## 6 Verification by Loadings Measured in a Mountain Bike
### Detecting Stationarity/Non-Stationarity by the Run Test
If \(r\) falls inside the acceptance region as in Equation (22), the sequence of \(\psi_{i}\) is presumed to be truly random, and accordingly, \(z(t)\) is classified as stationary. Conversely, if \(r\) falls outside, \(z(t)\) is classified as non-stationary. An important parameter in the run test is the number of segments, as it directly determines the segment time length \(T_{s}\). The important role of \(T_{s}\) has been emphasized in [38]. In that study, it was discovered--though only empirically--that too long a value of \(T_{s}\) has the effect of smoothing local variations in the rms value, if present, up to the point of hiding the presence of underlying trends. This effect can be understood quite intuitively by considering that the rms value of each segment, \(\psi_{i}\), is computed as a time average of the signal in that segment. It is then presumed that the value \(\psi_{i}\) fully represents the signal rms value in that segment, regardless of whether the actual rms value is in fact constant within the segment. Any local fluctuation of the rms value _within_ each segment is averaged out, indeed. As a direct consequence, the longer the segment length, the greater the averaging effect in each segment, with the risk of smoothing or even hiding any non-stationary fluctuation--the run test may classify the signal as stationary though it is not. Conversely, too short a segment length, while increasing the sensitivity to local temporal variations, tends also to enhance the presence of irregular fluctuations of rms values across adjacent segments that, being only due to a sampling variability, may erroneously indicate a false non-stationarity. Regrettably, the interesting observations made in [38] are not used to suggest practical recommendations for the optimal choice of \(T_{s}\). For this reason, the run test is here applied by choosing two different values: \(T_{s}=5\) s and \(T_{s}=10\) s. Figure 9 illustrates the run test applied to one of the measured records belonging to the presumed \"stationary\" set. When \(T_{s}=10\) s, there are \(n_{+}=n_{-}=15\) observations above and below the median; from statistical tables, the upper and lower limits in Equation (22) are \(r_{15,0,975}=10.7\) and \(r_{15,0,025}=21.3\); for a 5% significance level, \(\beta=0.05\). Since the observed number of runs is \(r=15\), and it falls within the acceptance region of the test (10.7, 21.3), the load record is classified as stationary. The same conclusion is also confirmed by choosing a shorter window length, for which in general the verdict of the run test tends to be in favor of non-stationarity. For a window length \(T_{s}=5\) s, there are \(n_{+}=n_{-}=30\) observations above and below the median, while the observed number of runs is \(r=27\), which again falls within the acceptance region (23.5, 38.5) defined, for 5% significance, as a function of \(n_{+}\). The load record is again classified as stationary. When applied to all other records of the \"stationary\" set, the run test (with both window lengths) classified them as being truly stationary. When the run test is applied, instead, to the set of switching measured loading, the outcome is opposite. For the case of \(T_{s}=5\) s (see Figure 10a), the observed number of runs \(r=11\) falls outside the acceptance region (23.5, 38.5) defined by the same number of observations \(n_{+}=n_{-}=30\) as above. The stress-time history is classified as non-stationary. Nothing changes if the window length increases to \(T_{s}=10\) s, see Figure 10b. With a longer window length, the number of runs decreases to \(r=7\) and again falls outside the acceptance region (10.7, 21.3) defined by the same observations \(n_{+}=n_{-}=15\) as above. The same outcome \"non-stationary\" was obtained for all measured switching records.