应用场景中成功测试，发动机测试台和轧机，后者包括一个真实发生的渐进式漂移（其在数据中的位置已知）。结果显示，与不应用拆分的情况相比，应用拆分时随时间推移的误差趋势线明显改善：平均绝对模型误差减少约三分之一（轧机）和约二分之一（发动机测试台）。分裂概念被集成到模糊系统的广义智能进化学习引擎（称为 Gen-Smart-EFS）中，并在两个实际应用场景中成功测试，发动机测试台和轧机，后者包括一个真实发生的渐进式漂移（其在数据中的位置已知）。结果显示，与不应用拆分的情况相比，应用拆分时随时间推移的误差趋势线明显改善：平均绝对模型误差减少约三分之一（轧机）和约二分之一（发动机测试台）。