Contextual mapping: Visualization of high-dimensional spatial patterns in a single geo-map

Highlights

• A generic method based Self Organizing Maps to encode the high dimensional spatial patterns into single numerical vector.

• The numerical vector makes it possible to visualize high dimensional patterns in a single geo-map, called contextual maps.

• Instead of rigid spatial clusters it produces a color-coded spectrum of changing high dimensional emergent patterns.

• It can be used in a hierarchically to combine several contextual maps, each representing a unique high dimensional context.

Abstract

In this study, we proposed a generic methodology for combining high-dimensional spatial data to identify and visualize the hidden spatial patterns in a single-layer geo-map. By using the less explored one-dimensional self-organizing maps, we showed how the high-dimensional data can be transformed into a spectrum of one-dimensional ordered numbers. These numbers (codes) can index a high-dimensional space with the important property that similar indices refer to similar high-dimensional contexts. Thus, the high-dimensional vectors will be attributed to single numbers, and this one-dimensional output can be easily rendered as a new single data layer in the original geographic map. As a result, it simultaneously identifies the main spatial clusters and visualizes the high-dimensional correlations (if any) in a single geographic map. Further, because the output of the proposed method is a set of ordered indices, there is no need to define a fixed number of clusters in advance.

Because these composite spatial layers are identified on the basis of the selected context (i.e., the selected features or aspects of the spatial phenomena), they are called contextual maps.

Finally, we showed the results of applying the proposed methodology to several synthetic and real-world data sets.

Introduction

With the current rapid growth in the amount of digital data, we must address the challenge of finding appropriate techniques to harness the power of these data streams. For example, in many cities across the world, no longer does anyone lack access to digital spatial maps; instead, the current challenge is, considering the amount and diversity of these digital data regarding different aspects of the cities, how one can picture his/her own map of the space as a combination of several factors of interest.

Toward this direction, there have been several interesting cases such as peoplemaps¹ or Livehood projects (Cranshaw, Schwartz, Hong, & Sadeh, 2012), which are explorations and mapping of activities within cities based on data available from online social networks. One of the cases most similar to our work is a project called Whereabout,² where by applying the K-means data-clustering algorithm to a collection of spatial data consisting of > 200 different aspects of each ward in the city of London, a fixed number of groups were created by grouping based on informational similarities (not physical locations). Then, on top of the classical map of London, people get an impression of different regions on the basis of their similarities in all of these categories. In a similar manner, but only based on demographic information, a new coding system of London called LOAC was developed (Longley & Singleton, 2014).

The classical clustering algorithms divide the high-dimensional data space into a predetermined number of groups, where each will be given a label (usually an arbitrary number). Then, these cluster labels attributed to each spatial data point can be visualized on the geographic map with a specified color code. However, despite the fact that standard clustering methods such as K-means are easy to use, they have some limitations in the domain of spatial pattern recognition. One of the main problems is that they divide the space into a small number of categories. Instead, it would be preferred to have a continuous and smooth changing pattern on top of the high-dimensional data. Further, one needs to select the number of clusters in advance, which is a critical decision (Tibshirani, Walther, & Hastie, 2001). In addition, in the context of spatial clustering, because the cluster labels are not ordered according to their high-dimensional similarities, the colored visualization of clusters in the geographic map is not directly helpful. Therefore, similar colors in a clustered geo-map do not necessarily refer to similar high-dimensional patterns. As a result, increasing the number of clusters with different colors may result in final spatial visualizations that are not helpful, but having too few clusters produces results that are too aggregated. One current solution to this problem is to create an RGB (red, green, blue) pattern after data clustering by reducing the high-dimensional vectors of the cluster centers to their first three principal components (Mahinthakumar, Hoffman, Hargrove, & Karonis, 1999). However, in addition to losing some information (by selecting only three principal components), the color interpretations will need an additional step.

The main hypothesis of this study is that if we find a method to sort the clusters in a way such that similar cluster indices refer to similar contexts (i.e., similar high-dimensional patterns), we can make a direct projection from high-dimensional spatial data to a one-dimensional vector and visualize the high-dimensional patterns in the geographical maps using a simple color spectrum. In this manner, by having many indices instead of dividing the high-dimensional data into a few distinct groups, one can create a spectrum of high-dimensional patterns that are visualized with a colored spectrum on spatial maps. Because the high-dimensional patterns would change gradually, this would also solve the problem of distinct cluster borders and the fixed number of clusters. As we show in Section 2, our proposed approach can be discussed from the viewpoint of dimensionality reduction and manifold learning (Bengio, Courville, & Vincent, 2013), where one of the best methods that satisfies these requirements is self-organizing maps (SOMs) (Kohonen, 2013).