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In recent times, urban road networks are faced with severe congestion problems as a result of the accelerating demand for mobility. One of the ways to mitigate the congestion problems on urban traffic road network is by predicting the traffic flow pattern. Accurate prediction of the dynamics of a highly complex system such as traffic flow requires a robust methodology. An approach for predicting Motorised Traffic Flow on Urban Road Networks based on Chaos Theory is presented in this paper. Nonlinear time series modeling techniques were used for the analysis of the traffic flow prediction with emphasis on the technique of computation of the Largest Lyapunov Exponent to aid in the prediction of traffic flow. The study concludes that algorithms based on the computation of the Lyapunov time seem promising as regards facilitating the control of congestion because of the technique’s effectiveness in predicting the dynamics of complex systems especially traffic flow.

In recent times, urban traffic road networks are faced with severe congestion problems as a result of the accelerating demand for mobility. The excessive congestion in the form of immense traffic jams on urban roads has hindered mobility along these roads. This is one of the major challenges encountered in most mega cities around the world with urban road networks and in turn has a serious effect on road users which includes economic, health, and environmental problem such as vehicle emission and air pollution, arising out of increased fuel consumption during the long periods of congestion. U.S. Bureau of Transport Statistics in 2007 recorded that, due to traffic congestion, Americans residing in urban areas were coerced to travel more 4.2 billion hours and spent about $87.2 billion in purchasing extra 2.8 billion gallons of fuel [

Urban planning and complex traffic network studies have been explored explicitly to potentially mitigate congestion and its associated problems on urban roads. Several efforts and studies have been made in time past by researchers on two major areas that affect urban traffic, namely, traffic flow modeling and prediction and information communications technology which is meant to give guidance to drivers through updated information about their desired routes [

One of the major concerns of traffic managers in traffic management system is traffic volume estimation, a major component of Intelligent Transport System (ITS), as it helps in the decision making and efficient traffic management planning when monitoring the current traffic flows in the road networks. Thus, to reduce the effect of congestion on urban road networks, accurate prediction of the Motorised Traffic Flow as well as traffic estimation is of paramount importance as it provides information on road accidents and level of congestion along the roads [

Based on the reports on experimental data found in literature, traffic flow patterns are highly predictable and often exhibit irregular and complex behaviours which changes abruptly when entering or leaving a congestion zone [

Several methods have been used in time past for short-term traffic flow prediction flows, including ARIMA-type models, Artificial Neural Networks, SARIMA models, Generalised Linear models, Nonparametric Statistical methods, Dynamic Neural Networks, Support Vector Regression models, and STARIMA models just to mention but a few. A brief review of some related work on Traffic Flow Prediction is presented below.

Catriona and Casper in [

Dauwels et al. in [

ARIMA is one of the most precise methods for traffic flow prediction when compared to other known methods. In particular, Seasonal ARIMA (SARIMA) models have been shown to perform better than the other traditional based models but often times it is faced with some restriction in applicability as a result of using huge historical database for model development. Kumar and Vanajakshi, having this background knowledge in their work in [

Previous studies have shown that ANN has stable and consistent performance even if there is an increase in the travel time interval for the traffic flow prediction. This was so evident in [

Chaos Theory is a novel science paradigm with numerous applications that have not been deeply explored and seems very promising with respect to the analysis and prediction of complex systems like traffic flows, although at the moment little empirical evidence exists to confirm this notion. It can be used to analyze the traffic flow patterns in urban road network by utilizing the intrinsic deterministic nature of the traffic flow in order to reduce congestion on urban road networks. In this paper, we report a systematic review of Chaos Theory and propose an approach to predicting Motorised Traffic Flow on Urban Road Networks based on Chaos Theory with emphasis on the Largest Lyapunov Exponent method for prediction, the most common, effective, and direct technique of analyzing the presence of Chaos in a given dynamical system. This work contributes to this research field in the sense that the proposed approach is different from other conventional models found in literature and serves as an alternative method for predicting Motorised Traffic Flow on Urban Networks. Also, the effectiveness of the Largest Lyapunov Exponent prediction method seems very promising in terms of prediction accuracy as well as reducing the congestion problems on urban network, although this is yet to be fully validated using computer based algorithm on empirical traffic flow data.

The layout of this paper is as follows. The congestion problem on urban road networks is introduced in Section

One of the applications of ITS as earlier mentioned is predicting road traffic volumes in order to make efficient traffic management and planning over a network as well as implementing road safety measures. [

Based on the study carried in [

Two consecutive vehicles (a follower-vehicle,

It can be observed that vehicle,

Trajectories of a two-car traffic stream (after [

Figure

In single-lane traffic (microscopic traffic model), vehicles always keep their relative order. However, for multilane traffic (macroscopic traffic model), this principle can no longer be obeyed due to overtaking manoeuvres, resulting into irregular vehicle trajectories. If the same time-space diagram were to be drawn for several lanes (in multilane traffic), then some vehicles’ trajectories would suddenly appear or fade away at the point where there exists a change of lane.

Figure

A time-space diagram showing nonlinear trajectories of several vehicles where movements are bounded by three regions of measurement, that is,

In Figure

Traffic flows are subject to variations over numerous time scales, namely, yearly, monthly, weekly, and daily. It also varies directionally as well as from place to place. Aside the fact that roads carry different volumes of traffic, the characteristics of the vehicles using these roads also change depending on the road facility [

The Federal Highway Administration (FHA) in 1996 reported that most truck travel falls into one of two basic time-of-day patterns, namely, a pattern that is centered on travel during the business hours of a day (working hours) and a pattern that shows almost constant travel all day through (twenty-four-hour day). Figure

Vehicle volume distributions by classification of vehicles in California in 1996 (source: [

As can be seen in Figure

A close observation at Figure

Moreover, time-of-day patterns usually differ significantly with respect to places at any specific location as a result of local trip generation patterns that differ from the norm. For example, a city with night clubs or recreational facilities will generate an abnormal amount of traffic during the night hours or other hours of operation because that city is very active late at night. Also, in heavily congested urban areas, the commute period traffic volume peaks flatten out and can last three or more hours.

The same study also revealed that there exists a large difference in daily patterns of the ordinary vehicle categories and typical trucks since truck travels are mainly business motivated as opposed to ordinary vehicles whose drivers have several travel objectives. Figure

Day of the week traffic variations in California in 1996 (source: [

It is evident from the graph that the day-of-week traffic variations are highly responsible for the traffic congestion that comes in form of jams on urban roads. A good example is the stampede observed along Kwame Nkrumah circle in Accra, Ghana, whose immense traffic jams are estimated to have caused annual losses of about $125 million to travelers along this road in 2014 (monetary value of lost time during traffic jam), as pointed out by traffic experts of the Ghana Institute of Engineers [

To mitigate this problem of congestion on urban roads, it is very necessary to carry out a substantial traffic estimation which requires a method of high precision to forecast a complex entity such as traffic flow. This is the main reason for proposing an alternative way of addressing complex systems like traffic flows, using effective techniques based on Chaos Theory (which studies dynamic systems) to analyse and predict traffic flow patterns.

Several systems exist in everyday life that evolve with time. Such systems are difficult to predict accurately on long-term scale even with robust statistical prediction models. Examples of such system include weather, turbulent fluids (flowing across planes), population infected by epidemic, and stock market indices and they are generally referred to as dynamical systems [

These systems are said to exhibit “Chaos.” Chaos in a simple term refers to any state of confusion or disorder that is showing the absence of some kind of particular order. Many work exists in literature that addresses dynamical systems as well as the chaotic behaviour. Kiel and Elliott in [

A Chaotic system can be described as one that is complex, aperiodic (it never exactly repeats), and sensitive to its initial conditions. Chaos Theory is novel Science paradigm in the field of nonlinear analysis which is used to describe the realms of nonrepeating and highly complex dynamic systems. This discipline is accredited to a meteorologist from the Massachusetts Institute of Technology (MIT), Muhmoudabadi [

Chaotic systems have a number of distinctive characteristics which are used to describe the dynamic evolution of such systems. These characteristics include the following.

Point attractor: a system is said to have a point attractor if the system evolves to a fixed point, for example, a single singing pendulum bob (see Figure

Limit cycle: if the system is cyclic and its position in the cycle can be predicted, then the system is said to have a limit cycle, for example, planetary motions (see Figure

Limit torus: a system that has a limit torus is similar to that of a limit except that the system’s trajectories are bounded within a region of a ring torus; for example, the “halo” ring of planet Jupiter is a torus composed of mainly dust particles in motion (see Figure

Strange attractor: if a system takes an aperiodic irregular shape and never repeats itself in time, the system is said to have a strange attractor. Such an attractor can also be described as a limit region (object with fractional (fractal) dimension) within phase space which is ultimately occupied by all trajectories of a dynamical system. Examples of such strange attractors include the famous Lorenz attractor illustrated in Figure

Different types of attractors constructed in 2-dimensional phase space; (a) point attractor, (b) limit cycle, (c) limit torus, and (d) strange attractor (after [

Mandelbrot’s plot that is self-replicating according to some predetermined rule such that the boundary of the set has fractal dimensions (drawn in a 2-dimensional complex plane) (after [

Other examples of shapes in nature with fractal dimensions include coastlines and slow flakes [

To summarize the properties of Chaotic systems, we note that there are two important characteristics that make chaotic systems very complex and our focus is on these characteristics:

The strange attractor, which contains a large number of unstable system trajectories.

The ergodicity (ergodicity is a system behaviour that is averaged over time and space for all the system’s states) in the dynamics of the system trajectories. In other words, as the system evolves temporarily, a small neighbourhood of every point in one of the unstable orbits within the attractor is visited [

In meteorology, Chaos Theory is used to predict slight changes in weather, air, and aerosol movements in the atmosphere and so forth as studied by Lorenz in the late 1960s [

Although Chaotic systems have good characteristics that are suitable for analysis of complex behaviours, there are significant factors that hinders one from accurately predicting the behaviour of complex system. They include sensitive dependence on initial conditions which are in most cases unknown as most assumptions made often lead to error, the current stage of this “new” discipline of science (just half a century old) as one is not yet very sure of how much data is required to precisely reconstruct phase space and determine the fractal dimension of a given system (discussed in Section

Chaos systems can be controlled in order to reduce computational errors due to the adverse effects of the above limitations. Shewalo et al. in [

Having established the fact that Chaotic systems exhibit nonlinearity property with time evolution in previous sections, we now briefly describe the Chaotic Time Series and how it can be applied in the prediction of dynamical systems such as traffic flow based on the nonlinearity concept.

Phase space dynamics can be used to analyse and make predictions of dynamical systems. Nonlinear processes resulting in higher dimensional objects (called attractors when drawn in phase space) are characterised by nonlinear time series that intrinsically describe the behaviour of the system under study [

One can make prediction for a given time series using phase space techniques which is often referred to as the determinism test of a system. Such techniques are based on the fundamental fact that trajectories in close proximity asymptotically approach each other within the phase space [

Taking a look at traffic systems in particular, they highly depend on human and physical factors in a given road facility and this even becomes more complicated due to the presence of immeasurable quantities such as traffic laws and social codes. Nevertheless traffic flow patterns are deterministic and their time series have been found to be nonlinear [

Since Chaotic systems exhibit a nonlinearity property, developing a Chaos prediction model for a given dynamical system is based on nonlinear time series analyses which mainly involves two steps, that is,

reconstruction of the phase space from a given data set,

developing of a methodology for predicting the phase space dynamics.

Let

A diffeomorphism is a map between manifolds (smooth space system states), which is differentiable and has a differentiable inverse.

This simply means that all the information about the system’s complex

During reconstruction, new space states are created that are (in the sense of diffeomorphisms) equivalent to the original space states so that the relevant geometrical properties of the system are always preserved. The set of reconstructed trajectories,

Therefore the matrix

Now, suppose we have a scalar observed nonlinear times series, say from empirical traffic data,

Illustration time series,

Figure

Parameters (topological parameters) such as the dimension of

Another method of determining

The concept of bin in a histogram will help in understanding how the information is obtained. We define a bin of a histogram intuitively with the following example. The bar graph of a histogram simply shows how many data points fit within a certain range. That range is called the bin (sometimes called the bin width). See Figure

An example of a histogram plot illustrating arbitrary bins,

For instance, suppose we want to plot a histogram graph after counting the number of cars passing through a certain area per hour. Using histogram chart in Figure

Thus, we can compute the above probabilities (

AMI against

To obtain the most appropriate value of

Next, we compute the embedding dimension,

Now, suppose 2 points,

In the same way,

The authors of [

Consider Figure

A 2-dimensional plot of the Hénon attractor showing

The above procedure is repeated for all possible pairs of points in dimensions of ascending order until the fraction of FNN drops to zero (or gets close to zero), a process usually termed as “unfolding” of the attractor. The percentage of FNN should drop to zero when the appropriate embedding dimension,

For a given dynamical system such as traffic flow, a suitable value of

Plot of the percentage of FNN against embedding dimension,

Normally, the value of

We now discuss the different methodologies for prediction of Chaotic system’s behaviour having discussed the topological parameters of the attractor.

Literature suggests that it is very necessary to check for Chaos in a given data set before predictions are made. The reason for the check is that there might be presence of random data, which are often assumed to be chaotic, in the data set.

There are several methods used to test for Chaos in a time series data set of a dynamical system. The following methods covered in this work were briefly discussed. They include computation of the (i) Correlation Dimension,

The Correlation Function,

If the time series is characterized by an attractor, then

Plot of the Correlation Dimension,

If

Now we define an upper bound for the

The Hurst Exponent,

If a time series vector,

Therefore, one can determine whether the time series data is randomly distributed or not. This is obtained through the square root relation between increments after a certain time interval as follows:

Reference [

For

Having established that the exponential divergence of nearby trajectories is the hallmark of Chaotic behaviour as explained by [

If the exponents are arranged in descending order such that

The length of the principle axis of spectrum is proportional to

The area determined by 2 principle axes is proportional to

The volume of the first

To understand the above relationships, we compute the he Euclidean distance between 2 points in phase space. Suppose that originally we have 2 points in phase space that is

Our focus is mainly on the Largest Lyapunov Exponent (LLE),

A number of

A plot of the Stretching Factor,

The least squares approach gives a smooth line (fit) on Figure

If

The Lyapunov time,

In the same way, if

Practically in traffic flow analysis, the one-dimensional traffic flow time series data is replaced with

This study have shown how Chaos Theory can be used in the analysis of dynamical systems via a systematic review of the characteristic features of Chaotic system. In particular, it showed how Chaos Theory can be used for Motorised Traffic Flow Time Series Prediction in Urban Transport Network based on the the method of computation of the Largest Lyapunov Exponent,

In order to make a complete and robust prediction model for traffic flow, there is need to develop a computer based algorithm that will compute the time delay, embedding dimension, and Lyapunov time of a real time series from empirical traffic flow data. Thus, the validation aspect of the proposed approach and comparison with other known conventional models for traffic flow prediction especially in the area of prediction accuracy is still in progress and left for our future work so as to enable us have access to available traffic flow data sets. Moreover, there is need to come up with a concrete relationship (most preferably a mathematical equation) that links the Lyapunov time with traffic flow so as to aid in proper traffic predictions. The effect of noise on traffic flow data as well as determining the type of noise and magnitude is also an important area to look into in our future work. Thus, by effectively incorporating all these into the present work, it is believed that the proposed approach will be useful in reducing the congestion problem on urban traffic networks.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The support of Dr. Abbas Mahmoudabadi of the Road Maintenance and Transport Organization, Tehran, Iran, is appreciated. The financial support of the African Institute for Mathematical Sciences (AIMS), Senegal, is hereby acknowledged.