pyActigraphy.analysis.SSA.trajectory_matrix

SSA.trajectory_matrix()

Trajectory matrix of the signal. Time-series \(x=(x_0,x_1,\dots,x_n,\dots,x_{N−1})^\intercal\), with length N, represents the signal under analysis. The mapping of this signal into a matrix A, of dimension L × K , assuming \(L \leq K\), is called immersion, and can be defined as:

\[\begin{split}A = \begin{bmatrix} x_{0} & x_{1} & x_{2} & \dots & x_{K-1} \\ x_{1} & x_{2} & x_{3} & \dots & x_{K} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{L-1} & x_{L} & x_{L+1} & \dots & x_{N-1} \end{bmatrix}\end{split}\]

where L is the window length, or embedding dimension, and \(K = N − L + 1\). A is a Hankel matrix, called the trajectory matrix.