## 3 Materials and Methods
### Interpretability
#### 3.7.3 Building the ITR  
As we indicated above, our proposal provides a novel mechanism to understand why the decision process works for an individual transaction in a \(CFDIS\). Accordingly, the method is agnostic from the mechanism that we use to obtain the latent space; in this way, if a more powerful mechanism appears in the state of the art, it is compatible with said proposal. That is, if instead of using an autoencoder we use another algorithm, through STE and generating random samples by perturbations around the instance, we can also determine the contribution for each feature. Once the mechanism is decided, we can use this black box to obtain the weights of each feature. In this work we use autoencoder approaches to obtain the latent space, and, from there, the weights.  
Considering the weights obtained using STE, through ITR, we build a ranking of individual transactions. This ranking captures the order of features, allowing us to know for each individual transaction which features are the most influential in the decision process. Formally, it can be expressed as follows.  
**Definition 2**.: _An \(ITR_{i}\) for the transaction \(t_{i}\) participating into the \(CFDIS\) is an estimation \(\Delta_{i}^{t}\) of its more representative features \(\rho_{i}\), such that:_  
\[ITR_{i}=\Delta_{i}^{t}=(F^{i},\succ_{O_{i}^{t}}) \tag{4}\]  
_where:_  
* \(F^{i}\subset F\) _is a subset of features used in the decision process in the_ \(t_{i}\)_;_
* \(\succ_{O_{i}^{t}}\) _is an ordering function, such that_ \(O_{i}^{t}:F^{i}\times t_{ij}\times Enc\rightarrow\mathbb{R}\) _assigns a value to a certain feature in_ \(F^{i}\) _taking into account the result of applying a lineal classifier in the latent space using autoencoder to a transaction_ \(t_{ij}\)_._  
**Example 2**.: _Let us illustrate this definition by the following example. For instance, let \(\mathbf{F}^{1}=\{\text{living beyond means},\text{lack of transaction trail, car loans}\}\) be the set of features, which are made up of transactions \(\mathbf{t_{1}}\), \(\mathbf{t_{2}}\), and \(\mathbf{t_{3}}\) with different weights obtained using STE:_  
* \(\mathbf{t_{1}}:(\text{car loans})=5.2\succ(\text{living beyond means})=4.9\succ( \text{lack of transaction trail})=4.7\)
* \(\mathbf{t_{2}}:(\text{car loans})=3.9\succ(\text{living beyond means})=3.7 \succ(\text{lack of transaction trail})=3.2\)
* \(\mathbf{t_{3}}:(\text{lack of transaction trail})=4.2\succ(\text{living beyond means})=3.7\succ(\text{car loans})=3.1\)  
_Then for the transactions \(\mathbf{t_{1}}\) and \(\mathbf{t_{2}}\) we can see have the same \(ITR\) (car loans) \(\succ(\text{living beyond means})\succ(\text{lack of transaction trail})\) and \(\mathbf{t_{3}}\) have different properties with other \(ITR\) (lack of transaction trail) \(\succ(\text{living beyond means})\succ(\text{car loans})\)._  
We can see the process to calculate ITR summarized as shown in Algorithm 3.  
```
0: Training set in real space \(\mathbf{X}\), number of features \(L\), number of transactions \(k\)
1: Calculate weights for all instances \(W_{i}\gets STE(\mathbf{X}_{i})\) with \(i=1,\dots,k\).
2: Generate individual ranking for each transaction.
3:for\(b\gets 1\) to \(k\)do
4: Depending on the weights of each feature, we obtain its numerical position in the significance ranking, where the highest weight is the first in the ranking and the lowest is the last. \(ITR_{b}=generateRanking(W_{b})\).
5:endfor
```  
**Algorithm 3** Interpretability. Obtain individual transaction rankings