# Unequal Interval Dynamic Traffic Flow Prediction with Singular Point Detection
## 1 Introduction  
Traffic flow predictions have been used widely as metrics in traffic management, e.g., planning routes for commuters, alleviating traffic congestion, and improving traffic operation efficiency [1; 2]. Depending on the real-time and historical traffic information [3], many prediction models have been proposed to forecast traffic flow [4], including linear prediction models (autoregression, multivariate adaptive regression splines), neural network models, echo state, support vector machine, adaptive filter, deep learning system, etc. Prediction accuracy depends on the parameter adjustment and model training. However, researchers also pointed out that training models have drawbacks, i.e., scenario limitation and training cost [5]. Analysis of a traffic flow signal is suitable for solving the bottleneck of fixed-interval prediction to capture the dynamics of a traffic flow [4]. The singular point of a time-series signal is an intrinsic property of traffic flow, and it can reflect changes in a continuous process. The singular point of a traffic flow signal labels a nonstationary traffic flow status, which might be caused by a traffic congestionor accident [6]. In wavelet multiresolution analysis, an important decision is to select the decomposition level [7]. [PERSON] et al., proposed a hybrid model that combined the discrete wavelet decomposition and prophet framework [8]. The model decomposed the traffic flow from low frequency to high frequency and then was trained to predict the components. The adjusted components were reconstructed to complete the prediction. [PERSON] et al., gave a nonparametric dynamic time-delay recurrent wavelet neural network model to predict the traffic flow [9]. The concept of wavelet frame was introduced and exploited in the model to provide flexibility in the design of wavelets and to add extra features. Wavelet denoising and phase space reconstructions improve the prediction model in terms of accuracy [10]. Singular point detection is an effective approach to provide the basis for traffic flow status division, and singular points should be redetected to keep the prediction accuracy. The precision of singular point detection is balanced by the accuracy and computing time of further traffic flow prediction. It is apparent that a wavelet transform is an effective method for multiscale signal analysis, which can detect traffic flow signals' singular points. However, it is still challenging to decide when the traffic flow prediction should be dynamically updated to balance the prediction accuracy and computing time. In this paper, we aim at developing a method for efficiently and dynamically updating traffic flow prediction by analyzing the singular points in real-time and historic traffic flow signals. The overall architecture contains three subprocesses, i.e., (1) traffic flow analysis that detects the multiscale singular points via an improved wavelet transform, (2) unequal interval decision considering the accuracy and computing time comprehensively, and (3) traffic flow prediction based on a Vasicek model involving a matrix decomposition method. The work in this paper can realize the accurate and rapid dynamic traffic flow prediction in the road segment scenario, which acquires its continuous historical and real-time traffic flow data. The main contributions of this paper are emphasized as follows:  
* The paper gives a framework to use singular point detection to analyze the traffic flow signal. An improved wavelet transform method is proposed to detect singular points dynamically based on the wavelet's adjustable scale. Based on the position of the singular points, the traffic flow signal is divided into unequal slots in time series, which is determined via the multiscale. * Depending on the real-time observation and historical traffic data, we propose an algorithm to predict the position of the next singular point in the observed process based on similarity measurement via a weighted function.