Continuous Logit Model Research Articles

Finite-Mixture Continuous Logit Model: Formulation and Application for Shopping Start Time Choice of Non-Commuters

As a continuous generalization of the multinomial logit (MNL) model, the continuous logit (CL) model can be used for continuous response variables (e.g., departure time and activity duration). However, the existing CL model requires the calculation of numerical integrals to obtain the choice probabilities; it thus takes a long time to estimate the model parameters, particularly when the sample size is large. In this paper, we formulate the finite-mixture CL (FMCL) model as a new continuous choice model by combining the finite-mixture method and the CL model, in which the continuous distributional function of the finite mixture is embedded in the CL model. As a result, the individual choice probability can be obtained directly by computing the probability density of the continuous distribution function; this avoids calculation of the integral but still obeys the random utility maximization (RUM) principle. Simulation experiments are conducted to demonstrate the capability of the model. In an empirical study, the proposed model is applied for non-commuters’ shopping activity start time using the expectation-maximization (EM) algorithm based on Shanghai Household Travel Survey data. The results show that the FMCL model developed in this paper can greatly reduce the model estimation time (10,048 observations requiring only 3 min) of the CL model, and the model also has a more intuitive interpretation of model coefficients, directly reflecting variable effects on time-of-day choice. These two advantages can greatly enhance the practical value of the proposed modeling method.

Autoregressive continuous logit: Formulation and application to time-of-day choice modeling

In this paper, we formulate the autoregressive continuous logit model as a novel continuous choice model capable of representing correlations across alternatives in the continuous spectrum. We formulate this model by considering two approaches: combining a discrete-time autoregressive process of order one (i.e., a linear stochastic difference equation) with the continuous logit model, which leads to the discrete-time autoregressive continuous logit; and combining a continuous-time autoregressive process (i.e., a linear stochastic differential equation), known in the stochastic process literature as Ornstein-Uhlenbeck process with the continuous logit model, which leads to the continuous-time autoregressive continuous logit. The autoregressive nature of the model is in its error structure, allowing correlations in unobserved heterogeneity. For both approaches, we study their properties numerically. We also compare both approaches to highlight their relation to each other.

Urban land use, sorting, and population density: A continuous logit model

This paper analyzes land use, sorting, and population density in a polycentric city. To this end, the paper develops a continuous spatial choice logit model of a closed polycentric city with multiple income classes and endogenous land use where households select their workplace and residence locations probabilistically. In contrast to the classic urban model with deterministic location choices, the continuous logit model predicts incomplete segregation of citizens who work in different business centers, and, therefore, cross commuting. The relative size of income elasticity of both land demand and commuting costs determines income sorting of individuals working in a particular business district. Due to heterogeneous preferences for residence locations and workplaces, this may not hold for the aggregate spatial pattern of multiple income classes. Finally, individual land use may decrease as distance between workplace and residence increases.

Advance purchase behaviors of air passengers: A continuous logit model

Modeling the advance purchase behaviors of air passengers is essential when airlines develop revenue management strategies. Therefore, this study empirically investigates advance purchase behaviors based on the air ticket transaction data by using a continuous logit model. The estimation results show that advance purchase behaviors are significantly affected by price, price uncertainty, time of day (morning, afternoon and evening flight), days of week (flight on Friday), months of year (peak or off-peak seasons), and consecutive holiday. Accordingly, different pricing strategies should be used for different flights to maximize revenue.

A model of bus bunching under reliability-based passenger arrival patterns

If bus service departure times are not completely unknown to the passengers, non-uniform passenger arrival patterns can be expected. We propose that passengers decide their arrival time at stops based on a continuous logit model that considers the risk of missing services. Expected passenger waiting times are derived in a bus system that allows also for overtaking between bus services. We then propose an algorithm to derive the dwell time of subsequent buses serving a stop in order to illustrate when bus bunching might occur. We show that non-uniform arrival patterns can significantly influence the bus bunching process. With case studies we find that, even without exogenous delay, bunching can arise when the boarding rate is insufficient given the level of overall demand. Further, in case of exogenous delay, non-uniform arrivals can either worsen or improve the bunching conditions, depending on the level of delay. We conclude that therefore such effects should be considered when service control measures are discussed.

A continuous spatial choice logit model of a polycentric city

This paper analyzes a closed linear polycentric city where households probabilistically select their workplace and residence locations. The study utilizes a continuous logit model to describe household location choices. In contrast to the classic urban model with deterministic location choices, the continuous logit model predicts noticeable direct effects of more than one workplace on land rents, asymmetry of the land rent schedule around secondary business districts, incomplete segregation of citizens who work in different business centers and, therefore, cross-commuting, and incomplete segregation of workers and farmers.

A Model of Bus Bunching under Reliability-based Passenger Arrival Patterns

If bus service departure times are not completely unknown to the passengers, non-uniform passenger arrival patterns can be expected. We propose that passengers decide their arrival time at stops based on a continuous logit model that considers the risk of missing services. Expected passenger waiting times are derived in a bus system that allows also for overtaking between bus services. We then propose an algorithm to derive the dwell time of subsequent buses serving a stop in order to illustrate when bus bunching might occur. We show that non-uniform arrival patterns can significantly influence the bus bunching process. With case studies we find that, even without exogenous delay, bunching can arise when the boarding rate is insufficient given the level of overall demand. Further, in case of exogenous delay, non-uniform arrivals can either worsen or improve the bunching conditions, depending on the level of delay. We conclude that therefore such effects should be considered when service control measures are discussed.

A continuous logit hotelling model with endogenous locations of consumers

This paper analyzes location and price choices of firms and subsequent location choices of consumers in a linear city model when consumers have different perceptions of locations and firms. The study utilizes a continuous logit model to describe consumers’ location and supplier choices. A unique subgame perfect Nash equilibrium exists where firms make positive profits and the minimum differentiation principle applies. For identical firms, prices converge to marginal costs as heterogeneity of consumers’ perceptions disappears.

Continuous Logit Polycentric City Model

This paper analyzes a closed, essentially linear polycentric city with homogenous households who probabilistically select their workplace and residence locations. The study utilizes a continuous logit model to describe household location choices. In contrast to the classic urban model with deterministic location choices, the continuous logit model predicts noticeable direct effects of more than one workplace on land rents, asymmetry of the land rent schedule around secondary business districts, incomplete segregation of citizens who work in different business centers and, therefore, cross commuting, and incomplete segregation of workers and farmers. JEL-Code: R130, R140, D110.

The continuous cross-nested logit model: Formulation and application for departure time choice

Discrete choice models, like the multinomial logit (MNL), have long been recognized for their ability to capture a wide array of transport-related choice phenomena. However, a number of choices are continuous response variables (e.g., location, departure time, activity duration, and vehicle usage). This paper introduces the continuous cross-nested logit (CCNL) model. The CCNL model results from generalizing the discrete cross-nested logit (CNL) model for a continuous response variable, much like the continuous logit model emerges by generalizing the MNL. The model is formulated and shown to come from the generalized extreme value (GEV) class of models. In addition, the structure of utility correlations is presented. The model’s parameters are estimated for a work-tour departure time context using Bayesian estimation techniques and San Francisco Bay Area data. Empirical results suggest model predictions that are very similar to the continuous logit, but it out-performs the continuous logit in terms of out-of-sample prediction with these data. The CCNL also allows a more flexible choice behavior to emerge. Finally, a simple welfare example is illustrated and a number of model extensions are presented.

Empirical Investigation of Continuous Logit for Departure Time Choice with Bayesian Methods

Numerous models of travel timing have been calibrated and reported in the literature. Some studies have treated time as a discrete variable by using familiar discrete choice methods, whereas others have treated time in a continuous fashion. Both approaches offer distinct advantages. Here a continuous logit model of work tour departure time choice is estimated; this model offers the advantage of a continuous-time response. A random utility maximization structure is used to capitalize on the key advantages of both main approaches to the modeling of travel timing. Bayesian techniques are used to estimate model parameters, and estimation results suggest a variety of predictive densities for departure times across different individuals. In addition, ordinary least squares regression models are used to estimate travel times and their variance across times of day for the auto and transit modes. These network variables are used to inform estimation of the continuous logit model of departure time. The results are meaningful for multiple applications, and the continuous logit can readily be extended to a two-dimensional choice construct, such that the departure and return times can be modeled simultaneously. In addition, Bayesian estimation techniques allow for the utility function to take any number of forms, which may offer greater predictive ability.

Continuous spatial choice: The continuous logit model and distributions of trips and urban densities

The continuous generalization of the discrete logit model is derived from random utility theory. The vontinuous logit model is then applied to derive spatial distributions of residential locations, workplaces and trips. Different distributions are obtained for a wide range of assumptions about the spatial distributions of elemental alternatives, or choice opportunities, and transportation level of service attributes. Some of these distributions have previously been proposed and fitted to data. The developments of this paper add a choice theory interpretation to existing models and suggest new improved models.