The geovisualizations discussed so far, in their basic forms, represent one attribute, e.g., Cartograms are representations of a single attribute in the datasets. These representations become limiting for datasets with multiple attributes. To resolve this, one can additionally overlay cartographic maps or cartograms with glyphs having certain characteristics. Thus, one can visualize more variables using a composition of the aforementioned techniques.

Hargrove et al. [11] have shown how different color channels namely red, green, and blue, can correspond to three different variables and this multivariate heat map is used on map visualization. However, providing the spatial context for the visual representation of multiple attributes becomes a challenge at times. One of the strategies to circumvent this issue is the use of composite visualizations. Composite visualizations contain more than one visualizations which are to be used simultaneously. The taxonomy of composite visualizations [15] is based on the differences in the composition operator. The juxtaposed view is the most popular type of composite visualization, which is a juxtaposition of several visualizations. For multivariate geovisualizations, several single attribute map visualizations can be juxtaposed, an example of which is shown in [11], as well.

Another type of composite visualization is superimposed views. Geospace [23] is an example of overlaying different GIS layers, where each layer corresponds to an attribute or variable, and transparency is used to composite different layers, which is done to ensure that data in any layer is not inadvertently occluded.

Yet another example of composite visualizations is juxtaposing map-based visualizations with spatial-context-free multivariate visualizations, such that only the mapbased views preserve the spatial context. Popularly used multivariate visualizations are scatterplot matrices, parallel coordinates plot, and matrix visualization (Figure 6).

Fig. 6: Multivariate visualizations which can be juxtaposed with maps: (Left) scatterplot matrices, (middle) parallel coordinate plots, and (right) corrgram. These visualizations are for the multivariate automobile dataset, for demonstration purposes. Image courtesy:https://queue.acm.org/detail.cfm?id=1805128 [13] and http://www.infovis.info/index.php?words=exploratory

**Scatterplot matrices ** lay out the scatter plots of pairwise variables in a matrix format. Each row of the matrix corresponds to a variable. The columns of the matrix correspond to the same set of variables, as the rows. The ordering of the variables in the rows and columns is the same. An element of the matrix is a scatterplot corresponding to the variables in the corresponding row and column of the matrix. A variant of the scatterplot matrix relevant for geovisualization is the one with embedded maps, called “multiform bivariate matrix with space fill and map” [25]. The idea here is to fill the upper triangular part of the matrix with maps, and lower triangular one with scatterplots. The upper triangular part gives the spatial context and the lower triangular shows overall correlation between variables taken pairwise.

**Parallel coordinates plots (PCP)** are used for identifying trends in overall data, as well as correlation between any two variables. Each of the parallel axes in the PCP corresponds to a single attribute or variable. A “polyline” in a PCP corresponds to a row in a table or log, and is an alternate representation to a point in the Cartesian coordinate plot. PCP is effective only if the axes can be swapped so that we can bring two axes together for correlation analysis.

**Table or matrix visualization **is a visual representation of the tabular or matrix format of the data where the variables and the entities (e.g. locations in geospatial data) are rows and columns, or vice versa. In the visualization, each matrix element is colored using a transfer function for the value represented in the element, thus giving a heatmap.

The matrix can be constructed in two different ways – two-way two-mode representation, and two-way one-mode representation. Two-mode representation is where the rows and columns represent different sets of objects. The one-mode representation implies that the same set of objects are used in both rows and columns, and their ordering is the same in both row and column. An example of the one-mode representation is the corrgram which is used for studying the correlation between two variables, given in the row and column. The corrgram additionally uses glyphs and color to indicate value and sign of the correlation. One-mode representation also includes association matrix (between variables) and similarity matrix (between objects). Matrix visualization is effective only if it allows the user to perform seriation or reordering of variables [21], which is a permutation of the ordering of variables. Seriation maintains the restriction that the ordering followed in rows as well as columns is the same permutation.

These visualizations are made effective with specific user interactions. Brushing and linking are relevant user interactions for parallel coordinates plot and scatterplot matrices. Brushing is equivalent to using a lasso to subselect a few set of points, and linking implies updating different visualizations based on highlighting the “brushed” sub selection. A rendering technique called edge bundling reduces the clutter of overlapping edges in the parallel coordinate plot.

There are other ways of encoding visualization channels (such as color, texture, geometry, etc.) with variables. Artistic rendering can also be used for multivariate visualization, e.g. in [34], where the painterly rendering styles, e.g. brush stroke length, brush thickness, etc. can be used to encode different variables, such that the final rendering shows the spatial trends in co-occurrences of different variables.

Reducing the complexity of multivariate data involves use of data mining algorithms. Jin et al. [16] have used multivariate clustering using self organizing maps (SOMs). Colors assigned based on clusters in SOM are used for choropleth map visualization. Inspired by the same methodology of multivariate clustering, Zhang et al. [36] compare clustering results using choropleth maps, parallel coordinates plot, rose plots, and scatter plots for principal component analysis (PCA). While these works are listed as geovisualizations, the use of data mining (SOMs, PCA, etc.) makes them geovisual analytics applications.

## 4. Geovisual Analytics: A Case Study

In this section, we discuss a case study of a set of applications of geovisual analytics for similar datasets. Geovisual analytics is governed by the nature of the dataset, its complexity, its analytical goals, and its stakeholders. Hence, we briefly explain these aspects of the two tools covered in the case study for analysis of transport networks.

**4.1 Taxi Trajectories (Transport Networks)**

We consider two visual analytics systems used for taxi trajectories, namely, VAIT [22] and TrajGraph [14]. VAIT is designed to be a scalable tool for statistical analysis, and TrajGraph for graph-based analysis. VAIT is targeted to improve the profits and productivity of the taxi services. TrajGraph is used for studying traffic congestion using taxi trajectory data.