Exponential FD on M25 Data

Figure 5.10: ACF for 2 inlet BC times for FD and BC sampling for the Exponential FD for simulated data using a parallel tempering sampler. The 2 time points represent the extremes of the ACF plots of the BCs.

5.2 Exponential FD on M25 Data

We now run the sampler developed in section (5.1) on the M25 dataset to as-sess its performance. We employ a parallel tempering sampler as in chapter 4 but now with 4 inverse-temperatures ([0.44, 0.58, 0.76, 1]) and 4 within-temperature moves. For the within-within-temperature proposals we propose a new FD and new BCs with probabilities 0.3 and 0.7 using a Gaussian proposal for the FD and Gibbs blocks for the BCs. The covariance matrices and blocks for each temperature are given in the appendix in section (??) and tables (B.20), (B.21), (B.22), and (B.23).

We show in figure (5.11) the traces for the FD parameters which seem to mix well.

The R.hat for α and β is 1.007, and the delay time for these two parameters is 3K.

We show in figure (5.12) the FD samples plotted with M25 flow data against 3 density estimates: density from speed, from occupancy, and estimated in BCs. To obtain the latter we used the mean BCs (both inlet and outlet) and picked out the time points that correspond to measurements. We then plotted the M25 flow data at those time points against the density in the BC means. We first observe that density estimated in the BCs does not agree with density estimated from occupancy but is similar to density

estimated from speed. In terms of the wave speeds, the free flow wave speeds implied by all three density estimates seem to agree, but the congested wave speeds do not. The congested flow wave speed in the fitted model seems to be in between the wave speed implied by the two other density estimates. As the congested flow waves in the fitted model (in figure (5.15)) seem to agree with the waves in M25 data (in figures (3.3) and (3.4)), this suggests that estimating the density in LWR rather than estimating it in a preprocessing step yields a better fit of the wave speeds.

Figure 5.11: Trace plots for the FD parameters α and β along with the log posterior. The samples are from FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. The 3 colours denote the 3 MCMC chains with a burnin of 150K

5.2. Exponential FD on M25 Data 173

Figure 5.12: FD samples plotted with M25 flow and three density estimates: density from occupancy, density from speed, and density from the mean BCs. The samples are from FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. We note how the density estimated in the BCs seems to agree with density estimated from speed rather than density from occupancy. However the congested flow wave speed in the fitted model seems to be in between the congested flow wave speeds implied by the two other density estimates.

We plot the outlet and inlet BC samples in figures (5.13) and (5.14). The outlet samples look similar to figure (4.25) in chapter 4, but we note how the inlet samples have changed from figure (4.26). If we plot the output of LWR in the x − t plane using the posterior mean parameters in figure (5.15), we can see that the congested flow waves cross the domain and reach the inlet BC (as they do in data). The can explain the change in inlet samples; the high density waves that reach the inlet constrain the likelihood. The BC have excellent R.hat diagnostics (close to 0) as can be seen from figures (5.16) and (5.17), and we show representative traces for the outlet and inlet BCs in figure (5.18) and (5.19). Furthermore, the delay times of the BCs are generally between 500 and 2K which is slightly faster than the simulated data. We also show the extreme ACF plots in figures (5.20) and (5.21).

We inspect the fit of the model by plotting residuals in figure (5.22). Although the

variance does not seem to be increasing with flow as would be expected of a Poisson error model, there is no apparent structure to the residuals; this suggests a good fit of the model.

Figure 5.13: Outlet BC samples from FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. The 3 colours denote the 3 MCMC chains.

Figure 5.14: Inlet BC samples from FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. The 3 colours denote the 3 MCMC chains.

5.2. Exponential FD on M25 Data 175

Figure 5.15: Output of LWR in the x −t plane with posterior mean FD and BCs for FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler.

The congested flow waves cross the domain as they do real data (in figures (3.3) and (3.4)).

Figure 5.16: R.hat values for the outlet BC from FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. All the time points are well below the recommended limit of 1.1

Figure 5.17: R.hat values for the inlet BC from FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. All the time points are well below the recommended limit of 1.1

Figure 5.18: Trace plots for 3 outlet BC times for FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. The 3 time points were chosen to be representative of the remaining trace plots of the BCs.

5.2. Exponential FD on M25 Data 177

Figure 5.19: Trace plots for 3 inlet BC times for FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. The 3 time points were chosen to be representative of the remaining trace plots of the BCs.

Figure 5.20: ACF for 2 outlet BC times for FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. The 2 time points represent the extremes of the BC ACF plots.

Figure 5.21: ACF for 2 inlet BC times for FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. The 2 time points represent the extremes of the BC ACF plots.

Figure 5.22: Residuals for FD and BC sampling for the Exponential FD for M25 data using a parallel tempering sampler. There is no apparent structure which suggests a good fit to data.