## 2 Materials and Methods
### Extracting Multiple Basis Functions to Form an Average Basis Function  
As examples of waveform segments and their information content vary in the time evolution of \(x\) trajectories, resulting basis functions \(p(\tau)\) also vary. These variations are illustrated in Figure 4 which shows a long waveform sample (\(\approx\)50 s with \(\approx\)160 bits of symbolic content) where smaller 8-bit segments of symbolic content are captured to generate waveform templates. These templates are held in memory and compared against all other consecutive 8-bit segments that exist in the waveform sample \(x(\tau)\). This comparison'scans' and captures instances of the Hamming distance criterion being met between the stored template \(x_{\text{template}}(\tau)\) and the sliding comparison segments \(x_{\text{compare}}(\tau)\). Algorithmically, this scanning first consists of creating templates with the following steps:  
1. set a global scanning index \(q\) to iterate template storage over the long waveform sample \(x(\tau)\),
2. store the \(q^{\text{th}}\) template segment of \(x(\tau)\) to form the \(q^{\text{th}}\) instance of \(x_{\text{template}}(\tau)\),
3. extract the symbolic content of the \(q^{\text{th}}\) instance of \(x_{\text{template}}(\tau)\). Next for a single iteration of \(q\), the \(q^{\text{th}}\) template is compared throughout the waveform by the following nested scan with the steps:  
1. set a scanning index \(k\) to iterate over the long waveform sample \(x(\tau)\)
2. store the \(k^{\text{th}}\) template segment of \(x(\tau)\) to form the \(k^{\text{th}}\) instance of \(x_{\text{compare}}(\tau)\)
3. extract the symbolic content of the \(k^{\text{th}}\) instance of \(x_{\text{compare}}(\tau)\)  
Figure 4: (**a**) Two long waveform segments from state variable x after being time-regularized. (**b**) Two examples of waveform segments with symbolic content that meets the Hamming distance criterion and their resulting instances of basis function \(p(\tau)\). (**c**) Two examples of waveform segments similar to Figure 4b with the exception that the summed peak is not inverted. Cases for Figure 4b,\(c\) must be accounted for when designing basis function extraction algorithms. * perform a test of the Hamming distance criterion between the symbolic content of the \(k^{\text{th}}\) instance of \(x_{\text{compare}}(\tau)\) and the symbolic content of the \(q^{\text{th}}\) instance of \(x_{\text{template}}(\tau)\). For cases where the Hamming distance criterion is met, an instance of the basis function \(p(\tau)\) is stored. Note, as shown in Figure 4, that some instances of \(p(\tau)\) may have an inverted peak and should be corrected to have a positive peak. This process is continued until the \(k\) index is exhausted throughout the long waveform sample \(x(\tau)\). Upon completion of the final \(k\) index, the global index \(q\) is incremented, and the nested scan is repeated. As the global index \(q\) is exhausted, the long waveform sample \(x(\tau)\) has served to populate many instances of the basis function \(p(\tau)\) which can be averaged. This process is highlighted with an example waveform in Figure 4. Two specific instances of the Hamming distance criterion being met for global index \(q\) and scanning index \(k\) and lead to two specific examples of a basis function \(p(\tau)\). The first basis function instance \(p(\tau)\) is illustrated by Figure 4a, and obtained from the global index \(q=2\) (highlighted in red and labeled as _Template 1_). The Hamming distance criterion is met in this instance for a scanning index \(k=45\) (highlighted in red and labeled _Template 1_ _Match_). Note that \(x_{\text{template}}(\tau)-x_{\text{compare}}(\tau)\) results in a negative peak, as illustrated in the left plots of Figure 4b, so it first inverted before being stored as an instance of \(p(\tau)\). Similarly, another instance of \(p(\tau)\) is illustrated by Figure 4a, but this time is obtained from the global index \(q=5\) (highlighted in blue and labeled as _Template 2_).