TimeCluster用PCA等价于线性系统子空间识别，可直接提取动态系统状态空间。
TimeCluster with PCA is Equivalent to Subspace Identification of Linear Dynamical Systems
- 用滑动窗口构造汉克尔矩阵，再通过PCA（SVD）提取低维子空间
- 与经典子空间识别方法得出的主方向完全一致，嵌入结果相同
- 适合做时序系统建模、在线分析及噪声数据中的趋势可视化
TimeCluster是一种用于发现长多变量时间序列结构的可视化分析技术，通过将数据的重叠窗口投影到低维空间实现。本文证明，当使用主成分分析（PCA）作为降维手段时，该过程在数学上等价于经典的线性子空间识别方法（即块汉克尔矩阵加奇异值分解，SVD）。两种方法从时间序列中提取出相同的低维线性子空间。我们首先回顾TimeCluster方法和子空间系统识别理论，接着说明时间序列的滑动窗口矩阵构成汉克尔矩阵，对它进行PCA（通过SVD）即可恢复与子空间识别相同的主方向。因此，TimeCluster的聚类坐标与子空间识别方法一致。我们在合成与真实动态信号上进行了实验，验证了两种嵌入结果的一致性。最后，探讨了这一等价性带来的新机遇，包括基于识别状态空间的预测、流式/在线扩展、引入外部输入的可视化以及在污染数据中显示底层趋势的鲁棒方法。
原文摘要 · Abstract (English)
TimeCluster is a visual analytics technique for discovering structure in long multivariate time series by projecting overlapping windows of data into a low-dimensional space. We show that, when Principal Component Analysis (PCA) is chosen as the dimensionality reduction technique, this procedure is mathematically equivalent to classical linear subspace identification (block-Hankel matrix plus Singular Vector Decomposition (SVD)). In both approaches, the same low-dimensional linear subspace is extracted from the time series data. We first review the TimeCluster method and the theory of subspace system identification. Then we show that forming the sliding-window matrix of a time series yields a Hankel matrix, so applying PCA (via SVD) to this matrix recovers the same principal directions as subspace identification. Thus the cluster coordinates from TimeCluster coincide with the subspace identification methods. We present experiments on synthetic and real dynamical signals confirming that the two embeddings coincide. Finally, we explore and discuss future opportunities enabled by this equivalence, including forecasting from the identified state space, streaming/online extensions, incorporating and visualising external inputs and robust techniques for displaying underlying trends in corrupted data.
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