非平稳过程的伸缩算子和诱导形空间流形表示与刻画
Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds
摘要：
我们通过由膨胀引起的几何引入了随机过程的新观点。给定过程的伸缩矩阵是由旋转矩阵合成得到的，它以一种压缩的方式包含了过程的测度信息。特别有趣的是，膨胀矩阵的获得与底层过程的平稳性无关。当过程是平稳的，它符合Naimark膨胀，只有一个旋转矩阵的计算，当过程是非平稳的，一组旋转矩阵的计算。特别地，可能出现在某些类信号中的相关函数的周期性被传输到扩张矩阵集合。这些旋转矩阵可以根据信号的采样或重新缩放而任意地彼此接近，被视为信号的区别特征。为了研究这个矩阵序列，并且在重新缩放信号的可能性的指导下，与膨胀的理论结果一起使用的正确几何框架是流形上的曲线空间，即位于基流形上的所有曲线的集合。给出了曲线空间的一个度量，并导出了曲线空间的测地线方程。一般结果适用于基流形是旋转矩阵的李群的更具体的情况。曲线的形状的概念可以形式化为由曲线空间的商空间和递增的双同态给出的曲线的等价类的集合。曲线空间中的度量自然地延伸到形状空间，并使形状之间的比较成为可能。
英文摘要：
We have introduce a new vision of stochastic processes through the geometry induced by the dilation. The dilation matrices of a given processes are obtained by a composition of rotations matrices, contain the measure information in a condensed way. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, it coincides with the Naimark Dilation and only one rotation matrix is computed, when the process is non-stationary, a set of rotation matrices are computed. In particular, the periodicity of the correlation function that may appear in some classes of signal is transmitted to the set of dilation matrices. These rotation matrices, which can be arbitrarily close to each other depending on the sampling or the rescaling of the signal are seen as a distinctive feature of the signal. In order to study this sequence of matrices, and guided by the possibility to rescale the signal, the correct geometrical framework to use with the dilation's theoretic results is the space of curves on manifolds, that is the set of all curve that lies on a base manifold. To give a complete sight about the space of curve, a metric and the derived geodesic equation are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices. The notion of the shape of a curve can be formalized as the set of equivalence classes of curves given by the quotient space of the space of curves and the increasing diffeomorphisms. The metric in the space of curve naturally extent to the space of shapes and enable comparison between shapes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
