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A Modified Late Arrival Penalised User Equilibrium Model and Robustness in Data Perturbation

Manlan Li, Huifu Xu

TL;DR

The LAPUE model based on the new penalty function has a unique equilibrium and is stable with respect to (w.r.t.) small perturbation of probability distribution under moderate conditions and the theoretical analysis of statistical robustness is carried out.

Abstract

In this paper, we revisit the LAPUE model with a different focus: we begin by adopting a new penalty function which gives a smooth transition of the boundary between lateness and no lateness and demonstrate the LAPUE model based on the new penalty function has a unique equilibrium and is stable with respect to (w.r.t.) small perturbation of probability distribution under moderate conditions. We then move on to discuss statistical robustness of the modified LAPUE (MLAPUE) model by considering the case that the data to be used for fitting the density function may be perturbed in practice or there is a discrepancy between the probability distribution of the underlying uncertainty constructed with empirical data and the true probability distribution in future, we investigate how the data perturbation may affect the equilibrium. We undertake the analysis from two perspectives: (a) a few data are perturbed by outliers and (b) all data are potentially perturbed. In case (a), we use the well-known influence function to quantify the sensitivity of the equilibrium by the outliers and in case (b) we examine the difference between empirical distributions of the equilibrium based on perturbed data and the equilibrium based on unperturbed data. To examine the performance of the MLAPUE model and our theoretical analysis of statistical robustness, we carry out some numerical experiments, the preliminary results confirm the statistical robustness as desired.

A Modified Late Arrival Penalised User Equilibrium Model and Robustness in Data Perturbation

TL;DR

The LAPUE model based on the new penalty function has a unique equilibrium and is stable with respect to (w.r.t.) small perturbation of probability distribution under moderate conditions and the theoretical analysis of statistical robustness is carried out.

Abstract

In this paper, we revisit the LAPUE model with a different focus: we begin by adopting a new penalty function which gives a smooth transition of the boundary between lateness and no lateness and demonstrate the LAPUE model based on the new penalty function has a unique equilibrium and is stable with respect to (w.r.t.) small perturbation of probability distribution under moderate conditions. We then move on to discuss statistical robustness of the modified LAPUE (MLAPUE) model by considering the case that the data to be used for fitting the density function may be perturbed in practice or there is a discrepancy between the probability distribution of the underlying uncertainty constructed with empirical data and the true probability distribution in future, we investigate how the data perturbation may affect the equilibrium. We undertake the analysis from two perspectives: (a) a few data are perturbed by outliers and (b) all data are potentially perturbed. In case (a), we use the well-known influence function to quantify the sensitivity of the equilibrium by the outliers and in case (b) we examine the difference between empirical distributions of the equilibrium based on perturbed data and the equilibrium based on unperturbed data. To examine the performance of the MLAPUE model and our theoretical analysis of statistical robustness, we carry out some numerical experiments, the preliminary results confirm the statistical robustness as desired.
Paper Structure (23 sections, 11 theorems, 99 equations, 11 figures, 2 tables)

This paper contains 23 sections, 11 theorems, 99 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Watling's LAPUE model and its variation
  3. LAPUE model
  4. A new penalty function for lateness and the MLAPUE model
  5. Sample average approximation and perturbation of distribution
  6. Statistical robustness of MLAPUE
  7. Single data perturbation
  8. Breakdown point analysis
  9. All data perturbation
  10. Numerical tests
  11. Example 1: a simple 2-OD network
  12. Example 2: the Nguyen and Dupuis network
  13. Concluding remarks
  14. Perliminaries
  15. Proof of section \ref{['sec2:MLPAUE']}
  16. ...and 8 more sections

Key Result

Theorem 2.1

Consider the marginal path travel time density function for path $r$ and write it as $\psi_r(c_r;\mu_r)$ to denote its (partial) parameterisation by the mean path travel time $\mu_r$. Assume: (a) the functions are well-defined, continuous and non-decreasing; (b) $\theta_1>0$ and $\theta_2\geq 0$ in defi:pathdisutility-random; (c) the mean arc travel time functions $t^P_a(\boldsymbol{v})$ are cont

Figures (11)

  • Figure 1: (Color online) (a)-(b) Parameterized penalty function with parameter $t=0.5$ and $t=1$.
  • Figure 2: Data perturbations
  • Figure 3: Test network 1 ($\theta_0=0,\theta_1=1,\theta_2=2,\tau_1=27,\tau_2=22$)
  • Figure 4: (Color online) (a) $F_{Q_a}(x)$ and $F(x;1500,5)$. (b) Empirical distributions of $v_{1}(P_M)$ and $v_{1}(Q_M)$ based on $L=500$ simulations with parameter $t=0.01$.
  • Figure 5: (Color online) (a)-(b) Empirical distributions of $v_{1}(P_M)$ and $v_{1}(Q_M)$ based on $L=10$ and $L=30$ simulations with parameter $t=0.01$.
  • ...and 6 more figures

Theorems & Definitions (24)

  • Theorem 2.1: Existence and uniqueness of LAPUE Watling06
  • Proposition 2.1: Existence and uniqueness of MLAPUE
  • Proposition 2.2: Properties of the new disutility function
  • Definition 2.1: Strong regularity
  • Theorem 2.2: Approximation of MLAPUE to LAPUE
  • Theorem 2.3
  • Theorem 3.1
  • Example 3.1
  • Proposition 3.1
  • Theorem 3.2: Stability of MLAPUE
  • ...and 14 more