*3.1. Distributional Representation*

All the data of a store *si* can be stored in a matrix *L*(*si*) ∈ <sup>R</sup>*<sup>m</sup>*×*K*, in which the columns represent the *K* KPIs, and the rows represent the *m* users that completed the survey for a specific store *si* (Table 1). Then, each cell contains the value of a KPI for a customer.


**Table 1.** Matrix representing store *si*. Each column refers to a KPI, and each row refers to a customer.

Each column of *L*(*si*) can be considered to be a sample of the data related to a KPI. A column *k* can then be represented as a one-dimensional histogram *h* (*si*) *k* , whose support space [*zk*, *uk*] can be divided into *η* bins. The weight of each bin is given by the number of customers of the sample, whose score for the specific KPI falls into that bin. Figure 3 shows an example of the histograms associated with three different stores regarding a KPI.

**Figure 3.** Three different stores represented as univariate histograms. KPI values are on the x-axis, and their relative frequencies are on the y-axis.

As each histogram represents a single KPI, it is possible to compute the distance between two stores as the distance between the two histograms given by the same KPI. This representation naturally extends to multi-dimensional histograms. Characterizing a store using all KPIs, each store is represented as a *K*-dimensional histogram. For instance, considering two KPIs, the supports of the two-dimensional bins are squares, and the weights of the bins are the number of customers whose *KPIi* and *KPIj* scores fall into that bin. The natural representation is a heatmap, as shown in Figure 4.

**Figure 4.** Two different stores represented as bivariate histograms. KPI values related to *KPIi* and *KPIj* are on the x-axis and y-axis, respectively, and each bin is colored by their relative frequencies.

Since histograms are instances of discrete probability distributions, the stores become elements in a probabilistic space. Another characterization of stores in this probabilistic space can be obtained by representing the matrices *L*(*si*) as point clouds. Figure 5 displays an example of point cloud representation. On the left, one KPI for two stores is shown, and on the right, a plot of the same two stores for two KPIs is shown.

**Figure 5.** Point cloud representations of two stores. The **left** plot considers one KPI: KPI values are on the x-axis, and the absolute frequency is on the y-axis. The **right** plot considers two KPIs: KPI values are on the x-axis and y-axis, and each point represents a user.

The set of all KPIs is denoted as *S*. The power set of *S* is the set of all subsets, including the empty one and *S* itself. If *S* has cardinality *K*, then the number of subsets is 2*K*. All subsets but the empty one can be regarded as a description of a store. Therefore, the analysis can be performed on each element (except the empty one) of the power set of *S*.

A subset of cardinality *k* = 1, ... , *K* is associated with store *k*'s KPIs, which can be analyzed as *k* one-dimensional histograms or one *k*-dimensional histogram. The informational value of the two approaches is different, and the computational cost is also very different, as it increases with *k*. To mitigate this cost, one can choose the most significant KPIs using feature selection methods, as outlined in Section 5.1.

The histogram is a convenient representation of the *m* × *K* matrix *L*(*si*) in a space <sup>R</sup>*d*, where *d* = *<sup>η</sup><sup>K</sup>*, with *η* representing the number of bins. It is important to remark that *d* does not depend on the number of users *m* and can be reduced by considering an element of the power set *S* of cardinality *k* < *K* or a smaller number of bins.
