Complexity

Volume 2017 (2017), Article ID 3515272, 16 pages

https://doi.org/10.1155/2017/3515272

## Forecasting the Short-Term Traffic Flow in the Intelligent Transportation System Based on an Inertia Nonhomogenous Discrete Gray Model

^{1}College of Science, Wuhan University of Technology, Wuhan 430070, China^{2}College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{3}School of Business Administration, Zhejiang University of Finance & Economics, Hangzhou 310018, China

Correspondence should be addressed to Xinping Xiao

Received 15 April 2017; Accepted 28 May 2017; Published 3 July 2017

Academic Editor: Bo Zeng

Copyright © 2017 Huiming Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The traffic-flow system has basic dynamic characteristics. This feature provides a theoretical basis for constructing a reasonable and effective model for the traffic-flow system. The research on short-term traffic-flow forecasting is of wide interest. Its results can be applied directly to advanced traffic information systems and traffic management, providing real-time and effective traffic information. According to the dynamic characteristics of traffic-flow data, this paper extends the mechanical properties, such as distance, acceleration, force combination, and decomposition, to the traffic-flow data vector. According to the mechanical properties of the data, this paper proposes four new models of structural parameters and component parameters, inertia nonhomogenous discrete gray models (referred to as INDGM), and analyzes the important properties of the model. This model examines the construction of the inertia nonhomogenous discrete gray model from the mechanical properties of the data, explaining the classic NDGM modeling mechanism in the meantime. Finally, this paper analyzes the traffic-flow data of Whitemud Drive in Canada and studies the relationship between the inertia model and the traffic-flow state according to the data analysis of the traffic-flow state. A simulation accuracy and prediction accuracy of up to 0.0248 and 0.0273, respectively, are obtained.

#### 1. Introduction

Traffic-flow theory is the basic theory of the intelligent transportation system, that is, the use of mathematical and mechanical laws to study the laws of road traffic-flow theory [1]. By analyzing the relationship between the parameters of the traffic-flow system, it can seek to establish the most rational model to analyze the changes in traffic flow [2], providing a theoretical basis for rational planning and efficient traffic management. The study of traffic-flow theory promotes the interdependence and interaction of dynamics, applied mathematics, fluid mechanics, and traffic engineering.

Traffic-flow short-term forecasting for the intelligent transportation system to provide traffic information is an important basis for traffic analysis [3, 4] and control [5]. Short-term traffic-flow forecasting has been widely researched by scholars at home and abroad, who have obtained many research results [6–8], and many theories and methods [9, 10] have been applied to the study of short-term traffic forecasting. The results of this study can be applied directly to the advanced traffic information system and traffic management system, which can provide real-time and effective information for walkers, realize the route planning, reduce the travel time of the traveler, alleviate road congestion, reduce pollution, save energy, and so on. Traffic-flow forecasting is also based on the dynamic acquisition of traffic-flow time-series data to predict the future traffic-flow status data.

The traffic-flow characteristics can be described by the traffic-flow state, and the traffic flow exhibits different characteristics in different states. In the study of urban traffic-flow parameter models, the traffic state is divided into free flow, congested flow, and jam flow. Usually, the traffic-flow rate, speed, and occupancy rate are considered as parameters of the resulting traffic state. The interval and forecast period of time-series data for short-term traffic-flow parameters are shorter, usually within 15 minutes. There are many methods for short-term traffic-flow forecasting: chaos theory [11], time series [12, 13], neural networks [14, 15], nonparametric regression [16, 17], gray prediction [18], and other methods [19, 20].

However, the short-term traffic-flow system has a large degree of similarity with the fluid system with respect to basic dynamic characteristics and, at the same time, a high degree of uncertainty. It is difficult to accurately grasp the roles of the system factors and mechanisms due to poor information. If the time interval for collecting traffic is 5 minutes, only 12 groups of data are obtained in one hour, resulting in a small sample size. Therefore, it is reasonable to study the inertial gray model by using the gray system with less data and a poor information system combined with the mechanical properties of traffic-flow data.

The gray system theory was put forward by Deng [21]. After 30 years of development, a framework of system analysis and evaluation [22], model prediction [23–25], and decision control [26] has been established as the main technical system. The gray prediction model is the core component of gray system theory. Since its introduction, the gray prediction model has been widely studied and continuously developed and optimized [27–30]. GM, as the core gray prediction model, has also been improved [25, 31, 32] and has been widely used in various fields. However, in practical applications and the theoretical research process, GM is not fully suitable for fitting homogenous exponential series. The problem of transforming the GM from discrete form to continuous form is solved by the proposed discrete gray model [33]. At the same time, many scholars have extended the properties and optimization of model parameters. However, the discrete gray model, like the classical GM, can only solve the problem of exponential growth order, and sequences with exponential growth are very rare in real life; comparatively speaking, more original sequence data conform to nonexponential growth laws. The discrete gray model of the approximate nonhomogenous exponential sequence extends the application range of the discrete model to approximate nonhomogenous exponential sequences [34], which enhances the applicability of the discrete gray model.

However, the gray prediction model of the classical GM, discrete gray model (DGM), and approximate nonhomogenous discrete gray model (NDGM) is used as the modeling mechanism of the least-squares method. These models do not describe the modeling process from the point of view of the data. Professor Deng proposed the inertia GM in [21], emphasizing that inertia is the quality of the material mass of a temperament, which is the abstract amount that has to be considered when researching material movement and thought movement. At the same time, he suggested that the data are generated by the thought movement in [35], and the value of the thinking process is much greater than the value of a certain function. It can be said that the number of sequences is the formation of thinking or things and that the sequence in different minds and the processes of forming different things have different meanings. He noted that, in the GM, can reflect the velocity in mechanics, the accumulating generation operator (AGO) can reflect the deposition of this process, and can reflect the background, while at the same time representing the inertia GM from the force resolution of the data. Traffic-flow theory and fluid systems have a high degree of similarity, as they both have the same basic dynamic characteristics.

Therefore, this paper introduces the basic concepts and properties of mechanics distance, acceleration, force combination, and resolution in traffic-flow data and studies the inertia model, which is adapted to short-term traffic forecasting. At the same time, the model structure clearly shows the formation process of the INDGM using understandable structure parameters and manifestation of component parameters. This model is also closely related to the classical NDGM, and the modeling mechanism of the classical NDGM from the mechanical decomposition of the data is illustrated. Finally, the paper analyzes the state data of traffic flow and appropriately selects the inertia model for traffic-flow data of Whitemud Drive in Canada, which can effectively improve the simulation and forecasting effect of short-term traffic flow.

This paper is organized as follows. In Sections 2, the basic concepts and properties of mechanics in the data vector are introduced. In Sections 3, the NDGM is introduced; the INDGM is put forward using the mechanics decomposition of the data, and an important property of the model is studied. In Sections 4, traffic-flow data from Canada is used for the fitting analysis in the empirical study. The conclusion of this study is discussed in Section 5.

#### 2. Basic Concepts and Properties of Mechanics in the Data Column

Inertia is the temperament of the mass quality of the reaction material, and it is also the property of the energy system. The social, economic, technical, military, transportation, ecology, and agriculture are generalized energy systems. This section mainly introduces the basic concepts and mechanical properties of the mechanics in the data column.

*Definition 1 (see [21]). *Regarding distance, the following definitions are given:(1)The measure of the position difference between two points is called distance.(2)Let be a proposition. The distance measure under is the journey length.

*Definition 2 (see [21]). *Assume that the sequence is an original data sequence under equal-interval proposition . If , the sequence is an acceleration sequence under equal-interval proposition . Let be the symbol of data ; that is,Then, let be a nonnegative real number. is the number consistent with if the following are satisfied:(1),(2),(3),(4) = Value , for ,where = value under criterion .

Let be the proposition. The distance measure under becomes as follows:(1) is called the incentive coefficient of the sequence ,(2), which is the inverses of , is called the inertial coefficient of the sequence , where ,(3) is called the external force of the sequence at the th moment (zone).

*Definition 3 (see [21]). *Let be the original sequence and let be the acceleration sequence of .(1)If the external force of at point satisfies , then . This relationship is called the unit-incentive relationship, and the external force sequence under is called the external force sequence of unit incentive.(2)If under the criterion , that is, , then we call the relationshipthe incentive relationship corresponding to the incentive external force sequence .

*Definition 4 (see [21]). *Let be the original sequence and let be the acceleration sequence of . ThenLet be the incentive and let be the transformation , whereand then is called the force transformation of data and

*Definition 5 (see [21]). *Let be the original sequence and let be the acceleration sequence of . Suppose that the external force sequence under incentive satisfiesthen(1) is called the first-order combination of , where ,(2) is called the second-order combination of , where (3) is called the third-order combination of , where (4) is called the force space ( for short), where is called the first-order force element, is called the second-order force element, and is called the third-order force element.

*Definition 6 (see [21]). * of the original sequence , (Accumulating Generation Operator), and average sequence *,* addressed as all order element, constitute the decomposition transform of data.

*Definition 7 (see [21]). *Let be the original sequence, let of be , and let the average sequence of be ,Then one has the following:(1)If transformation satisfies , then the inverse transformation is called the decomposition transform of .(2)If transformation satisfies , then the inverse transformation is called the decomposition transform of .(3)If transformation satisfies , then the inverse transformation is called the decomposition transform of .(4) is called the force-transformation set.If satisfies , then the inverse transformation is called the decomposition transform of .

According to [21], theorems related to the force-decomposition transform can be stated as follows.

Theorem 8 (see [21]). *(1) Let be the decomposition transform of ,Then**(2) Let be the decomposition transform of Then*

#### 3. Inertia Nonhomogenous Discrete Gray Model (INDGM)

The gray model is one of the core components of gray system theory. It is characterized by its simple calculation, which is superior to traditional prediction methods. The nonhomogenous discrete gray model (NDGM) is constructed based on the approximate nonhomogenous index trend. This section introduces the relevant information about the NDGM and inertia nonhomogenous discrete gray model.

##### 3.1. Nonhomogenous Discrete Gray Model (NDGM)

Assume that the sequenceis an original data sequence. is an 1-AGO sequence,where

*Definition 9. *Assuming that sequences and are defined as in (15) and (16), respectively, the equationsdefine the first-order gray system-prediction model including a variable, referred to as the NDGM [34]. Then the recurrence function is defined as follows:

##### 3.2. Parameter Space of the NDGM

Assuming that sequences and are defined as in (15) and (16), respectively, the equationis called the acceleration sequence of the original sequence, where Thenis called the external force sequence of the original sequence, where . Letwhere and are, respectively, first-order external force combination and second-order external force combination of . Force decompositions of are as follows:

*Definition 10. *The NDGM can be defined by Definition 9. Then one has the following: (1) is called the principle parameter space of NDGM, are called principle parameters, and is the I-order parameter packet of the NDGM.(2)Let , let be structure parameters of NDGM, and let be a structure parameter model or structure model.(3)Let where are called component parameters of NDGM, is called the component-parameter space or component-parameter model of NDGM.Then, second-order parameter of the component-parameter space is solved by substituting (23) into , which yieldswhere ;where ;where ,where ;where ;where

Substituting (23) into (24) for the structural parameters yieldsSimilarly,

##### 3.3. Inertia Nonhomogenous Discrete Gray Model (INDGM)

Assuming that sequence is defined by (15), the acceleration sequence is defined as follows: and the acceleration sequence can be represented as is the external force of original sequence on point . Let .

The incentive coefficient is , and the inertia coefficient is . At the same time, Based on (32)–(35), Definition 11 can be given.

*Definition 11. *LetwherewithThen(1) is called the competency model of NDGM;(2) is called the first-order inertia (incentive) model of NDGM, denoted as the FINDGM;(3) + is called the second-order inertia (incentive) model of NDGM, denoted as the SINDGM;(4) + is called the third-order inertia (incentive) model of NDGM, denoted as the TINDGM.

##### 3.4. Properties of the INDGM

Theorem 12. *The competency model and the FINDGM of the NDGM inertia model do not exist.*

*Proof. *Definition 11 and (29)–(34) can be substituted into the expressions for and . Then . From Definition 11, parts (1) and (2), it is clear that the competency model and the first-order inertia model of the NDGM inertia model do not exist.

Theorem 13. *The TINDGM is equivalent to the NDGM.*

*Proof. *From the definition of the structure parameters of the NDGM, From (32)–(35), the TINDGM is equivalent to the NDGM, which is equivalent to the NDGM. Thus, TINDGM is equivalent to the NDGM.

From Definition 11, the four INDGM are named according to the magnitude of the exponent of the inertial coefficient. The exponent of the inertia coefficient is closely related to the mechanical decomposition of the data. Meanwhile, from Theorem 12, the competency model and the FINDGM of the inertia NDGM do not exist, and from Theorem 13, the TINDGM is equivalent to the NDGM. Thus, from the point of view of the mechanical decomposition of the data, the classical NDGM is an evolutionary process with four inertial models, from the competency model, the FINDGM, and the SIDGM to the TINDGM; however, the FINDGM and SINDGM do not exist, which implies that there does not exist a mechanical decomposition like this for the NDGM. At the same time, the modeling mechanism of the NDGM is obtained according to the data mechanical-decomposition process of the exponent of the inertia coefficient from low to high.

Theorem 14. *The restored value of the INDGM iswhere**The same sequence has different meanings under different thoughts and processes of things. At the same time, the data have different mechanical decompositions, depending on the data source. Next, we examine whether changes in the inertia coefficient for the same data affect the accuracy of the model.*

Theorem 15. *For the same sequence and different decompositions of the data, the second-order parameters of the component parameters and structure parameters of the model will change, but the values of the model parameters are invariant; that is, the simulation accuracy of the model is not influenced by the choice of decomposition.*

*Proof. *Assume that sequence has an incentive coefficient of the form . Then under two different incentive coefficients, Therefore, the force element satisfies ,Thus, the first-order combination of the force element satisfies ; and It follows that the second-order combination of the force element satisfies .

Now, assuming that the component parameters of inertia exponent are the component parameters of inertia exponent are , where . From (26),Comparing (50) with (51), (48) impliesSimilarly,By (51), we haveSimilarly,(1)Competency model does not include , so does not change with changes in .(2)The FNDGM is , where for and , , substituting (54)-(55) into yieldsThus, does not change with changes in the inertia parameter. Similarly, it can be shown that do not change with changes in the inertia parameter.

By substituting (54)–(59) into the SINDGM and TINDGM, it can be similarly proved that model parameters , do not change with the model inertia parameter . Therefore, the form of the mechanical decomposition of the data does not affect the accuracy of the model.

According to the modeling mechanism of the proposed INDGM, the flow chart of the new model is presented in Figure 1.