Purpose:
characterize the evolution of dynamical systems. In this paper, a novel method based on epsilon-recurrence networks is proposed for the study of the evolution properties of dynamical systems.
Methodology:
1. convert time series to a high-dimensional recurrence network and a corresponding low-dimensional recurrence network.
network dimension L represents the number of state vectors that form a node in the network.
phase space reconstruction based on Takens‘ embedding theorem. -----------> a series of state vectors R1, R2, ..., Rn` can be obtained. ----------------> construct a high dimensional recurrence network (RN) and a low dimensional RN. 每个结点代表着a segment of the phase space trajectory. distance matrix Dl between nodes can be obtained by equation 2, which reflects the distance between segments of the phase space trajectories. ---------> obtain the adjacency matrix.
The construction of the network is highly dependent on the threshold, ? , which should be tailored to specific questions that need to be solved. --------------> choose a fixed link density.
Therefore, the similarity between the two networks can reflect the evolution properties of the studied dynamical systems. ?why?
?? 结果不明白.
Basic knowledge:
1. phase space 相空间重构
如果把一个时间序列看成是由一个确定性的非线性动力系统产生的, 要考虑的是以下反问题: 如何有时间序列来恢复并刻画原动力系统.
The fundamental starting point of many approaches in nonlinear data analysis is the construction of a phase space portrait of the considered system. The state of a system can be described by its state variables $x^1(t), x^2(t), ... ,x^d(t)$, for example the both state variables temperature and pressure for a thermodynamic system. The d state variables at time t form a vector in a d-dimensional space which is called phase space. The state of a system typically changes in time, and, hence, the vector in the phase space describes a trajectory representing the time evolution, the dynamics, of the system. The shape of the trajectory gives hints about the system; periodic or chaotic systems have characteristic phase space portraits.
The observation of a real process usually does not yield all possible state variables. Either not all state variables are known or not all of them can be measured. However, due to the couplings between the system‘s components, we can reconstruct a phase space trajectory from a single observation u_i by a time delay embedding (Takens, 1981): 由时间序列恢复原系统最常用的方法是利用Takens的延迟嵌入定理.
where $m$ is the embedding dimension and $\tau$ is the time delay (index based; the real time delay is $\tau\,\Delta t$). This reconstruction of the phase space is called time delay embedding. The phase space reconstruction is not exactly the same to the original phase space, but its topological properties are preserved, if the embedding dimension is large enough (the embedding dimension has to be larger then twice the phase space dimension, or exactly m > 2 d + 1). And this reconstructed trajectory is sufficient enough for a subsequent analysis.
Now we look at the phase space portrait of an harmonic oscillation, like an undamped pendulum. First we create the position vector y1 and the velocity vector y2
x = 0 : pi/10 : 6 * pi; y1 = sin(x); y2 = cos(x);
The phase space portrait
plot(y1, y2)
xlabel(‘y_1‘), ylabel(‘y_2‘)
2. 非线性时间序列预测.
基本方法:
局域预测法: 局部平均预测法, 局部线性预测法,局部多项式预测法.
全局预测法: 神经网络, 小波网络, 遗传算法.