Progressive Glimmer: Expanding Dimensionality in Multidimensional Scaling

TL;DR

The proposed Progressive Glimmer algorithm is a progressive multidimensional scaling (MDS) algorithm that allows for increasing the number of dimensions in spatio-temporal data and provides more stable results, leading to visually consistent results for progressive rendering and making the approach applicable to streaming data.

Abstract

Progressive dimensionality reduction algorithms allow for visually investigating intermediate results, especially for large data sets. While different algorithms exist that progressively increase the number of data points, we propose an algorithm that allows for increasing the number of dimensions. Especially in spatio-temporal data, where each spatial location can be seen as one data point and each time step as one dimension, the data is often stored in a format that supports quick access to the individual dimensions of all points. Therefore, we propose Progressive Glimmer, a progressive multidimensional scaling (MDS) algorithm. We adapt the Glimmer algorithm to support progressive updates for changes in the data's dimensionality. We evaluate Progressive Glimmer's embedding quality and runtime. We observe that the algorithm provides more stable results, leading to visually consistent results for progressive rendering and making the approach applicable to streaming data. We show the applicability of our approach to spatio-temporal simulation ensemble data where we add the individual ensemble members progressively.

Figures (4)

• Figure 1: Runtime and stability of Glimmer and Progressive Glimmer. The computation time is substantially faster for progressive Glimmer (a), where the indicated times are for one step of progressive Glimmer. The stress over the number of included dimensions (b) reveals a sensitivity to the initial condition but also rapid improvements confirmed in the Shephard diagrams (c).
• Figure 2: Comparing stress of Glimmer and Progressive Glimmer for different overlaps of a random data set. The data points of Glimmer are connected for better interpretability but are computed independently.
• Figure 3: Progressive visualization for temporal data of one ensemble member of the MPI-GE dataset. The evolution of stress (a) does not show substantial differences between using temporal or random order when adding the time steps. The scatterplots for different numbers of time steps $N$ show that the variations for randomly adding dimensions (b) are small. When adding the steps in temporal order (c), the shape between the first two examples shown here varies substantially, while the following results are more similar.
• Figure 4: When adding the data in chunks of 1128 time steps (one ensemble member), more iterations are required to obtain smaller stress values (a). The result for five ensemble members for a maximum of 100 iterations (b) and a maximum of 500 iterations (c) show structural differences.