## 5 Experimental Results  
We now present the experimental results of our approach and compare the performance of the proposed architecture with some of the recent deep learning architectures proposed for spatio-temporal forecasting. In particular, we compare the performance of our model against the following architectures. * FCN: Fully Connected Network;
* RNN-FCN: Recurrent Neural Network followed by a fully connected layer;
* T-GCN: Temporal graph convolution network for traffic prediction as proposed in [14];
* STGCN: Spatio Temporal Graph Convolution Network for traffic prediction, as proposed in [21];* ST-RGAN: Spatio Temporal Residual Graph Attention Network as proposed in [15]. As mentioned earlier, the output of our model is the distribution of wait times, \(f_{WT}(wt)\), for a given input traffic pattern, signal timing parameters. Since we are trying to predict the distributions, a softmax operation is applied to the model output to convert it into a probability distribution. All the models are trained with the Adam optimizer and Mean Square Error (MSE) as the loss function. The different terms used to describe input, output variables is shown in Figure 4. Each of the waveforms aggregate counts for five-second intervals (the level of aggregation was chosen so that the number of vehicles in each interval is very small--generally less than three, but also large enough to keep the size of the network to be small). Table 2 shows that our methods are better for all the error measures: Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Square Error (MSE). These results show that InterTwin is significantly better than FCN, STGCN, and T-GCN in terms of mean square error for MOE prediction. The key advantage of predicting distributions is that different summary statistics such as mean, median, and percentile values can be derived. Figure 5 shows actual vs. predicted scatter plot of the 50 th, 70 th percentile of wait time computed from the distribution. Figure 6 shows actual versus predicted distribution of wait time for a given inflow waveform, for different green time splits. These plots show that the models are able to capture the interrelationship between input traffic and signal timing parameters. We trained separate models each requiring different input traffic patterns for different Signal Timing waveform (SIG) using:  
1. Only Inflow Traffic Waveforms (INF);
2. Only Advanced and Stopbar Waveforms (ADV and STP). Table 3 shows the training and test errors for models with different input combinations. The accuracy is high when stopbar, advance waveforms are used as input compared to using only an inflow waveform. An important practical advantage of the model with only stopbar and advance waveforms as input is that both of these waveform are easily available in recorded controller logs at each intersection that support ATSPM. Thus, this model can be very effectively used to infer wait time distributions from recorded controller log data  
Figure 4: A typical intersection with 8 different directions of vehicular movement (phases 1 to 8). Vehicle waveforms are observed at Stop bar and Advance detectors (STP, ADV). STP, ADV, and INF corresponds to the traffic waveform at stopbar detector, advanced and inflow (500 m away from the intersection) aggregated at a 5-s interval. SIG corresponds to signal timing at a 5-s interval. STP, ADV, and SIG are typically available in ATSPM data for every intersection. in real world. Unfortunately, this model cannot be used directly for computing MOEs for different signal timing plans. The latter is generally required for optimization purposes. For optimization, it is more practical to use the INF waveform and SIG waveforms as input as this model allows for varying signal timing plans. As discussed earlier, INF waveforms for each approach can be computed using neural networks that use ADV and STP waveforms along with SIG waveforms.