An enhanced SPSA algorithm for the calibration of Dynamic Traffic Assignment models
Introduction
Dynamic Traffic Assignment (DTA) models are capable of capturing the complex dynamics in a traffic system to estimate and predict the transition of traffic states and the evolution of recurrent and non-recurrent congestion (Ben-Akiva et al., 2001, Mahmassani, 2001, Ziliaskopoulos et al., 2004, Tampère et al., 2010). In their core, DTA systems model two major components: demand and supply, as well as their interaction (Ben-Akiva et al., 2002a, Ben-Akiva et al., 2002b). Demand models mainly focus on dynamic origin–destination (OD) flows, drivers’ departure time and route choice behaviors (Antoniou et al., 1997, Ashok and Ben-Akiva, 2000, Ashok and Ben-Akiva, 2002, Ben-Akiva and Bierlaire, 2003). Supply models load traffic into the network and decide vehicles’ movement patterns. Extensive research has been undertaken on DTA models (see e.g. Peeta and Ziliaskopoulos, 2001, Viti and Tampère, 2010, Tampère et al., 2010, and Chiu et al., 2011), which can be divided into two main streams: analytical DTA models based on mathematical programs and simulation-based DTA based on computer simulation. Simulation-based DTA has more detailed modeling abilities by capturing individual drivers’ behaviors and the sophisticated interactions between the demand and supply, and therefore has been applied more frequently in real-world traffic systems to evaluate network performance, including flow, density, speed, travel time, and queues. There are numerous examples of successful adoption of simulation-based DTA for traffic planning (see, e.g., Ben-Akiva et al., 2007, Rathi et al., 2008, Balakrishna et al., 2008, Balakrishna et al., 2009, Sundaram et al., 2011, Florian et al., 2001, Barcelo and Casas, 2006, Ziliaskopoulos et al., 2004, Tampère et al., 2010), and real-time traffic systems to provide consistent traffic prediction to travelers for route planning (e.g. Paz and Peeta, 2009) and to traffic control centers for control strategy generation (see, e.g., Ben-Akiva et al., 1997, Mahmassani, 2001, Antoniou, 2004, Wen et al., 2006, and Antoniou et al., 2011a). The implementation of simulation-based DTA systems in large-scale highly congested urban networks has been made possible by research in scalable DTA algorithms and parallel DTA simulation (Wen, 2009).
To successfully apply a DTA model to a real-world traffic network to replicate the traffic dynamics accurately, a great number of parameters need to be calibrated with local traffic surveillance data, including measured time-dependent segment flow and travel times. This process is detailed in Antoniou et al. (2011b). The off-line calibration of a simulation-based DTA model is essentially the process of determining the model parameters in both the demand side (time-depend OD flows, route choice models, etc.) and the supply side (e.g. segment-specific speed density relationship parameters and capacities for a mesoscopic supply model) of the DTA, given a set of initial parameter values, to minimize the error between the simulated output and observed values (surveillance data). This calibrated set of parameters forms a historical database, which is the fundamental of the DTA for future planning or real-time applications in the network (Balakrishna, 2006). On-line calibration deals with the dynamic adjustment of the off-line calibrated parameters utilizing real-time surveillance data as they become available during the application of a DTA model (Antoniou et al., 2007).
Conventional calibration methodologies estimate parameters in the demand side and supply side separately in a sequential and iterative manner (see, e.g., Ben-Akiva et al., 2002a, Ben-Akiva et al., 2002b). OD estimation, for example, has been one of the most active fields in the DTA calibration research. Researchers find that the traditional generalized least square (GLS) framework proposed by Cascetta et al. (1993) has limitations under congested conditions, when the linear assumption on the assignment matrix fails to represent the complex relationship between OD flows and sensor counts. Moreover, DTA applications on large-scale networks require more computationally efficient methodologies for OD estimation. To address these challenges, researchers have proposed a multitude of novel frameworks. Hazelton (2008) views the OD estimation problem in a statistical perspective with a multivariate model for the link counts. The author describes the day-to-day evolution of OD demands with a modest number of parameters and proposes a Bayesian approach to inference. Frederix et al. (2011) introduce the use of marginal computation (MaC) and the use of kinematic wave theory principles to estimate the often non-linear relationships between OD flows and link flows under congested conditions. MaC efficiently estimates the relationships by perturbing the OD flows and calculating the sensitivity of link flows to the perturbation. The derived relationships were used in the dynamic OD estimation problem and showed its advantage over traditional OD estimation approaches that rely on linear assignment matrices. Djukic et al. (2012) apply principal component analysis (PCA) to pre-process OD data. This methodology can significantly reduce the size of time series of OD demand without sacrificing much of the accuracy, which leads to a significant reduction of computational costs.
Simultaneous demand and supply calibration framework has received significant attention and become the state-of-the-art (Balakrishna, 2006, Balakrishna et al., 2007a). Compared with sequential estimation, the joint demand–supply calibration is more efficient in terms of the use of the data and may significantly save computational time. Its aggregate nature makes it an ideal methodology regardless of the specific model category and it has already been successfully applied in both microscopic traffic simulation applications (Balakrishna et al., 2007b) and mesoscopic simulation-based DTA systems (Ben-Akiva et al., 2012). Most importantly, being able to capture the correlation between the demand side and the supply side, the joint calibration was found to lead to better results than sequential estimation (Balakrishna, 2006).
The simultaneous demand–supply calibration problem is formulated as an optimization problem over a vector of DTA model parameters that aims to minimize the objective function – a scalar value of the summed distance between the fitted and true (observed) measurement values. This formulation makes it easy to incorporate any type of traffic data, including traditional loop detector counts as well as counts and travel time from emerging sources such as point-to-point counts and automatic vehicle identification (AVI) systems (Dixon and Rilett, 2005, Zhou and Mahmassani, 2006, Vaze et al., 2009, Antoniou et al., 2011). In the context of simulation-based DTA, the fitted measurements are obtained directly from the simulation output, which makes the optimization problem a non-linear, non-analytical problem with a potentially huge set of parameters and considerable system noise. Simultaneous perturbation stochastic approximation (SPSA) is an iterative gradient-free optimization algorithm designed for stochastic problems. It was initially proposed by Spall, 1992, Spall, 1998a, Spall, 1998b and successfully applied in the optimization of a variety of systems where accurate system models are not available. SPSA efficiently approximates the gradient with only two successive measurements of the objective function (independently of the number of parameters) and therefore significantly saves computational time for large-scale problems over traditional gradient methods (e.g. finite-differences stochastic approximation, FDSA), whose computational time directly depends on the problem size. These characteristics make SPSA a suitable solution algorithm for the calibration problem and it has been shown to be efficient in the calibration of real-world DTA systems (Balakrishna et al., 2007a) and microscopic simulation models (e.g. Lee and Ozbay, 2009).
However, as SPSA only uses the aggregated error between model output and observed measurements, a great amount of information is lost; for example, the location and time interval of different measurements. Also, approximating the gradient for each parameter using the aggregate error in the whole network across the entire simulation period introduces noise from uncorrelated measurements. This noise grows rapidly with the size of network and the number of intervals. This shortcoming influences the accuracy and convergence rate significantly when the size of the target network and the number of simulation intervals are extremely large. To solve this problem, this paper proposes an enhanced SPSA algorithm, termed Weighted SPSA (or W-SPSA), which incorporates the known spatial and temporal correlation between parameters and measurements to minimize the noise generated by uncorrelated measurements. This modification improves the performance of SPSA, especially when it is applied to large-scale networks with large number of time intervals.
The remainder of this paper is structured as follows: Section 2 provides a detailed review of the problem formulation and the SPSA algorithm. Section 3 proposes W-SPSA, an enhancement to the original SPSA. In Section 4, rigorous tests on large-scale, yet synthetic, data are used to demonstrate the performance comparison between SPSA and W-SPSA, and illustrate the characteristics of the proposed enhanced algorithm. Section 5 demonstrates the performance and technical details of applying W-SPSA to real-world DTA systems with a case study of the entire expressway network in Singapore. Section 6 concludes the paper with insights obtained and directions for further research.
Section snippets
Problem formulation
Let the time period of interest be divided into intervals h = 1, 2, …, H. is a vector of parameters in the DTA model that need to be calibrated, and may be time-dependent or constant over time, depending on the type of parameter. It is also the decision vector of this optimization problem. and are vectors of time-dependent observed traffic measurements and corresponding simulated traffic measurements, respectively, including sensor count, speed measurement, travel
Shortcomings of the application of SPSA to DTA calibration
The SPSA algorithm has been successfully applied to the calibration of DTA models and provided satisfactory results in a number of case studies (Balakrishna, 2006, Balakrishna et al., 2006, Ma et al., 2007, Lee and Ozbay, 2008, Vaze et al., 2009, Ben-Akiva et al., 2012, Paz et al., 2012). However, it was found through a case study on the whole Singapore expressway system that, although SPSA kept its advantage of high computational efficiency, the accuracy performance of SPSA deteriorated when
Experimental design
A synthetic test system was built to systematically compare the performance of SPSA and W-SPSA, and to investigate several properties of these two algorithms. Linear functions were used to represent the relationships between parameters to be calibrated (x = {xi}) and simulated measurements (y = {yi}). Observed measurements were calculated using the following (known) analytical functions and the true values of parameters:where β = {βij} are coefficients of the linear functions.
This system
Case study
In this section a case study of the entire expressway network in Singapore is discussed to demonstrate the performance of W-SPSA in a real-world large-scale DTA calibration problem. Section 5.1 introduces the simulation-based DTA model used in the case study. Network and data in this case study are introduced in Section 5.2. Section 5.3 shows details of the calculation of weight matrix, while Section 5.4 describes the choice of the algorithmic parameter values. Calibration results are presented
Conclusions
The topic of calibration of large-scale simulation models is an active research field. This paper presents an enhancement to SPSA, a well-established solution algorithm of the state-of-the-art joint demand–supply calibration of DTA models. The original SPSA algorithm is generalized by incorporating spatial and temporal correlations with a weight matrix to reduce the gradient approximation error and make the algorithm work better, especially for DTA systems applied to sparsely correlated and
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