# Simple Degree-of-Freedom Modeling of the Random Fluctuation Arising in Human-Bicycle Balance
## 5 Identification Results
### KS Test  
Up to this point, we have found that our simple model (9) provides high reproducibility of the PDF shapes of the human-bicycle fluctuation. However, this does not directly support the stochastic reliability of our model. Therefore, in this section, we conduct a statistical-hypothesis test called the KS test [12; 13] on our results to check their stochastic reliability.  
As described in [12], the statistic of the one-sample KS test is provided by  
\[Z(n,E)=n^{1/2}E, \tag{22}\]  
where \(E\) is the KS distance already given in (16), and \(n\) is the number of samples. On the other hand, the two-sample KS test employs the following statistic:  
\[Z(n_{1},n_{2},E)=\left(\frac{n_{1}n_{2}}{n_{1}+n_{2}}\right)^{1/2}E, \tag{23}\]  
where \(n_{1}\) and \(n_{2}\) are the numbers of two independent samples. Under the null hypothesis (the sample follows a given distribution or the two samples follow the same distribution) and for large \(n\) (or \(n_{1}\) and \(n_{2}\)), the random variable \(Z\) follows the following CDF [12; 13]:  
\[F(z):=\mathrm{Prob}(Z>z)=2\sum_{i=1}^{\infty}(-1)^{i-1}\exp(-2i^{2}z^{2}), \tag{24}\]  
where the lowercase \(z\) denotes a value of \(Z\). The value of \(F(z)\) is called the \(p\)-value of the observed \(z\). If this \(p\)-value is smaller than a pre-defined \(\alpha\) called a significance level, the null hypothesis is rejected.  
In Figure 6, the solid curve shows \(F(z)\). The small circles plot the \(p\)-values between \(P_{hum}^{(s)}\) and \(P_{sim}^{(s)}\) at \(z=Z(n_{1},n_{2},E)\) using \(E=E(\mathbf{p}^{(s)})\) listed in Table 2, the measured data length \(n_{1}=I\times N=18,001\times 5\), and the simulated \(n_{2}=I\times N^{\prime}=n_{1}\). Under the significance level \(\alpha=0.01\), the null hypothesis is rejected for \(s=1,5\), and \(8\); that is to say that three of the eight measurements cannot be said to follow our simulated distributions. On the other hand, the cross marks show the results between \(P_{hum}^{(s)}\) and \(P_{Gauss}^{(s)}\) at \(z=Z(n,E)\) using the \(E\) values listed in Table 2 and \(n=n_{1}\). In this Gaussian case, the null hypothesis is rejected for all \(s\); i.e., no measurements can be said to follow these Gaussian distributions.  
Given the above, we conclude in terms of statistical hypothesis testing that our simple model (9) can simulate the time series of human-bicycle fluctuations much better than the Gaussian distribution.  
Figure 6: Kolmogorov–Smirnov (KS) testing results. The solid curve plots \(F(z)\), the KS statistic cumulative distribution function (CDF). The small circles indicate the \(p\)-values between measured \(P_{hum}^{(s)}\) and our proposed \(P_{sim}^{(s)}\), and the cross marks indicate those between \(P_{hum}^{(s)}\) and Gaussian \(P_{Gauss}^{(s)}\).  
Note that our proposed model can separately fit the measurements of individual participants; it does not provide a general description of human-bicycle fluctuation. Conversely, it provides a means of mechanical parameterization of individual difference. The obtained parameter vector \(\mathbf{p}\) is useful for comparing individuals and seeing how they differ, in a mechanical manner.