Tensor alternating least squares grey model and its application to short-term traffic flows

Highlights

• The tensor alternating quadratic GM(1,1) model was proposed.

• High-dimensional tensor multi-mode is used to represent traffic flow data.

• The organic combination of tensor alternating quadratic method and grey model.

• The new model can effectively predict the short-term traffic flow.

Abstract

Traffic flow data, as an important data source for the research and development of intelligent transportation systems, contain abundant multi-mode features. In this paper, a high-dimensional multi-mode tensor is used to represent traffic flow data. The Tucker tensor decomposition least squares algorithm is used to establish the tensor alternating least squares GM (1,1) model by combining the modelling mechanism of the grey classical model GM (1,1) with the algorithm, and the modelling steps are obtained. To demonstrate the effectiveness of the new model, first, the multi-mode traffic flow data are represented by the tensor model, and the correlation of the traffic flow data is analysed. Second, two short-term traffic flow prediction cases are analysed, and the results show that the performance of the GM (1, 1) model based on the tensor alternating least squares algorithm is obviously better than that of the other models. Finally, the original tensor data and the approximate tensor data during the peak period from 8:00 to 8:30 a.m. for six consecutive Mondays are selected as the experimental data, and the effect of the new model is much better than that of the GM (1,1) model of the original tensor data.

Introduction

Intelligent transportation systems (ITS) are a type of traffic control information management system that has been developed in recent years. Short-term traffic flow forecasting is an important part of ITS. Improving short-term traffic flow forecasting precision can provide support for intelligent transportation systems. Traffic flow data, as an important data source for the research and development of ITS, contain abundant multi-mode features. Tensors are generalizations of vector and matrix models that can represent multi-dimensional data and have multi-mode characteristics. At the same time, traffic data are observed from different modes of week, day, space and time, showing significant multi-mode features and strong multi-correlation [1], [2]. The use of tensors to represent multi-dimensional data with multi-modal features [3] can overcome the shortcomings of vector and matrix data forms that have difficulty characterizing multi-dimensional features [4]. Therefore, to improve the accuracy of short-term traffic flow predictions, the multi-mode characteristics of traffic data and the overall trends of traffic flow data can be fully considered to improve the accuracy of short-term traffic flow predictions.

With the rapid development of ITS, researchers have performed extensive research on traffic flow predictions. Pattern information is often divided into the vector data stream form, matrix data stream form and tensor data stream form as research objects for short-term traffic flow predictions. Short-term traffic flow predictions more commonly use vector data flow than other forms: Stephanedes [5] used a historical average model and the autoregressive integral moving average (ARIMA) model [6] to predict short-term traffic flow. Nonlinear prediction methods, support vector regression (SVR) models [7] and support vector machine theory [8] have been used to mine the nonlinear characteristics of traffic data. Subsequently, spectral analysis [9] prediction models, chaos prediction models [10], neural networks [11] and other model types have exhibited improved prediction effects on high-quality historical sample databases.

Short-term traffic flow predictions in the form of matrix data flow: multivariate time series prediction models regard the traffic data from different spatial points as interrelated variables and construct the traffic data as multivariate time series [12]. Multi-section short-term traffic flow predictions are mainly based on the traffic flow data from multiple sections of the road. Based on a time series of traffic data, Cheng et al. [13] reconstructed daily and weekly time series of traffic data to predict the short-term traffic flow. Guo et al. [14] used the Kalman filter to predict the seasonal autoregressive real-time traffic flow. Zhang et al. [15] analysed the periodic trend, certainty and volatility of traffic data using spectral analysis technology, ARIMA and the generalized autoregressive conditional heteroskedastic (GARCH) model. Hong [16] proposed a seasonal SVR prediction model. Yang et al. [17] created a coupled prediction of time series data and cross-section data at intersections, and these methods achieved good results.

Short-term traffic flow predictions in the form of tensor data flow: traffic data are observed from different modes of the week, day, space and time, and the traffic data show significant multi-mode features. To improve the accuracy of short-term traffic flow predictions, the multi-mode characteristics of traffic data are fully considered. For example, Tan et al. [18] reconstructed the daily time series and weekly time series of traffic data to predict the short-term traffic flow based on the time series of traffic data. Tan et al. [19] expressed traffic flow data with a tensor model and proposed a new method for filling data with a dynamic tensor. Duan et al. [20], [21] used the multi-mode characteristics of traffic flow data to expand the traffic flow data by using the “week-day-time” mode and establish a coupled tensor multi-mode model and the grey prediction model for tensor dynamics, and this method achieved better results than previous methods.

The above prediction models are usually based on large sample sizes and cannot be used to solve small-scale problems. At the same time, short-term traffic flow forecasting uses road traffic flow state data that are dynamically acquired to infer data on the future traffic flow state. The time scale of the traffic flow data is generally limited to 15 min. The traffic flow data in the previous period obviously have the greatest impact on the next period if one hour is taken as the time scale. The interval between records is 5 min; thus, only 12 groups of data are collected in one hour. Due to the daily time scales, traffic flow data are composed of small sample sizes. In addition, there are uncertain factors in the traffic system, such as vehicle characteristics, weather factors and the abnormal loss of traffic flow data. The traffic flow is an integral value that can approximately determine the range of true or abnormal data according to the traffic conditions and measured values in the front and back periods, and its range is a discrete set. These uncertainties within a certain interval or a set of numbers reflect the characteristics of the grey system. Simultaneously, the grey prediction model has strong adaptability and can adequately handle mutation parameters.

The grey prediction model [22] is an important part of grey theory. After more than 30 years of development, this model has been widely used in agriculture, industry, society, economics, transportation, energy, medicine and other fields [23], [24], [25], [26], [27], [28], [29], [30], [31]. With much research, the classic grey prediction model, the GM (1,1) model, has been extended to different types of models, such as GM (1, N) [32], DGM (1, 1) [33], NDGM (1,1) [34], [35], ONGM (1,1) [36], ENGM (1,1) [37] and other new prediction models. At the same time, GM (1,1) [38], [39], [40], [41], [42], [43], [44], [45] has been studied from multiple perspectives such as data accumulation, optimization of background values, model properties, and modelling mechanisms, which has promoted the development and perfection of the theoretical system of the grey prediction model. Short-term traffic flow prediction is an important application of the grey model [17], [20], [21], [34], [35], [36], [37], [38], [39]. Hsu et al. [46] proposed an adaptive GM (1,1) model for traffic prediction of an intersection without a detector. Guo et al. [47] established a grey nonlinear delay GM (1,1) model for short-term traffic flow. Bezuglov and Comert [48] established a GM (1,1) model and a grey Verhulst model with Fourier error correction, which achieved good prediction results in the prediction of short-term traffic flow speed and travel time. Xiao et al. [49] proposed a seasonal GM (1,1) rolling prediction model based on the cycle truncation accumulated generating operation (CTAGO). According to the law of vehicle conservation, Xiao et al. [50] established a new dynamic grey prediction model to predict the short-term traffic flow. Lu et al. [51] used the nonlinear grey Bernoulli equation to obtain the grey prediction model for traffic flow prediction and achieved good results. According to the mechanical characteristics of traffic flow data, Duan et al. [52] applied the grey inertia model to predict short-term traffic flow and determine the traffic flow state.

However, the abovementioned short-term traffic flow prediction methods (vector data flow, matrix data flow, tensor flow data flow), as well as the following grey prediction methods, care about only the nonlinearity, volatility, periodicity and other characteristics of a single time series change or spatial change at the target road section. These methods fail to make full use of the multi-mode characteristics of traffic flow data to deeply determine the temporal correlation of traffic flow data, especially the “weekday time” traffic flow data characteristics, which to some extent affect the prediction. At the same time, the above models directly usually use the collected traffic flow data in the model. These models do not make full use of the multi-mode characteristics of traffic flow and fail to grasp the overall trends of traffic flow, which also affects the prediction by the model.

Therefore, to better mine the multiple pattern characteristics of traffic data, such as weeks, days and hours, this paper uses tensors to represent the multi-pattern characteristics of traffic flow data and uses the tensor alternating least squares method to fully mine the traffic data. Combined with the modelling mechanism of the classic GM (1,1) model, a grey prediction model of tensor traffic multiplication is proposed. This paper discusses the tensor representation of traffic flow data in the minute, hour, day and week modes and analyses the correlation of the data. At the same time, cases are given to analyse the validity of the GM (1,1) model for tensor alternating multiplication.

In the full text, the different abbreviations are for the different grey prediction models. Abbreviations and their meanings are provided in Table 1.

The remaining chapters of this paper are arranged as follows. Part 2 introduces the basis of tensor algebra and the theory of the tensor Tucker decomposition model. The third part establishes the tensor alternating least squares grey forecasting model, and the fourth part describes a case study and provides a comparative discussion. Part 5 presents the conclusion.