Picturing agreement between clustering solutions using multidimensional unfolding: An application to greenhouse gas emissions data

When evaluating a clustering solution, we often have to compare alternative solutions – e.g., to address clustering stability or external validity. Each comparison essentially relies on a contingency table referring to a pair of (crisp) clustering solutions. These data is commonly used as an input to: (1) an assignment problem, to match the clusters of the two partitions; (2) determine several indices of agreement; (3) represent the two partitions in a two-dimensional map resorting to Correspondence Analysis. We propose using the Multidimensional Unfolding (MDU) technique to picture the cross-classification data between two partitions, complementing a clustering evaluation analysis and overcoming some limitations of the traditional approaches (1) to (3). This approach relies on a new similarity measure that excludes agreement between clusters due to chance alone. The resulting MDU map is very easy to interpret, picturing agreement between clustering solutions: the further apart are the clusters (represented by points) from the two partitions, the larger the (Euclidean) distances between the corresponding points. Two applications illustrate the relevance of this approach: an application to a data set on UCI Machine Learning Repository to access clustering external validity; and an application to greenhouse gas emissions data to address the temporal stability of clustering solutions, the clusters of European countries, which have homogeneous sources of pollutant emissions, being compared over three years.