*Article* **The "Unreasonable" Effectiveness of the Wasserstein Distance in Analyzing Key Performance Indicators of a Network of Stores**

**Andrea Ponti 1,2,\*, Ilaria Giordani 1,3, Matteo Mistri 1, Antonio Candelieri 2 and Francesco Archetti 3**

	- 20126 Bicocca, Italy

**Abstract:** Large retail companies routinely gather huge amounts of customer data, which are to be analyzed at a low granularity. To enable this analysis, several Key Performance Indicators (KPIs), acquired for each customer through different channels are associated to the main drivers of the customer experience. Analyzing the samples of customer behavior only through parameters such as average and variance does not cope with the growing heterogeneity of customers. In this paper, we propose a different approach in which the samples from customer surveys are represented as discrete probability distributions whose similarities can be assessed by different models. The focus is on the Wasserstein distance, which is generally well defined, even when other distributional distances are not, and it provides an interpretable distance metric between distributions. The support of the distributions can be both one- and multi-dimensional, allowing for the joint consideration of several KPIs for each store, leading to a multi-variate histogram. Moreover, the Wasserstein barycenter offers a useful synthesis of a set of distributions and can be used as a reference distribution to characterize and classify behavioral patterns. Experimental results of real data show the effectiveness of the Wasserstein distance in providing global performance measures.

**Keywords:** Wasserstein distance; customer experience; key performance indicators
