W–SPSA in Practice: Approximation of Weight Matrices and Calibration of Traffic Simulation Models

Abstract

The development and calibration of complex traffic models demands parsimonious techniques, because such models often involve hundreds of thousands of unknown parameters. The Weighted Simultaneous Perturbation Stochastic Approximation (W-SPSA) algorithm has been proven more efficient than its predecessor SPSA (Spall, 1998), particularly in situations where the correlation structure of the variables is not homogeneous. This is crucial in traffic simulation models where effectively some variables (e.g. readings from certain sensors) are strongly correlated, both in time and space, with some other variables (e.g. certain OD flows). In situations with reasonably sized traffic networks, the difference is relevant considering computational constraints. However, W-SPSA relies on determining a proper weight matrix (W) that represents those correlations, and such a process has been so far an open problem, and only heuristic approaches to obtain it have been considered.This paper presents W-SPSA in a formally comprehensive way, where effectively SPSA becomes an instance of W-SPSA, and explores alternative approaches for determining the matrix W. We demonstrate that, relying on a few simplifications that marginally affect the final solution, we can obtain W matrices that considerably outperform SPSA. We analyse the performance of our proposed algorithm in two applications in motorway networks in Singapore and Portugal, using a dynamic traffic assignment model and a microscopic traffic simulator, respectively.

Demand Calibration of Multimodal Microscopic Traffic Simulation using Weighted Discrete SPSA

“…Although the SPSA altgorithm has been successfully applied to the calibration of DTA models, it has been observed that its accuracy deteriorates as the problem scale increases in terms of network size and time intervals due to a gradient 'approximation error' (18). To decrease this approximation error, researchers (18,19) have introduced the concept of a weight matrix ( ) to incorporate the spatial and temporal correlations between OD parameters and measurements which are determined by network topology and route choice behavior. Accordingly, each component ( + × ℎ 1 −1 , + ×(ℎ 2 −1) ) in the weight matrix represents the relative correlation between the p th OD parameter at the interval ℎ 1 and the m th individual sensor measurement in the interval ℎ 2 .…”

“…The SPSA algorithm has received significant attention because of efficiency (only two evaluations of the objective function are required in each iteration), applicability (simple to implement in high-dimensional real-world problems), accuracy (properly handles stochasticity and measurement errors), and has been applied to the off-line demand calibration problems by various researchers (13,14,15,16). However, recent work has identified an issue of unstable convergence of the SPSA algorithm in large-scale applications arising from the gradient approximation (17,18,19). To overcome this, the c-SPSA (cluster-wise SPSA) and W-SPSA (Weighted SPSA) algorithms were proposed.…”

“…The c-SPSA (17) approximates the gradient within a small number of homogeneous clusters one at a time. The W-SPSA (18,19) incoporates spatio-temporal correlations between parameters and measurements in approximating the gradient thereby minimizing the impact of uncorrelated measurements. The enhanced algorithms outperform the original SPSA showing improved robustness and convergence.…”

Dispatching and Rescheduling Tasks and Their Interactions with Travel Demand and the Energy Domain: Models and Algorithms

“…Other major contributions are provided by [217,218] which addressed the simultaneous adjustment of a dynamic traffic demand matrix by means of a gradient approximation approach representing a variant of the simultaneous perturbation stochastic approximation (SPSA) path search optimisation method proposed by [219,220]. Further variants of the SPSA approach are W-SPSA proposed by [221,222] and c-SPSA presented by [223]. Similarly, [224] proposed a method based on the use of linear approximations of the assignment matrix in optimisation iterations and tested several specific solution algorithms which differ in the search direction.…”

Towards a generic benchmarking platform for origin–destination flows estimation/updating algorithms: Design, demonstration and validation

“…Recent extensions, such as W-SPSA(Lu et al, 2015;Antoniou et al, 2015;Cantelmo et al, 2014b), and c-SPSA(Tympakianaki et al, 2015) aim to overcome this particular limitation.…”