Temporal Evolution of Short-Term Urban Traffic Flow: A Nonlinear Dynamics Approach

Abstract

Abstract:  Recognizing temporal patterns in traffic flow has been an important consideration in short-term traffic forecasting research. However, little work has been conducted on identifying and associating traffic pattern occurrence with prevailing traffic conditions. We propose a multilayer strategy that first identifies patterns of traffic based on their structure and evolution in time and then clusters the pattern-based evolution of traffic flow with respect to prevailing traffic flow conditions. Temporal pattern identification is based on the statistical treatment of the recurrent behavior of jointly considered volume and occupancy series; clustering is done via a two-level neural network approach. Results on urban signalized arterial 90-second traffic volume and occupancy data indicate that traffic pattern propagation exhibits variability with respect to its statistical characteristics such as deterministic structure and nonlinear evolution. Further, traffic pattern clustering uncovers four distinct classes of traffic pattern evolution, whereas transitional traffic conditions can be straightforwardly identified.

1 INTRODUCTION

Current approaches for modeling and forecasting (short-term) traffic are based on the straightforward argument that traffic variables (e.g., volume, occupancy, and speed) exhibit dependence on previous measurements; as such, previous information—that is, measurements from previous time intervals—can be used to model and generate a variable's subsequent value(s). This approach prevails in most traffic prediction research (Davis et al., 1991; Clark et al., 1993; Kwon and Stephanedes, 1994; Hamed et al., 1995; Smith and Demetsky, 1997; Williams et al., 1998; Park et al., 1999; Zhang, 2000; Williams, 2001; Abdulhai et al., 2002; Stathopoulos and Karlaftis, 2003; Ishak and Alecsandru, 2004; Jiang and Adeli, 2005; Xie and Zhang, 2006).

Traffic flow's temporal dependence is crucial to the effectiveness of a prediction system regarding its adaptability to shifting traffic conditions (Smith and Oswald, 2003). A system that has “captured” traffic's evolution, in essence “knows” the traffic patterns, is more likely to predict accurately, and adjust to variable traffic conditions (the term traffic pattern describes the manner in which sequential measurements propagate in time and space—Kerner, 2004a). This improvement in prediction stems from the realization that temporal patterns in traffic data are related to traffic flow's multiregime and transitional behavior. This behavior has been reported in previous studies for both freeways and signalized arterials (Chowdhury et al., 2000; Helbing, 2001; Nagatani, 2002; Kerner, 2004b), and has been associated with queue formation and dissipation (Vlahogianni et al., 2007).

A number of earlier studies have used data-driven approaches to uncover patterns before predicting traffic variables (Danech-Pajouh and Aron, 1991; Van Der Voort et al., 1996; Chen et al., 2001; Yin et al., 2002; Ishak and Alecsandru, 2004). These approaches combine a layer of pattern clustering and input dimensionality reduction with a subsequent statistical or neural network technique to generate predictions. Recently, the temporal patterns of traffic volume were studied using wavelets (Jiang and Adeli, 2004); the idea was to decompose a time series of volume into a set of subseries and obtain a simpler temporal structure. The literature suggests that traffic propagates in patterns and that, when basing the analysis on patterns, predictions are improved compared to those from other prediction algorithms (Ishak and Alecsandru, 2004). However, the effectiveness of these approaches has only considered mean predictive accuracy while disregarding the manner in which traffic's temporal patterns are related to prevailing traffic conditions as well as to the conditions under which shifts in patterns occur.

A question that naturally arises is how to distinguish the statistical behavior of traffic patterns with respect to perceptible/measurable traffic information; further, it is important to identify the boundary conditions associated with the transitions in traffic pattern-based evolution. A system that can identify and relate statistical behavior to traffic conditions should be both accurate, as it is asked to generate predictions based on a clear statistical behavior, as well as adaptable, as it can adjust the predictions to the changes in traffic conditions and particularly to extreme event occurrence (incidents) or the onset of congestion. In this article, we provide a statistical framework for clustering traffic flow conditions (joint consideration of volume and occupancy) with respect to the statistical behavior of traffic flow's pattern-based evolution. The basic approach employed is illustrated in Figure 1; based on raw volume and occupancy data, the statistical characteristics of traffic's evolution are identified and then associated to prevailing traffic flow conditions.

Figure 1

Flow diagram of the proposed multilayer traffic pattern recognition system.

2 IDENTIFYING  TRAFFIC  PATTERNS

Identifying traffic's temporal patterns involves reconstructing traffic flow's evolution in the phase–space through the series of volume and occupancy; phase–space is a vector space such that a point in the space of the system specifies the state of the system and vice versa (Kantz and Schreiber, 1997). In this way, the evolution of a system is studied through the temporal evolution of the points in the specific vector space. Consider a traffic volume V{t} and occupancy O{t} series that are unfolded (or embedded) in a multivariate phase–space that is representative of the original traffic system. The vectors created have the following form:

(1)

Each of the above vectors is characterized by two parameters: the embedding delay (τ) and the embedding dimension (m) (Kantz and Schreiber, 1997). In traffic, embedding procedures involve recreating the manner in which traffic flow propagates (the pattern of traffic) by representing traffic variables such as volume, occupancy, and speed in the vector-space (one can follow the evolution of traffic by studying the trajectories of the unfolded traffic parameters in the vector-space). These parameters define the look-back time window [(m− 1)τ] that contains useful information for prediction purposes.

A common approach for determining the embedding delay τ is the first local minimum of the mutual information function as proposed by Fraser and Swinney (1986). Literature indicates that mutual information is more robust than the autocorrelation function in revealing both linear and nonlinear relationships in data (Abarbanel, 1996). The selection of the minimum embedding dimension m of the reconstructed phase–space is based on the false nearest neighbor algorithm that examines the behavior of near neighbors under changes in the embedding dimension from m to m+ 1 (Kennel et al., 1992); both techniques are applied in traffic volume series and discussed in detail in Vlahogianni et al. (2006).

2.1 Analysis of traffic's short-term temporal evolution

Traffic flow studied through the time series of its variables (volume, occupancy, and so on) encompasses dynamic characteristics. Traffic's temporal evolution can be described by its structure (deterministic or stochastic) in a time window of study and by the evolution of its structure in time (Kantz and Schreiber, 1997). The structure describes the (geometric) relation of a sequence of traffic states (pattern) in the study window. The structure is characterized by its evolution in time and thus can be persistent in time, cyclic, and so on.

Sequential traffic states can be close, suggesting that traffic has a recurrent behavior, or can be spatially (geometrically) far. The geometric closeness depends on a predefined threshold of distances ɛ_i under which states are said to be recurrent (Eckmann et al., 1987); recurrence of a state in the phase–space is given as (Marwan et al., 2007):

(2)

where N is the number of states x_i in the time window of study, m is the embedding dimension, ɛ is a threshold of distances, and ∥·∥ a norm (in our case the Euclidean norm). Equation (2) provides a matrix of recurrences known as the recurrence plot (RP) and is the basis for recurrent quantitative analysis (RQA) proposed by Zbilut and Webber (1992).

RPs can be extended to a bivariate formulation known as the cross-recurrence plots (CRPs). In the case of traffic flow, when both volume and occupancy are jointly investigated, Equation (2) becomes:

(3)

Equation (3) describes the coupling of volume and occupancy in time; coupling—the cross-correlation of volume and occupancy—has been a focal point of the short-term traffic flow forecasting literature (Persaud and Hall, 1989; Stathopoulos and Karlaftis, 2003; Kamarianakis and Prastakos, 2003; Vlahogianni et al., 2005). The proposed approach for identifying the characteristics of traffic flow pattern evolution provides a framework for treating nonstationary and nonlinear processes, well established in several fields of research such as biology, physics, and economics (Zbilut, 2006). Cross-recurrence quantitative analysis (CRQA) is considered as more robust for studying couplings than conventional statistical approaches such as linear cross-correlations (Shockley et al., 2002; Zbilut, 2004), because it is independent of constraining statistical assumptions and limitations, filtering, linear detrending, and data transformations (Zbilut, 2004).

CRQA is based on the density of recurrent points of the joint consideration of volume and occupancy in the time window of study (Zbilut et al., 1998). From recurrent states, some occur in a predetermined manner (deterministically), or stochastically (isolated recurring states) (Gao and Cai, 2000); the deterministic states form parallel structures in the cross-recurrence plot. Moreover, the length of these structures determines the degree of nonlinearity in the system. Following the mathematical formulations of Marwan et al. (2007), consider a sliding time window W_T of N measurements of traffic variables—for example, volume and occupancy—updated every T, and let R be the percentage of recurrent states:

(4)

where N is the number of states in the window of study and R_i,j is estimated from Equation (3). In time window W_T, the deterministic structure of traffic flow is quantified by the %DET statistic, which is the percentage of points that form diagonal lines parallel to the main diagonal in the recurrence plot:

(5)

where l is the length of the line parallel to the main diagonal, with l_min the minimum threshold under which a line is considered as a deterministic structure (usually equal to 2 as explained in Webber and Zbilut, 2005) and P(l) is the frequency distribution of the lengths l of the diagonal structures in the CRP: P(l) ={l_i; i= 1, 2, …N}.

Traffic's temporal evolution is related to the persistence of traffic flow's structure in the selected time window; if the structure persists in the study window, a cyclic behavior can be assumed whereas, in the opposite case, the structure is nonlinear (Kantz and Schreiber, 1997); in such nonlinear systems the deterministic behavior collapses exponentially (Gao and Cai, 2000). The temporal evolution of the deterministic structure of traffic flow is quantified by the variable L_max that equals the maximum duration of the parallel movement in the window of study W_T and can be shown mathematically to be inversely proportional to the largest positive Lyapunov exponent, suggesting that low values for L_max are an indication of chaos (Trulla et al., 1996). Here, L_max is calculated as (Marwan and Kurths, 2002):

(6)

Both the structure and its evolution can be used to specify what is called “patterns in traffic”; they can also be used to identify the statistical behavior of the changes in temporal traffic patterns by following their temporal evolution.

Several points should be made regarding the statistical characterization of traffic patterns. First, the concepts of determinism and nonlinearity are attributes of the pattern-based evolution of traffic flow; this clearly suggests that traffic flow, when studied in the form of a series of volume and occupancy, probably exhibits different behavior than when studied through its patterns. Second, the notions of determinism and stochasticity are based on the temporal evolution of traffic and the geometrical (in the phase–space) relation of traffic states; this suggests that when states are recurrent and “close,” overall patterns can be characterized as deterministic, whereas when states are found to be recurrent but isolated, the overall pattern can be characterized as stochastic.

The implementation of the CRQA involves a procedure based on two stages (Figure 1): (a) the selection of the sliding time window of study, and (b) the quantification of recurrences CRQA. One approach to determine the time window of study is to iteratively search for the optimum sliding time window that, for a fixed threshold ɛ, includes adequate degrees of freedom for statistically analyzing recurrences and calculating recurrence statistics (Zbilut, 2006). For highly nonstationary data, the literature indicates that the values of ɛ for which %R is kept below 5% are the optimal choice (Zbilut et al., 2002; Webber and Zbilut, 2005).

3 TRAFFIC  PATTERN  CLUSTERING

The final stage of the proposed approach involves clustering traffic patterns in groups that have common traffic and statistical behavior; following the previous analysis of traffic dynamics, the available information for clustering is separated into two distinct categories: (a) information on the evolution of the deterministic and nonlinear statistical properties of traffic flow patterns (%DET, L_max), and (b) information on average values of volume and occupancy in the temporal window of study. In cases where outliers or intense fluctuations may distort clustering results, the literature suggests using a two-stage approach to improve on computational requirements and accuracy (Xu and Wunsch, 2005); this two-stage approach is based on the concept of providing an additional level of processing that acts as an information abstraction/data reduction level. This approach is widely used because of its robustness to missing values and to fluctuating data (Lampinen and Oja, 1992).

The proposed clustering approach consists of two levels of processing: (a) Data reduction level, and (b) Final clustering level. The first level involves a data reduction process for representing a four-dimensional space (,%DET and L_max) into a simpler two-dimensional mapping. The proposed two-stage clustering approach based on Kohonen self-organizing map (KSOM) has been found to be more accurate and computationally efficient than conventional partitioning techniques such as k-means (Vesanto and Alhoniemi, 2000; Kuoa et al., 2006). The data reduction process involves the use of a KSOM that consists of an input space where traffic temporal patterns are presented in the network in the form of vectors such as and a two-dimensional lattice of output neurons. Its output neuron represents a prototype vector M; the purpose of KSOM's training is to generate a structure of prototype vectors topologically ordered in a two-dimensional map in such a way that the original features of the high-dimensional input space are preserved. During training, patterns are iteratively mapped to the output prototypes; prototypes “compete” among themselves and the “winner” exhibits the smaller Euclidean distance—from x—between all output neurons; the training algorithm is explained in detail in Kohonen (2001). The process is repeated until a certain distance criterion is reached and the position of prototypes in the lattice cannot be modified further.

The second level is the final stage of clustering and involves the implementation of a simple k-means algorithm to cluster the M prototypes by the KSOM. The k-means is one of the most straightforward unsupervised learning algorithms in clustering. As the algorithm exhibits sensitivity to initial cluster selection, it must be run repeatedly (Webb, 2002). Moreover, k-means works with a fixed number of clusters so that the resulting clustering needs to be evaluated in terms of the optimal number of clusters. For this, the Davies–Bouldin index (D–B index; Davies and Bouldin, 1979), a relative index of cluster validity, is calculated for different values of cluster numbers. Small values for the D–B index occur for a solution with low variance within clusters and high variance between clusters; therefore, a choice is made concerning the number of clusters at which this index attains its minimum value (Hruschka and Natter, 1999).

4 IMPLEMENTATION AND FINDINGS

4.1 The data

The available data come from an extended data set of volume and occupancy measurements from arterial links in the center of Athens (Greece); data are collected by loop detectors (mid-block) located 90 m from the stop-line. To demonstrate the abilities of the chosen methodology, from the extensive arterial network, a single major arterial of 1.1 km length is extracted that is controlled by three loop detectors as can be seen in Figure 2. The specific area experiences significant inflows and outflows, as well as uncontrolled demand (mid-block or side street traffic) inducing complexity to the distribution of traffic flow along the arterial links. Prior to the analysis of traffic flow patterns, a simulation was conducted to test the efficiency of current coordinated signalization plans. The efficiency was judged based on the portion of upstream traffic in the busiest part of the cycle (Husch and Albeck, 2004); values up to 45 designate that flow arrivals are uniformly distributed across cycle, whereas values near 100 indicate greatly platooned traffic flow. A relevant efficiency of the existing signal coordination plans was observed (portion of upstream traffic in the busiest part of the cycle equals on average 64) indicating that, despite the existence of substantial inflows and outflows of traffic, the roadway maintains a relatively smooth operation suitable for flow estimation testing and prediction.

Figure 2

Schematic representation of the set of arterial links under study.

However, due to the existence of several uncontrolled intersections between the main signalized intersections in the study area, as well as uncertainty regarding signalization's synchronization with the traffic flow measurements, information regarding the signalization was discarded from the analysis of traffic patterns. Knowledge regarding the manner in which traffic evolves in the study area stems from traffic volume and occupancy measurements per 90-second intervals (average cycle length); these variables will be used to identify traffic flow's pattern-based evolution.

The proposed approach to identifying traffic flow's pattern-based evolution is purely data driven without considering signalization phases. This suggests that our primary focus is on the joint consideration of volume and occupancy, whereas data requirements focus on the completeness of the occupancy–volume relationship particularly with regards to different traffic flow conditions. Figure 3 depicts the volume and occupancy time series for a typical weekday in the study area. As can be observed, time series from all three locations under study exhibit temporal variability. Interestingly, the series of traffic volume have, on average, similar temporal evolution, whereas occupancy's temporal evolution differs among the three control locations.

Figure 3

Time series of volume and occupancy for three sequential locations during a typical day.

The next step is to reconstruct volume and occupancy series by applying the mutual information and the false nearest neighborhood algorithms. The reconstruction process is the following: mutual information and false nearest neighborhood algorithms are first applied to various 1-day time series of volume and occupancy; then, the same approach is used to shrinking time windows until the series reaches a 1-hour duration. The resulting values of embedding delay and dimension characterize the pattern of traffic flow within a sliding hour (Equation (1)). The above process yielded τ= 1 and m= 5 as the embedding parameters for both volume and occupancy, suggesting that the dynamics of traffic should be studied—following Equation (1)—through the following volume and occupancy vectors:

(7)

4.2 Statistical characteristics of temporal traffic patterns

In this case study, an iterative preliminary process of CRQA analysis in time window W_T of different durations (30 to 90 minutes) was implemented; for this we used threshold values ɛ_i ranging from 20% to 40% of the mean distance separating traffic flow states in the reconstructed phase–space, to select the optimum values of W_T and ɛ_i (the criterion was to stabilize (%)R in low levels (∼5%) and provide valid statistical recurrences). This preliminary process shows that the minimum extent of the time window of study W_T is 1 hour; this suggests that the deterministic and nonlinear characteristics of traffic flow should be studied in a 40 × 40 recurrences matrix (Equation (4)) updated every one interval.

The resulting series of %DET and L_max provide a measure of traffic's temporal evolution in the selected time window. The use of a sliding window in which the values of %DET and L_max are calculated, also encompasses information on transitions in traffic because it provides constantly updated traffic information (windows “sliding” in time). Figure 4 depicts the series of occupancy (O) volume (V), mean occupancy (mean O), and mean volume (mean V) in the sliding windows W_T, as well as of %DET and L_max. Variable behavior is observed for the variables that describe traffic pattern-based dynamics (%DET and L_max).

Figure 4

Series of mean values of occupancy (%) and volume (veh/90seconds),%DET, and L_max for every 1-hour sliding window updated every 90 seconds (typical day).

4.3 Two-level clustering implementation and validation

The first level of traffic clustering was to develop a KSOM to produce the set of prototypes. The map developed is a 2 × 90 grid, suggesting 180 prototype neurons of the following form: w_j=[meanV_j, meanO_j, % DET_j, L_maxj]. The network was trained in two adaptive phases: the self-organizing and the fine-tuning phase. During self-organization, the network was trained with a learning rate η(n) of initial value η₀= 0.1 decreasing to 0.01. The initial neighborhood width σ₀ was set to 14. During fine-tuning, the learning rate was kept low (about 0.01) and the neighborhood width was decreased to 1; average quantization error was 0.03.

In the second level, the prototypes produced were clustered by a k-means algorithm. As k-means needs a prefixed number of clusters to be applied, 11 different values of C_i (2 ≤i≤ 12), where C_i is the number of clusters, were calculated and a clustering validation procedure was undertaken. The calculated D–B validity index (Davies and Bouldin, 1979) for each value of C_i indicates that i= 4 provides optimal data partitioning. Further, the two-level clustering technique was tested against two other clustering strategies: (a) a simple k-means algorithm using only statistical information in the form of %DET and L_max and (b) a simple k-means algorithm using both statistical (%DET, L_max) and traffic information (mean values of volume and occupancy in the sliding study windows W_T). The Davies–Bouldin validity was calculated for the two additional strategies; results from all three strategies are shown in Figure 5 (i= 4 gives again optimal partitioning for all strategies). The partitionings provided by the three tested clustering approaches are compared via the relative conditional entropy (Vesanto and Alhoniemi, 2000). The calculation of the entropy describing the uncertainty in between different partitionings indicates that the proposed two-level approach to clustering traffic patterns is “better”—in terms of the information provided to the knowledge of clusters—than the other strategies tested.

Figure 5

Three applied clustering strategies.

Results for the four identified regions of traffic flow with respect to the statistical characteristics of traffic evolution are demonstrated in Table 1; further, Figure 6 depicts the time series of volume (vehicle/hour) and mean occupancy (percentage) according to the traffic flow area volume and occupancy measurements they belong to. It is evident that when flow settles at high levels of volume and occupancy its evolution seems to be stable in terms of frequency of transitions. On the other hand, in high values of volume and medium values of occupancy, traffic flow has a variable evolution with respect to each statistical characteristic; this evolution is decomposed into two distinct overlapping traffic flow areas.

Table 1.
Summary table of basic area characteristics of resulting patterns

Figure 6

Time series of traffic volume (veh/hour) and occupancy (%) for location 3 during the 6 a.m. to 3 p.m. time period.

4.4 Traffic areas of characteristic statistical behavior

From a traffic perspective, the results of the clustering process are of interest; first, the three basic traffic areas of a volume-occupancy diagram can be identified: free-flow conditions (unqueued conditions—area I), synchronized conditions (area II and III) that reflect traffic flow evolution near capacity where occupancy rises fast whereas volume stabilizes at high oscillating values, and congested conditions (area IV) where occupancy and volume oscillate at high values. According to the numerical results presented in Table 1, although free-flow and congestion demonstrate clear statistical behavior (cyclic weakly deterministic and cyclic strongly deterministic, respectively), the area in the middle is divided into two subareas with diverging statistical behavior; area II exhibits a stochastic structure whereas area III has a deterministic structure, with both areas exhibiting strongly nonlinear characteristics.

Interestingly, when traffic approaches congestion, it demonstrates patterns that are unstable. The strongly nonlinear characteristics are indicative of the oscillating nature observed by the series of flow and occupancy; instability and chaotic-like behavior have also been reported by previous studies based on stochastic microscopic traffic flow modeling through a sequence of traffic lights with both fixed and irregular signalization characteristics (Nagatani, 2005, 2008). In this article, although the reasons why such traffic behavior occurs cannot be readily identified, two structures are reflected; first, there is the deterministic structure that dominates the area of synchronized flow; second, a stochastic short-term evolution in synchronized flow can be identified, probably encompassing nonrecurrent incidents. Further, the observed instability of traffic flow near capacity is indicative of the considerable transitional nature of traffic flow near congestion.

An important insight gained by clustering traffic patterns is that transitional conditions can be quantitatively characterized. Boundary traffic volume and occupancy values along with information on the %DET and L_max statistics can lead to recognizing patterns in traffic; for example, in the specific study area, observing the resulting clustering reveals that congestion is reached through a nonlinear deterministic behavior (area III). On the other hand, traffic flow “leaves” free-flow conditions either through a nonlinear deterministic manner (and moves to area III) or a sudden stochastic shift (and moves to area II). Moreover, the study of traffic's propagation between areas can uncover the duration of various traffic phenomena. For example, congestion for the urban arterial under study is found to last, on average, about 45 minutes; transitive states near congestion last much less, with mean duration in area III at approximately 12 minutes. Moreover, transitions between areas III and II are frequent, with a shift approximately every 7.5 minutes and maximum time period between shifts 45 minutes.

Finally, Figure 7 depicts the resulting traffic flow areas in the volume–occupancy relationship in the three locations of interest during the 4-hour morning peak period. As can be observed, traffic flow in each location has a distinct temporal behavior; traffic flow in location 1 seems to exhibit intense shifts to extreme traffic flow conditions compared to the dynamics of downstream locations. The dependence of each location on the near upstream or downstream location is a critical issue for further study due to its immediate effects on the implementation of short-term forecasting algorithms.

Figure 7

A graphical representation of the resulting traffic clustering in the volume–occupancy relationship for the three locations under study.

5 CONCLUSIONS

Short-term traffic forecasting practice has, for some time, indicated the need to uncover and model patterns of traffic to improve on its predictions. However, previous efforts on the subject did not uncover traffic patterns with respect to their statistical behavior or the conditions under which they occur. In this article we offered a methodology for identifying different traffic flow patterns, the traffic conditions under which these patterns occur, and their temporal evolution characteristics. Results from the analysis of traffic's temporal evolution indicate that traffic pattern propagation exhibits intense variability with respect to its statistical characteristics such as deterministic structure and nonlinear evolution. Further, clustering of traffic patterns uncovers four distinct classes of traffic pattern evolution.

The proposed multilayer traffic pattern recognition strategy presents several interesting features:

• 1

  Multivariate modeling (joint consideration of volume and occupancy) of traffic variables;

• 2

  Identification and statistical characterization of the temporal pattern-based evolution of traffic flow;

• 3

  Generation of clusters of traffic patterns that exhibit similar statistical characteristics with respect to their temporal evolution;

• 4

  Identification of transitional traffic conditions where a shift in pattern occurs.

The proposed approach can be considered as transferable to the degree that its data-driven nature permits; given a new set of data that reflects new topological/highway characteristics, the proposed approach can reveal and numerically approximate traffic patterns and traffic conditions for the new area under study. Finally, the proposed approach is purely dynamic both in its concept as in its operation; it can result in a recursive procedure for recognizing the dynamic dependence of traffic (the traffic pattern) in any time interval. This is an important feature that needs to be considered when deploying dynamic prediction structures operating in real time.