An investigation of incident frequency, duration and lanes blockage for determining traffic delay

Journal of Advanced Transportation, Fall, 2009 by Yi "Grace" Qi, Hualiang "Harry" Teng, David R. Martinelli

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

which takes the form of the negative binomial distribution with parameter (1/[a.sub.i][r.sub.ij]), where [a.sub.i]=1/[[theta].sub.i] and [r.sub.ij] = [[lambda].sub.ij]/([[lambda].sub.ij] 1/[a.sub.i]) which determines that the variance to mean ratio is greater than 1. It can be seen from Equation (5) that negative binomial becomes Poisson distribution, when [a.sub.i] [right arrow] 0. Whether or not Poisson distribution is appropriate can be determined by testing the hypothesis of [a.sub.i] = 0. In Figure 2, the t-test statistic for [a.sub.i] is denoted as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In a zero-inflated frequency model, it is assumed that [y.sub.iij] = 0 with probability [q.sub.ij] and [y.sub.ij] follows distribution f([y.sub.ij]) with probability 1 - [q.sub.ij]. In a zero-inflated Poisson (ZIP) model, f([y.sub.ij]) takes the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

In the zero-inflated negative binomial (ZINB) model, f([Y.sub.ij]) is expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

As in the Poisson and negative binomial regression models, we still assume [[lambda].sub.ij] = exp([beta]' [X.sub.ij]) for f([y.sub.ij])conditioned on the values of [X.sub.ij]. The testing of whether a zero-inflated incident state is more appropriate than the non-zero-inflated incident state is based on a test statistic proposed by Vuong, an expression of which can be found in Greene (1997). Asymptotically, Vuong's statistic, as denoted as [v.sub.ZIP] and [x.sub.ZINB] in Figure 2 for ZIP and ZINB models respectively, is distributed as standard normal, so its value can be compared to the critical value of the standard normal distribution (Lee and Mannering 1999). In addition, the estimate of a in Equation (7) can be used to determine whether a ZIP model can be used for modeling the data if the ZINB model is not appropriate. Its t-test statistic value is represented as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Figure 2.

Given the introduction of these non-linear regression models, the decision tree shown in Figure 2 is described below. The process of choosing a model starts with the estimation of a NB model. A decision is made on whether to fit a ZINB regression model or a ZIP model based on a t-test on a, as expressed in Equation (5). Following the left branch, which is the case when the t-test statistic is greater than a threshold, e.g., 1.95 with 95% confidence, a ZINB model needs to be fitted. A test based on Vuong's statistic needs to be conducted. If Vuong's statistic is greater than a chosen threshold, an additional test needs to be performed based on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Otherwise, it can be determined that NB is the right model for the incident frequency data. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is greater than a given threshold, it is the Z1NB model that needs to be fitted for the incident frequency data. Otherwise, the ZIP model should be chosen for the data. Returning to where the left branch start, where a t-test for the value of a is performed in the fitted NB model, a ZIP model needs to be fired if the t-statistic is smaller than the chosen threshold. Then, a test based on Vuong's statistic needs to be performed. If it is greater than a given threshold, the final chosen model should be ZIP. Otherwise, it should be a Poisson model. It can be seen that there are two paths to reach a decision of ZIP model, one is on the left through fitting a ZINB model, the other is on the fight through fitting a ZIP model. For Poisson and NB models, there is only one path to reach them, which are the left and right paths, respectively.