4. Empirical Illustration 19
4.3 SML-GHK Simulator
In this section, results of the analysis of dynamic Tobit models with lagged latent dependent variables by using the SML-GHK simulator for our traffic data will be presented. Firstly, the latent volume values are simulated by generating uniform random numbers and applied to GHK simulator. The parameters of interests will then be estimated by maximizing the unbiased likelihood estimator numerically
L =ˆ 1 R
R r=1
T t=1
f (yt|yt−1, yt−1∗(r))It
P (It= 0|yt−1, yt−1∗(r))1−It
Two models are proposed as follows.
• Model 4 : yt∗ = λ1yt−1∗ + βxt+ α + t t = 1, . . . , 120
• Model 5 : yt∗ = λ1yt−1∗ + λ2yt−2∗ + βxt+ α + t t = 2, . . . , 120
Let the number of simulations R = 15 in these cases. The latent volume, y35∗ and y97∗ , for two models above can then be estimated and the results have reported in Table 5 , the SDs of simulated volume are calculated as well. The estimated param-eters and their SDs are summarized in Table 6. As it indicates, the traffic volume yt−1 and yt−2 affect the present volume in the adverse direction.The prediction and cumulative prediction summaries are listed in Table 7 and Table 8 representatively.
ARPE and ARCPE both are smaller for Model 5. The time series plots of predic-tion and cumulative predicpredic-tion are shown in Figure 14 to Figure 17. The results of residual analysis, summarized in Table 9, indicate that both models are adequate.
The times series plots of residuals have been plotted in Figure18 and Figure 19.
Table 5: Simulation Estimation with SML-GHK simulator where R = 15.
Model 4 Model 5
Latent volume Mean SD Mean SD
y35∗ 16.8244 1.6308 17.1131 1.6767 y97∗ 17.1570 1.7882 17.0346 1.7135
Table 6: Estimated parameters and the corresponding stan-dard deviations (SD) with SML-GHK simulator.
Model 4 Model 5
Parameter Estimate SD Estimate SD
λ1 -0.0370 0.0048 -0.0335 0.0047
λ2 -0.2215 0.0024
β 0.0920 0.0026 0.0850 0.0024
α 6.5605 0.0086 8.5090 0.0074
σ 3.1385 0.0071 3.0365 0.0053
Table 7: Descriptive statistics of prediction compared with observation by SML-GHK simulator.
Model 4 Model 5
Observation Prediction Prediction
Mean 7.9250 7.8574 7.9183
SD 3.3460 1.1706 1.3030
Max 15 12.8021 12.2905
Min 1 6.4969 5.4005
ARPEa 0.5462 0.5409
aAverage relative prediction error = T1 T
t=1|yt−ˆyt| yt .
Table 8: Descriptive statistics of cumulative prediction com-pared with observation by SML-GHK simulator.
Model 4 Model 5
Cumulate Cumulate Cumulate
Observation Prediction Prediction
Mean 491.4083 463.7327 462.2012
SD 277.8814 273.5527 274.0060
Max 951 942.8892 942.2777
ARCPEa 0.0696 0.0664
aAverage relative cumulative prediction error = T1 T
t=1|Yt− ˆYt| Yt .
Table 9: Residual analysis for Model 4 and 5 with SML-GHK Simula-tor.
Model 4 Model 5
Testing Method Statistic p-value statistic p-value
Ljung-Boxa 24.397 0.2255 13.459 0.8568
McLeod-Lia 18.026 0.5857 19.967 0.4600
Jarque-Bera(normality)b 0.8467 0.6549 1.0543 0.5903
a The test statistic has asymptotically a χ2 distribution with 20 degrees of freedom.
bThe test statistic has asymptotically a χ2 distribution with 2 degrees of freedom.
Figure 14: Time series plot of prediction compared with observation for Model 4 (ARPE = 0.5462).
Figure 15: Time series plot of prediction compared with observation for Model 5 (ARPE = 0.5409).
Figure 16: Time series plot of cumulative prediction compared with cumulative observation for Model 4 (ARCPE = 0.0696).
Figure 17: Time series plot of cumulative prediction compared with cumulative observation for Model 5 (ARCPE = 0.0664).
Figure 18: Time series plot of residuals for Model 4.
Figure 19: Time series plot of residuals for Model 5.
5. Conclusion and Discussion
This study proposed two methodologies to deal with the traffic censored data and our goal is to find out the cumulative traffic flow predictions. The Newton-Raphson optimization algorithm and SML-GHK simulator have been adopted to overcome our problem under different assumptions. We introduce the Poisson regression dynamic Tobit models and solve the MLE of parameters by using NR algorithm. The Tobit models with lagged latent dependent variables are also used to obtain the SML estimator via SML-GHK simulator. It can be found that similar results of predicted volume and cumulative predicted volume are obtained by different techniques under different models. The following Table 10 shows ARPEs and ARCPEs under five models.
Table 10: The ARPE and ARCPE from NR al-gorithm and SML-GHK simulator.
ARPEa ARCPEb
Model 1 0.5576 0.0622
Model 2 0.5370 0.0693
Model 3 0.5387 0.0706
Model 4 0.5462 0.0696
Model 5 0.5409 0.0664
aAverage relative prediction error.
bAverage relative cumulative prediction error.
This is no strong evidence to discern between models, but it is recognized that both NR algorithm and SML-GHK simulator result in satisfactory cumulative pre-dicted traffic flow. The advantage of using Poisson regression is that the discreteness nature of traffic flow has been taken into consideration, while through SML-GHK simulator the potential censored data can be recovered. Hopefully these two choices will be helpful to design the signal timing plan and ease traffic congestion problem more or less.
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