## 3 Materials and Methods
### Interpretability
#### 3.7.1 Problem Formulation  
The following lines describe the credit fraud detection interpretation system (CFDIS). Figure 3: Fine-tuning process. (**a**) Represents the autoencoder architecture. (**b**) Represents the encoder with an additional softmax layer. In this stage, we freeze the encoder layers and fit, allowing the gradient to backpropagate only through the softmax layer. (**c**) We again train the encoder and softmax layer, unfreezing the encoder layers. (**d**) We remove the softmax layer and we train the encoder and decoder freezing the encoders layers. **Notation 1**. : _Let \(CFDIS=\langle\mathbf{T},\mathbf{F},\mathbf{C}\rangle\) be an interpretability system for CFD, where:_  
* \(\mathbf{T}=\{\mathbf{t}_{1},\dots,\mathbf{t}_{n}\}\) _is the set of transactions participating to the system;_
* \(\mathbf{F}=\{\mathbf{f}_{1},\dots,\mathbf{f}_{m}\}\) _is the set of relevant features in a transaction;_
* \(\mathbf{C}=\{\mathbf{c}_{1},\dots,\mathbf{c}_{g}\}\) _is the set of classes in a CFD (fraud, non-fraud)._  
Thus, we formally define a transaction \(\mathbf{T}_{ij}\in\mathbf{T}\) as \(\mathbf{T}_{ij}=\langle\mathbf{t}_{i},\mathbf{c}_{j},\mathbf{W}_{ij},\mathbf{F}\rangle\) where the transaction \(\mathbf{t}_{i}\) is classified as \(\mathbf{c}_{j}\) and its interpretability is based on the numerical weights \(\mathbf{W}_{ij}\) for all features in \(\mathbf{F}\). Typically, \(\mathbf{c}_{j}\) is defined as the classes for fraud or non-fraud, assuming that they are known. We define a ranking function \(\succ\) as the ordering of a subset of features according to their contribution to the decision process. Taking into consideration this definition, we make the following assumption:  
**Assumption 1**. : _Each transaction \(\mathbf{t}_{i}\in\mathbf{T}\) has a set of ordered features in the decision process, denoted by \(\boldsymbol{\rho}_{i}=(\mathbf{F}^{i}_{i}\!\succ\!\!_{O_{i}})\), with \(\mathbf{F}^{i}\subseteq\mathbf{F}\) and representing its features. This means that the transaction \(t_{i}\) has a partial ordering of a subset of features \(\mathbf{F}^{i}\subseteq\mathbf{F}\) according to a certain ordering function \(\succ_{O_{i}}\), such that \(\succ_{O_{i}}\): \(\mathbf{F}^{i}\rightarrow\mathbf{R}\) allows the transaction \(t_{i}\) to assign a value to a certain feature in \(\mathbf{F}^{i}\), representing its weight in the decision process, regarding this particular transaction._  
**Example 1**. : _For instance, let \(\mathbf{F}^{1}=\{\text{living beyond means, lack or absence of transaction trail, car loans}\}\) be the set of features, which are made up of transaction \(\mathbf{t}_{1}\)'s informative features in the decision process. Transaction \(\mathbf{t}_{1}\) runs its ordering function \(\succ_{O_{i}}\), obtaining the weight \(\mathbf{W}_{ij}\):_  
* \(\succ_{O_{i}}(\text{Living beyond means})=4.9\)_._
* \(\succ_{O_{i}}(\text{Lack or absence of transaction trail})=4.7\)_._
* \(\succ_{O_{i}}(\text{Car loans})=5.2\)_._  
_Then, the most representative features for \(\mathbf{t}_{1}\) in the decision process are \(\boldsymbol{\rho}_{1}=(\text{car loans}\succ\text{living beyond means} \succ\text{lack or absence of transaction trail})\)._  
Therefore, if in a transaction the weight of feature 1 is higher than the weight of feature 2, we can infer that the decision process is being more influenced by feature 1. **Definition 1**. : _Using the aforementioned notation, the problem that we tackle in this work is defined as follows:_  
1. _An interpretability system for CFD is represented by_ \(CFDIS=\langle\mathbf{T},\mathbf{F},\mathbf{C}\rangle\)_;_
2. _A set_ \(\{\boldsymbol{\rho}_{i}\}\) _of internal ordered weights_ \(W_{ij}\) _for each feature_ \(\mathbf{F}\)_;_
3.