*2.1. CPE*

Based on the permutation entropy and IOTA, Shi et al. proposed the CPE to analyze the information interactions between financial time series [39]. The implementation process is as follows:


$$k\_{t} = \sum\_{i=1}^{d-2} \sum\_{j=i+1}^{d-1} \Theta[(G\_{t}(j+1) - G\_{t}(i))(G\_{t}(i) - G\_{t}(j))] \tag{1}$$

where Θ[*x*] is the Heaviside function:

$$\Theta[\mathbf{x}] = \begin{cases} \mathbf{1}, \ x > 0 \\ \mathbf{0}, \ x \le 0 \end{cases} \tag{2}$$

4. According to this method, all state vectors of the time series are traversed, and the number of the intersections of each state vector can be expressed as a unique integer *z*, *z* ∈ [0, *R*], *R* = (*d* − 1)(*d* − 2)/2 is the maximum possible number of intersections. For all the *R* + 1 possible values for the integer *zi*, *i* = 0, 1, ... , *R* of intersection points *kt* in each state vectors, its probability can be obtained by

$$p(z\_i) = \frac{\#\{k\_t | k\_t = z\_i\}}{N - (d - 1)\tau} \tag{3}$$

where 1 ≤ *t* ≤ *N* − (*d* − 1)*τ*, 0 ≤ *i* ≤ *R*, *#* is the number of elements in the set. Then, after obtaining the probability distribution set *P* = {*p*(*zi*), *i* = 1, . . . , *R*}, CPE is defined as:

$$H\_{x \to y}(d, \tau) = -\sum\_{i=0}^{R} p(z\_i) \log\_2 p(z\_i) \tag{4}$$

According to the above definition, the greater the coupling strength between the two time series, the smaller the CPE. For two random time series, the entropy value reaches the theoretical maximum log2(*R* + 1).
