## 3 Results
#### Discussion  
These results validate the effectiveness of extracting basis functions using the proposed algorithm. In previous sections, three independent, non-averaged basis functions were given by Figures 3 and 4. The bit statistics and averaged waveforms for the \(x\) variable in respect to the natural dynamics of the chaotic Lorenz oscillator and uniformly distributed representations of bit sequences in Figure 5 along with difference between the average basis function for natural Lorenz bit statistics and uniform bit statistics (which is small). Finally, we report extracted waveforms from all three state variables \(x\), \(y\), and \(z\) in Figure 6 with natural Lorenz bit statistics. The extracted and averaged basis function \(<p(\tau)>\) for the state variable \(x\) is in high agreement with [PERSON]'s work [21]. The underlying statistics of the bit sequences chosen to construct \(<p_{x}(\tau)>\) have a quantifiable impact on the shape of the resulting basis function as shown by the data in Figure 5b. If the contributions of these bit sequences are not  
Figure 5: Results of processing a 5000 [s], \(\approx\)5,000,000 timestep chaotic Lorenz waveform \(x(t)\) via Runga-Kutta (MATLAB ODE45) integration with the parameters \(\sigma^{\prime}=\sigma-\Delta\approx 10,\varrho^{\prime}=\rho-\Delta\approx 28\) and \(\beta^{\prime}=\beta-\Delta\approx\frac{8}{3}\) with \(\Delta=10^{-6},dt=0.001\) and length 5000 [s]. (**a**) Bit statistics for each 8-bit long segment used to establish that the Hamming distance criterion was met. The red histogram shows the natural density of symbolic content provided by the \(x\) state variable. This red histogram constitutes the symbol sequences that are used to average the basis function \(<p(\tau)>\). The blue histogram shows uniformly distributed bit sequences (all bit sequences are represented evenly with \(\approx\)200 examples of each) and constitutes the average basis function \(<\rho(\tau)>\). (**b**) Averaged basis functions in various regions of the red histogram from (**a**). (**c**) The resulting averaged basis functions for both natural bit density (solid red) and uniformly distributed bit density (dashed blue). Below each of these averaged basis functions show their small difference. statistically balanced, the resulting, average basis function may be distorted. Interestingly, the difference between averaged basis function \(<p(\tau)>\) with natural bit sequences and the averaged basis function \(<\rho(\tau)>\) with uniform bit sequences was small. This is likely due to the fact that the natural bit statistics are dominated by sequences with histogram peaks that are'statistically balanced.' To the author's knowledge, these bit statistics have not been previously reported in regard to constructing basis functions for chaotic Lorenz waveforms. Similarly, novel data are given by Figure 2 that shows the natural timing statistics of chaotic Lorenz peaks for the time evolution of state variable \(x\). Finally, the bit statistics and extracted average basis functions for all three state variables given in Figure 6 indicate a well-known correlation between state variables \(x\) and \(y\) as their extracted basis functions are qualitatively similar with similar bit sequence statistics. Interestingly, the details of state variable \(z\) do not clearly lend themselves to waveform synthesis as readily as the results from state variables \(x\) and \(y\). First, we note the non-uniformity and imbalance of bit sequence statistics involved with constructing the average basis function \(<p_{z}(\tau)>\). Second, we note that the forward-time tail of \(<p_{z}(\tau)>\) does not tend to cleanly resolve to zero amplitude. In fact, for some collections of bit sequences, \(<p_{z}(\tau)>\) becomes ill-behaved in amplitude and would inject unwanted inter-symbol interference if used to reconstruct a chaotic time series of state variable \(z\).