# Unequal Interval Dynamic Traffic Flow Prediction with Singular Point Detection
## 4 Singular Point Detection and Prediction via Wavelet Transform
### Singular Point Detection via Multiscale Wavelet Transform  
According to the aforementioned discussion, the singular point of a traffic flow timing signal reflects the intrinsic traffic condition. The trend of a traffic flow is relatively stable between two successive singular points, and the parameters within this interval fluctuate via a certain predictable range. Once a singular point appears in the traffic flow signal, it means that a traffic congestion occurs in this road segment due to heavy traffic or a sudden traffic accident, which leads the previous relatively stable status to be broken. The parameters in the new trend and situation need to be predicted again. Therefore, this section mainly discusses how to detect the singular points of a traffic flow timing signal and filtrate the singular points via a multiscale. A wavelet transform can be used to highlight the local characteristics of a signal in time and frequency [27]. The crossing zero points and extreme points can better describe the details of signals, which play an important role in analyzing the signal's local character. Therefore, the conversion of the original traffic flow timing signal through a wavelet transform can reflect the comprehensive performance in a multiscale, especially its potential singular points of the traffic flow. Based on the definition of wavelet transform, the original signal can be transformed as  
\[\mathrm{WF}_{a}x(t)=\frac{1}{a}\int x(\tau)\varphi(\frac{t-\tau}{a})d\tau=x(t )*\varphi_{a}(t), \tag{1}\]  
where \(x(t)\) is the original signal representing the traffic flow timing signal. The wavelet \(\varphi_{a}(t)\) denotes \(\frac{1}{a}\varphi(\frac{t}{a})\). A wavelet transform can be used as the original signal's output after an impulse response that denotes \(\varphi_{a}(t)\). Furthermore, using the wavelet transform to detect a signal's singular points, the multiscale wavelet \(\varphi_{a}(t)\) depends on the low-pass smoothing function \(\theta(t)\), which should satisfy the following two conditions [28]:  
Figure 1: Illustration of the system process. * The order of smoothing and derivation of a signal does not influence the wavelet transform performance. * The first-order derivative and second-order derivative of this low-pass smoothing function \(\theta(t)\) should be a band-pass function, which denotes \[ \begin{split}\int\theta(t)dt&\
eq 0,\\ \Theta(\Omega=0)&\
eq 0.\end{split} \] (2) Therefore, when \(\theta(t)\) satisfies the above two conditions, \[\theta^{(1)}(t)=\frac{d\theta}{dt},\theta^{(2)}(t)=\frac{d^{2}\theta}{dt^{2}}\] (3) can be the basic wavelets in the transform. Furthermore, based on the scale property of the Fourier transform [29], the scale changes of \(\frac{d\theta}{dt}\) and \(\frac{d^{2}\theta}{dt^{2}}\) do not affect the performance, which denotes \(\varphi_{a}(t)\) in Equation (1) as \[\varphi_{a}(t)=\theta_{a}^{(1)}(t)=\frac{1}{a}\theta^{(1)}(\frac{t}{a}),\] (4) or \[\varphi_{a}^{{}^{\prime}}(t)=\theta_{a}^{(2)}(t)=\frac{1}{a}\theta^{(2)}( \frac{t}{a}).\] (5)  
The selection of the wavelet scale parameter \(a\) further influences singular point detection [28]. The wavelet with a small scale \(a\) detects more singular points but increases the probability of false detection because of the superposition effect. The wavelet with a large scale \(a\) detects few and sharp singular points. Parts of singular points are ignored and smoothed as noise of the signal. For instance, the traffic flow on the specific road segment has a sharp increment, which can be detected as a singular point. The algorithm decides whether a reprediction is needed. Based on the selected scale, the algorithm regards this change of traffic flow as a normal disturbance if it relates to a weak singular point. This case will be ignored and released by itself. Conversely, if it is a strong singular point, the algorithm judges that it cannot resume in a short time and will repredict the traffic flow tendency with an unequal interval.