# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 4 Multiscale Analysis with the Scale-Dependent Lyapunov Exponent (SDLE)  
SDLE is developed for better distinguishing chaos from noise and for better characterizing complex data, especially through obtaining the defining parameters of the data [38,189]. SDLE is closely related to two other methods, the time-dependent exponent curves [73,74,79,81] and the finite size Lyapunov exponent [190,191,192]. SDLE was first introduced in [38,189], and has been further developed in [193,194] and applied to characterize EEG [143], HRV [195,196], financial time series [76], Earth's geodynamo [197], precipitation dynamics [198], sea clutter [199], THz imagery [200], and evaluate randomness [99]. As with the presentation of AFA, here, we will only present the key elements of the method; a concrete example of combining this method with a machine-learning approach (random forest) for distinguishing epileptiform discharges from normal electroencephalograms can be found in [PERSON] et al. [201]. SDLE is based on the evolution of vectors in a high-dimensional phase space. If initially the data are a time series, then one needs to obtain a suitable phase space using delay coordinates, as explained before. If the original data are a scalar random process, then the main advantage of the embedding procedure is to obtain a self-similar vector process from the original self-affine process. This is because \(x\) and \(t\) have different units and therefore have to be scaled differently in order for them to look \"alike\". All the components of a vector are of the same nature, and therefore can be stretched or shrunk with the same fashion. Consequentially, whenever a truly random time series is analyzed, the specific value of the embedding dimension \(m\) is not important. Often ensuring \(m>1\) is sufficient. After a phase space is obtained, one can examine the evolution of an ensemble of trajectories. Denote the initial distance between two nearby trajectories by \(\epsilon_{0}\). We further denote their _average distance_ at time \(t\) by \(\epsilon_{t}\), and that at \(t+\Delta t\) by \(\epsilon_{t+\Delta t}\). A schematic showing how a small distance between two nearby trajectories grows with time is shown in Figure 29. With this setting, we can examine the relation between \(\epsilon_{t}\) and \(\epsilon_{t+\Delta t}\), where \(\Delta t\) is assumed to be small. When \(\Delta t\to 0\), we have,  
\[\epsilon_{t+\Delta t}=\epsilon_{t}e^{\lambda(\epsilon_{t})\Delta t}, \tag{103}\]  
where \(\lambda(\epsilon_{t})\) is the SDLE given by  
\[\lambda(\epsilon_{t})=\frac{\ln\epsilon_{t+\Delta t}-\ln\epsilon_{t}}{\Delta t}. \tag{104}\]  
Figure 28: AFA of power load data: (**a**) an example of \(\log_{2}F(w)\) vs. \(\log_{2}w\) for the load data of an arbitrarily chosen day, (**b**) temporal variation of the Hurst parameter (red) and the rescaled temperature (black). Equivalently, we can write,  
\[\frac{d\epsilon_{t}}{dt}=\lambda(\epsilon_{t})\epsilon_{t}. \tag{105}\]  
Now that we have introduced SDLE, we can better understand the classic algorithm for computing the largest Lyapunov exponent \(\lambda_{1}\) discussed earlier [75]. That algorithm assumes \(\epsilon_{t}\sim\epsilon_{0}e^{\lambda_{1}t}\) and then through averaging estimates \(\lambda_{1}\) by \((\ln\epsilon_{t}-\ln\epsilon_{0})/t\). This assumption may not even hold for true chaotic signals. This is reminded in the detail of the schematic plot shown in Figure 29 --\(\epsilon_{t+\delta t}\) may be smaller than \(\epsilon_{t}\). As already mentioned, a fundamental difficulty with this assumption is that for any type of noise, when \(\epsilon_{0}\) is small (which is the case when nearest neighbors are used), \(\lambda_{1}\) can always be positive, leading to misinterpreting noise as chaos. The reason is simple: \(\epsilon_{t}\) will rapidly converge to the most probable distance between the constructed vectors, and thus will almost be surely larger than \(\epsilon_{0}\).