neural network for gap acceptance at stop‐controlled intersections

N E U R A L N E T W O R K F O R G A P A C C E P T A N C E AT

S T O P - C O N T R O L L E D I N T E R S E C T I O N S

By Prahlad D. Pant, ~ Member, ASCE, and Purushothaman Balakrishnan 2

(Reviewed by the Highway Division)

ABSTRACT; The behavior of gap acceptance by vehicles at intersections with stop signs involves the complex interaction of numerous geometric, traffic, and envi- ronmental factors. Several methods, including empirical analysis, and theoretical, logit, and probit models have been used to estimate gap acceptance at stop-controlled intersections. In the past, neural networks have been used to examine problems involving complex interrelationship among many variables and found to perform better than conventional methods. This paper describes the development of a neural network and a binary-logitmodel for predicting accepted or rejected gaps at rural, low-volume two-way stop-controlled intersections. The type of control, the turning movements in both the major and minor directions, size of gap, service time, stop type, vehicular speed, queue in the minor direction, and existence of vehicle in the opposite approach were found to influence the driver's decision to accept or reject a gap. The results of the neural network and the binary-logit model were compared with the observations recorded in the field. The results revealed that the neural network correctly predicted a higher percentage of accepted or rejected gaps than the binary-logit model.

INTRODUCTION

Intersections that carry a low volume of traffic are generally controlled by stop signs. The stop signs can be located on either all approaches (four- way stop control), or approaches only in the major directions (two-way stop control). Stop signs installed at two-way stop-controlled intersections (TWSCI) give priority to drivers in the major directions to cross the intersection without any interruption from drivers in the minor directions. When stop signs fail to provide the safe movement of vehicles and the installation of a traffic signal is not warranted, an intersection-control beacon may be used as a traffic-control device to supplement the stop signs. An intersection- control beacon is suspended over the center of an intersection and flashes yellow in the nonstopped directions and red in the stopped directions.

The delay and capacity analysis of TWSCI is influenced by the number and size of gaps in the major directions and the acceptance of these gaps by the drivers in the minor directions. Many researchers have developed delay and capacity equations for analysis of unsignalized intersections (Tan- ner 1962; Dunne and Buckley 1972; Plank 1982; Troutbeck 1988) and have examined the various factors concerning gap acceptance, including their effects upon the calculation of capacity and delay (Cowan 1971; Hewitt 1983, 1985).

XAssoc. Prof., Dept. of Civ. and Envir. Engrg., Univ. of Cincinnati, 741 Baldwin Hall #71, Cincinnati, OH 45221-0071.

2Grad. Asst. Dept. of Civ. and Envir. Engrg., Univ. of Cincinnati, 741 Baldwin Hall #71, Cincinnati, OH.

Note. Discussion open until November 1, 1994. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on June 18, 1993. This paper is part of the Journal of Transportation Engineering, Vol. 120, No. 3, May/June, 1994. �9 ISSN 0733-947X/94/0003-0432/$2.00 + $.25 per page. Paper No. 6412.

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OBJECTIVE

The primary objective of this study is to develop a neural network for predicting the gap-acceptance behavior of drivers at rural, low-volume TWSCI. The study identifies various parameters that influence a driver's decision to accept or reject gaps at stop-controlled intersections, those with intersection- control beacons as well as those without intersection-control beacons. In the past, neural networks have been used to examine problems involving complex interrelationships among many variables. Recent interest in neural computing and successful application of this technique in other engineering disciplines have prompted this study. The estimates of gap acceptance obtained from the neural network are compared with those from a binary- logit model to examine the suitability of the neural-network technique for further application in traffic engineering.

BACKGROUND

At an intersection controlled by stop signs, drivers in the minor directions are required to stop completely before proceeding into the intersection and completing their maneuvers based on gaps available in the major direction. Gap is the elapsed-time interval between arrivals of successive vehicles in the major direction at a specified reference point in the intersection area. The most crucial factor determining the behavior of drivers in the minor direction is the availability and duration of gaps in the major direction and the acceptance of these gaps by the drivers. The minimum time interval just sufficient for the driver to make the maneuver is called the critical gap.

Gap-acceptance behavior of drivers depends on driver characteristics, roadway geometry, gap size, waiting time, weather, etc. The simplest form of gap acceptance was originally explained by a step function (Adams 1936; Raft 1951; Tanner 1951). This function indicates that each driver is consistent in accepting gap sizes greater than a critical value 0") and rejecting gap sizes less than T. This is represented as

f ( g ) = I-t(g - r (la)

H ( g - ~) = 0; for g < -r; ( lb)

H ( g - -r) = 1; for g > = "r (lc)

where g = gap; and �9 = critical gap. Solberg and Oppenlander (1966) used probit analysis for the statistical

treatment of the gap-acceptance phenomenon. The acceptance or rejection of a time gap is represented by an all-or-nothing, or binomial response, which is dependent on the size of the gap. The probit of the expected proportion accepting a time gap is described by the following linear equation:

Y = 5.0 + I ( X _ I*) (2) cr

where Y = probit of proportion-accepting time gap; X = logarithm of time gap; I* = mean tolerance distribution; and cr = standard deviation of tolerance distribution.

The mean-gap and lag-acceptance times were estimated. Lag is the time interval between the arrival of the minor-stream vehicle and the arrival of the next major-stream vehicle at a marked point, that is, the portion of the gap remaining when the minor-stream vehicle appears. Solberg and Op-

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penlander indicated that there was no significant difference between the gap and lag-acceptance times, hence gap and lag can be treated together. However, other studies (Wagner 1966; Uber 1978) revealed that critical gaps obtained by using only gap-acceptance times were often overestimated and the usage of only lag data resulted in information loss. Ashworth (1968) used a theoretical approach to relate gap acceptance over the population. Miller (1971) evaluated nine different methods to estimate gap acceptance parameters. A comparative study by Hewitt (1985) showed that the maximum-likelihood method was complicated, but that it gave an accurate es- timation of mean and standard deviation.

The speed of the vehicle in the major stream has been a controversial factor in the study of gap acceptance and critical gap. Simulation studies by Sinha and Tomiak (1971) and Brilon (1988) indicated that the critical gap increases as the speed increases. On the contrary, experimental studies on gap acceptance by Ashworth and Bottom (1977) revealed a negative effect of speed on gap acceptance.

Among other studies, Solberg and Oppenlander (1966) examined the effects of community size on gap acceptance. Tsongos (1969) showed that there was no difference in gaps between day and night. Studies by Ashton (1971) and Maze (1981) found that the length of queuing time did not affect gap acceptance. However, Ashworth and Bottom (1977) found that a driver waiting for a longer time accepted a shorter gap than others. A study by Polus (1983) found that mean gaps and lags may be influenced by the type of sign control (yield versus stop).

Maze (1981) modeled gap-acceptance behavior using a logit model which closely approximates the probit model. The simple dichotomous choice logit- function form is

1 P - [1 + e r(x)] (3)

and its linear transformation is

F ( x ) : e (e/'-P) (4)

where P = cumulative probability of accepting a gap; x = variables affecting gap acceptance; and F ( x ) = linear function.

A study by Blumenfeld and Weiss (1979) classified gap acceptance in two groups, namely, consistent and inconsistent. If the driver's behavior was assumed consistent, the average delay could be estimated correctly. How- ever, the variance of delay could be grossly overestimated. Catchpole and Plank (1986) used consistency in the behavior of drivers and homogeneity between drivers to classify gap-acceptance behavior in four categories: consistent and homogeneous (M1), consistent and nonhomogeneous (M2), inconsistent and homogeneous (M3), and inconsistent and nonhomogeneous (M4). They found that M4 is the most sophisticated and accurate model to represent the true gap-acceptance behavior, which is also less practical, mathematically intractable, and difficult to quantify.

Kyte et al. (1991) found that the size of the accepted gap is affected by the length of time a vehicle has been delayed, the flow rate on the conflicting approaches, and the directional movement of the subject vehicle. Fitzpatrick (1991) used a logit model to determine the effect of vehicle type (truck, passenger car) on gap-acceptance behavior. The study concluded that the gap-acceptance behavior of truck drivers should be considered at intersections with a significant volume of truck traffic.

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Thus, several methods, including empirical analysis, in addition to logit, probit, and theoretical models, have been used to estimate gap acceptance at stop-controlled intersections. There is sufficient evidence available in other engineering applications that neural networks have performed better than many statistical methods. Hence, this study will develop a neural network for gap acceptance and compare its performance with a binary-logit model.

NEURAL NETWORK

A neural network can be characterized as a computational model which is composed of a highly interconnected mesh of nonlinear elements and whose structure is inspired by the biological nervous system. The different units in the neural network are analogous to the neuron in the human brain, receiving inputs from neighbors or external sources. These inputs are used to compute an output signal which is propagated to other units. The output signal is transformed by a transfer function before it is communicated to other units. The transfer functions normally used are sigmoid, hyperbolic, or sine functions. The strength of the connection between units is called weight.

One of the most successful neural-net paradigms is the feed-forward neural network and the associated backpropagation (BP) algorithm. The backpropagation algorithm is a form of supervised learning in which correct outputs for a particular pattern of inputs are required to teach correct behavior to the network. The network learns using a gradient-descent algorithm. A feed- forward neural network has a layered structure and consists of an input layer, output layer, and at least one hidden layer. The number of units in the input or output is equal to the number of variables in each layer. The input units are connected to the hidden units, which are connected, in turn, to the output units. All units in the network receive a constant source of input from the bias which is analogous to a ground in an electrical circuit.

The net input to a unit j is given by

xj = ~ wqxi + bi (5) i

where xi = output from the previous layer, wq -- weight of the connection between unit i and j, and bi = bias.

The weights of the network are randomly initialized within a particular range prior to the start of the training process. For each input pattern (x~,, alp), the computation of an output at the output layer is based on the output from the hidden layer using (5). The calculated output vector (%) is compared with the desired output vector (alp). An error function is defined as the difference between % and dp, as follows:

1 E = - (6)

The error computed at the output layer is backpropagated through the network, and the weights (wq) are modified according to their contributions to the error function

OE Vwq = - c ~ - - (7)

OWq

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where c~ = learning rate. The weights can be updated after every single presentation or after a certain number of patterns, called epoch size, are presented. The algorithm can be summarized as follows:

Step 1

�9 Decide network architecture. �9 Randomize the weights within a particular range. �9 Specify the threshold error or the stopping criterion.

Step 2

�9 Compute the output at the output layer for a given input sample. �9 Compute the error at the output using (6). �9 Stop training if the threshold error is met.

Step 3

�9 For each output unit j, compute

8k = (dK - ok)f'(Xk) (8)

�9 For hidden unit j, compute

gj = f '(xj) ~'. 8kWjk (9) k

Step 4 The weights are updated by

Vwq(n + 1) = e~gjoi + momentum.Vwq(n) (10)

The momentum term is used to speed up the convergence by allowing rapid decrease in the error. The method of adaption of weights is called the learning rule and the parameters governing the learning rule are called the learning schedule (Eberhart et el. 1990; Krose et el. 1991; Wasserman 1990).

BINARY-LOGIT MODEL

A binary-logit model is an individual choice model. It has two alternative outputs from which an individual can choose. The desirability of choosing an alternative is usually based on a linear combination function known as the utility expression. A linear-utility expression can be expressed as

U, = a + [31X1 + [32X2 + . . . . . . + [3nX, (11)

where i = 1, 2, 3 . . . (various alternatives); Ui = desirability of choosing a particular alternative; a = constant; )(1, X=, . . Xn = independent variables; and [31, [32, . . [3n = weight coefficients.

To predict whether a particular alternative will be chosen or not, the utility function should be transformed into a probability. The logit model transforms the utility function into probability of choice i in the following manner:

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1 Pk(i) = 1 + e-~,k (12)

The coefficients of the utility expression are estimated by the maximum- likelihood method that obtains these coefficients through an iterative search process throughout the data set.

DATA COLLECTION AND PREPROCESSING

The data collected for the study (Pant et al. 1992) "Development of Guidelines for Installation of Intersection Control Beacons" were used for developing the neural network and the binary-light model described in this paper. The data were collected at 16 low-volume, two-way stop controlled intersections in rural Ohio highways. Of these intersections, eight were controlled by stop signs, and the remaining eight were controlled by stop signs as well as intersection-control beacons. The intersection-control beacons were not temporary installations. Sometimes intersection-control beacons are used as an interim device before the installation of traffic signals. The posted speed limit on the roadways in the major directions was 55 mph. Only one lane existed at each approach to the intersection, both in the major and minor directions, and there was no separate lane for left-turning vehicles. The data were collected for 3 -4 h during day and night at each intersection. Six video cameras were used to collect the data at each intersection. Four cameras were placed along the minor directions and two along the major directions to record all vehicular movements at, or near, the intersection (Fig. 1). The cameras recorded a time code on each frame at the rate of 30 frames per second. The information was analyzed by playing the video tapes in a videocassette recorder that displayed the time code on each frame. The following information about each vehicle in the major and minor directions of each intersection was extracted within an accuracy of 1/30 of a second:

Major Direction

�9 Vehicular speed �9 Turning movements: left, through, and right

Minor Direction

�9 Size of gap �9 Service time (i.e., the time spent by the lead vehicle at or near the

stopline before accepting the gap) �9 Existence of vehicle in the opposite approach �9 Stopping (i.e., does the driver stop completely before entering the

intersection or is it a rolling stop?) �9 Turning movements: left, through, and right �9 Presence of queue in the minor direction �9 Gap: accepted or rejected

METHODOLOGY

The data were partitioned in two groups, one for developing the model and other for testing the model. The neural network and the logit model were separately developed, tested, and compared.

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B

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FIG. 1. Location of Cameras for Data Collection

TABLE 1. Total Sample Size

Rejected Accepted Total sample Intersection cases cases size

(1) (2) (3) (4)

Control Type: Stop plus Beacon 1,471 1,471 2,942 Control Type: Stop 1,144 1,144 2,288

(a) Total Number of Vehicles

Stop and Stop plus Beacon I 2,615 I 2,615 [ 5,230

Neural N e t w o r k M o d e l The development process of a neural network involves two distinct phases,

learning and testing. During the learning process, the network is presented with the training data. After the network is trained, it is presented with the testing data to determine how the network has generalized the problem. The data consisted of 5,230 vehicles observed at 16 intersections (Table 1). The data were divided in two sets, training and test-data sets. The training data was prepared by randomly selecting 4,000 of 5,230 vehicles (76%), and the test data consisted of the remaining 1,230 vehicles (24%).

The output variable was coded (1,0) for accepted gap and (0,1) for rejected gap. All input variables, except vehicular speed, service time, and

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gap size, were of categorical type and were coded in the one-of-N following forms:

1. Input variables a. Control type

(1) Stop (1,0) (2) Stop plus Beacon (0,1)

b. Turning movement in major direction (1) Left (1,0,0) (2) Thru (0,1,0) (3) Right (0,0,1)

c. Turning movement in minor direction (1) Left (1,0,0) (2) Thru (0,1,0) (3) Right (0,0,1)

d. Queue in minor direction (1) Yes (1,0) (2) No (0,1)

e. Stop type (1) Complete (1,0) (2) Rolling (0,1)

f. Existence of vehicle in opposite approach (1) Yes (1,0) (2) No (0,1)

g. Size of utilized gap (s) h. Speed in major direction (mph) i. Service time (s)

2. Output variable a. Driver's response

(1) Accepted (1,0) (2) Rejected (0,1)

NeuralWorks (NeuralWare 1991) was used to develop the neural network using the backpropagation technique. The input layer had 17 processing units, and the output layer had two processing units. These units represented all variables just listed. The neural network had one hidden layer with three processing elements (shown in Fig. 2). The epoch size 16 was chosen. The initial weights were randomized within the range -0 .3-0.3 . The sigmoid function was used as the transfer function. The input data was presented to the neural network in the range 0-1. The output range was chosen slightly smaller than the range of the sigmoid function (0.2-0.8). The threshold error was set at 0.05 and the training was stopped after the threshold error criterion was satisfied. The best result was obtained with three units in the hidden layer. The learning coefficient and momentum rate were 0.85 and 0.90, respectively. The network converged to the desired error level after 10,000 iterations. The distributions of the weights and the error graph are shown in Fig. 3. In Fig. 3, the weights have moved away from zero in both directions, and further training may result in poor generalization.

After the neural network was trained, it was used to run the test data. The outcomes, gap accepted or gap rejected, were compared with the actual observations recorded at the sites. The outputs produced by the neural network were not exactly 0 or 1 since the computations involved in the

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FIG. 3. Weight Distribution and Error Graph

TABLE 2. Performance of Neural-Network Model on Test Data

Allowable error between Number of vehicles estimated output and field data correctly estimated Percentage of success

(1) (2) (3)

0.05 0.10 0.15 0.20 0.25

947 1,000 1,029 1,044 1,076

77 81.3 83.7 84.9 87.5

backpropagation algori thm are nonlinear. Hence , the error differences of 0.05, 0.10, 0.15, 0.20, and 0.25 were used to calculate the number of vehicles that had accepted or re jected gaps. For example , if the er ror difference was 0.05, an output of 0.95 was considered an accepted gap and an output of 0.05 was considered a re jec ted gap. The results showed that the neural network correctly predic ted 947 vehicles out of 1,230 (77%) for an error difference of 0.05, 1,000 vehicles (81%) for an error difference of 0.10, 1,029 vehicles (84%) for an error difference of 0.15, 1,044 vehicles (85%) for an error difference of 0.20, and 1,076 vehicles (88%) for an error dif-

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TABLE 4. Performance of Binary-Logit Model on Test Data

Allowable error between Number of vehicles estimated output and field data correctly estimated Percentage of success

(1) (2) (3)

0.05 0.10 0.15 0.20 0.25

710 838 905 950 975

57.7 68.1 73.6 77.2 79.3

TABLE 5. Comparison of Percentage of Success

Error difference (1)

0~05 0.10 0.15 0.20 0.25

Neural network (%) (2)

77 81.3 83.7 84.9 87.5

Binary Iogit model (%) (3)

57.7 68.1 73.6 7%2 79.3

Difference in performance (neural

network - Iogit model) (%) (4)

19.3 13.2 10.1 7.64 8.2

ference of 0.25. Table 2 shows the results obtained for different error differences.

Binary Logit Model "SST--Statist ical Software Tools" (Dubins and Rivers 1988) was used

to estimate the coefficients of the linear utility expression by using the maximum likelihood method. Correlation coefficients among the input variables were calculated, If the correlation coefficient between two independent variables was high, it would create a potential problem in linear model building (Anderson et al. 1990). Table 3 shows the correlation ma- trices among the input variables. It shows no significant linear association among the input variables. The utility function for the logit model was estimated as follows:

Ui = - (0 .47567-CTYP) + (0.44116.UGAP) + (0.02333.SPD)

+ (0.95405.MJTR1) + (1.87415.MJTR2) + (O.08779.SRVT)

- (O.09484.MNTR1) + (1.73174.MNTR2) - (6.13626.STYP)

+ (0.40523.QUE) + (0.33206.EXIS) (13)

where U; = utility function of accepted gap; CTYP = control type (1 = Stop, 0 = Stop plus Beacon); UGAP = utilized gap (s); SPD = speed in major direction (mph); MJTR1, MJTR2 = turning movements in major directions [(0,0) = through movements; (1,0) = left movement; and (0,1) = right movement]; S R V T = service time (s); MNTR1, MNTR2 = turning movement in minor directions [(0,0) = through movement; (1,0) = left movement; and (0,1) = riglat movement]; QUE = queue in minor direction (1 = Yes, 0 = No); S T Y P = stop type (1 = Complete, 0 = Rolling); and

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1250

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E ~ *~ 750

L

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_~ 500

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0.05 0.1 0.15 0.2

Error difference

I I

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Nem'a] Network l Logit Model Model

FIG. 4. Comparison of Performance

EXIS = existence of vehicle in opposite approach (1 = Yes, 0 = No). The coefficients of all variables in the model were tested at the 0.05 level of significance. The results showed that all variables were significant.

The probability of accepted gap was calculated using (12). Since the driver has only two choices, accept or reject a gap, the probability of rejected gap is

Prob(rejected gap) = 1 - Probability(accepted gap) (14)

The probabilities of both choices were calculated for the test-data set. Next, the number of vehicles correctly predicted by the logit model for error differences of 0.05, 0.10, 0.15, 0.20, and 0.25 were calculated. The calcu- lations showed that the logit model correctly predicted 709 vehicles out of 1,230 (57%) for an error difference of 0.05,836 vehicles (68%) for an error difference of 0.10, 903 vehicles (73%) for an error difference of 0 .15,942 vehicles (77%) for an error difference of 0.20, and 977 vehicles (79%) for an error difference of 0.25. Table 4 shows the performance of the binary- logit model for different error differences.

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It is evident from Table 5 that the neural network correctly predicted a higher percentage of vehicles than the binary-logit model at each error difference. When the desired difference was low (for example, 0.05 or 0.10), the neural network outperformed the binary-logit model by more than 14%. When the error difference was increased to 0.25, the accuracy of the binary- logit model increased to 79%. However, the neural network performed better than the logit model with an accuracy of 88%, an improvement of about 9%. These results are graphed in Fig. 4.

SUMMARY AND CONCLUSIONS

This study has described the development of a neural network and a binary-logit model for predicting accepted or rejected gaps at rural, low- volume, two-way stop-controlled intersections. The study examined several important variables for estimating the gap acceptance behavior of drivers at stop-controlled intersections. The data were divided into two sets, with one set used to develop the models and the other set used to examine the accuracy of the models. The estimates obtained from the models were compared with the actual observation recorded in the field. The study found that the type of control (stop-versus-stop plus intersection-control beacon) had an effect on the driver's decision to accept or reject gap. Other significant variables included the vehicular speed, turning movements (left, through, and right) in both the major and minor directions, size of gap, service time, stop type (rolling or complete), queue in the minor direction, and the existence of vehicle in the opposite approach. The neural network performed better than the logit model by correctly estimating a higher percentage of accepted or rejected gaps. While the neural network developed in this study was based on data collected at rural, low-volume intersections, the technique can be used for developing models for unsignalized intersections in urban areas with relatively high traffic volume. It could also be used for estimating critical gap accurately in the future.

ACKNOWLEDGMENTS

The data used in the development of the neural network was collected for a project titled "Development of Guidelines for Installation of Inter- section Control Beacons" funded by the Ohio Department of Transportation in cooperation with the Federal Highway Administration (FHWA). The assistance provided by William Edwards, Mohammad Khan, and Rodger Dunn at the Ohio Department of Transportation is gratefully acknowledged. The data were collected and analyzed by a team of graduate students led by Yongjin Park and assisted by Subbarao Neti, Wei Sun, and Nasser Obaidat at the University of Cincinnati. The writers express sincere appre- ciation to these individuals.

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