# Analysis of Wind Speed Characteristics Along a High-Speed Railway
## 4 Predictability Analysis  
The characteristics of wind itself affect the predictability of wind speed. Generally, the statistical properties of wind speed remain constant over time. However, the length of the wind speed data series influences these properties when analyzing historical wind speeds. Using fewer historical samples often leads to an \"underfitted\" model, while using an extensive dataset can result in an \"overfitted\" model and an unnecessarily prolonged modeling period. In this section, we assess the predictability of historical wind speed and quantify it using the normalized cumulative period (NCP) graph and Bartlett's test.  
Determine the periodogram function (\(\hat{f}(\omega_{j})\)) for a given stationary time series \((x_{1},x_{2},\ldots,x_{n})\), which represents the time series' spectral density at each frequency and is derived using the formula below:  
\[\hat{f}(\omega_{j})=\frac{1}{n}\bigg{|}\sum_{t=1}^{n}x(t)\exp\bigl{(}2\pi i(t- 1)\omega_{j}\bigr{)}\bigg{|}^{2}, \tag{2}\]  
where \(\omega_{j}=(j-1)/n\) is the j-th frequency \((j=1,\ldots,\mathrm{q})\), \(n\) is the length of the time series, \(\mathrm{q}=\frac{n}{2}+1\), \(x(t)\) is the time series data at time \(t\), and \(\hat{f}(\omega_{j})\) is the periodic spectrum of frequency \(\omega_{j}\). The frequency distribution of a time series is graphically represented by the periodogram.  
Figure 6: Turbulence intensity and mean wind speed.  
NCP is the accumulation based on the periodogram function (\(\hat{f}(\omega_{j})\)):  
\[\hat{F}(\omega_{k})=\frac{\sum_{j=1}^{k}\hat{f}(\omega_{j})}{\sum_{j=1}^{q}\hat {f}(\omega_{j})}, \tag{3}\]  
where \(\hat{F}(\omega_{k})\) is the NCP at frequency \(\omega_{k}\), \(k=1,\ldots,\text{q}\). If the power spectral density of \(\omega_{k}\) is uniformly distributed over the frequency of the NCP, the randomness of \(\omega_{k}\) can be identified. Data from a random time series is consistently dispersed across the whole frequency range and is not centered on a small number of particular frequencies. Therefore, it can be inferred that it follows a random walk if the confidence interval is drawn between \((0,0)\) and \((0.5,1)\) as a straight line in NCP. Confidence interval lines are drawn at the vertical distance \(\pm\frac{K_{e}}{n-1/2}\) straight line connecting the upper and lower ranges of \((0,0)\) and \((0.5,1)\). The 99% confidence interval of the cumulative periodogram is determined by \(K_{e}\), which is set to 1.63 [39]. Figure 7 presents the NCP of 3600 random numbers. The NCP (bold blue line) remains within the 99% confidence interval, indicating that the 3600-number sequence conforms to a random walk and demonstrates a lack of predictability.  
[PERSON]'s test is a common method for examining the null hypothesis that the data come from white noise [40]. To calculate the [PERSON]'s test statistic B and \(p\)-value, the formula is as follows:  
\[B=\sqrt{q}\max_{1\leq k\leq q}\left|\hat{F}(\omega_{k})-\frac{k}{q}\right|,k= 1,\ldots,\text{q}, \tag{4}\]  
\[p-\text{value}=1-\sum_{j=-\infty}^{\infty}(-1)^{j}e^{-2B^{2}\hat{f}}, \tag{5}\]  
\(\hat{F}(\omega_{k})\) is the NCP at frequency \(\omega_{k}\), \(\omega_{k}=(k-1)/n\) is the kt-h frequency, and \(\text{q}=\frac{n}{2}+1\). [PERSON]'s test statistic \(B\) is defined as the deviation of \(\hat{F}(\omega_{k})\) from the NCP straight line. The null hypothesis of white noise is rejected if the \(p\)-value calculated from [PERSON]'s test statistic B is less than the specified significance level \(\alpha\).