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A Copula-Based Joint Model to Capture the Interaction between Mode and Departure Time Choices in Urban Trips
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Transportation Research Procedia 00 (2018) 000–000
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Transportation Research Procedia 41 (2019) 722–730
mobil.TUM 2018 "Urban Mobility – Shaping the Future Together" - International Scientific Conference on Mobility and Transport
A Copula-Based Joint Model to Capture the Interaction between Mode and Departure Time Choices in Urban Trips Seyedehsan Seyedabrishamia*, Arash Rasa Izadia a
Tarbiat Modares University, Tehran, Iran
Abstract Mode and departure time choices are two effective decisions on the urban travel demand. Previous studies have different assumptions for modeling these decisions. Some studies assume that mode and departure time choices are independent and other studies consider nested logit as a joint structure. Comparison of joint and independent models shows that these decisions are interrelated. This paper uses a copula-based joint modeling framework to depict this interrelation. To achieve better fitted model several copula functions including product, AMH and Frank are used. For mode choice and departure time choice models, multinomial logit and bivariate logit models are used, respectively and finally, a BL-MNL joint model is used to create linkage between mode and departure time choices. The data used in this study is drawn from the origin-destination data of Qazvin-Iran in 2010. Estimated copula dependence parameters with high statistical significance, approve the interrelation between error terms of two models. © 2019 The Authors. Published by Elsevier Ltd. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review scientific committee of the mobil.TUM 2018 conference. Peer-reviewunder underresponsibility responsibilityofofthethe scientific committee of the mobil.TUM18. Keywords: Type your keywords here, separated by semicolons ;
1. Introduction The continual growth in urban travels, in particular, traveling with the private vehicle is one of the major reasons that causes urban traffic congestion and related problems such as noise and environmental pollution, energy dissipation and safety reduction. As the quality of human life is improved, they need more comfort and convenience, and in travel behavior scope, this leads to more use of personal vehicles. In such condition, changing transportation infrastructures seems to be necessary to service the vehicles. The high cost of constructions and operations necessitates planner to
* Corresponding author. E-mail address: [email protected] 2352-1465 © 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the mobil.TUM 2018 conference.
2352-1465 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the mobil.TUM18. 10.1016/j.trpro.2019.09.120
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find better options. There are policies and planning along with the development of transportation infrastructures that can response the future requirements. To solve traffic congestion, it is important to determine the factors causing this problem. A major part of morning and evening traffic congestion during the peak hours is caused by using the private vehicle for the commutes. These trips are done for work and educational purposes. Commutes are considered as mandatory trips that principally it is hard to cancel or postpone them, but for non-mandatory trips it is possible to set another plans that lead to the traffic congestion reduction, for example, improving public transportation efficiency and active transport during peak hours and using the private vehicle during the off-peak hours. Mode and departure choices have a direct effect on the traffic condition. It is crucial to develop an understanding of how these decisions can be modeled. It then boils down to identifying the appropriate modeling specifications that can reflect the behavior of decision makers with regard to mode and departure time choice. This study includes six sections as follows. Section 2 presents an overview of the literature review of mode and departure time choice models. Section 3 presents the model structure, followed by section 4 discussing used data. Sections 5 elaborates on the findings of this research. Finally, section 6 presents a summary and conclusion. 2. Literature review Generally, the relationship between mode and departure time choices has been less studied in the literature (Bhat 1998a; Habib and Sasic 2014). There are many mode choice models in the context of the four-step modeling where departure time is significantly overlooked (Bhat 1998b; Ermagun, Hossein Rashidi, and Samimi 2015). There is also a notable number of studied looked at departure time in the context of tour-based and activity-based modeling structures where the effect of mode choice has not been sufficiently studied (Tringides 2004). Some models considered a weak linkage between these two decisions. This linkage was handled by using one of these choices as an exogenous variable in the choice of the other one. For example, Lim and Sivaramakrishnan (Lim and Srinivasan 2014) used the chosen mode as an independent variable in the departure time choice model for starting and finishing time of social-recreational tours. Bhat (Bhat 1998b) used a multi-dimensional logit model to analyze mode and departure time choices for homebased social-recreational tours. This model is developed by using home-based social-recreational trips from the family regional survey of San Francisco in 1990. Also this paper employed a multinomial probit formulation as a flexible formula which considers the covariance among the unobserved attributes of the alternatives (Bhat 1998b). Ding et al. (Ding et al. 2014) used nested logit and cross-nested logit models to estimate mode and departure time choices. These models were estimated by using data from Maryland in 2007-2008 for passengers who had work trips. The results show that a cross-nested model provides better results compared to the nested and multinomial logit models. Based on the discussion presented above, some previous neglect the correlation between these decisions and some other studies assumed a sequential hierarchy for these two travel attributes. There is no reason to consider any type of sequence between the two decisions because people choose Mode and departure time simultaneously. This paper by using copula formulations modeled these decisions in a joint closed form structure. Copula has been used in previous transportation and travel behavior studies. For example, Ermagun et al. (Ermagun, Hossein Rashidi, and Samimi 2015) show the superiority of a copula-based model for modeling mode choice and escort decisions for school trips. In another study, Rasaizadi and Kermanshah (Rasaizadi and Kermanshah 2018) study the interrelation between mode and number of stops using a joint modeling framework. Bhat and Sener (Bhat and Sener 2009) mentioned that “copula-based approach is a simple and powerful approach that can accommodate spatial error correlation across observational units without imposing restrictive distribution assumption on the dependency structures between the error components. Another importance of using copula is that it results in a closed form without any intensive computational and infeasibility for large sample sizes. Also, this approach is straightforward to apply using a standard and direct maximum likelihood inference procedure”.
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3. Methodology This paper employed a copula-based joint model to recognize factors that have simultaneous effect on the mode and departure time choices. The mode choice is modeled using multinomial logit formulation. The bivariate-logit formulation is taken for departure time model. In this study AMH, Frank and Product copulas are used to reach betterfitted model. 3.1. Model structure The mode choice is modeled by using a multinomial logit model. Let q be the index for individuals and i be the index for the mode. Let h�� be the latent utility acquired by individual q for choosing travel mode i (Ben-Akiva and Lerman 1985; Hensher, Rose, and Greene 2005). (1) hqi x qi qi Where x�� is the column vector of exogenous variables specified to mode i and � is the corresponding column vector of parameters to be estimated. ��� represents an idiosyncratic error term. Assume that the ��� values are identically and independently extreme-value distributed with location parameter of zero across alternative i and individual q. According to utility theory, individual q selects alternative i if and only: (2) hqi max j i hqi Let ��� be a dummy variable; r�� � � if the ith mode is chosen by qth individual and r�� � � otherwise. By combining equation (1) and equation (2):
x qi i max j i hqj
The equation (3) can be rewritten:
x qi max j i hqj qi
Define:
v qi
max
j i
(3)
(4)
hqj qi
(5)
So r�� � � if and only if x qi v qi .
The implied marginal distribution of ��� can be obtained from equation (5) and from the distribution assumption on the ��� (Ben-Akiva and Lerman 1985; Hensher, Rose, and Greene 2005):
Fi ( x qi ) Pr(v qi X qi )
exp(x qi ) , j 1, 2,..., J exp(x qj )
(6)
j
The binomial logit formulation is used to model the departure time choice. The choice set includes traffic peak hours and off-peak hours. Assume that q is the index of individuals, k is the index of departure time choices and ��� is the utility of alternative k for the person q. Define (Ben-Akiva and Lerman 1985; Hensher, Rose, and Greene 2005): (7) u�� � γz�� � ��� where ��� is the vector of independent variables, � is a vector of parameters to be estimated and ��� is the random error term of the utility function. ��� follows an iid extreme value distribution, with location parameter equal to 0 and scale parameter equal to 1. Person q selects option k if its utility is more than the other one. (8) u�� � u�� ��� � �� If the systematic utility of alternative l is considered to be equal to zero, equation (8) can be written as: (9) γz�� � ��� � ���
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� � � � � ���� (10) Also, � will be defined as � � � � � , then: � � � ���� (11) So, S�� = 1 if and only if � � � ���� . The random variable of � which is resulted from the differentiation of two random variables with the iid extreme value distribution follows a logistic distribution. The marginal distribution of � is presented in equation (12) (Hensher, Rose, and Greene 2005; Ben-Akiva and Lerman 1985): G� �� ���� � �
����� ���� �
(12)
������� ���� �
The joint probability that person q chooses mode i at departure time k can be written as: ������ � �; ��� � �� � ������ � ���� ; τ��� � � ���� � � ������ � ���� � � ������ � ���� ; τ��� � � ���� � (13) To calculate this probability function, a bivariate distribution function between the error terms of each model is needed. Copula is a function that approximates a cumulative distribution functions by using the marginal density functions (Trivedi and Zimmer 2007; Portoghese et al. 2011). Equation (13) can be rewritten: (14) ������ � �; ��� � �� � u�� � ���� �u�� ; u�� � � �� ����� � � ���� ��� ����� �; G� �� ���� �� In which; u�� � �� ����� �, u�� � G� �� ���� �. Also, F and G are marginal distribution functions of mode choice and departure time models, respectively. ��� is copula dependence parameter that shows the correlation between the utility error terms of mode i and departure time k. Frank and AMH† copulas are used for joint modeling and to show the importance of dependence parameter as they can accommodate positive and negative values for the correlation parameters. A product copula represents an independent model with no correlations between the error terms. Table 1 shows some attributes of these copula functions (Trivedi and Zimmer 2007; Nelsen 1999). Copula Product Frank AMH
Table 1: Attributes of copulas ��u� ; u� � u� u� ���� � ���e���� � �� �e �� ��� �� � � e�� � � u� u� �� � � � u� ��� � u� ����
range of θ � ���� �� ������
3.2. Estimation procedure Let 1[.] be an indicator function taking the value of unity if the expression in parenthesis is true and 0 otherwise. Also, define: (15) M qik 1[ rqi 1] 1[s qk 1] The log-likelihood function for the estimation of parameters takes the form:
log L
Q
I
K
rqi 1, s qk 1)]) ( M qik log[Pr(
(16)
q 1 i 1 k 1
All parameters in the model are consistently estimated by maximizing the log-likelihood function. The parameters to be estimated in the joint model are the � ��� � vectors and dependence parameters of the best-fitted copula. Maximizing is accomplished by using R-Studio programming. Numerical methods are used to maximizing loglikelihood function.
†
Ali-Mikhail-Haq copula
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5
4. Data
Frequency
Used data of this study is drawn from the origin-destination travel data of Qazvin, a city in Iran, conducted in 2010. There are 6,603 of non-work home-based trips in this dataset. The choice set of the mode choice model includes private vehicle, taxi, bus, and active transportation (bike and walk). These choices have been considered based on available modes to individuals. Thus, for individuals who do not own a private vehicle, the private vehicle mode is omitted from their choice set. The departure time choice set includes traveling in peak hours and off-peak hours. In order to determine peak and off-peak hours, the frequency diagram of the departure time of all trips, including work and non-work trips is drawn and time interval with the highest frequency is considered as peak hours. Work trips are considered in the calculation of peak hours because these trips play an important role in traffic congestion during peak hours. The frequency graph of trips is shown in figure 1. 10 5 0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time
Figure 1: Frequency of trips in a day As this graph shows, 6-8 AM, 12-13 PM, and 17-18 PM are the peak points of the graph. 6-8 AM have been considered as the morning peak hours and 12-13 PM is considered as returning trips of students and 17-18 PM is considered as returning trips of workers. Table 2 provides an overview of the explanatory variables which are used in the model. The frequency of models choice sets is presented in table 3. Info. Age Sex Job
Education Driving license Travel distance Travel time Walking time to the bus station Travel aim
Variable AGE 5-18 AGE 19-30 AGE 31-41 AGE>41 SEX ADM.JOB SERV.JOB EDU.JOB OTHER LOW.EDU MED.EDU HIGH.EDU DL LGH TT
Table 2: Variables definition and frequency
Definition 1= if age is between 6 to 18, 0= otherwise 1= if age is between 19 to 30, 0= otherwise 1= if age is between 31 to 41, 0= otherwise 1= if age is more than41, 0= otherwise 1= if sex is male, 0= otherwise 1= if the job is an administrative job, 0= otherwise 1= if the job is a service job, 0= otherwise 1= if the job is an educational job, 0= otherwise 1= if the job is another job, 0= otherwise 1= if education level is high school, 0= otherwise 1= if a person has Associate's or Bachelor's degree, 0= otherwise 1= if a person has Master's Degree or Doctorate, 0= otherwise 1= if a person has a driving license, 0= otherwise Distance between origin and destination (km) Travel time between origin and destination (min)
Share 10.84 22.94 30.99 25.24 38.30 12.42 13.40 14.42 59.76 49.45 37.70 12.87 48.81 2.88 6.73
WT
Walking time to the bus station (min)
7.5
SHOPPING RECREATIONA L VISITING OTHER
1= if travel aim was shopping, 0= otherwise
36.84
1= if travel aim was recreation, 0= otherwise
16.67
1= if travel aim was visiting, 0= otherwise 1= if travel aim was other aims, 0= otherwise
26.39 20.10
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Land use
ORG-EDU ORG-ADM ORG-BUSIN DES-EDU DES-ADM DES-BUSIN
Origin education area, a thousandth of square meters Origin administrative area, a thousandth of square meters Origin business area, a thousandth of square meters Destination education area, a thousandth of square meters Destination administrative area, a thousandth of square meters Destination business area, a thousandth of square meters
727
5.86 7 12.78 4.67 10.46 21.63
Table 3: Frequency of models choice sets
Alternatives Private vehicle Taxi Bus Active Peak hours Off-peak hours Has car
Share 1684 2019 980 1920 1845 4758 4792
Frequency 25 31 15 29 28 72 73
5. Results In the departure time choice model, the off-peak hour is considered as base alternative with the utility of zero and coefficients of peak hours are computed relative to this zero utility of the off-peak hour option. Each copula function presents a different description of the correlation between the error terms of the two models. The joint model using the Frank copula has larger log-likelihood than the AMH copula so it selected as the final model for which the detailed results are presented later. Also, this model has a larger log-likelihood value than the model developed based on the product copula that shows the importance of considering the inter-relationship. Table 4 shows the joint model results based on the Frank copula. Table 5 shows the details of each model. Table 4: joint model results based on the frank copula Variable Copula Dependence Parameter Constants Personal info. AGE 5-18 AGE 19-30 AGE 31-41 AGE>41 SEX ADM.JOB SERV.JOB EDU.JOB OTHER DL
Personal vehicle 0.111 (0.345) -1.100 (-10.031) -0.365 (-3.766) 1.911 (20.281) 0.432 (3.744) -
MNL Taxi 1.320 (4.748) 0.243 (1.560)
-0.194 (-1.172) -0.150 (2.003) -
BL Off-peak hours
Peak hours
-
-
-
0.173 (1.778)
0.801 (5.105) -
-
-
-
-
-
0.445 (5.931) -
0.580 (6.178) -
-
-
-
-0.218 (-2.072) -
Bus 1.002 (3.851) -0.681 (-4.494)
Active -0.412 (-1.321) 1.488 (11.489)
-0.423 (-4.326) -
-
-
-
0.377 (2.616) -
-
-
-
-
-1.033 (-8.699)
-1.379 (-11.866)
-
-0.615 (-6.968) -0.910 (-8.512)
0.241 (3.678) -
728
LOW.EDU MED.EDU HIGH.EDU Travel info. TT LGH SHOPPING RECREATIONAL VISITING
Seyedehsan Seyedabrishami et al. / Transportation Research Procedia 41 (2019) 722–730 Author name / Transportation Research Procedia 00 (2018) 000–000
-
0.531 (6.531) -
0.472 (4.922) -
-
-
-
-
-
-
-
-0.766 (-4.466)
-0.678 (-4.992)
-
-0.205 (-2.080)
-
-
-
-
-
-
-
-
-0.337 (-27.585) -
-
-1.029 (-10.289) -1.029 (-7.930) -
-
-0.146 (-1.573) -
-
-
-0.190 (-1.915) -
-
0.057 (4.324) 0.681 (10.236) -
-
-
0.570 (7.858) -
0.014 (2.938) -
-
-
-
-
0.160 (1.859) -
0.776 (8.949) 0.304 (3.194)
0.353 (3.208) -
OTHER
-
Land use info. ORG-EDU
-
-
-
ORG-ADM
-
-0.004 (-1.682) -
-
-0.029 (-5.266) -
ORG-BUSIN DES-EDU
-0.005 (-2.195) -
DES-ADM
-
DES-BUSIN
7
-0.009 (-5.898)
-
-
-
-
-
0.002 (1.334) -
-
-
-
-
-
-
-
-0.030 (-6.973) 0.007 (3.902) -
Table 5: Details of goodness-of-fit of the developed models Model log likelihood value Based on Frank copula -10009.94 Based on AMH copula -10032.65 Based on product copula -10039.34 Only with constants -12096.67 5.1. Copula dependence parameters analysis The t-value statistics for the copula dependence parameters show that they have strong significance level for the taxi, bus and active transport. In other words, it seems to be strong inter-dependency on using traffic peak hours and taxi, bus and active transport. Regarding the sign of copula parameters, there are some correlated unobserved factors having the mutual effect on choosing the bus, taxi or active and peak hours. For example, for positive copula dependence parameter sign estimated for bus and peak hours, there are some unobserved factors that simultaneously increase choosing probability of bus and traffic peak hours. 5.2. Analysis of estimated coefficients Results show that individuals between 6 to 18 years old have less tendency to use taxi and have more tendency to use active transport for their non-work trips. For individuals between 18 to 30 years old, personal vehicle and bus are
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not preferable alternatives. Also, these individuals prefer to choose traffic peak hours. Personal vehicle and taxi are not attractive alternatives for individuals over 41 years old compared to bus and active transport. Men are more likely to choose the personal vehicle and active transport rather than women. The probability of choosing a personal vehicle and bus is increased for individuals with ADM.job and EDU.jobs. Taxi is not a preferred option for individuals with OTHER jobs. Peak hours have the least utility for individuals with ADM.jobs and highest utility for OTHER jobs. A personal vehicle is a preferable option for individuals that have driving license regarding negative coefficient estimated for the other three modes. Individuals with the low level of income have more tendency to use bus and taxi. A personal vehicle is an attractive option for individuals with the medium level of income. People with the high level of income have less tendency to choose the bus, active and peak hours. By increasing the travel time, the probability of choosing active transport is decreased. Also by increasing the length of trips the probability of choosing peak hours is increased. The probability of choosing a personal vehicle is decreased for shopping and recreational trips. Taxi seems to be a more appropriate option for visiting and other trips. People prefer to use the bus for their visiting trip and this option is not an attractive option for shopping trips. Active transport has the least utility for recreational trips. Individuals have more tendency to choose peak hour for their shopping and visiting trips. As the origin and destination business area are increased the probability of choosing a personal vehicle is decreased. Also, the probability of choosing a taxi is decreased as origin administrative area and destination educational area are increased. Individuals more likely choose active transport if origin educational and business area are increased. If destination administrative area is decreased or destination business area is increased, the probability of choosing peak hour is increased. 6. Sensitivity Analysis In this section, a sensitivity analysis is presented for the estimated parameters. Formally, elasticity defined as a unit less measure that describes the relationship between a percentage change of some independent variables (i.e. an attribute of an alternative or the socio-demographic attributes of a decision maker) and some percentage change in the quantity demanded (Hensher, Rose, and Greene 2005). This paper calculates disaggregate elasticities (for individuals) for continuous variables including travel time, travel distance, walking time, and land use variables. Then aggregate elasticities are obtained by using the average of disaggregate elasticities. Table 6 shows elasticities for the joint model using Frank copula. Table 6: shows elasticities for the joint model using Frank copula
0.17
TAXIOFF PEAK 0.27
BUSOFF PEAK 0.28
ACTIVEOFF PEAK -1.97
WT
-0.04
-0.06
-0.06
Variable
PERS -OFF PEAK
TT
PERSPEAK
TAXIPEAK
BUSPEAK
ACTIVEPEAK
0.18
0.34
0.31
-1.93
0.24
-0.04
-0.08
-0.07
0.23
ORG-EDU
-0.02
-0.03
-0.03
0.06
-0.02
-0.03
-0.03
0.06
ORG-ADM
0.01
-0.02
0.01
0.01
0.01
-0.02
0.01
0.01
ORG-BUSIN
-0.04
0.00
0.00
0.03
-0.04
0.00
0.00
0.03
DES-EDU
-0.02
-0.14
-0.01
-0.02
0.08
-0.01
0.14
0.12
DES-ADM
0.02
0.02
0.02
0.02
-0.04
-0.06
-0.06
-0.05
DES-BUSIN
-0.11
0.03
0.03
0.03
-0.11
0.04
0.03
0.03
LGH
0.03
0.03
0.03
0.05
-0.09
-0.13
-0.13
-0.11
As it can be seen from table 6, for instance, 1% increase in travel time in peak and off-peak hours has the positive influence on personal vehicle selection. Here, 1% increase in destination administrative land use in off-peak hours, the probability of personal vehicle is decreased by 0.04%. Also, it can be said that active mode of transport is more
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sensitive to increase in travel attributes such as travel time and distance and it can be related to this mode is appropriate for short distance trips. 7. Conclusion This study focused on mode choice and departure time choice as two decisions that are expected to have some level of interaction. This interaction was studied by considering the correlation between unobservable random variable. Considering a multinomial logit model for mode choice and a binomial logit model for the time of day, the interaction is approximated using some copula (Frank, AMH, and product) functions. One of the most important findings of this study is to confirm that the interdependency between the two decisions reflected by the highly statically significance copula dependence parameters. The joint model provided a goodness-of-fit to the data with interpretable parameters, which shows the significance of using the proposed joint model for capturing the interaction between these two decisions.
References Ben-Akiva, Moshe E, and Steven R Lerman. 1985. Discrete choice analysis: theory and application to travel demand (MIT press). Bhat, Chandra R. 1998a. 'Accommodating flexible substitution patterns in multi-dimensional choice modeling: formulation and application to travel mode and departure time choice', Transportation Research Part B: Methodological, 32: 455-66. Bhat, Chandra R. 1998b. 'Analysis of travel mode and departure time choice for urban shopping trips', Transportation Research Part B: Methodological, 32: 361-71. Bhat, Chandra R, and Ipek N Sener. 2009. 'A copula-based closed-form binary logit choice model for accommodating spatial correlation across observational units', Journal of Geographical Systems, 11: 243-72. Ding, Chuan, Binglei Xie, Yaowu Wang, and Yaoyu Lin. 2014. 'Modeling the joint choice decisions on urban shopping destination and travel-toshop mode: A comparative study of different structures', Discrete Dynamics in Nature and Society, 2014. Ermagun, Alireza, Taha Hossein Rashidi, and Amir Samimi. 2015. 'A joint model for mode choice and escort decisions of school trips', Transportmetrica A: transport science, 11: 270-89. Habib, Khandker M Nurul, and Ana Sasic. 2014. 'A GEV model with scale heterogeneity for investigating the role of mobility tool ownership in peak period non-work travel mode choices', Journal of choice modelling, 10: 46-59. Hensher, David A, John M Rose, and William H Greene. 2005. Applied choice analysis: a primer (Cambridge University Press). Lim, Kwang-Kyun, and Sivaramakrishnan Srinivasan. 2014. "Modeling the Choice of Time-of-Day for Joint Social-Recreational Tours." In Transportation Research Board 93rd Annual Meeting. Nelsen, Roger B. 1999. 'Introduction.' in, An Introduction to Copulas (Springer). Portoghese, Alessandro, Erika Spissu, Chandra R Bhat, Naveen Eluru, and Italo Meloni. 2011. 'A copula-based joint model of commute mode choice and number of non-work stops during the commute', International Journal of Transport Economics/Rivista internazionale di economia dei trasporti: 337-62. Rasaizadi, A, and M Kermanshah. 2018. 'Mode choice and number of non-work stops during the commute: Application of a copula-based joint model', SCIENTIA IRANICA, 25: 1039-47. Tringides, Constantinos A. 2004. 'Alternative formulations of joint model systems of departure time choice and mode choice for non-work trips'. Trivedi, Pravin K, and David M Zimmer. 2007. 'Copula modeling: an introduction for practitioners', Foundations and Trends® in Econometrics, 1: 1-111.
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Transportation Research Procedia 41 (2019) 722–730
mobil.TUM 2018 "Urban Mobility – Shaping the Future Together" - International Scientific Conference on Mobility and Transport
A Copula-Based Joint Model to Capture the Interaction between Mode and Departure Time Choices in Urban Trips Seyedehsan Seyedabrishamia*, Arash Rasa Izadia a
Tarbiat Modares University, Tehran, Iran
Abstract Mode and departure time choices are two effective decisions on the urban travel demand. Previous studies have different assumptions for modeling these decisions. Some studies assume that mode and departure time choices are independent and other studies consider nested logit as a joint structure. Comparison of joint and independent models shows that these decisions are interrelated. This paper uses a copula-based joint modeling framework to depict this interrelation. To achieve better fitted model several copula functions including product, AMH and Frank are used. For mode choice and departure time choice models, multinomial logit and bivariate logit models are used, respectively and finally, a BL-MNL joint model is used to create linkage between mode and departure time choices. The data used in this study is drawn from the origin-destination data of Qazvin-Iran in 2010. Estimated copula dependence parameters with high statistical significance, approve the interrelation between error terms of two models. © 2019 The Authors. Published by Elsevier Ltd. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review scientific committee of the mobil.TUM 2018 conference. Peer-reviewunder underresponsibility responsibilityofofthethe scientific committee of the mobil.TUM18. Keywords: Type your keywords here, separated by semicolons ;
1. Introduction The continual growth in urban travels, in particular, traveling with the private vehicle is one of the major reasons that causes urban traffic congestion and related problems such as noise and environmental pollution, energy dissipation and safety reduction. As the quality of human life is improved, they need more comfort and convenience, and in travel behavior scope, this leads to more use of personal vehicles. In such condition, changing transportation infrastructures seems to be necessary to service the vehicles. The high cost of constructions and operations necessitates planner to
* Corresponding author. E-mail address: [email protected] 2352-1465 © 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the mobil.TUM 2018 conference.
2352-1465 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the mobil.TUM18. 10.1016/j.trpro.2019.09.120
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find better options. There are policies and planning along with the development of transportation infrastructures that can response the future requirements. To solve traffic congestion, it is important to determine the factors causing this problem. A major part of morning and evening traffic congestion during the peak hours is caused by using the private vehicle for the commutes. These trips are done for work and educational purposes. Commutes are considered as mandatory trips that principally it is hard to cancel or postpone them, but for non-mandatory trips it is possible to set another plans that lead to the traffic congestion reduction, for example, improving public transportation efficiency and active transport during peak hours and using the private vehicle during the off-peak hours. Mode and departure choices have a direct effect on the traffic condition. It is crucial to develop an understanding of how these decisions can be modeled. It then boils down to identifying the appropriate modeling specifications that can reflect the behavior of decision makers with regard to mode and departure time choice. This study includes six sections as follows. Section 2 presents an overview of the literature review of mode and departure time choice models. Section 3 presents the model structure, followed by section 4 discussing used data. Sections 5 elaborates on the findings of this research. Finally, section 6 presents a summary and conclusion. 2. Literature review Generally, the relationship between mode and departure time choices has been less studied in the literature (Bhat 1998a; Habib and Sasic 2014). There are many mode choice models in the context of the four-step modeling where departure time is significantly overlooked (Bhat 1998b; Ermagun, Hossein Rashidi, and Samimi 2015). There is also a notable number of studied looked at departure time in the context of tour-based and activity-based modeling structures where the effect of mode choice has not been sufficiently studied (Tringides 2004). Some models considered a weak linkage between these two decisions. This linkage was handled by using one of these choices as an exogenous variable in the choice of the other one. For example, Lim and Sivaramakrishnan (Lim and Srinivasan 2014) used the chosen mode as an independent variable in the departure time choice model for starting and finishing time of social-recreational tours. Bhat (Bhat 1998b) used a multi-dimensional logit model to analyze mode and departure time choices for homebased social-recreational tours. This model is developed by using home-based social-recreational trips from the family regional survey of San Francisco in 1990. Also this paper employed a multinomial probit formulation as a flexible formula which considers the covariance among the unobserved attributes of the alternatives (Bhat 1998b). Ding et al. (Ding et al. 2014) used nested logit and cross-nested logit models to estimate mode and departure time choices. These models were estimated by using data from Maryland in 2007-2008 for passengers who had work trips. The results show that a cross-nested model provides better results compared to the nested and multinomial logit models. Based on the discussion presented above, some previous neglect the correlation between these decisions and some other studies assumed a sequential hierarchy for these two travel attributes. There is no reason to consider any type of sequence between the two decisions because people choose Mode and departure time simultaneously. This paper by using copula formulations modeled these decisions in a joint closed form structure. Copula has been used in previous transportation and travel behavior studies. For example, Ermagun et al. (Ermagun, Hossein Rashidi, and Samimi 2015) show the superiority of a copula-based model for modeling mode choice and escort decisions for school trips. In another study, Rasaizadi and Kermanshah (Rasaizadi and Kermanshah 2018) study the interrelation between mode and number of stops using a joint modeling framework. Bhat and Sener (Bhat and Sener 2009) mentioned that “copula-based approach is a simple and powerful approach that can accommodate spatial error correlation across observational units without imposing restrictive distribution assumption on the dependency structures between the error components. Another importance of using copula is that it results in a closed form without any intensive computational and infeasibility for large sample sizes. Also, this approach is straightforward to apply using a standard and direct maximum likelihood inference procedure”.
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3. Methodology This paper employed a copula-based joint model to recognize factors that have simultaneous effect on the mode and departure time choices. The mode choice is modeled using multinomial logit formulation. The bivariate-logit formulation is taken for departure time model. In this study AMH, Frank and Product copulas are used to reach betterfitted model. 3.1. Model structure The mode choice is modeled by using a multinomial logit model. Let q be the index for individuals and i be the index for the mode. Let h�� be the latent utility acquired by individual q for choosing travel mode i (Ben-Akiva and Lerman 1985; Hensher, Rose, and Greene 2005). (1) hqi x qi qi Where x�� is the column vector of exogenous variables specified to mode i and � is the corresponding column vector of parameters to be estimated. ��� represents an idiosyncratic error term. Assume that the ��� values are identically and independently extreme-value distributed with location parameter of zero across alternative i and individual q. According to utility theory, individual q selects alternative i if and only: (2) hqi max j i hqi Let ��� be a dummy variable; r�� � � if the ith mode is chosen by qth individual and r�� � � otherwise. By combining equation (1) and equation (2):
x qi i max j i hqj
The equation (3) can be rewritten:
x qi max j i hqj qi
Define:
v qi
max
j i
(3)
(4)
hqj qi
(5)
So r�� � � if and only if x qi v qi .
The implied marginal distribution of ��� can be obtained from equation (5) and from the distribution assumption on the ��� (Ben-Akiva and Lerman 1985; Hensher, Rose, and Greene 2005):
Fi ( x qi ) Pr(v qi X qi )
exp(x qi ) , j 1, 2,..., J exp(x qj )
(6)
j
The binomial logit formulation is used to model the departure time choice. The choice set includes traffic peak hours and off-peak hours. Assume that q is the index of individuals, k is the index of departure time choices and ��� is the utility of alternative k for the person q. Define (Ben-Akiva and Lerman 1985; Hensher, Rose, and Greene 2005): (7) u�� � γz�� � ��� where ��� is the vector of independent variables, � is a vector of parameters to be estimated and ��� is the random error term of the utility function. ��� follows an iid extreme value distribution, with location parameter equal to 0 and scale parameter equal to 1. Person q selects option k if its utility is more than the other one. (8) u�� � u�� ��� � �� If the systematic utility of alternative l is considered to be equal to zero, equation (8) can be written as: (9) γz�� � ��� � ���
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� � � � � ���� (10) Also, � will be defined as � � � � � , then: � � � ���� (11) So, S�� = 1 if and only if � � � ���� . The random variable of � which is resulted from the differentiation of two random variables with the iid extreme value distribution follows a logistic distribution. The marginal distribution of � is presented in equation (12) (Hensher, Rose, and Greene 2005; Ben-Akiva and Lerman 1985): G� �� ���� � �
����� ���� �
(12)
������� ���� �
The joint probability that person q chooses mode i at departure time k can be written as: ������ � �; ��� � �� � ������ � ���� ; τ��� � � ���� � � ������ � ���� � � ������ � ���� ; τ��� � � ���� � (13) To calculate this probability function, a bivariate distribution function between the error terms of each model is needed. Copula is a function that approximates a cumulative distribution functions by using the marginal density functions (Trivedi and Zimmer 2007; Portoghese et al. 2011). Equation (13) can be rewritten: (14) ������ � �; ��� � �� � u�� � ���� �u�� ; u�� � � �� ����� � � ���� ��� ����� �; G� �� ���� �� In which; u�� � �� ����� �, u�� � G� �� ���� �. Also, F and G are marginal distribution functions of mode choice and departure time models, respectively. ��� is copula dependence parameter that shows the correlation between the utility error terms of mode i and departure time k. Frank and AMH† copulas are used for joint modeling and to show the importance of dependence parameter as they can accommodate positive and negative values for the correlation parameters. A product copula represents an independent model with no correlations between the error terms. Table 1 shows some attributes of these copula functions (Trivedi and Zimmer 2007; Nelsen 1999). Copula Product Frank AMH
Table 1: Attributes of copulas ��u� ; u� � u� u� ���� � ���e���� � �� �e �� ��� �� � � e�� � � u� u� �� � � � u� ��� � u� ����
range of θ � ���� �� ������
3.2. Estimation procedure Let 1[.] be an indicator function taking the value of unity if the expression in parenthesis is true and 0 otherwise. Also, define: (15) M qik 1[ rqi 1] 1[s qk 1] The log-likelihood function for the estimation of parameters takes the form:
log L
Q
I
K
rqi 1, s qk 1)]) ( M qik log[Pr(
(16)
q 1 i 1 k 1
All parameters in the model are consistently estimated by maximizing the log-likelihood function. The parameters to be estimated in the joint model are the � ��� � vectors and dependence parameters of the best-fitted copula. Maximizing is accomplished by using R-Studio programming. Numerical methods are used to maximizing loglikelihood function.
†
Ali-Mikhail-Haq copula
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4. Data
Frequency
Used data of this study is drawn from the origin-destination travel data of Qazvin, a city in Iran, conducted in 2010. There are 6,603 of non-work home-based trips in this dataset. The choice set of the mode choice model includes private vehicle, taxi, bus, and active transportation (bike and walk). These choices have been considered based on available modes to individuals. Thus, for individuals who do not own a private vehicle, the private vehicle mode is omitted from their choice set. The departure time choice set includes traveling in peak hours and off-peak hours. In order to determine peak and off-peak hours, the frequency diagram of the departure time of all trips, including work and non-work trips is drawn and time interval with the highest frequency is considered as peak hours. Work trips are considered in the calculation of peak hours because these trips play an important role in traffic congestion during peak hours. The frequency graph of trips is shown in figure 1. 10 5 0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time
Figure 1: Frequency of trips in a day As this graph shows, 6-8 AM, 12-13 PM, and 17-18 PM are the peak points of the graph. 6-8 AM have been considered as the morning peak hours and 12-13 PM is considered as returning trips of students and 17-18 PM is considered as returning trips of workers. Table 2 provides an overview of the explanatory variables which are used in the model. The frequency of models choice sets is presented in table 3. Info. Age Sex Job
Education Driving license Travel distance Travel time Walking time to the bus station Travel aim
Variable AGE 5-18 AGE 19-30 AGE 31-41 AGE>41 SEX ADM.JOB SERV.JOB EDU.JOB OTHER LOW.EDU MED.EDU HIGH.EDU DL LGH TT
Table 2: Variables definition and frequency
Definition 1= if age is between 6 to 18, 0= otherwise 1= if age is between 19 to 30, 0= otherwise 1= if age is between 31 to 41, 0= otherwise 1= if age is more than41, 0= otherwise 1= if sex is male, 0= otherwise 1= if the job is an administrative job, 0= otherwise 1= if the job is a service job, 0= otherwise 1= if the job is an educational job, 0= otherwise 1= if the job is another job, 0= otherwise 1= if education level is high school, 0= otherwise 1= if a person has Associate's or Bachelor's degree, 0= otherwise 1= if a person has Master's Degree or Doctorate, 0= otherwise 1= if a person has a driving license, 0= otherwise Distance between origin and destination (km) Travel time between origin and destination (min)
Share 10.84 22.94 30.99 25.24 38.30 12.42 13.40 14.42 59.76 49.45 37.70 12.87 48.81 2.88 6.73
WT
Walking time to the bus station (min)
7.5
SHOPPING RECREATIONA L VISITING OTHER
1= if travel aim was shopping, 0= otherwise
36.84
1= if travel aim was recreation, 0= otherwise
16.67
1= if travel aim was visiting, 0= otherwise 1= if travel aim was other aims, 0= otherwise
26.39 20.10
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Land use
ORG-EDU ORG-ADM ORG-BUSIN DES-EDU DES-ADM DES-BUSIN
Origin education area, a thousandth of square meters Origin administrative area, a thousandth of square meters Origin business area, a thousandth of square meters Destination education area, a thousandth of square meters Destination administrative area, a thousandth of square meters Destination business area, a thousandth of square meters
727
5.86 7 12.78 4.67 10.46 21.63
Table 3: Frequency of models choice sets
Alternatives Private vehicle Taxi Bus Active Peak hours Off-peak hours Has car
Share 1684 2019 980 1920 1845 4758 4792
Frequency 25 31 15 29 28 72 73
5. Results In the departure time choice model, the off-peak hour is considered as base alternative with the utility of zero and coefficients of peak hours are computed relative to this zero utility of the off-peak hour option. Each copula function presents a different description of the correlation between the error terms of the two models. The joint model using the Frank copula has larger log-likelihood than the AMH copula so it selected as the final model for which the detailed results are presented later. Also, this model has a larger log-likelihood value than the model developed based on the product copula that shows the importance of considering the inter-relationship. Table 4 shows the joint model results based on the Frank copula. Table 5 shows the details of each model. Table 4: joint model results based on the frank copula Variable Copula Dependence Parameter Constants Personal info. AGE 5-18 AGE 19-30 AGE 31-41 AGE>41 SEX ADM.JOB SERV.JOB EDU.JOB OTHER DL
Personal vehicle 0.111 (0.345) -1.100 (-10.031) -0.365 (-3.766) 1.911 (20.281) 0.432 (3.744) -
MNL Taxi 1.320 (4.748) 0.243 (1.560)
-0.194 (-1.172) -0.150 (2.003) -
BL Off-peak hours
Peak hours
-
-
-
0.173 (1.778)
0.801 (5.105) -
-
-
-
-
-
0.445 (5.931) -
0.580 (6.178) -
-
-
-
-0.218 (-2.072) -
Bus 1.002 (3.851) -0.681 (-4.494)
Active -0.412 (-1.321) 1.488 (11.489)
-0.423 (-4.326) -
-
-
-
0.377 (2.616) -
-
-
-
-
-1.033 (-8.699)
-1.379 (-11.866)
-
-0.615 (-6.968) -0.910 (-8.512)
0.241 (3.678) -
728
LOW.EDU MED.EDU HIGH.EDU Travel info. TT LGH SHOPPING RECREATIONAL VISITING
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0.531 (6.531) -
0.472 (4.922) -
-
-
-
-
-
-
-
-0.766 (-4.466)
-0.678 (-4.992)
-
-0.205 (-2.080)
-
-
-
-
-
-
-
-
-0.337 (-27.585) -
-
-1.029 (-10.289) -1.029 (-7.930) -
-
-0.146 (-1.573) -
-
-
-0.190 (-1.915) -
-
0.057 (4.324) 0.681 (10.236) -
-
-
0.570 (7.858) -
0.014 (2.938) -
-
-
-
-
0.160 (1.859) -
0.776 (8.949) 0.304 (3.194)
0.353 (3.208) -
OTHER
-
Land use info. ORG-EDU
-
-
-
ORG-ADM
-
-0.004 (-1.682) -
-
-0.029 (-5.266) -
ORG-BUSIN DES-EDU
-0.005 (-2.195) -
DES-ADM
-
DES-BUSIN
7
-0.009 (-5.898)
-
-
-
-
-
0.002 (1.334) -
-
-
-
-
-
-
-
-0.030 (-6.973) 0.007 (3.902) -
Table 5: Details of goodness-of-fit of the developed models Model log likelihood value Based on Frank copula -10009.94 Based on AMH copula -10032.65 Based on product copula -10039.34 Only with constants -12096.67 5.1. Copula dependence parameters analysis The t-value statistics for the copula dependence parameters show that they have strong significance level for the taxi, bus and active transport. In other words, it seems to be strong inter-dependency on using traffic peak hours and taxi, bus and active transport. Regarding the sign of copula parameters, there are some correlated unobserved factors having the mutual effect on choosing the bus, taxi or active and peak hours. For example, for positive copula dependence parameter sign estimated for bus and peak hours, there are some unobserved factors that simultaneously increase choosing probability of bus and traffic peak hours. 5.2. Analysis of estimated coefficients Results show that individuals between 6 to 18 years old have less tendency to use taxi and have more tendency to use active transport for their non-work trips. For individuals between 18 to 30 years old, personal vehicle and bus are
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not preferable alternatives. Also, these individuals prefer to choose traffic peak hours. Personal vehicle and taxi are not attractive alternatives for individuals over 41 years old compared to bus and active transport. Men are more likely to choose the personal vehicle and active transport rather than women. The probability of choosing a personal vehicle and bus is increased for individuals with ADM.job and EDU.jobs. Taxi is not a preferred option for individuals with OTHER jobs. Peak hours have the least utility for individuals with ADM.jobs and highest utility for OTHER jobs. A personal vehicle is a preferable option for individuals that have driving license regarding negative coefficient estimated for the other three modes. Individuals with the low level of income have more tendency to use bus and taxi. A personal vehicle is an attractive option for individuals with the medium level of income. People with the high level of income have less tendency to choose the bus, active and peak hours. By increasing the travel time, the probability of choosing active transport is decreased. Also by increasing the length of trips the probability of choosing peak hours is increased. The probability of choosing a personal vehicle is decreased for shopping and recreational trips. Taxi seems to be a more appropriate option for visiting and other trips. People prefer to use the bus for their visiting trip and this option is not an attractive option for shopping trips. Active transport has the least utility for recreational trips. Individuals have more tendency to choose peak hour for their shopping and visiting trips. As the origin and destination business area are increased the probability of choosing a personal vehicle is decreased. Also, the probability of choosing a taxi is decreased as origin administrative area and destination educational area are increased. Individuals more likely choose active transport if origin educational and business area are increased. If destination administrative area is decreased or destination business area is increased, the probability of choosing peak hour is increased. 6. Sensitivity Analysis In this section, a sensitivity analysis is presented for the estimated parameters. Formally, elasticity defined as a unit less measure that describes the relationship between a percentage change of some independent variables (i.e. an attribute of an alternative or the socio-demographic attributes of a decision maker) and some percentage change in the quantity demanded (Hensher, Rose, and Greene 2005). This paper calculates disaggregate elasticities (for individuals) for continuous variables including travel time, travel distance, walking time, and land use variables. Then aggregate elasticities are obtained by using the average of disaggregate elasticities. Table 6 shows elasticities for the joint model using Frank copula. Table 6: shows elasticities for the joint model using Frank copula
0.17
TAXIOFF PEAK 0.27
BUSOFF PEAK 0.28
ACTIVEOFF PEAK -1.97
WT
-0.04
-0.06
-0.06
Variable
PERS -OFF PEAK
TT
PERSPEAK
TAXIPEAK
BUSPEAK
ACTIVEPEAK
0.18
0.34
0.31
-1.93
0.24
-0.04
-0.08
-0.07
0.23
ORG-EDU
-0.02
-0.03
-0.03
0.06
-0.02
-0.03
-0.03
0.06
ORG-ADM
0.01
-0.02
0.01
0.01
0.01
-0.02
0.01
0.01
ORG-BUSIN
-0.04
0.00
0.00
0.03
-0.04
0.00
0.00
0.03
DES-EDU
-0.02
-0.14
-0.01
-0.02
0.08
-0.01
0.14
0.12
DES-ADM
0.02
0.02
0.02
0.02
-0.04
-0.06
-0.06
-0.05
DES-BUSIN
-0.11
0.03
0.03
0.03
-0.11
0.04
0.03
0.03
LGH
0.03
0.03
0.03
0.05
-0.09
-0.13
-0.13
-0.11
As it can be seen from table 6, for instance, 1% increase in travel time in peak and off-peak hours has the positive influence on personal vehicle selection. Here, 1% increase in destination administrative land use in off-peak hours, the probability of personal vehicle is decreased by 0.04%. Also, it can be said that active mode of transport is more
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sensitive to increase in travel attributes such as travel time and distance and it can be related to this mode is appropriate for short distance trips. 7. Conclusion This study focused on mode choice and departure time choice as two decisions that are expected to have some level of interaction. This interaction was studied by considering the correlation between unobservable random variable. Considering a multinomial logit model for mode choice and a binomial logit model for the time of day, the interaction is approximated using some copula (Frank, AMH, and product) functions. One of the most important findings of this study is to confirm that the interdependency between the two decisions reflected by the highly statically significance copula dependence parameters. The joint model provided a goodness-of-fit to the data with interpretable parameters, which shows the significance of using the proposed joint model for capturing the interaction between these two decisions.
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