基于符号分析的时间序列数据降阶马尔可夫建模
Symbolic Analysis-based Reduced Order Markov Modeling of Time Series Data
摘要：
本文提出了一种用于时间序列数据紧凑表示的降阶马尔可夫建模技术。在这项工作中，基于符号动力学的工具已被用来推断一个近似的生成马尔可夫模型。首先通过划分信号的连续测量空间来符号化时间序列数据，然后使用符号动力学对离散序列数据进行建模。在所提出的方法中，时间记忆的符号序列的大小估计从所得到的随机阵对应于一阶马尔可夫模型的符号序列的频谱特性。然后，层次聚类被用来表示相应的全状态马尔可夫模型的状态，以构建具有非确定性代数结构的降阶或大小马尔可夫模型。随后，通过利用贝叶斯推断规则从原始模型识别降阶马尔可夫模型的参数。最终的模型是使用信息理论的标准选择。作为示例，在两个不同的数据集上阐述和验证了所提出的概念。第一个例子分析了一组压力数据从旋流稳定燃烧室，其中控制协议用于诱导火焰不稳定性。推导出的马尔可夫模型的复杂性的变化表示系统的操作条件如何从稳定的燃烧状态变化到不稳定的燃烧状态。在第二个例子中，数据集是从美国宇航局的旋转轴上的轴承特性的数据库。我们表明，即使在一个非常小的状态空间，降阶模型能够实现相当的性能，所提出的方法提供了灵活性，在选择一个最终的模型表示和学习。
英文摘要：
This paper presents a technique for reduced-order Markov modeling for compact representation of time-series data. In this work, symbolic dynamics-based tools have been used to infer an approximate generative Markov model. The time-series data are first symbolized by partitioning the continuous measurement space of the signal and then, the discrete sequential data are modeled using symbolic dynamics. In the proposed approach, the size of temporal memory of the symbol sequence is estimated from spectral properties of the resulting stochastic matrix corresponding to a first-order Markov model of the symbol sequence. Then, hierarchical clustering is used to represent the states of the corresponding full-state Markov model to construct a reduced-order or size Markov model with a non-deterministic algebraic structure. Subsequently, the parameters of the reduced-order Markov model are identified from the original model by making use of a Bayesian inference rule. The final model is selected using information-theoretic criteria. The proposed concept is elucidated and validated on two different data sets as examples. The first example analyzes a set of pressure data from a swirl-stabilized combustor, where controlled protocols are used to induce flame instabilities. Variations in the complexity of the derived Markov model represent how the system operating condition changes from a stable to an unstable combustion regime. In the second example, the data set is taken from NASA's data repository for prognostics of bearings on rotating shafts. We show that, even with a very small state-space, the reduced-order models are able to achieve comparable performance and that the proposed approach provides flexibility in the selection of a final model for representation and learning.
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