## 3 Challenges of Using SDR with Dynamic Process Data
### Standard Isomap and Dynamic Process Data  
While the Isomap trajectories in Figure 2(b) diverge from the initial time point, close inspection reveals that the trajectories exhibit nondivergent behavior in the earlier steps, which is expected since the morphologies are expected to evolve in a similar fashion at the beginning of the simulation.  
We only applied Isomap to the early stage data represented by the first 30 time steps of each trajectory (the threshold was selected by the domain expert). The results are shown in Figure 2(c). The results here are more encouraging as one can clearly observe that data points for all trajectories cluster together before quickly diverging. This observation points us to the key deficiency of Isomap when dealing with dynamic process data. When data instances exhibit strong temporal correlations, the neighborhood computation for any data instance in Isomap is heavily dominated by other instances belonging to the same trajectory. Thus, Isomap cannot capture the relationships across different trajectories and the reduced representation is dominated by the time dimension, as can be seen in Figure 2(b).  
To further study this point, we consider the pairwise distance matrix \(\mathbf{D}\) that contains the distance between every pair of data instances. The rows and columns of the matrix are ordered by the trajectory index and time, as shown in Figure 3(a). Consider any pair of row (and column) indices, \(i\) and \(j\), such that the trajectory index that contains the \(i\)th data instance is \(I\), and the trajectory index that contains the \(j\)th data instance is \(J\) and \(\tau_{i}\), and \(\tau_{j}\) denotes the time index for each instance within the corresponding trajectory. Then, row \(i\) precedes row \(j\) in the matrix \(\mathbf{D}\) if either \(I<J\) or \(\tau_{i}\leq\tau_{j}\) if \(I=J\).  
We consider another visualization of this matrix (See Figure 3(b)), where the row ordering is retained, but each row contains the sorted distances (increasing order) of the corresponding data instance and all other instances in the dataset. We can observe that for the majority of instances, the first several nearest neighbors are always from the same trajectory. This is problematic because the  
Figure 3: Low-dimensional (\(d=3\)) representation of six trajectories with fixed \(\chi=3.0\) and variable \(\phi\in\{0.50,0.52,0.54,0.56,0.58,0.6\}\) obtained using (a) PCA, (b) Isomap with \(k=8\), (c) Isomap with \(k=8\) using only the first 30 time steps of each pathway. (Please view in color).  
ability of Isomap to learn an accurate description of the underlying manifold depends on how well the neighborhood matrix captures the relationship _across_ the trajectories. We refer to this relationship as _cross-talk_, or _mixing_ among the trajectories. For any given point, the desired effect would be that the nearest neighborhood set contains points from multiple trajectories. However, the sorted neighborhood matrix indicates a lack of mixing, which essentially means that the Isomap algorithm does not consider information from other trajectories when learning the shape of the manifold in the neighborhood of one trajectory.