# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Essentials of Chaos Theory
#### 2.3.5 Chaotic Time Series Analysis
Since the 1980s, a number of excellent methods have been proposed to optimally determine \(m\) and \(\tau\). Below we describe two approaches, which have been extensively tested and are very systematic. **(1) False nearest-neighbor method**: This is a geometrical method. Consider the situation in which an \(m_{0}\)-dimensional delay reconstruction is embedded but an \((m_{0}-1)\)-dimensional reconstruction is not. Passing from \(m_{0}-1\) to \(m_{0}\), self-intersection in the reconstructed trajectory is eliminated. This feature can be quantified by the sharp decrease in the number of nearest neighbors when \(m\) is increased from \(m_{0}-1\) to \(m_{0}\). Therefore, the optimal value of \(m\) is \(m_{0}\). More precisely, for each reconstructed vector \(V_{i}^{(m)}=[x(t_{i}),x(t_{i}+\tau),x(t_{i}+2\tau),\cdots,x(t_{i}+(m-1)\tau)]\), its nearest neighbor \(V_{j}^{(m)}\) is found (to ensure unambiguity, here the superscript \((m)\) is used to emphasize that this is an \(m\)-dimensional reconstruction). If \(m\) is not large enough, then \(V_{j}^{(m)}\) may be a false neighbor of \(V_{i}^{(m)}\) (something like both the north and south poles are mapped to the center of the equator, or multiple different objects have the same shadow). If embedding can be achieved by increasing \(m\) by 1, then the embedding vectors become  
\[V_{1} =[x(t_{1}),x(t_{1}+\tau),x(t_{1}+2\tau),...,x(t_{1}+(m-1)\tau],\]1 Embedding of the harmonic oscillator. \[V_{1} =[x(t_{1}),x(t_{1}+\tau),x(t_{1}+2\tau),...,x(t_{1}+(m-1)\tau],\]2 (38) where \(R_{T}\) is a heuristic threshold value. [PERSON] [72] recommends \(R_{T}=15\). After \(m\) is determined, \(\tau\) can be obtained by minimizing \(R_{f}\). While this method is intuitively appealing, it should be pointed out that it works less effectively in the noisy case. Partly, this is because nearest neighbors may not be well defined when data have noise. **(2) Time-dependent exponent curves**: This is a dynamical method developed by [PERSON] and [PERSON] [73; 74]. The basic idea is that false neighbors will fly apart rapidly if we follow them on the trajectory. Denote the reconstructed trajectory by \(V_{1}^{(m)}\), \(V_{2}^{(m)},\cdots\). If \(V_{i}^{(m)}\) and \(V_{j}^{(m)}\) are false neighbors, then it is unlikely that points \(V_{i+k}^{(m)}\), \(V_{j+k}^{(m)}\), where \(k\) is the evolution time, will remain close neighbors. That is, the distance between \(V_{i+k}^{(m)}\) and \(V_{j+k}^{(m)}\) will be much larger than that between \(V_{i}^{(m)}\) and \(V_{j}^{(m)}\) if the delay reconstruction is not an embedding. The metric recommended by [PERSON] and [PERSON] is  
\[V_{1} =[x(t_{1}),x(t_{1}+\tau),x(t_{1}+2\tau),...,x(t_{1}+(m-1)\tau],\]3  
Here, for simplicity, the superscript \((m)\) in the reconstructed vectors is no longer indicated. The angle brackets denote the average of all possible \((V_{i},V_{j})\) pairs satisfying the condition  
\[V_{1} =[x(t_{1}),x(t_{1}+\tau),x(t_{1}+2\tau),...,x(t_{1}+(m-1)\tau],\]4  
where \(\epsilon_{i}\) and \(\Delta\epsilon_{i}\) are more or less arbitrarily chosen small distances. Geometrically speaking, Equation (40) defines a shell, with \(\epsilon_{i}\) being the diameter of the shell and \(\Delta\epsilon_{i}\) the thickness of the shell. When \(\epsilon_{k}=0\), the shell becomes a ball; in particular, if the embedding dimension \(m\) is 2, then the ball is a circle. Note that the computation is carried out for a series of shells, \(i=1,2,3,\cdots\), and \(\Delta\epsilon_{i}\) may depend on the index \(i\). With this approach, the effect of noise can be greatly suppressed. As a rule of thumb, [PERSON] and [PERSON] find that for a fixed small \(k\), the minimal \(m\) is such that when further increasing \(m\), \(\Lambda(m,L,k)\) no longer decreases significantly. After \(m\) is determined, \(L\) can be chosen by minimizing \(\Lambda(m,L,k)\). Now that we have determined an optimal embedding, we can discuss how to estimate the largest positive Lyapunov exponent, dimension, and Kolmogorov entropy of chaotic attractors.