## 5 Forecasting Results  
In total, seven models have been estimated. Table 2 presents a summary of the validation results for each of the models. In the table the error rates and average error distance are mentioned for the 1 st, 5 th and 20 th prediction of the sequences. Here, the Nth prediction identifies the prediction of the Nth visited cell in a particular cell sequence given that all previous locations (1 until N-1 th) are specified as input. We expect that the models that incorporate a memory of previous steps predict the cell location of steps later in the sequence better than the location of steps early on in the sequence. The table illustrates that the Markov chain and RNN-GRU model structures are slightly better at predicting the next cell at the end of a sequence, than the beginning of a sequence. Contrary to our expectation, the difference in the error rates for the 1 st, 5 th and 20 th predictions of a particular model vary only slightly. This suggests that only the previous one or two visited cells influence the current cell visited. At the same time, the fact that the RNN-GRU outperforms the Markov Chain models at any stage of a cell sequence. RNN-GRU's includes more complex conditional probability distributions to model transitions between cells. Therefore, the outperformance suggests that long-term dependencies exist in the movement behavior of the visitors that are more easily captured by the RNN-GRU model. The table illustrates that the two goodness-of-fit measures indicate two distinct model structures. That is, the RNN-GRU predicts the next cell most often correctly. We expect the low error rate to be the consequence of the incorporation of the memory of earlier steps in the sequence. However, if the wrong next cell is predicted, the Markov chain is more likely to predict another next cell near the cell that was actually visited. Given that the 2 nd-order Markov chains have a very high error rate and the average error distance of the RNN-GRU is almost similar to the error distance of the 3 rd order Markov chain, the RNN-GRU is chosen to be the best model amongst the tested models to predict the cell sequence behavior of the visitors at Mysteryland. The RNN-GRU model predicts the correct next \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline \multirow{2}{*}{**Model Network**} & \multirow{2}{*}{**Division Method**} & \multicolumn{3}{c}{**Error Rate (\(E^{m}\))**} & \multicolumn{3}{c}{**Average Error Distance (\(D^{m}\))**} \\  & & **1** & **5** & **20** & **1** & **5** & **20** \\ \hline Shallow neural network & 1 st order & 0.9780 & 0.9427 & 0.9492 & 366.52 & 309.38 & 322.64 \\ Shallow neural network & 2 nd order & 0.9468 & 0.9420 & 0.9535 & 469.34 & 363.64 & 305.14 \\ Shallow neural network & 3 rd order & 0.9604 & 0.9752 & 0.9722 & 451.42 & 456.68 & 464.93 \\ Markov chain & 1 st order & 0.7756 & 0.7077 & 0.7204 & 170.82 & 150.82 & 150.95 \\ Markov chain & 2 nd order & 0.6197 & 0.5786 & 0.5705 & **135.73** & **128.69** & **121.78** \\ Markov chain & 3 rd order & 0.6589 & 0.6576 & 0.6227 & 160.25 & 155.29 & 131.93 \\ RNN-GRU & Inf. & **0.2772** & **0.3436** & **0.2676** & 136.42 & 167.87 & 124.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of cross-validation results for distinct model structures, where the bold numbers identify the best model according to the goodness-of-fit metrics. Figure 5: Visualization of the average transition probabilities towards other cells for (**a**): cell 10 and (**b**) cell 50.  
cell in approximately 82.3% of the time. Besides that, when the model predicts the wrong location, in many cases, a cell adjoining the correct cell is identified. However, these seven model structures have been trained in a very specific manner. The use of a training set with a random drawn selection of cell sequencing featuring crowd movements on both days and all hours of the event does limit the usage of the model for real-time prediction purposes.