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Time Gap Modeling under Mixed Traffic Condition: A Statistical Analysis
JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 12, Issue 6, December 2012 Online English edition of the Chinese language journal
Cite this article as: J Transpn Sys Eng & IT, 2012, 12(6), 7284.
RESEARCH PAPER
Time Gap Modeling under Mixed Traffic Condition: A Statistical Analysis DUBEY Subodh Kant, PONNU Balaji, ARKATKAR Shriniwas S* Civil Engineering Department, Birla Institute of Technology & Science Pilani-333031, Rajasthan, India
Abstract: This paper attempts to model vehicular time gap, which is defined as the time interval between any two successive arrivals of vehicles at a reference point of measurement on a road segment. Such an approach is justified under the non-lane-based heterogeneous traffic conditions prevailing in developing countries such as India, characterized by many “zero” time gaps due to simultaneous arrivals within a given road width. In addition, time gap data are characterized by a significant amount of data in the tail region due to long headways. Nevertheless, many researchers of time gap modeling have used light-tailed distributions that modeled time gaps satisfactorily due to two reasons: (a) The tail data was merged into a single bin; and (b) goodness-of-fit tests such as the Chi-square test, which has many limitations, were used. Further, some researchers have suggested different distributions for the same range of traffic flows, leading to ambiguity in distribution selection. In addition, bin size, which dictates the degree of fit of any distribution, has been ascribed very less importance in time gap modeling. Hence, this paper tries to consolidate and standardize the existing research in time gap modeling research by addressing all these issues. Two new distributions, namely Generalized Pareto (GP) and Generalized Extreme Value (GEV) with better tail modeling properties, have been proposed along with other conventional distribution to model vehicular time gaps over a wide range of flow from 550 vph to 4,100 vph. Two types of goodness-of-fit tests, namely Area-based and Distance-based tests, have been used. It has been found from the study that GP distribution fits the time gap data well (overall and tails) up to a flow range of 1,500 vph based on both kinds of tests, and GEV fits the data well for the flow levels above 1,500 vph based on the area test only. Key Words: highway transportation; vehicular time gap; heterogeneous traffic flow; bin size; K-S tests; A-D tests
1
Introduction
Traffic conditions prevailing in developing countries such as India, China, Bangladesh, and Sri-Lanka are heterogeneous; that is, they are mixed in nature and comprise several categories of vehicles with varying dimensions, maneuvering capabilities, and speed. There are broadly nine to ten categories of vehicles ranging from slow moving vehicles such as cycles, motorcycles, three wheelers (auto-rickshaw), and pick-ups to fast moving vehicles such as cars, vans, and 2-axle and 3-axle trucks. There is an imperfect or no-lane discipline, with vehicles rarely following the lane markings and not strictly following any leading vehicle unlike under homogeneous traffic conditions. Further, there can be more than one vehicle arriving at a point on the road at any given instant, leading to zero time gaps. The follower headway
concept that measures the time interval between two successive vehicles in a traffic lane or a single file traffic stream as they pass a reference point on the roadway is not apposite to heterogeneous traffic conditions. In heterogeneous traffic, vehicles move based on the entire road width and could follow more than one vehicle. Hence, under such conditions, a better approach involves considering the time gap which is the time interval between consecutive vehicles passing a reference line on the entire road width, rather than the lane-based approach. Many researchers[1,2] have used the entire road space-based arrivals at a reference line on the road for the modeling of time gap under heterogeneous conditions. Vehicular time gap incorporates both following and non-following interactions that are typical of a heterogeneous traffic scenario in developing countries such as India. In spite of this advantage, there are a few challenges in time gap
Received date: Jan 12, 2012; Revised date: Feb 27, 2012; Accepted date: May 8, 2012 *Corresponding author. E-mail: [email protected] Copyright © 2012, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(11)60233-X
DUBEY Subodh Kant et al. / J Transpn Sys Eng & IT, 2012, 12(6), 7284
200
200 150
100
Frequency
Frequency
150
TailData data=20% Tail = 20 %(approx) (approx)
50
100
TailData data=1% Tail =1 % (approx) (approx) 50 0
0 0
5
10
15
20
25
30
35
40
45
TimeGap gapClass classInterval, interval sec (s) Time
(a) Flow level: 1020 vph
1
2
3
4
5
6
7
8
9
10 11 12
TimeGap gapClass class Interval, interval (s) Time sec
(b) Flow level: 4100 vph
Fig. 1 Rank frequency plot of time gap on log-log scale
modeling. Due to the presence of both fast and slow moving, small and large vehicles, time gaps may range from 0 to 25 s, with a significant amount (0–20%) of data in the tail regions, as shown in Fig. 1. Hence, the modeling of both zero time gaps and in the tail regions assumes paramount importance and leads to erroneous results when neglected. In a nut shell, it becomes imperative to perform statistical modeling of the entire road width-based time gaps under heterogeneous traffic conditions coupled with better tail modeling. Vehicular time gaps under heterogeneous traffic conditions are significantly different from those of time headways under homogeneous traffic conditions, as they both measure two different quantities[2]. Nevertheless, many researchers of time gap modeling have adopted distributions such as exponential, gamma, erlang, and lognormal, which have been used by researchers of headway modeling. The researchers of heterogeneous traffic conditions have also referred to the entire road width-based inter-arrival rate as “headway,” though it is not the same as the lane-based follower headway that prevails under homogeneous conditions. Under homogeneous conditions, Al-Ghamdi[3] studied exponential, shifted exponential, and erlang distribution for headways and established boundaries such as low traffic (less than 400 vph), medium traffic (400 to 1,200 vph), and high traffic (more than 1,200 vph). Similarly, in time gap research in heterogeneous traffic, Kumar and Rao[4] studied negative exponential distribution for flow ranges varying from about 100 vph to 200 vph. Chandra and Kumar[5] analyzed the headways on urban roads in India and suggested hyperlang distribution for a flow range of 900–1,600 vph. Arasan and Koshy[6] have proposed negative exponential for all flow ranges while considering the sampling approach. The reason for the[3–6] light-tailed distributions just cited to have modeled time gap data is the application of the Chi-square goodness-of-fit test, which is not only a weak powered test (Steele et al.[7]) but also
one that ascribes no specific importance to tail data. From the literature just cited, it is also obvious that different authors on time gap modeling have used different distributions for the same flow range and vice versa, which necessitates streamlining of the existing literature. Some authors in time gap research have also used heavy-tailed distributions. Ramanayya[8] proposed exponential distribution for flows up to 500 vph, shifted exponential distribution for 500–650 vph, and lognormal for higher flow levels. Yin et al.[9] reported that lognormal distribution gives the best fit regardless of traffic conditions with a maximum flow level of 617 vph. Even distributions such as lognormal that gave a better fit than their light-tailed counterparts could only do so because of two reasons: (a) merging of the data in the tails into a single bin and (b) using the goodness-of-fit test such as the Chi-square test in most of the situations. The merging of data in the tail, that is, combining two or more bins into a single bin, could result in the loss of a significant amount of information and can also lead to a non-robust modeling. One more interesting fact is that the researchers of both headway and time gap modeling have neglected the importance of bin size compared with distribution performance. Dey and Chandra[10] have used arbitrary bin sizes in their work. Arasan and Koshy[5] used Sturges’ rule that uses range of the data for bin size calculation and which is applicable only to a data set with a maximum of 100 observations[11]. Sahoo et al.[12] chose an arbitrary bin size of three seconds for modeling time gap data and proposed negative exponential for a maximum flow of 850 vph. A detailed discussion about the consequences of using non-optimal bin size has been presented in Section 5. Some authors used vehicle-specific headway models that considered the effect of traffic composition. Hoogendoom and Bavy[13] used Branston’s General Queuing model (GQM) for headways that were aggregated according to vehicle type and
DUBEY Subodh Kant et al. / J Transpn Sys Eng & IT, 2012, 12(6), 7284
period of the day and proposed Pearson-III-based mixed-type GQM. Ye and Zhang[14] proposed four headway distributions for the combinations of car–truck, truck–car, truck–truck, and car–car as Erlang, shifted negative exponential, shifted negative exponential, and negative plus shifted negative exponential, respectively. However, one difficulty involved in performing vehicle-specific time gap modeling is the numerous combinations that are formed from 10 vehicle categories available in countries such as India. Therefore, this research work restricts its scope to modeling the time gap with a high accuracy only at an aggregate level. This approach would be pertinent from both the modeling and implementation point of view. The primary objectives that are desired to be accomplished through this study are as follows: (1) to model time gap data using non-composite distributions with high importance to tail data over a wide range of flows; (2) to investigate the effect of bin size on the degree of goodness of fit of various distributions used for the modeling of time gaps; (3) to assess the statistical validity of probability distributions used for modeling time gap data using high-powered goodness-of-fit tests (Steele et al.[7])
2
Methodology
With the aim of accomplishing the objectives of this study just cited, the following steps were adopted. The time gap data for the purpose of analysis was collected from two locations in India where flows ranging from 550 vph to 4,100 vph were observed. Five well-known distributions in the field of headway and time gap modeling, namely Negative Exponential, Gamma, Weibull, Normal, Lognormal. Two new distributions, namely Generalized Pareto and Generalized Extreme Value, were used to model these flows. Further, these distributions were plotted using four different bin-size rules, namely Sturges’, Scott’s, Freedman-Diaconis’, and Modified Sturges’. Goodness-of-fit tests such as Chi-square, Kolmogorov-Smirnov (K-S), and Anderson-Darling (A-D) tests were performed on each of the distributions to ascertain their statistical validity. 2.1 Generalized Pareto (GP) distribution The Generalized Pareto distribution is based on the power law whose probability density function with shape parameter k0, scale parameter , and threshold parameter are given by Eq. (1) as follows: y
f (x | k,V ,T )
x T · § 1 ·§ ¸ ¨ ¸¨1 k V ¹ © V ¹©
1
1 k
(1)
For0 or for
for
x T ] §1· f ( x | 0, V , T ) ¨ ¸ exp[ V V © ¹
(2)
The shape parameter (k) of the generalized Pareto distribution assumes all three types of values, namely (a) if the tail of the data decreases exponentially, then it is zero; (b) if the tails decrease as a polynomial, then it is positive; (c) if the tails are finite, then it is negative. 2.2 Generalized extreme value (GEV) distribution In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions that are developed within the extreme value theory. The probability density function for the GEV distribution with location parameter μ, scale parameter , and shape parameter k 0 is given by Eq. (3) as follows: f ( x | k, P,V )
y
1 1 ª º 1 x P · k »§ x P · k §1· § « ¨ ¸ exp« ¨1 k ¸ ¨1 k ¸ (3) V ¹ »© V ¹ ©V ¹ © «¬ »¼
For 1+k(x–)/>0, k>0 corresponds to Type II case; whereas k<0 corresponds to Type III case. If the limit for k tends to 0, corresponding to the Type I case, the density is given by Eq. (4) as follows: (4) § x P · x P º §1· ª y
f (0, P, V ) ¨ ¸ exp« exp¨ © ©V ¹ ¬
V
¸ ¹
V
» ¼
2.3 Bin-size rules If a variable assumes a large number of values, then it is easier to handle and present the data by grouping the values into classes. In this study, three well-known bin-size rules, namely Sturges’, Scott’s, and Freedman–Diaconis’, were employed (a) to demonstrate the effect of bin size on the degree of fit of the distribution and (b) to find the bin size that captures the nature of the data to the best extent. In addition, Sturges’ rule was modified by replacing range by Inter Quartile Range (IQR), which is called “Modified Sturges’ Rule” in this paper. 2.3.1 Sturges’ rule[11] Given the interval (a, b), an equally spaced histogram or frequency diagram can be constructed using the bin counts in K equally spaced intervals or bins. The width of each bin is denoted by h (Eq. (5)), where h
ba 1 log 2 (n)
(5)
2.3.2 Scott’s rule[15] Scott’s rule, a formula (Eq. (6)) that provides the bin width for an equally spaced histogram with continuous data: 1
h 3.5V x n 3
(6)
where h is the bin width, and x is the estimate of the standard deviation. This rule gives a minimum mean integrated square error (MISE). MISE is a measure of the goodness of fit for a histogram with the underlying rate that generates observable events. It is defined as the difference between the estimated density and true density of a histogram[15,16]. With a chosen density function, it is difficult to find a single bin width that may be closest to the true density for all values of x. The bin
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width that does not give a smooth histogram for some intervals of the data points may be smooth or optimal for some other interval, resulting in multiple optimization points. However, the practical difficulty of constructing locally adaptive estimators makes the single bin width case the most important. Numerical integration of the deviations of the fitted histogram with a chosen bin width over the entire range of the data points would give a reasonable estimate of the error. This is called the “integrated squared error” (ISE). Since the ISE varies not only from sample to sample but also with the number of data points, the average value of the ISE is called the “mean integrated squared error” (MISE). The bin size that gives the least MISE is said to be optimal[17]. 2.3.3 Freedman–Diaconis’ rule[18] The bin width according to this rule is given by Eq. (7): 1
2 IQR n 3
h
(7)
where IQR is the inter-quartile range of the data (which is the difference between the 75th percentile value minus the 25th percentile value estimated from the cumulative distribution function). This rule also minimizes MISE. 2.3.4 Modified Sturges’ rule To avoid the presence of outliers in the data, the inter-quartile range of the data as defined earlier was substituted for the range in Sturges’ rule, as given in Eq. (8): h
IQR 1 3.22 log 2 ( n)
size on the degree of fit of various distributions. As an example, the generated histograms for a flow of 1,473 vph using two of the four rules are provided in Figs. 2 and 3. Fig. 3, which pertains to Freedman–Diaconis’ rule, represents the multi-peaked nature of the data; whereas Fig. 2, which is based on Sturges’ rule, does not capture the same. The bin size that minimizes MISE is called the “optimal bin size,” and it can accurately capture the inherent or underlying nature of the data[16]. Both Scott’s and Freedman–Diaconis’ rules minimize MISE and, hence, may be considered to give an optimal bin size compared with Sturges’ or Modified Sturges’ rule. Moreover, since the sample size in any time gap data is generally more than 100 (assuming very less traffic), Sturges’ rule should not be resorted to for the determination of bin size. Modified Sturges’ rule, which uses IQR instead of range to calculate bin size, also captures the multi-peaked nature of data. It has been found from the study that Modified Sturges’ rule converges to Freedman–Diaconis’ rule at higher ranges of flow. 2.5 Goodness-of-fit tests Goodness-of-fit tests are useful in ascertaining the degree of fit of the chosen distribution to the observed data in the field. These tests are essentially based on either of the two
600 500
2.4 Effect of bin size on distribution of data Researchers of both headway and time gap modeling have not given much emphasis to the determination of the bin size that is used for calculating theoretical frequencies. It is important to choose the correct bin size for the following reasons: (a) if too small a bin size is chosen, then the bar height of each bin suffers significant statistical fluctuations due to the paucity of samples in each bin; (b) if too large a bin size is chosen, then the histogram cannot represent the true shape of the underlying distribution because in this case the resolution is not very good; (c) changing the bin size could alter the degree of fit of the candidate distributions. Sturges’ rule[11], which is one of the binning rules frequently used in headway and time gap-related research[6] in the past, has the following disadvantages compared with using the “range”: (a) It can be used only for data that are normally distributed, and time gap or headway data are not often distributed normally; (b) it cannot be used when the sample size is more than 100, which again is not always the case; (c) it does not minimize MISE. Among the four bin-size rules used in this study, two rules, namely Scott’s and Freedman–Diaconis’, minimize MISE, and the other two rules, namely Sturges’ and Modified Sturges’, do not take MISE into account. The objective behind using all these four rules is to demonstrate the effect of bin
400
Frequency
(8)
300 200 100 0
5
10 15 Time gap class intervals (s)
20
Fig. 2 Histogram plotted based on Sturges’ rule 300
Frequency
250 200 150 100 50 0
5
10
15
20
Time gap class intervals (s)
Fig. 3 Histogram plotted based on Freedman Diaconis’ rule
DUBEY Subodh Kant et al. / J Transpn Sys Eng & IT, 2012, 12(6), 7284
distribution elements, namely the probability density function (PDF) and the cumulative distribution function (CDF). Tests based on PDF are called “area tests,” because they are applicable to binned data that represent an area in the histogram. These tests are done based on the difference between observed and estimated frequencies based on the calculated bin sizes. Tests based on CDF are called “distance tests,” as their statistics are based on deviations between observed and estimated data points. They are conducted based on differences between cumulative distribution function (CDF) and empirical distribution function (EDF) for a particular distribution. Chi-square test is an “area test”; K-S can be either area or distance based, whereas A-D is a “distance test” only[19]. An attractive feature of the Chi-square goodness-of-fit test is that it can be applied to any binned data irrespective of the distribution it comes from. The Chi-square test is based on the Chi-square distribution and assesses the degree of fit of the chosen distribution to the observed data. The approximation becomes better as the expected bin frequencies grow larger and may be considered inappropriate for tables with very small expected bin frequencies (chance of committing Type II error is high). However, there are certain “rules of thumb” to overcome this problem, which may have some undesirable consequences. For instance, if we merge two or more bins together, this can destroy the evidence of non-independence, if any, present in data and lead to erroneous inferences. In addition, the Chi-square test has not been designed for structural and sampling zeroes[20]; a situation in which one can observe zero frequency in either an observed or an estimated data frequency table, thus making the Chi-square statistic become infinite. Hence, statistical tests such as K-S and A-D, which are distance based (no binning problem and correlation, if any, present in data is preserved) and more powerful[7] than Chi-square test, should be adopted. The critical values of the K-S test can be found in several texts[21]. These critical values have been computed for distributions whose parameters were assumed beforehand to be independent of the data. However, when the distribution parameters are estimated from the data, these critical values should be modified[21]. Such a procedure is called an “adaptive” K-S procedure[21] and uses a critical value for an error of ' that is four times (level of significance) which we are testing. 2.5.1 Goodness of fit: procedure for area test (K-S)[22] Let O(i) be the observed frequency and E(j) be the estimated frequency, where i=j=1, 2, ···, n. The various tests that are involved in the conduction of the area-based K-S test are given as follows: Step 1: Calculate the difference between the observed and estimated frequencies for each of the class intervals as D(p)=O(i)–E(j). Step 2: Sort the D(p) values, which are estimated in Step 1, in the ascending order. Step 3: Get the
maximum value from the data, which are sorted out in Step 2, which can be denoted as D (maximum value from the sorted D(P)). This D is the calculated K-S statistic value from the given data or population estimated. Step 4: Calculate F1=sum of observed frequency and F2=sum of estimated frequency. Step 5: Then, A1=F1+F2 and A2=F1×F2. Step 6: Calculate C.I.=Sqrt (A1/ A2). Step 7: The critical value (CV), which is to be compared with the K-S statistic (D), can be estimated at various levels of significance as follows: (1) 1.22×C.I. at a 10% significance level; (2) 1.36×C.I. at a 5% significance level; (3) 1.48×C.I. at a 25% significance level; (4) 1.63×C.I. at a 1% significance level. Step 8: Compare the K-S statistic (D) with CV for a given level of significance: If DCV, then data do not follow the given distribution. 2.5.2 Goodness of Fit: Distance-based K-S Test Procedure[21] The K-S test procedure involves the comparison between CDF and EDF for a particular distribution. If the difference is larger than the critical value, the chosen distribution is rejected. For carrying out this test, sort the data first and then estimate the parameters of the distribution from the data (null hypothesis). Then, obtain the CDF(F0) and EDF(Fn) values at each data point i. In the distance-based K-S, the maximum distance | F0–Fn| is discovered. Then, define D+=Fn–F0 and D– =F0–Fn–1 for every data point Xi. The K-S statistic is D=Maximum of all D+ and D– (0); for each data point, i=1,···, n. 2.5.3 A-D Test Procedure[23] Since A-D test is a distribution-specific test, explaining the procedure for all the seven distributions would be beyond the scope of this paper. However, for the purpose of illustration, a sample A-D test procedure for the exponential distribution is given below. The A-D goodness-of-fit test for exponential distribution has the following functional form (Eq. (9)). AD
º ª 1 2i «¦ i n {ln(1 exp( z i )) z ( n1i ) } n» ¼ ¬
(9)
where zi=[x(i)/*] is the ith sorted sample value, n is sample size, ln is the natural logarithm, and subscript i refers to data points from 1 to n. Asterisk (*) denotes exponential parameter. AD* (1 0.2 / n ) AD
OSL 1 /{1 exp[0.1 1.24 ln( AD*) 4.48( AD*)]}
The OSL (observed significance level) is used for testing the exponential assumption. If OSL<0.05, then the exponential assumption is rejected and vice versa.
3
Data collection
The time gap data were collected by videography, considering the entire width of the road intended for one
DUBEY Subodh Kant et al. / J Transpn Sys Eng & IT, 2012, 12(6), 7284
Table 1 Vehicle composition (%) of the observed flows
n
(1 k )¦ i 1
Vehicle category
n
V ( xi T )
550
1021
1473
2125
2633
4100
2-Wheeler
16.00
13.20
12.00
59.58
46.79
71.00
¦
3-Wheeler
3.45
2.50
4.00
8.94
5.39
10.32
1 k
Cars
56.73
27.80
30.12
27.67
31.98
16.98
ª k ( xi T ) º ln «1 ¦ » V ¬ ¼ i 1
Buses
11.82
21.00
21.40
1.36
3.80
0.19
L.C.V
4.00
15.10
12.23
2.45
11.00
1.09
Trucks
8.00
20.40
20.25
0.00
1.04
0.42
direction of flow from two locations in India. The first set of observations was made on the National Highway (NH) 48, which is a four-lane divided road that connects the cities of Bangalore and Mangalore in South India for 7 consecutive days, where traffic flows ranged from 550 to 1,473 vph during the observation period. The second set of observations was made on Anna Salai, a major arterial of the city of Chennai, for 3 consecutive days, where the flow was found to vary between 2,125 and 4,100 vph during the observation period. The study locations were chosen in such a way that they were far away from any intersection, bus stops, or parking lots, and, hence, there was no hindrance to the traffic flow on the selected road section. Vehicle composition of the six different flows observed at both these locations is given in Table 1. Time gap data from videos were extracted at a rate of 25 frames per second with the aim of achieving a high accuracy.
4
wL wV
Flow (vph)
Data analysis
4.1 Generalized Pareto distribution[24,25] The parameters of all the seven distributions were estimated using the Maximum Likelihood technique. As an examples, the parameter estimation procedure adopted for the Generalized Pareto distribution has been explained next. The cumulative distribution function for generalized Pareto distribution is
n
i 1
Since the log-likelihood function is unbounded with regard to , an ML estimator cannot be obtained for . Therefore, the minimum value of the data set is assigned as the value of and is considered the location or the threshold parameter. Solving Eqs. (10) and (11) leads to Eqs. (12) and (13), as mentioned next. 4.2 Fitting of distributions The time gap data ranging from 550 vph to 4,100 vph were fitted to seven different statistical distributions as mentioned in Section 2. The fitted distribution for a flow level of 1,473 vph is shown in Fig. 4 as an example. First, the statistical significance of the distributions has been tested using area-based tests, namely, Chi-square and K-S tests. Since the parameters of all distributions were estimated from the data itself, an adaptive K-S test procedure was used. As per this procedure, the test was conducted at a significance level of significance () of 0.05, but K-S statistics were compared with critical value at a level of significance ’=4×=4×0.05=0.2. It should be noted that in all the cases, fitting has been done without merging the tail data. The A-D test is distribution specific and is used to check whether a data sample emerges from a population with a specific distribution[23]. It is a modification of the K-S test and assigns more weight to tails, and it is appropriate for evaluating the fit of a particular distribution with a greater emphasis to the tail region.
i 1
§
k ( xi T ) · ¸ V ¹ 2 k
Generalized extreme value
Generalized pareto Lognomal Normal Weibull
Density
0.25 0.20 0.15
0 5
(1 k )¦ i
Observed time gap Exponential Gama
0.45
0.05
n
¦ ln¨©1
(13)
nk
0.10
To estimate the parameter, differentiate the log-likelihood function w.r.t each parameter and equate it to zero
wL wk
(12)
n 1 k
0.35
§1 k · n ª k º n ln(V ) ¨ ¸¦ ln 1 ( xi T ) » © k ¹ i 1 «¬ V ¼
xi T § k ( xi T ) · 1 V ¨1 ¸ V © ¹ k
V
(11)
0
0.40
The general log-likelihood function of GP is
n
( xi T )
n
1
ª k ºk F ( x) 1 «1 ( x T )» if k z 0 ¬ V ¼ §T x · F ( x) 1 exp¨ ¸ if k 0 © V ¹
L ( x, k , V , T )
V
k ( xi T ) § k ( xi T ) · V ¨1 ¸ V © ¹ k 2
10
15
20
25
Time gap (s) 0
(10)
Fig. 4 Time gap distributions fitted for traffic flow of 1,473 vph
DUBEY Subodh Kant et al. / J Transpn Sys Eng & IT, 2012, 12(6), 7284
Table 2 Results of area-based tests for a flow of 1,473 vph Bin-size Rule Time gap distribution (1)
Freed-Diaconis’
Scott’s
Sturges’
Modified Sturges’
Bin size=0.45
Bin size=0.78
Bin size=1.98
Bin size=0.23
2 value (2)
K-S value (3)
2 value (4)
K-S value (5)
2 value (6)
K-S value (7)
2 value (8)
K-S value (9)
GEV
74.860*
0.037
72.310*
0.038
44.230*
0.029
63.520*
0.041
Gen-Pareto
62.220*
0.037
28.150
0.017
2.090
0.013
75.980*
0.040
Lognormal
94.720*
0.051*
79.780*
0.042
63.070*
0.037
87.290*
0.052*
Normal
N.A.
0.184*
N.A.
0.153*
212.390*
0.146*
N.A.
0.155*
Exponential
55.610*
0.039
48.720*
0.020
5.250
0.018
66.540*
0.041
Weibull
55.070*
0.036
33.470
0.023
5.880
0.021
65.690*
0.038
Gamma
53.330*
0.030
35.920
0.026
8.450
0.022
62.510*
0.033
Values denoted by * are above the critical value; GEV: General Extreme Value; Gen-Pareto: General Pareto distribution; 2: Chi-square. The term ‘N.A.’ denotes that the values are unreasonably larger than the critical values and hence are not mentioned here.
The values obtained by Chi-square and K-S tests, for each of the bin-size rules, were calculated and were compared against their critical values for each of the six different flows. These values for the flow of 1,473 vph are shown in Table 2 as an example. The table also mentions whether a specific distribution fits the data for a given bin size for a given area-based goodness-of-fit test. The best and second best-fit distributions as identified from the area-based K-S test are given in Table 3. Similarly, the distance-based goodness-of-fit tests, namely K-S and A-D, were carried out for all the flow levels, and their results were compared with the area-based test results obtained earlier. As an example, the K-S and A-D statistics obtained for the flow of 1,473 vph are given in Table 4. The procedure that is used to conduct the A-D test for other distributions along with their critical values can be found elsewhere[23,26,27]. The best-fit and second best-fit distributions as identified through these tests are summarized in Table 5.
5
Results and discussion
This section presents a detailed discussion on the empirical investigations carried out for time gap modeling over a wide range of traffic flow levels under heterogeneous traffic conditions and also the inferences based on the results obtained through this study. 5.1 Inference from area-based tests Among all the seven distributions, Generalized Pareto (GP) and generalized extreme value (GEV) distributions have been found to be the best for modeling the time gap data in the observed ranges of flow. Both these distributions have a slower decay rate than other distributions such as Exponential, Erlang, Weibull, Gamma, and Lognormal; though GP decays at a slower rate than GEV. This explains the fit of both these distributions in different ranges of flow. In this study, when the flows are lesser than or equal to 1,473 vph, GP fits the data better than that of GEV. The reason behind this are that at
lower flow levels, there are much data in the tail region of the distribution and GP assigns more weight to tails than that of GEV. Large time gaps (15 s or more) can also be observed due to an alternating pattern of a platoon of vehicles during one time interval and no vehicle in the next. Such an alternating pattern is called “bursting”[28]. When the observed flows in the study are above 1,473 vph, it was found that GEV models the data better than GP as the amount of data in the tail region is lesser at these flow levels.
Table 3 Ranking summary of distributions per area based K-S test No
Flow (vph)
1
550
2
1021
3
1473
4
2125
5
2633
6
4100
Best fit distribution
Second best fit distribution
Exponential & Gen. Pareto Generalized Pareto
Weibull & Gamma Exponential
Generalized Pareto Generalized extreme value Generalized extreme value Generalized extreme value
Exponential -
Table 4 Results of distance-based tests for a flow of 1,473 vph Time gap Distribution (1) Lognormal
K-S test D-value AD value (2) (3) 0.058*
10.845*
A-D test AD* value (4)
OSL (5)
-
-
Normal
0.169*
82.736*
-
-
Weibull
0.045*
2.117
2.128
0.00003*
Exponential
0.049*
2.140
2.150
0.00002*
Generalized Pareto
0.038
2.211
-
-
Generalized extreme value
0.099*
32.619*
-
-
Gamma
0.039*
2.108*
-
-
OSL: Observed significance level; Values denoted by * are above the critical value D is K-S statistic.
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Table 5 Ranking summary of distributions as per A-D and K-S tests S. No
Flow (vph)
Best fit Distribution
Second best fit distribution
1
550
Generalized Pareto
Weibull
2
1021
Generalized Pareto
-
3
1473
Generalized Pareto
-
4
2125
-
-
5
2633
-
-
6
4100
-
-
120 Estimated time gap 100 Observed time gap
Frequency
80 60 40 20 0 5
10
15
20
25
Time gap class intervals (s)
Fig. 5 Performance of Gamma distribution at 550 vph with Freedman–Diaconis’
This is because that at flow rates as high as 1,473 vph and above, the time gaps are not in a random state but rather proceed toward a constant state. In this study, GEV has modeled time gaps for flows of 2,125, 2,633, and 4,100 vph as per the area test, which checks the degree of fit based on the entire bin rather than on individual data points. It is interesting to note that exponential distribution models time gaps for a flow of 550 vph based on area tests, which can be explained by the random time gap state prevailing at this flow level. 5.2
Distance-based statistical inference
Distance-based goodness-of-fit tests are more rigorous than area-based tests, as the former checks the fit at each data point and can reject the null hypothesis if the difference between the CDF and EDF is larger than the critical value. In this study, it has been found that Generalized Pareto Distribution models the time gap data for flows below 1,473 vph based on distance tests also. Thus, GP can be said to model time gaps for all heterogeneous flows below 1,473 vph with confidence with a better tail modeling than other distributions. In addition, exponential distribution, which has been found to model time gaps at a flow level of 550 vph, is rejected at this flow level by a distance-based test. This demonstrates the inability of exponential distribution to model time gap data that contain a large amount of data in its tail regions. It has also been found that no distribution could model flows above 1,473 vph according to distance-based tests. Data in the tail is not
significant as the flows are in a constant state as mentioned earlier. Hence, the reason for none of the distributions to work in this range is due to the lack of fit in the non-tail region of the distribution. One solution to this problem would be to either investigate other non-composite distributions or use mixture models that may offer a better fit due to higher flexibility in terms of its shape parameters. 5.3 Disadvantages of area-based tests Chi-square, one of the most frequently used tests in time gap and headway modeling has many disadvantages as a goodness-of-fit test. First, it poses many constraints as mentioned earlier and second, it gives out erroneous conclusions in many circumstances. It was found from the study that at a flow level of 550 vph and Freedman–Diaconis’ bin rule, Chi-square declared Gamma distribution to model the time gap satisfactorily. However, there are significant differences between the observed and estimated frequencies (as per Gamma and Freedman–Diaconis’ rule) at many locations in the x axis (between 1 s & 3 s, 6 s & 8 s, 10 s & 12 s and 15 s & 17 s time gaps) as shown in Fig. 5. On the contrary, the A-D test and distance-based K-S tests declared Gamma not to model the data satisfactorily, thus conforming to Fig. 5. Similar results were also observed for area-based K-S. Hence, distance-based tests such as A-D and K-S (non-binned) should be preferred compared with area-based tests. Given such a non-robust nature of area-based tests, it would be rather apt to question the necessity of optimum bin size as explained elsewhere in this paper. However, it is pertinent as most of the existing research on time gap modeling has used binned data for analysis and has also used tests such as the Chi-square to assess the goodness of fit as mentioned earlier. Hence, emphasizing a scientific procedure for bin size estimation in terms of optimum bin size assumes a great importance. In addition, the analysis of data according to optimum bin size enables us to know about the underlying nature (in terms of a distribution) of the data. Hence, this exercise can assist us in short listing the distributions that would serve as the best candidate for the time gap data. It can be also noted that this technique is similar to that employed for finding the PDF of the data by estimating kernel density.
6
Conclusions
The following are the important findings that could be made from this study: (1) General Pareto distribution models time gap data up to a flow level of 1,473 vph by K-S (both distance and area based) and A-D tests, better than light-tailed distributions such as Exponential, Gamma, Weibull, and Normal and heavy-tailed distribution such as Lognormal and Generalized Extreme Value. Such a superior fit of Generalized Pareto is due to the fact that its tails decay at a slower rate than that of others, and, hence, it models flows below 1,473 vph which have a large
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amount of data in the tails. (2) General extreme value distribution models the time gap data from the flow level of 2,125 to 4,100 vph better than the other distributions (including General Pareto), only according to the area-based test. No distribution considered in the study is able to model the time gap data for flow levels above 1,473 vph based on distance-based tests. Hence, for modeling time gap data for flows above 1,473 vph, other non-composite or mixture models should be investigated. (3) Exponential distribution ones the most widely-used models the time gap data only at a flow of 550 vph and that too, only as per area-based tests. This distribution could not model time gaps satisfactorily for this flow level as per distance-based tests. (4) Modeling of tail data assumes a high importance in time gap modeling under heterogeneous traffic conditions. Hence, distributions such as Generalized Pareto and Generalized Extreme Value that assign high probability to the tail compared with other distributions were able to model the data. Their slow decay rate explains the reasons for their better performance. (5) Sturges’ should be replaced by other bin size rules such as Scott’s and Freedman-Diaconis’ for finding class intervals. This is due the restrictions posed by Sturges’ rule, which are such as: (a) The data should be normally distributed, which is not always the case with time gap data; (b) the sample size should be between 10 and 100; (c) the range of the data, which is highly sensitive to outliers and (d) does not take MISE into account. (6) Area-based goodness-of-fit tests such as the Chi-square should be avoided and robust tests such as A-D and K-S (distance based) should be adopted. The Chi-square test (a) does not work when observed frequencies fall below a certain value or when there are either structural or sampling zeros in the frequency table, and (b) can give out misleading results. In addition, the future scope of the study may concern: (1) Development of singular/mixture models for time gap modeling for flows above 1,473 vph. (2) Extending distributions such as Generalized Pareto and Generalized Extreme Value to model headways of homogenous traffic conditions.
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Cite this article as: J Transpn Sys Eng & IT, 2012, 12(6), 7284.
RESEARCH PAPER
Time Gap Modeling under Mixed Traffic Condition: A Statistical Analysis DUBEY Subodh Kant, PONNU Balaji, ARKATKAR Shriniwas S* Civil Engineering Department, Birla Institute of Technology & Science Pilani-333031, Rajasthan, India
Abstract: This paper attempts to model vehicular time gap, which is defined as the time interval between any two successive arrivals of vehicles at a reference point of measurement on a road segment. Such an approach is justified under the non-lane-based heterogeneous traffic conditions prevailing in developing countries such as India, characterized by many “zero” time gaps due to simultaneous arrivals within a given road width. In addition, time gap data are characterized by a significant amount of data in the tail region due to long headways. Nevertheless, many researchers of time gap modeling have used light-tailed distributions that modeled time gaps satisfactorily due to two reasons: (a) The tail data was merged into a single bin; and (b) goodness-of-fit tests such as the Chi-square test, which has many limitations, were used. Further, some researchers have suggested different distributions for the same range of traffic flows, leading to ambiguity in distribution selection. In addition, bin size, which dictates the degree of fit of any distribution, has been ascribed very less importance in time gap modeling. Hence, this paper tries to consolidate and standardize the existing research in time gap modeling research by addressing all these issues. Two new distributions, namely Generalized Pareto (GP) and Generalized Extreme Value (GEV) with better tail modeling properties, have been proposed along with other conventional distribution to model vehicular time gaps over a wide range of flow from 550 vph to 4,100 vph. Two types of goodness-of-fit tests, namely Area-based and Distance-based tests, have been used. It has been found from the study that GP distribution fits the time gap data well (overall and tails) up to a flow range of 1,500 vph based on both kinds of tests, and GEV fits the data well for the flow levels above 1,500 vph based on the area test only. Key Words: highway transportation; vehicular time gap; heterogeneous traffic flow; bin size; K-S tests; A-D tests
1
Introduction
Traffic conditions prevailing in developing countries such as India, China, Bangladesh, and Sri-Lanka are heterogeneous; that is, they are mixed in nature and comprise several categories of vehicles with varying dimensions, maneuvering capabilities, and speed. There are broadly nine to ten categories of vehicles ranging from slow moving vehicles such as cycles, motorcycles, three wheelers (auto-rickshaw), and pick-ups to fast moving vehicles such as cars, vans, and 2-axle and 3-axle trucks. There is an imperfect or no-lane discipline, with vehicles rarely following the lane markings and not strictly following any leading vehicle unlike under homogeneous traffic conditions. Further, there can be more than one vehicle arriving at a point on the road at any given instant, leading to zero time gaps. The follower headway
concept that measures the time interval between two successive vehicles in a traffic lane or a single file traffic stream as they pass a reference point on the roadway is not apposite to heterogeneous traffic conditions. In heterogeneous traffic, vehicles move based on the entire road width and could follow more than one vehicle. Hence, under such conditions, a better approach involves considering the time gap which is the time interval between consecutive vehicles passing a reference line on the entire road width, rather than the lane-based approach. Many researchers[1,2] have used the entire road space-based arrivals at a reference line on the road for the modeling of time gap under heterogeneous conditions. Vehicular time gap incorporates both following and non-following interactions that are typical of a heterogeneous traffic scenario in developing countries such as India. In spite of this advantage, there are a few challenges in time gap
Received date: Jan 12, 2012; Revised date: Feb 27, 2012; Accepted date: May 8, 2012 *Corresponding author. E-mail: [email protected] Copyright © 2012, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(11)60233-X
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200
200 150
100
Frequency
Frequency
150
TailData data=20% Tail = 20 %(approx) (approx)
50
100
TailData data=1% Tail =1 % (approx) (approx) 50 0
0 0
5
10
15
20
25
30
35
40
45
TimeGap gapClass classInterval, interval sec (s) Time
(a) Flow level: 1020 vph
1
2
3
4
5
6
7
8
9
10 11 12
TimeGap gapClass class Interval, interval (s) Time sec
(b) Flow level: 4100 vph
Fig. 1 Rank frequency plot of time gap on log-log scale
modeling. Due to the presence of both fast and slow moving, small and large vehicles, time gaps may range from 0 to 25 s, with a significant amount (0–20%) of data in the tail regions, as shown in Fig. 1. Hence, the modeling of both zero time gaps and in the tail regions assumes paramount importance and leads to erroneous results when neglected. In a nut shell, it becomes imperative to perform statistical modeling of the entire road width-based time gaps under heterogeneous traffic conditions coupled with better tail modeling. Vehicular time gaps under heterogeneous traffic conditions are significantly different from those of time headways under homogeneous traffic conditions, as they both measure two different quantities[2]. Nevertheless, many researchers of time gap modeling have adopted distributions such as exponential, gamma, erlang, and lognormal, which have been used by researchers of headway modeling. The researchers of heterogeneous traffic conditions have also referred to the entire road width-based inter-arrival rate as “headway,” though it is not the same as the lane-based follower headway that prevails under homogeneous conditions. Under homogeneous conditions, Al-Ghamdi[3] studied exponential, shifted exponential, and erlang distribution for headways and established boundaries such as low traffic (less than 400 vph), medium traffic (400 to 1,200 vph), and high traffic (more than 1,200 vph). Similarly, in time gap research in heterogeneous traffic, Kumar and Rao[4] studied negative exponential distribution for flow ranges varying from about 100 vph to 200 vph. Chandra and Kumar[5] analyzed the headways on urban roads in India and suggested hyperlang distribution for a flow range of 900–1,600 vph. Arasan and Koshy[6] have proposed negative exponential for all flow ranges while considering the sampling approach. The reason for the[3–6] light-tailed distributions just cited to have modeled time gap data is the application of the Chi-square goodness-of-fit test, which is not only a weak powered test (Steele et al.[7]) but also
one that ascribes no specific importance to tail data. From the literature just cited, it is also obvious that different authors on time gap modeling have used different distributions for the same flow range and vice versa, which necessitates streamlining of the existing literature. Some authors in time gap research have also used heavy-tailed distributions. Ramanayya[8] proposed exponential distribution for flows up to 500 vph, shifted exponential distribution for 500–650 vph, and lognormal for higher flow levels. Yin et al.[9] reported that lognormal distribution gives the best fit regardless of traffic conditions with a maximum flow level of 617 vph. Even distributions such as lognormal that gave a better fit than their light-tailed counterparts could only do so because of two reasons: (a) merging of the data in the tails into a single bin and (b) using the goodness-of-fit test such as the Chi-square test in most of the situations. The merging of data in the tail, that is, combining two or more bins into a single bin, could result in the loss of a significant amount of information and can also lead to a non-robust modeling. One more interesting fact is that the researchers of both headway and time gap modeling have neglected the importance of bin size compared with distribution performance. Dey and Chandra[10] have used arbitrary bin sizes in their work. Arasan and Koshy[5] used Sturges’ rule that uses range of the data for bin size calculation and which is applicable only to a data set with a maximum of 100 observations[11]. Sahoo et al.[12] chose an arbitrary bin size of three seconds for modeling time gap data and proposed negative exponential for a maximum flow of 850 vph. A detailed discussion about the consequences of using non-optimal bin size has been presented in Section 5. Some authors used vehicle-specific headway models that considered the effect of traffic composition. Hoogendoom and Bavy[13] used Branston’s General Queuing model (GQM) for headways that were aggregated according to vehicle type and
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period of the day and proposed Pearson-III-based mixed-type GQM. Ye and Zhang[14] proposed four headway distributions for the combinations of car–truck, truck–car, truck–truck, and car–car as Erlang, shifted negative exponential, shifted negative exponential, and negative plus shifted negative exponential, respectively. However, one difficulty involved in performing vehicle-specific time gap modeling is the numerous combinations that are formed from 10 vehicle categories available in countries such as India. Therefore, this research work restricts its scope to modeling the time gap with a high accuracy only at an aggregate level. This approach would be pertinent from both the modeling and implementation point of view. The primary objectives that are desired to be accomplished through this study are as follows: (1) to model time gap data using non-composite distributions with high importance to tail data over a wide range of flows; (2) to investigate the effect of bin size on the degree of goodness of fit of various distributions used for the modeling of time gaps; (3) to assess the statistical validity of probability distributions used for modeling time gap data using high-powered goodness-of-fit tests (Steele et al.[7])
2
Methodology
With the aim of accomplishing the objectives of this study just cited, the following steps were adopted. The time gap data for the purpose of analysis was collected from two locations in India where flows ranging from 550 vph to 4,100 vph were observed. Five well-known distributions in the field of headway and time gap modeling, namely Negative Exponential, Gamma, Weibull, Normal, Lognormal. Two new distributions, namely Generalized Pareto and Generalized Extreme Value, were used to model these flows. Further, these distributions were plotted using four different bin-size rules, namely Sturges’, Scott’s, Freedman-Diaconis’, and Modified Sturges’. Goodness-of-fit tests such as Chi-square, Kolmogorov-Smirnov (K-S), and Anderson-Darling (A-D) tests were performed on each of the distributions to ascertain their statistical validity. 2.1 Generalized Pareto (GP) distribution The Generalized Pareto distribution is based on the power law whose probability density function with shape parameter k0, scale parameter , and threshold parameter are given by Eq. (1) as follows: y
f (x | k,V ,T )
x T · § 1 ·§ ¸ ¨ ¸¨1 k V ¹ © V ¹©
1
1 k
(1)
For
for
x T ] §1· f ( x | 0, V , T ) ¨ ¸ exp[ V V © ¹
(2)
The shape parameter (k) of the generalized Pareto distribution assumes all three types of values, namely (a) if the tail of the data decreases exponentially, then it is zero; (b) if the tails decrease as a polynomial, then it is positive; (c) if the tails are finite, then it is negative. 2.2 Generalized extreme value (GEV) distribution In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions that are developed within the extreme value theory. The probability density function for the GEV distribution with location parameter μ, scale parameter , and shape parameter k 0 is given by Eq. (3) as follows: f ( x | k, P,V )
y
1 1 ª º 1 x P · k »§ x P · k §1· § « ¨ ¸ exp« ¨1 k ¸ ¨1 k ¸ (3) V ¹ »© V ¹ ©V ¹ © «¬ »¼
For 1+k(x–)/>0, k>0 corresponds to Type II case; whereas k<0 corresponds to Type III case. If the limit for k tends to 0, corresponding to the Type I case, the density is given by Eq. (4) as follows: (4) § x P · x P º §1· ª y
f (0, P, V ) ¨ ¸ exp« exp¨ © ©V ¹ ¬
V
¸ ¹
V
» ¼
2.3 Bin-size rules If a variable assumes a large number of values, then it is easier to handle and present the data by grouping the values into classes. In this study, three well-known bin-size rules, namely Sturges’, Scott’s, and Freedman–Diaconis’, were employed (a) to demonstrate the effect of bin size on the degree of fit of the distribution and (b) to find the bin size that captures the nature of the data to the best extent. In addition, Sturges’ rule was modified by replacing range by Inter Quartile Range (IQR), which is called “Modified Sturges’ Rule” in this paper. 2.3.1 Sturges’ rule[11] Given the interval (a, b), an equally spaced histogram or frequency diagram can be constructed using the bin counts in K equally spaced intervals or bins. The width of each bin is denoted by h (Eq. (5)), where h
ba 1 log 2 (n)
(5)
2.3.2 Scott’s rule[15] Scott’s rule, a formula (Eq. (6)) that provides the bin width for an equally spaced histogram with continuous data: 1
h 3.5V x n 3
(6)
where h is the bin width, and x is the estimate of the standard deviation. This rule gives a minimum mean integrated square error (MISE). MISE is a measure of the goodness of fit for a histogram with the underlying rate that generates observable events. It is defined as the difference between the estimated density and true density of a histogram[15,16]. With a chosen density function, it is difficult to find a single bin width that may be closest to the true density for all values of x. The bin
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width that does not give a smooth histogram for some intervals of the data points may be smooth or optimal for some other interval, resulting in multiple optimization points. However, the practical difficulty of constructing locally adaptive estimators makes the single bin width case the most important. Numerical integration of the deviations of the fitted histogram with a chosen bin width over the entire range of the data points would give a reasonable estimate of the error. This is called the “integrated squared error” (ISE). Since the ISE varies not only from sample to sample but also with the number of data points, the average value of the ISE is called the “mean integrated squared error” (MISE). The bin size that gives the least MISE is said to be optimal[17]. 2.3.3 Freedman–Diaconis’ rule[18] The bin width according to this rule is given by Eq. (7): 1
2 IQR n 3
h
(7)
where IQR is the inter-quartile range of the data (which is the difference between the 75th percentile value minus the 25th percentile value estimated from the cumulative distribution function). This rule also minimizes MISE. 2.3.4 Modified Sturges’ rule To avoid the presence of outliers in the data, the inter-quartile range of the data as defined earlier was substituted for the range in Sturges’ rule, as given in Eq. (8): h
IQR 1 3.22 log 2 ( n)
size on the degree of fit of various distributions. As an example, the generated histograms for a flow of 1,473 vph using two of the four rules are provided in Figs. 2 and 3. Fig. 3, which pertains to Freedman–Diaconis’ rule, represents the multi-peaked nature of the data; whereas Fig. 2, which is based on Sturges’ rule, does not capture the same. The bin size that minimizes MISE is called the “optimal bin size,” and it can accurately capture the inherent or underlying nature of the data[16]. Both Scott’s and Freedman–Diaconis’ rules minimize MISE and, hence, may be considered to give an optimal bin size compared with Sturges’ or Modified Sturges’ rule. Moreover, since the sample size in any time gap data is generally more than 100 (assuming very less traffic), Sturges’ rule should not be resorted to for the determination of bin size. Modified Sturges’ rule, which uses IQR instead of range to calculate bin size, also captures the multi-peaked nature of data. It has been found from the study that Modified Sturges’ rule converges to Freedman–Diaconis’ rule at higher ranges of flow. 2.5 Goodness-of-fit tests Goodness-of-fit tests are useful in ascertaining the degree of fit of the chosen distribution to the observed data in the field. These tests are essentially based on either of the two
600 500
2.4 Effect of bin size on distribution of data Researchers of both headway and time gap modeling have not given much emphasis to the determination of the bin size that is used for calculating theoretical frequencies. It is important to choose the correct bin size for the following reasons: (a) if too small a bin size is chosen, then the bar height of each bin suffers significant statistical fluctuations due to the paucity of samples in each bin; (b) if too large a bin size is chosen, then the histogram cannot represent the true shape of the underlying distribution because in this case the resolution is not very good; (c) changing the bin size could alter the degree of fit of the candidate distributions. Sturges’ rule[11], which is one of the binning rules frequently used in headway and time gap-related research[6] in the past, has the following disadvantages compared with using the “range”: (a) It can be used only for data that are normally distributed, and time gap or headway data are not often distributed normally; (b) it cannot be used when the sample size is more than 100, which again is not always the case; (c) it does not minimize MISE. Among the four bin-size rules used in this study, two rules, namely Scott’s and Freedman–Diaconis’, minimize MISE, and the other two rules, namely Sturges’ and Modified Sturges’, do not take MISE into account. The objective behind using all these four rules is to demonstrate the effect of bin
400
Frequency
(8)
300 200 100 0
5
10 15 Time gap class intervals (s)
20
Fig. 2 Histogram plotted based on Sturges’ rule 300
Frequency
250 200 150 100 50 0
5
10
15
20
Time gap class intervals (s)
Fig. 3 Histogram plotted based on Freedman Diaconis’ rule
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distribution elements, namely the probability density function (PDF) and the cumulative distribution function (CDF). Tests based on PDF are called “area tests,” because they are applicable to binned data that represent an area in the histogram. These tests are done based on the difference between observed and estimated frequencies based on the calculated bin sizes. Tests based on CDF are called “distance tests,” as their statistics are based on deviations between observed and estimated data points. They are conducted based on differences between cumulative distribution function (CDF) and empirical distribution function (EDF) for a particular distribution. Chi-square test is an “area test”; K-S can be either area or distance based, whereas A-D is a “distance test” only[19]. An attractive feature of the Chi-square goodness-of-fit test is that it can be applied to any binned data irrespective of the distribution it comes from. The Chi-square test is based on the Chi-square distribution and assesses the degree of fit of the chosen distribution to the observed data. The approximation becomes better as the expected bin frequencies grow larger and may be considered inappropriate for tables with very small expected bin frequencies (chance of committing Type II error is high). However, there are certain “rules of thumb” to overcome this problem, which may have some undesirable consequences. For instance, if we merge two or more bins together, this can destroy the evidence of non-independence, if any, present in data and lead to erroneous inferences. In addition, the Chi-square test has not been designed for structural and sampling zeroes[20]; a situation in which one can observe zero frequency in either an observed or an estimated data frequency table, thus making the Chi-square statistic become infinite. Hence, statistical tests such as K-S and A-D, which are distance based (no binning problem and correlation, if any, present in data is preserved) and more powerful[7] than Chi-square test, should be adopted. The critical values of the K-S test can be found in several texts[21]. These critical values have been computed for distributions whose parameters were assumed beforehand to be independent of the data. However, when the distribution parameters are estimated from the data, these critical values should be modified[21]. Such a procedure is called an “adaptive” K-S procedure[21] and uses a critical value for an error of ' that is four times (level of significance) which we are testing. 2.5.1 Goodness of fit: procedure for area test (K-S)[22] Let O(i) be the observed frequency and E(j) be the estimated frequency, where i=j=1, 2, ···, n. The various tests that are involved in the conduction of the area-based K-S test are given as follows: Step 1: Calculate the difference between the observed and estimated frequencies for each of the class intervals as D(p)=O(i)–E(j). Step 2: Sort the D(p) values, which are estimated in Step 1, in the ascending order. Step 3: Get the
maximum value from the data, which are sorted out in Step 2, which can be denoted as D (maximum value from the sorted D(P)). This D is the calculated K-S statistic value from the given data or population estimated. Step 4: Calculate F1=sum of observed frequency and F2=sum of estimated frequency. Step 5: Then, A1=F1+F2 and A2=F1×F2. Step 6: Calculate C.I.=Sqrt (A1/ A2). Step 7: The critical value (CV), which is to be compared with the K-S statistic (D), can be estimated at various levels of significance as follows: (1) 1.22×C.I. at a 10% significance level; (2) 1.36×C.I. at a 5% significance level; (3) 1.48×C.I. at a 25% significance level; (4) 1.63×C.I. at a 1% significance level. Step 8: Compare the K-S statistic (D) with CV for a given level of significance: If D
º ª 1 2i «¦ i n {ln(1 exp( z i )) z ( n1i ) } n» ¼ ¬
(9)
where zi=[x(i)/*] is the ith sorted sample value, n is sample size, ln is the natural logarithm, and subscript i refers to data points from 1 to n. Asterisk (*) denotes exponential parameter. AD* (1 0.2 / n ) AD
OSL 1 /{1 exp[0.1 1.24 ln( AD*) 4.48( AD*)]}
The OSL (observed significance level) is used for testing the exponential assumption. If OSL<0.05, then the exponential assumption is rejected and vice versa.
3
Data collection
The time gap data were collected by videography, considering the entire width of the road intended for one
DUBEY Subodh Kant et al. / J Transpn Sys Eng & IT, 2012, 12(6), 7284
Table 1 Vehicle composition (%) of the observed flows
n
(1 k )¦ i 1
Vehicle category
n
V ( xi T )
550
1021
1473
2125
2633
4100
2-Wheeler
16.00
13.20
12.00
59.58
46.79
71.00
¦
3-Wheeler
3.45
2.50
4.00
8.94
5.39
10.32
1 k
Cars
56.73
27.80
30.12
27.67
31.98
16.98
ª k ( xi T ) º ln «1 ¦ » V ¬ ¼ i 1
Buses
11.82
21.00
21.40
1.36
3.80
0.19
L.C.V
4.00
15.10
12.23
2.45
11.00
1.09
Trucks
8.00
20.40
20.25
0.00
1.04
0.42
direction of flow from two locations in India. The first set of observations was made on the National Highway (NH) 48, which is a four-lane divided road that connects the cities of Bangalore and Mangalore in South India for 7 consecutive days, where traffic flows ranged from 550 to 1,473 vph during the observation period. The second set of observations was made on Anna Salai, a major arterial of the city of Chennai, for 3 consecutive days, where the flow was found to vary between 2,125 and 4,100 vph during the observation period. The study locations were chosen in such a way that they were far away from any intersection, bus stops, or parking lots, and, hence, there was no hindrance to the traffic flow on the selected road section. Vehicle composition of the six different flows observed at both these locations is given in Table 1. Time gap data from videos were extracted at a rate of 25 frames per second with the aim of achieving a high accuracy.
4
wL wV
Flow (vph)
Data analysis
4.1 Generalized Pareto distribution[24,25] The parameters of all the seven distributions were estimated using the Maximum Likelihood technique. As an examples, the parameter estimation procedure adopted for the Generalized Pareto distribution has been explained next. The cumulative distribution function for generalized Pareto distribution is
n
i 1
Since the log-likelihood function is unbounded with regard to , an ML estimator cannot be obtained for . Therefore, the minimum value of the data set is assigned as the value of and is considered the location or the threshold parameter. Solving Eqs. (10) and (11) leads to Eqs. (12) and (13), as mentioned next. 4.2 Fitting of distributions The time gap data ranging from 550 vph to 4,100 vph were fitted to seven different statistical distributions as mentioned in Section 2. The fitted distribution for a flow level of 1,473 vph is shown in Fig. 4 as an example. First, the statistical significance of the distributions has been tested using area-based tests, namely, Chi-square and K-S tests. Since the parameters of all distributions were estimated from the data itself, an adaptive K-S test procedure was used. As per this procedure, the test was conducted at a significance level of significance () of 0.05, but K-S statistics were compared with critical value at a level of significance ’=4×=4×0.05=0.2. It should be noted that in all the cases, fitting has been done without merging the tail data. The A-D test is distribution specific and is used to check whether a data sample emerges from a population with a specific distribution[23]. It is a modification of the K-S test and assigns more weight to tails, and it is appropriate for evaluating the fit of a particular distribution with a greater emphasis to the tail region.
i 1
§
k ( xi T ) · ¸ V ¹ 2 k
Generalized extreme value
Generalized pareto Lognomal Normal Weibull
Density
0.25 0.20 0.15
0 5
(1 k )¦ i
Observed time gap Exponential Gama
0.45
0.05
n
¦ ln¨©1
(13)
nk
0.10
To estimate the parameter, differentiate the log-likelihood function w.r.t each parameter and equate it to zero
wL wk
(12)
n 1 k
0.35
§1 k · n ª k º n ln(V ) ¨ ¸¦ ln 1 ( xi T ) » © k ¹ i 1 «¬ V ¼
xi T § k ( xi T ) · 1 V ¨1 ¸ V © ¹ k
V
(11)
0
0.40
The general log-likelihood function of GP is
n
( xi T )
n
1
ª k ºk F ( x) 1 «1 ( x T )» if k z 0 ¬ V ¼ §T x · F ( x) 1 exp¨ ¸ if k 0 © V ¹
L ( x, k , V , T )
V
k ( xi T ) § k ( xi T ) · V ¨1 ¸ V © ¹ k 2
10
15
20
25
Time gap (s) 0
(10)
Fig. 4 Time gap distributions fitted for traffic flow of 1,473 vph
DUBEY Subodh Kant et al. / J Transpn Sys Eng & IT, 2012, 12(6), 7284
Table 2 Results of area-based tests for a flow of 1,473 vph Bin-size Rule Time gap distribution (1)
Freed-Diaconis’
Scott’s
Sturges’
Modified Sturges’
Bin size=0.45
Bin size=0.78
Bin size=1.98
Bin size=0.23
2 value (2)
K-S value (3)
2 value (4)
K-S value (5)
2 value (6)
K-S value (7)
2 value (8)
K-S value (9)
GEV
74.860*
0.037
72.310*
0.038
44.230*
0.029
63.520*
0.041
Gen-Pareto
62.220*
0.037
28.150
0.017
2.090
0.013
75.980*
0.040
Lognormal
94.720*
0.051*
79.780*
0.042
63.070*
0.037
87.290*
0.052*
Normal
N.A.
0.184*
N.A.
0.153*
212.390*
0.146*
N.A.
0.155*
Exponential
55.610*
0.039
48.720*
0.020
5.250
0.018
66.540*
0.041
Weibull
55.070*
0.036
33.470
0.023
5.880
0.021
65.690*
0.038
Gamma
53.330*
0.030
35.920
0.026
8.450
0.022
62.510*
0.033
Values denoted by * are above the critical value; GEV: General Extreme Value; Gen-Pareto: General Pareto distribution; 2: Chi-square. The term ‘N.A.’ denotes that the values are unreasonably larger than the critical values and hence are not mentioned here.
The values obtained by Chi-square and K-S tests, for each of the bin-size rules, were calculated and were compared against their critical values for each of the six different flows. These values for the flow of 1,473 vph are shown in Table 2 as an example. The table also mentions whether a specific distribution fits the data for a given bin size for a given area-based goodness-of-fit test. The best and second best-fit distributions as identified from the area-based K-S test are given in Table 3. Similarly, the distance-based goodness-of-fit tests, namely K-S and A-D, were carried out for all the flow levels, and their results were compared with the area-based test results obtained earlier. As an example, the K-S and A-D statistics obtained for the flow of 1,473 vph are given in Table 4. The procedure that is used to conduct the A-D test for other distributions along with their critical values can be found elsewhere[23,26,27]. The best-fit and second best-fit distributions as identified through these tests are summarized in Table 5.
5
Results and discussion
This section presents a detailed discussion on the empirical investigations carried out for time gap modeling over a wide range of traffic flow levels under heterogeneous traffic conditions and also the inferences based on the results obtained through this study. 5.1 Inference from area-based tests Among all the seven distributions, Generalized Pareto (GP) and generalized extreme value (GEV) distributions have been found to be the best for modeling the time gap data in the observed ranges of flow. Both these distributions have a slower decay rate than other distributions such as Exponential, Erlang, Weibull, Gamma, and Lognormal; though GP decays at a slower rate than GEV. This explains the fit of both these distributions in different ranges of flow. In this study, when the flows are lesser than or equal to 1,473 vph, GP fits the data better than that of GEV. The reason behind this are that at
lower flow levels, there are much data in the tail region of the distribution and GP assigns more weight to tails than that of GEV. Large time gaps (15 s or more) can also be observed due to an alternating pattern of a platoon of vehicles during one time interval and no vehicle in the next. Such an alternating pattern is called “bursting”[28]. When the observed flows in the study are above 1,473 vph, it was found that GEV models the data better than GP as the amount of data in the tail region is lesser at these flow levels.
Table 3 Ranking summary of distributions per area based K-S test No
Flow (vph)
1
550
2
1021
3
1473
4
2125
5
2633
6
4100
Best fit distribution
Second best fit distribution
Exponential & Gen. Pareto Generalized Pareto
Weibull & Gamma Exponential
Generalized Pareto Generalized extreme value Generalized extreme value Generalized extreme value
Exponential -
Table 4 Results of distance-based tests for a flow of 1,473 vph Time gap Distribution (1) Lognormal
K-S test D-value AD value (2) (3) 0.058*
10.845*
A-D test AD* value (4)
OSL (5)
-
-
Normal
0.169*
82.736*
-
-
Weibull
0.045*
2.117
2.128
0.00003*
Exponential
0.049*
2.140
2.150
0.00002*
Generalized Pareto
0.038
2.211
-
-
Generalized extreme value
0.099*
32.619*
-
-
Gamma
0.039*
2.108*
-
-
OSL: Observed significance level; Values denoted by * are above the critical value D is K-S statistic.
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Table 5 Ranking summary of distributions as per A-D and K-S tests S. No
Flow (vph)
Best fit Distribution
Second best fit distribution
1
550
Generalized Pareto
Weibull
2
1021
Generalized Pareto
-
3
1473
Generalized Pareto
-
4
2125
-
-
5
2633
-
-
6
4100
-
-
120 Estimated time gap 100 Observed time gap
Frequency
80 60 40 20 0 5
10
15
20
25
Time gap class intervals (s)
Fig. 5 Performance of Gamma distribution at 550 vph with Freedman–Diaconis’
This is because that at flow rates as high as 1,473 vph and above, the time gaps are not in a random state but rather proceed toward a constant state. In this study, GEV has modeled time gaps for flows of 2,125, 2,633, and 4,100 vph as per the area test, which checks the degree of fit based on the entire bin rather than on individual data points. It is interesting to note that exponential distribution models time gaps for a flow of 550 vph based on area tests, which can be explained by the random time gap state prevailing at this flow level. 5.2
Distance-based statistical inference
Distance-based goodness-of-fit tests are more rigorous than area-based tests, as the former checks the fit at each data point and can reject the null hypothesis if the difference between the CDF and EDF is larger than the critical value. In this study, it has been found that Generalized Pareto Distribution models the time gap data for flows below 1,473 vph based on distance tests also. Thus, GP can be said to model time gaps for all heterogeneous flows below 1,473 vph with confidence with a better tail modeling than other distributions. In addition, exponential distribution, which has been found to model time gaps at a flow level of 550 vph, is rejected at this flow level by a distance-based test. This demonstrates the inability of exponential distribution to model time gap data that contain a large amount of data in its tail regions. It has also been found that no distribution could model flows above 1,473 vph according to distance-based tests. Data in the tail is not
significant as the flows are in a constant state as mentioned earlier. Hence, the reason for none of the distributions to work in this range is due to the lack of fit in the non-tail region of the distribution. One solution to this problem would be to either investigate other non-composite distributions or use mixture models that may offer a better fit due to higher flexibility in terms of its shape parameters. 5.3 Disadvantages of area-based tests Chi-square, one of the most frequently used tests in time gap and headway modeling has many disadvantages as a goodness-of-fit test. First, it poses many constraints as mentioned earlier and second, it gives out erroneous conclusions in many circumstances. It was found from the study that at a flow level of 550 vph and Freedman–Diaconis’ bin rule, Chi-square declared Gamma distribution to model the time gap satisfactorily. However, there are significant differences between the observed and estimated frequencies (as per Gamma and Freedman–Diaconis’ rule) at many locations in the x axis (between 1 s & 3 s, 6 s & 8 s, 10 s & 12 s and 15 s & 17 s time gaps) as shown in Fig. 5. On the contrary, the A-D test and distance-based K-S tests declared Gamma not to model the data satisfactorily, thus conforming to Fig. 5. Similar results were also observed for area-based K-S. Hence, distance-based tests such as A-D and K-S (non-binned) should be preferred compared with area-based tests. Given such a non-robust nature of area-based tests, it would be rather apt to question the necessity of optimum bin size as explained elsewhere in this paper. However, it is pertinent as most of the existing research on time gap modeling has used binned data for analysis and has also used tests such as the Chi-square to assess the goodness of fit as mentioned earlier. Hence, emphasizing a scientific procedure for bin size estimation in terms of optimum bin size assumes a great importance. In addition, the analysis of data according to optimum bin size enables us to know about the underlying nature (in terms of a distribution) of the data. Hence, this exercise can assist us in short listing the distributions that would serve as the best candidate for the time gap data. It can be also noted that this technique is similar to that employed for finding the PDF of the data by estimating kernel density.
6
Conclusions
The following are the important findings that could be made from this study: (1) General Pareto distribution models time gap data up to a flow level of 1,473 vph by K-S (both distance and area based) and A-D tests, better than light-tailed distributions such as Exponential, Gamma, Weibull, and Normal and heavy-tailed distribution such as Lognormal and Generalized Extreme Value. Such a superior fit of Generalized Pareto is due to the fact that its tails decay at a slower rate than that of others, and, hence, it models flows below 1,473 vph which have a large
DUBEY Subodh Kant et al. / J Transpn Sys Eng & IT, 2012, 12(6), 7284
amount of data in the tails. (2) General extreme value distribution models the time gap data from the flow level of 2,125 to 4,100 vph better than the other distributions (including General Pareto), only according to the area-based test. No distribution considered in the study is able to model the time gap data for flow levels above 1,473 vph based on distance-based tests. Hence, for modeling time gap data for flows above 1,473 vph, other non-composite or mixture models should be investigated. (3) Exponential distribution ones the most widely-used models the time gap data only at a flow of 550 vph and that too, only as per area-based tests. This distribution could not model time gaps satisfactorily for this flow level as per distance-based tests. (4) Modeling of tail data assumes a high importance in time gap modeling under heterogeneous traffic conditions. Hence, distributions such as Generalized Pareto and Generalized Extreme Value that assign high probability to the tail compared with other distributions were able to model the data. Their slow decay rate explains the reasons for their better performance. (5) Sturges’ should be replaced by other bin size rules such as Scott’s and Freedman-Diaconis’ for finding class intervals. This is due the restrictions posed by Sturges’ rule, which are such as: (a) The data should be normally distributed, which is not always the case with time gap data; (b) the sample size should be between 10 and 100; (c) the range of the data, which is highly sensitive to outliers and (d) does not take MISE into account. (6) Area-based goodness-of-fit tests such as the Chi-square should be avoided and robust tests such as A-D and K-S (distance based) should be adopted. The Chi-square test (a) does not work when observed frequencies fall below a certain value or when there are either structural or sampling zeros in the frequency table, and (b) can give out misleading results. In addition, the future scope of the study may concern: (1) Development of singular/mixture models for time gap modeling for flows above 1,473 vph. (2) Extending distributions such as Generalized Pareto and Generalized Extreme Value to model headways of homogenous traffic conditions.
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