Chapter 3 Travel as Response

Two different perspectives, individual and collective, can explain travel behavior and car use. When people contextualizing travel as a personal choice or decision-making, the traveler as a subject make mode choices, driving or not. When travel behavior is understood as a social phenomenon, researcher observe and understand all the trip distance, time, and distributions as a whole.

The two perspectives derived two schools of theory, Traveler choice and human mobility. In the school of Traveler choice, travel distance could be treat as an independent variable, a part of travel cost, or could be decided in the next step after mode choice , such as route choice. In the school of human mobility, driving distance grab more attentions.

3.1 Travel Variables

Three dimensions can reflect the degree of car use, travel mode, driving frequency, and driving distance. Previous studies commonly choose two metrics for measuring them, the share of auto trips (or other modes) and Vehicle Miles Traveled (VMT).

The share of mode is calculated by dividing the number of chosen mode over the total number of trips. The main travel modes, transit, bicycle, and walking are the alternatives to driving personal car. Given the same amount of travel demand, more active and transit modes means less car use. VMT is used to measure the travel distance made by a private vehicle. An integrated viewpoint is to treat the non-auto trips as zero-VMT. In this way, the probability distribution of VMT can comprehensively represents the travel behavior.

The smallest unit of VMT is recorded by trip from a daily travel survey. Then these records can be aggregated to personal or household daily VMT (DVMT). A traveler’s or household’s DVMT can account for the degree of automobile dependency by combining the number of trips and driving distance during a day. Given the survey day is randomly selected, DVMT can reflect the typical travel pattern in general.

Although, there are other approaches collect weekly, monthly, or longer VMT records by tracking car usage. The odometer records are more likely to represent the usage of vehicle rather than traveler’s behavior. It is not easy to acquire long-term VMT through survey-based method. The annual mileage and fuel efficiency information provided in some public data usually are estimated values using daily records and are not as accurate as DVMT.

On a personal scale, VMT relates to the economic cost of travel by car, while another dimension, travel time measuring the time cost of vehicle travel. For society as a whole, the total VMT measures the usage of road network. Thus, it acts as a major interest within the field of transportation, especially in the research of travel demand and infrastructure capacity.

And VMT highly correlated with the amount of fuel consumption, which is one of the main indicators of pollution and GHG emission. Since transportation is the second source of GHG emissions, it is also one of the priority issues involving sustainable development and climate change.

Previous research found that reducing VMT is instrumental in solving some urban problems and improving the qualities of urban life. The proportion of transportation cost in household expenditures is about 15 to 25 percent in the U.S. It is natural that urban studies try to figure out the relationship between VMT and some urban built-environment factors. Then urban or regional policies could identify the best practices to contribute VMT reduction.

3.2 Traveler Choice

Are ‘decision’ and ‘choice’ the same when discussing travel modes? Literally, a ‘choice’ is one decision given all available options at the same time. While ‘decision’ is a broader concept. A decision could be a schedule with a combination of many choices, such as modes, destination, and activities. A decision related to travel behavior could even include bicycle or car purchase, and relocation. This section will start from the theories of mode choice, then extend to a broader discussion of decision processes.

3.2.1 Rational Choice Theory

For prescriptive, analytical everyday decision-making, rationality is a basic assumption in reasoned behavior or rational choice theories (Edwards 1954; Von Neumann and Morgenstern 1944).

This category is also called ‘Normative Decision Theory,’ which assume people a traveler is an ideal decision maker who are full rational. It requires three necessary steps including information collection, utility evaluation, and choice making.

  • Expected Utility Theory (EUT)

Traditional economics focus on the utility evaluation and come up with the Expected Utility Theory (EUT) which is also called Consumer Choice Theory. The rule of EUT is Random Utility Maximization (RUM) (Ben-Akiva and Lerman 1985; McFadden 1973). This classical theory claims that customer always choose the one most appropriate by comparing the advantages and disadvantages of a range of alternatives, evaluating the benefits and costs of each possible outcome. Eventually travelers will select the optimal solution with the maximum ‘utility’ from the choice set.

In real life, Rational Choice Theory can not accurately describe the actual human behavior. Individuals do not often collect and analyse all the relevant information. They are not ‘ideal’ and are not able to calculate the utility for all possible alternatives with perfect accuracy. In many cases, the travel decision is not regarded as the ‘best’ one to achieve travelers’ desired objective. Many other theories were developmed to fix these issues.

3.2.2 Bounded Rational Behavior

Bounded rationality focused on the limitation of self-control (March and Simon 2005). In reality, individuals are behaving under many constraints including incomplete information, limited time, and cognitive capacity. The observed behaviors often are not optimal and are inconsistent with ‘pure’ rationality. Bounded rationality claims that, when people make decisions under constraints, heuristics and rules of thumb are more common than statistical inference. People are satisfied with a ‘good enough’ decision unless there is a definitively better alternative. The recently witnessed events would have stronger effects on an individual’s decision than others (Camerer, Loewenstein, and Rabin 2004).

3.2.3 Theory of Planned Behavior

In psychology, many theories and models are developed to explain people’s decision-making processes.5

Ajzen and Fishbein (1977) proposed the Theory of Reasoned Action (TRA) to understand people’s behavioral intentions and actual behaviors. They found two deciding psychological elements as attitudes and subjective norms. Ajzen (1991) adds a new part of Perceived Behavioral Control (PBC) and renames TRA as Theory of Planned Behavior (TPB).

Attitudes are personal evaluation and it means how people prefer or are against performing an activity. For example, a commuter might choose transit in spite of the longer travel time because this person believes that transit is an environment-friendly transport mode.

Subjective norm is the social pressure from others. In the example above, choosing transit is because of other people’s normative expectations rather than personal desirability.

PBC represents some nonvolitional factors such as time, budget, and resources. PBC is assessed by the individual’s perception of ease or difficulty of the behavior. PBC is one reason of the different between intentions and actual behaviors, which is called attitude-behavior gap (Kollmuss and Agyeman 2002; B. Lane and Potter 2007). In this case, a commuter might choose transit because this person is confident in catching the bus every day.

Based on RUM models, McFadden (2001) proposes a similar framework called the choice process including attitudes, perception, and preference. This framework is further developed to hybrid choice model (HCM) and non-RUM decision protocols (Ben-Akiva et al. 2002).

Two meta-analyses found that intentions to drive, perceived behavioral control, habits and past behavior play the primary roles in travel mode choice. Among these factors, PBC have the strongest effects on private car use. People don’t want to reduce the car use because they think it is very inconvenient. The effect of attitudes is modest while subjective norms have weak effect on car use (Lanzini and Khan 2017; Gardner and Abraham 2008).

3.2.4 Prospect Theory

Kahneman and Tversky (1979) introduced the ProspectTheory to study the impacts of biases. Prospect Theory is a descriptive theory with three main components: First, people are more sensitive to the sure things (e.g., the probability between 0.9 and 1.0, or between 0.0 and 0.1 ), while being indifferent to the middle range (e.g., from 0.45 to 0.55). Second, people care more about the change of overall proportion than the absolute values regardless of gains or losses. Third, people make choice based on a reference point, rather than the overall situation or worth. Economist also extend the theory of expected utility maximization to Behavioral Economics by address the influence of psychology on human behavior.

  • Regret Theory

Regret Theory introduces the notions of risk or uncertainty in decisions (Loomes and Sugden 1982). Psychological studies found that individuals will not only try to maximize the utility but to minimize the anticipation of regret. The fear of regret could affect people’s rational behavior. For example, A high risk of congestion in peak hours could encourage a commuter to choose transit mode. Likewise, a good reputation for punctuality can give traveler confidence in the rail system.

In addition to the traditional utility framework, a regret term is added to address the uncertainty resolution. The utility function on the best alternative outcome will be smaller after subtracting the regret term, which is an increasing, continuous and non-negative function.

  • Cognitive Bias

Another psychological factor, cognitive bias can result in judgement errors. For example, people treat potential gains and losses differently, that is called Loss Aversion. Loss Aversion suggests that the negative feeling about losses is greater than the positive response to gains (Tversky and Kahneman 1992). As a result, individual’s decisions may not be consistent with evidence and tend to pay additional costs to avoid losses.

3.3 Human Mobility

In Physics and Geography, travel distance and pattern are treated as an objective phenomenon. There is a long history of human mobility studies. The related theories try to use some statistical expressions to fit the aggregated trip distributions.

Gravity Law is a dominant theory in this field. Scholars have developed some more delicate forms of Gravity Law and found some mathematical relationship to other famous distribution laws. Some theories from different perspectives, like intervening opportunities also show strong ability for explaining travel patterns and regularities.

3.3.1 Distance Based Theories

  • Law of Migration

An early theory called Law of Migration by Ravenstein (1885) tried to explain the regional migration patterns. This found is based on observation rather than quantitative analysis. But it capture the fact that the direction of migration is toward the regional center with great commerce and industry. It also pointed out that distance is a primary factor for migrant. This theory inspired many studies on population movement consequently. Even today, socio-economic factors and distance-constraints are the essential parts in the relevant models and frameworks.

  • Zipf’s Law

Zip’s law is also called discrete Pareto distribution. It is found in linguistics to explain the inverse relationship between the frequency and rank of a word. The charm is that this rank-frequency distribution disclosed a universal law in many realms of society and physics, such as urban size, corporation sizes, cells’ transcriptomes and so on. Zipf interpreted the two competing factors as force of diversification and unification. The former produces larger amount of cases and the later tries to upgrade the rank. An equilibrium of the rank-frequency balance is controlled through a parameter \(\alpha\) in the exponent. For example, a city’s population size \(m\) has a negative power relationship to its rank \(r\) as below. (Visser 2013; Jiang, Yin, and Liu 2015; Rozenfeld et al. 2011; Gomez-Lievano, Youn, and Bettencourt 2012; Hackmann and Klarl 2020)

\[ m \sim 1 / r^{\alpha} \]

Zipf (1946) extended this expression to describe the traffic in both directions between two cities:

\[ t_{ij}\propto \frac{m_i m_j} {d_{ij}} \]

where \(t_{ij}\) represent the traffic flow of goods between two centers \(i\) and \(j\) with population sizes \(m_i\) and \(m_j\). \(d_{ij}\) is the distance from \(i\) to \(j\). Because Zipf’s formula has a same form with Newtonian mechanics (Newton 1848), people call this expression as Gravity Law.

  • Gravity Law

As the most influential theory, Gravity Law asserts that the amount of traffic flow between two centers is proportional to the product of their mass and inverse to their distance. The mass is often measured by population size.

\[\begin{equation} p_{ij}\propto m_i m_j f(d_{ij}), \qquad i\ne j \tag{3.1} \end{equation}\]

where \(p_{ij}\) is the probability of commuting between origin \(i\) and destination \(j\), satisfying \(\sum_{i,j=1}^n p_{ij}=1\). \(m_i\) and \(m_j\) are the population of two census units. The travel cost between the two places is represented as a distance decay function of \(d_{ij}\) .

Exponential and power are the two forms of the distance decay function with a parameter \(\lambda\) showed as below:

\[ f(d_{ij})=\exp(-\lambda d_{ij}) \] and

\[ f(d_{ij})={d_{ij}}^{-\lambda} \] The function implies that the movements between the origin and destination decays with their distance. In transportation modeling, a common form of gravity model is :

\[ T_{ij}= \alpha_i O_i \cdot \beta_j D_j \cdot f(d_{ij}) \]

where \(T_{ij}\) is the flow between \(i\) and \(j\). the two population are replaced by total tirp generation of origin \(O_i\) and total trip attraction of destination \(D_i\). \(\alpha_i\) and \(\beta_j\) are two constraining parameters to satisfy \(\sum_{i}^{n_i}T_{ij} = D_j\) and \(\sum_{j}^{n_j}T_{ij} = O_i\). It means that \(\alpha_i = [\sum_{j}^{n_j} \beta_j D_j \cdot f(d_{ij})]^{-1}\) and \(\beta_j = [\sum_{i}^{n_i} \alpha_i O_i \cdot f(d_{ij})]^{-1}\). Thus, this model is called as doubly constrained gravity model.

If it relieves the two constrains. this model will be simplified to single-constrained and unconstrained gravity model. By assuming \(\alpha\beta\) is an adjustment parameter irrelevant to locations \(i\) and \(j\) for controlling the total flows, this model will not guarantee that the attraction of a destination equals the sum of flow from all origins, and the generation of a origin equals the sum of flow to all destinations.

  • Power Law

Broadly speaking, Zipf’s law and Gravity Law have a common essence of power law, or scaling pattern. The Zipfian distribution is one of a family of power-law probability distributions. The power-law distribution also holds in many realms: urban size, population density, street blocks, building heights, etc.

The state-of-the-art studies of human mobility agree that travel behavior follows a power-law distribution at the population level (Barbosa et al. 2018). An example is Brockmann, Hufnagel, and Geisel (2006) use dollar bills to track travel habits and confirm this theory. It reflects the fact that both trip and land use, as two geographic variables, follow some Paretian-like distribution. Apparently, it conflicts with Gaussian thinking, the foundation frame of linear models based on the location and scale parameters (Jiang and Jia 2011; Y. Chen and Jiang 2018; Jiang 2018a, 2018b)

Meanwhile, the log-normal distribution may be asymptotically equivalent to a special case of Zipf’s law, which could support the logarithm transform in current VMT-density models (Saichev, Malevergne, and Sornette 2010).

3.3.2 Opportunity Based Theories

  • Law of Intervening Opportunities

Law of Intervening Opportunities by Stouffer (1940) developed the migration theory in a different direction. Stouffer proposed that “the number of people going a given distance is directly proportional to the number of opportunities at that distance and inversely proportional to the number of intervening opportunities.”

Comparing with gravity law, the number of intervening opportunities \(s_{ij}\) replaces the distance between origin and destination. For example, a resident living in location \(i\) is attracted to location \(j\) with \(s_{ij}\) job opportunities in between.

\[ p_{ij}\propto m_i \frac{P(1|m_i,m_j,s_{ij})}{\sum_{k=1}^n P(1|m_i,m_j,s_{ij})}, \qquad i\ne j \]

where the conditional probability \(P(1|m_i,m_j,s_{ij})\) can be expressed by Schneider (1959) as:

\[ P(1|m_i,m_j,s_{ij})=\exp[-\gamma s_{ij}] - \exp[-\gamma (m_j + s_{ij})] \]

  • Radiation Law

Simini et al. (2012) propose a radiation model express the probability of the destination \(j\) absorbing a person living in location \(i\) as below:

\[ P(1|m_i,m_j,s_{ij})= \frac{m_i m_j}{(m_i + s_{ij})(m_i + m_j + s_{ij})} \]

Or in transportation model it is expressed as:

\[ T_{ij}= O_i\cdot\frac{m_i m_j}{(m_i + s_{ij})(m_i + m_j + s_{ij})} \] To approximating the number of opportunities, \(s_{ij}\) is from the population within a circle centered at origin. The radius is the distance between \(i\) and \(j\). Then \(m_i + m_j + s_{ij}\) represents the total population within the circle, and \(m_i + s_{ij}\) is the total population within the circle but excluding \(j\), that is:

\[ T_{ij}= O_i\cdot\frac{m_i }{m_i + s_{ij}}\cdot\frac{m_j}{m_i + m_j + s_{ij}} \] The part of fraction converts to the product of two weights, the weights of origin and destination in the whole region. Although distance \(d_{ij}\) doesn’t appear in the expression of radiation model, it is still a determinant as in gravity model.

  • Distance Decay (hazard models)

Using the survival analysis framework, Yang et al. (2014) further extended this model by assuming a trip from origin to destination as a time-to-event process. Here time variable is replaced by the number of opportunities.

The survival function \(S(t)=Pr(T>t)\) represents the cumulative probability of the event not happened within a certain amount of opportunities. Choosing Weibull distribution as the survival function, \(S(t)=\exp[-\lambda t^\alpha]\) with scale parameter \(\lambda \in (0, +\infty)\). By assuming \(f(\lambda)=\exp[-\lambda]\) and integral on \(\lambda\), the derivation is:

\[\begin{equation} \tag{3.2} \begin{split} P(T>t)=&E\{\exp[-\lambda t^\alpha]\} \\ =&\int_0^{+\infty}\exp[-\lambda t^\alpha]\exp[-\lambda]d\lambda\\ =&\frac{1}{1+t^{\alpha}} \end{split} \end{equation}\]

By replacing \(t\) with \(m_i+s_{ij}\), the conditional probability is:

\[ \begin{aligned} P(1|m_i,m_j,s_{ij})= &\frac{P(T>m_i+s_{ij})-P(T> m_i+s_{ij}+m_j)}{P(T>m_i)} \\ =&\frac{[(m_i + s_{ij} + m_j)^{\alpha}-(m_i + s_{ij})^{\alpha}](m_i^{\alpha}+1)}{[(m_i + s_{ij} + m_j)^{\alpha}+1][(m_i + s_{ij})^{\alpha}+1]}\\ \end{aligned} \]

where \(\alpha\) is a parameter adjusting the effect of the number of job opportunities between origins and destinations.

A similar method can be found in Ding, Mishra, et al. (2017) ’s study. They use a multilevel hazard model to examine the effects of TAZ level and individual level factors with respect to commuting distance using the data of Washington metropolitan area.

Based on commuting data from six countries, Lenormand, Bassolas, and Ramasco (2016) found gravity law performs better than the intervening opportunities law. The reasons could be the circle with radius \(d_{ij}\) can not accurately represent the real influencing area, and the different between population and opportunities is not captured in this way.

3.3.3 Time Geography

In contrast to overall trip distribution, the movements of individuals are always research interest in geography. Hägerstraand (1970) proposed some concepts and tools in space and time to measure and understand the individual trajectories. This branch is called time geography. The famous “space-time aquarium/prism” is a 3D cube by adding temporal scales on the geographic space. It can capture the detailed structure and behavior of traveler.

A daily travel could include multiple trips and form a travel chain. The traveler may switch the sequence or adjust the routes to optimize the chain and minimize the travel costs. The daily total travel distance is the summation of every trip distances. The number of trips denotes as trip count. It exists but not so common that driving itself is the travel purpose, especially in daily life.

At individual level, time geography borrows some physical and mathematical concept and methods such as random walk, Brownian motion, and Levy flight

Along with the wide usage of Global Positioning System (GPS), high performance computer, and sophisticated algorithms, the high-resolution data being collected. The relevant studies also have a dramatic increase after 2005.

3.4 Probability Distributions

In the transportation field, there are some valuable studies to identify the distributions of trip variables. Based on some theoretical or empirical studies (Hellerstein and Mendelsohn 1993; Jang 2005; Bien, Nolte, and Pohlmeier 2011), scholars prove that trip generating frequency should not choose the linear regression models based on continuous functional forms. A zero-inflated negative binomial model is appropriate to solve the problems of over-dispersion and excess zero. This study implies that the diagnosis of variable distribution may be critical for regression modeling.

For continuous variables, it seems like choosing log-normal distribution for trip distance/time is a convention. Pu (2011) choose log-normal as prior assumption because a report called Future Strategic Highway Research Program (F-SHRP) (Cambridge Systematics, Texas Transportation Institute, Univ. of Washington, and Dowling Associates 2003; Associates and (US) 2013) says “the log-normal distribution is the closest traditional statistical distribution that describes the distribution of travel times.” Actually, the new version, SHRP2 says “formal tests (e.g., a Kolmogorov-Smirnov test) could be employed to evaluate the assumption and identify the sensitivity of the results to departures from this assumption.” (p. 130)

Meanwhile, Lin et al. (2012) validates the daily vehicle miles traveled (DVMT) follows a gamma distribution in the context of PHEV energy analysis. Based on the multidate (7-200 days) data sets from four countries, Plötz, Jakobsson, and Sprei (2017) found Weibull distribution is an overall good two-parameter distribution for daily VKT; while the log-normal estimates are more conservative. The studies on trip distance are still not conclusive. But the attention of three distributions is similar to the survival analysis, which is also called time-to-event analysis (Kleinbaum and Klein 2012). This shows a potential relation with the Distance-Decay, or travel-time-budget theories (Marchetti 1994). Similar to that, Kölbl and Helbing (2003) show a canonical-like energy distribution for short trips by modes, which imply “a law of constant average energy consumption for the physical activity of daily travel.” Some studies are not limited in parameter methods. Simini et al. (2012) propose a parameter-free model that predicts patterns of commuting.


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  1. CMDT=Cognitive moral development theory (Kohlberg, 1984),

    ITB=Ipsative theory of behavior (Frey, 1988),

    NAM=Norm activation model (Schwartz, 1977,Schwartz and Howard, 1981),

    SDT=Self-determination theory(Deci & Ryan, 1985),

    TAM=Technology acceptance model(Davis, 1989),

    TDM=Travel demand management measures,

    TNC=Theory of normative conduct (Cialdini et al., 1990,Cialdini et al., 1991),

    TPB=Theory of planned behavior(Ajzen, 1985,Ajzen, 1991),

    VBN=Value-belief-norm (Stern, 2000,Stern et al., 1999),

    MGB=Model of goal-directed behavior(Perugini & Bagozzi, 2001)↩︎