Embedding time series data in Euclidean space from DTW distances

核方法的优点之一是它们可以处理各种类型的对象，而不一定是具有固定数量属性的矢量数据。在本文中，我们开发了使用动态时间规整（DTW）距离的时间序列数据核。由于DTW距离是不满足三角不等式的伪距离，因此基于它们的核矩阵通常不是正半定的。利用半定规划方法来保证核矩阵的正定性。我们提出了邻域保持嵌入（NPE），一种SDP公式来获得最能保留时间序列数据局部几何形状的核矩阵。我们还提出了NPE的样本外扩展（OSE）。我们使用时间序列分类和时间序列嵌入两个应用来验证我们的方法。

One of the advantages of the kernel methods is that they can deal with various kinds of objects, not necessarily vectorial data with a fixed number of attributes.In this paper, we develop kernels for time series data using dynamic time warping (DTW) distances. Since DTW distances are pseudo distances that do not satisfy the triangle inequality, a kernel matrix based on them is not positive semidefinite, in general. We use semidefinite programming (SDP) to guarantee the positive definiteness of a kernel matrix. We present neighborhood preserving embedding (NPE), an SDP formulation to obtain a kernel matrix that best preserves the local geometry of time series data. We also present an out-of-sample extension (OSE) for NPE.We use two applications, time series classification and time series embedding for similarity search to validate our approach.