Interactive Visualization of SOM Clustering Results

The interactive visualization of clustering results provides tools for visual exploration of the data set. As a basis for the various data views, the visualization of trajectory samples and cell members in SOM grid is used. The SOM grid provides a representation of many trajectory fragments with a smaller number of trajectory prototypes.

The size of the SOM grid is an algorithm parameter. In each grid cell, the these prototypes can be displayed.

For the graphic representation of the patterns, we use their direct geometric representation when using geometric features for clustering and abstract feature vector representation as parallel coordinates or display of the trajectory nearest to the cell center in the other case (see Figure4.31for an illustration).

Based on the above mentioned SOM grid displays, four views are built (see following subsections for more details).

• Thegeneral viewallows to assess overall distribution of the patterns identified in the data set.

• Restricting the general view to a selected object results in a so calledobject-oriented view, which allows the analysis of the distribution of movement patterns occurring for a given object over the whole time period.

4.6. Visual Analysis of Two-Dimensional Time-Dependent Data Using SOM Clustering

• Thetime-oriented viewis an analogy to the object-oriented view, where the restriction has been posed on the time span. It shows the distribution of patterns for a selected time span over all objects in the data set.

• Pattern sequence viewis a comparative view on the time-dependent sequence of patterns of many objects simultaneously. It allows specifically to search for co-occurring and correlated patterns among time and objects.

The general view can be used for a wide variety of data sets. Time-oriented and sequence view are suitable for time-dependent data with multiple time spans and object oriented view is favorable for data sets with multiple objects.

In addition to these views, also the presentation of the so-calledcomponent planes(see Figure3.37) is pro-vided. It shows the values of the trajectory features across the SOM grid. The values are displayed as a heatmap.

Figure 4.30.: Visualization of component planes for analysis of feature distribution across the SOM grid. It shows the values of each feature in the cell center across the SOM grid using a heatmap matrix. White color indicates low values and orange color means high values.

(a) Geometry-based features

(b) Abstract features

Figure 4.31.: The SOM result view in cell grid. Top: Trajectories of the cell prototypes when using features with direct geometric representation. Bottom: Two types of result view when using abstract fea-tures. Bottom left: showing a trajectory nearest to the cell center. Bottom right: Abstract parallel coordinate view of the cell center.

4.6. Visual Analysis of Two-Dimensional Time-Dependent Data Using SOM Clustering

4.6.5.1. General view

The general view displays the distribution of movement patterns of all objects and over the full time horizon (see Figure4.32). In conjunction with the SOM clustering process, it yields an effective overview of the general-characteristic patterns in the data set. When using color coding for the background depending on the number of matches for a cell (cell members) (in the same way as in SOM result visualization), also the frequency of patterns can be explored.

Figure 4.32.: The general view visualizes the distribution of data movement patterns in the data set. Owing to the topology-preserving properties of the SOM algorithm, the map can be meaningfully interpreted in terms of pattern transitions. The background color (from white to yellow) indicates the frequency of the found patterns.

4.6.5.2. Object Oriented View

The object-oriented view (see Figure4.33) is obtained from the general view by restricting the set of sample trajectories to a selected object. Thereby, individual objects can be analyzed for occurrence of specific data movement patterns. Practically, due to the small number of matches per prototype, we directly overlay each matched sample to its prototype, and keep the patterns not matched by any sample in context by rendering only the prototype in a lighter color. The dashed line represents the median. For the matched prototypes, we use a coloring of the background to indicate the number of represented samples, scaling the color saturation proportional to the maximum occurring frequency.

Additional features in the object-oriented view support visual analysis of thetemporal sequence of patterns for a selected object. As each object trajectory fragment can be matched to a prototype trajectory, the sequence of data movements (i.e., the sequence of the fragments) can be visualized by connecting the respective SOM prototype positions. By restricting the distance between two consecutive movements in time, it is possible to filter for gradual or abrupt inter-temporal pattern transitions. This is demonstrated in Figure4.34, which shows abrupt inter-temporal pattern sequences for several objects, by filtering for a minimum grid distance of 14. The

Figure 4.33.: The object-oriented view is a version of the general view (see also Figure4.32) restricted to a given entity of interest. It shows the distribution of patterns occurring for a specific entity, and over the full time span.

view allows the identification of time spans where the object movement in the following time span roughly reversed. Filtering for small distances on the other hand would reveal periods of roughly recurring patterns over time, or smoother transitions thereof.

We state that depending on the selectivity of the filtering, overplotting effects could arise in this basic line-based view. A solution would be to rely on more advanced approaches for visualization of larger numbers of pattern connectors. To this end, an adoption of the edge bundle technique [Hol06] seems promising.

4.6.5.3. Time-Oriented View

A simple yet powerful view is achieved by filtering the set of sample trajectories by user-defined time subin-tervals. Thereby, the user can easily obtain an understanding of the distribution of patterns over time. This visualization technique follows the object-oriented view (see Figure4.35). It also uses the same background coloring scheme to show the number of samples for each pattern occurring in the selected time period.

4.6.5.4. Sequence View

The sequence view is a comparative view of movement patterns, for all entities over all time periods. The view is organized in a row-by-column scheme where each row refers to an object, each column refers to a given time fragment, and each cell contains the prototype representation of the actual movement sample. It provides an overview of the sequences of movement patterns over time and movement correlations between objects.

One main use case of this view is visual analysis for correlations, co-occurrences, and frequency of patterns.

For example, it allows to detect recurring patterns in individual time points or similarly moving entities over the time period. To this end, an automatic preprocessing of the sequence view can be undertaken using statistical evaluation methods. The results of this analysis are input for the visualization, controlling filtering and highlight-ing. In particular, we used a two-stage analysis scheme for automatic identification of possibly interesting view

4.6. Visual Analysis of Two-Dimensional Time-Dependent Data Using SOM Clustering

Figure 4.34.: Object-oriented views showing abrupt inter-temporal pattern transitions by connecting the SOM cells with consecutive trajectory fragments. Time tick scales are used to indicate the date of occur-rence of the pattern transition, relative to the global time scale (see bottom-right for a closeup).

Figure 4.35.: Time-oriented view shows the filtering of the general view to a particular time fragment. It shows the distribution of patterns for a specific time period fragment.

configurations. Firstly, the algorithm calculates the entropy measure [EM95] for the distribution of movement patterns across time fragments. In the second stage, an analysis of pattern frequency considering the grid-based pattern distances finds the most prominent patterns in the time spans of lowest entropy. These patterns represent those patterns in the identified time periods with similar data dynamics.

The results of the entropy analysis are presented to the user in form of thumbnail sequence views. To easily spot time periods with dominant market dynamics, a sorting of rows (objects) according to distance from the market trend is undertaken before the results are visualized. In the views, the prominent patterns are highlighted.

Our highlighting scheme assigns highlighting color saturation to reflect the similarity of each sequence pattern to the identified prominent pattern. Specifically, we assign three highlight color saturation grades centered around the selected pattern as shown in Figure4.36for two different patterns.

Figure 4.36.: Two trajectory patterns located at positions [6,6] (blue) and [0,2] (yellow) on the general view dis-played in Figure4.32. Neighboring patterns with grid distance 1 and 2 are highlighted at decreasing saturation. This color-coding is used in the sequence view shown in Figure4.37.

Figure4.37shows the sequence view with two selected patterns highlighted. It is the most detailed view which at the same time, suppresses small entity-specific detail and noisy patterns. The full sequence view supports visual analysis of the distribution and correlation of the two selected patterns over the full time period. Owing to the large amount of input data, the detailed sequence view requires a high resolution display. Therefore, the sequence view is also an interesting application for usage with large-scale displays such as the HEyeWall^R [HEy] or PowerWall [Pow] systems [TSB^∗08]. An illustration of such an application can be seen in Figure4.38.

4.6. Visual Analysis of Two-Dimensional Time-Dependent Data Using SOM Clustering

Figure 4.37.: The sequence view visualizing data movements of 83 objects during 66 time spans of observation.

Patterns [6,6] and [0,2] as selected by the user from a set of automatically generated candidate patterns are highlighted (see also Figure4.36).