## 3 Method
### Method Detailed Description
#### 3.2.3 First Instance Classification
According to GADHA, the Ensemble Learning Classification activity is performed based on several ETs of different types (heterogeneous, homogeneous non-parametric, homogeneous parametric), built randomly according to the following hyperparameters: number of triads to build, set of parametric algorithms, set of non-parametric algorithms and proportion of heterogeneous, homogeneous non-parametric, and homogeneous parametric ET to build, expressed by the proportions \(\lambda_{1},\lambda_{2}\), and \(\lambda_{3}\). Table 2 illustrates the Ensemble Triads hyperparameters. According to GADHA notation, the set \(\Gamma\) corresponds to the set of non-parametric functions \(\gamma(\overrightarrow{v})\), and the set \(\Theta\) is the set of parametric functions \(\theta(\overrightarrow{v})\). The feature space \(\overrightarrow{V^{i}}\) may have three different dimensions depending on which data acquisition is made from sensors. The function argument \(\overrightarrow{v}\) is a vector that can be any subspace feature, defined in Equation (1). \[ \begin{split}\Theta^{i}=\{\theta_{1}(\overrightarrow{v}_{1}),..., \theta_{i}(\overrightarrow{v}_{i})\mid\overrightarrow{v}_{i}\subset \overrightarrow{V^{i}},\ \overrightarrow{v}_{1}\
eq\overrightarrow{v}_{i},\ i\in\mathbb{N}\ and\ i>1\};\\ \Gamma^{j}=\{\gamma_{1}(\overrightarrow{v}_{1}),...,\gamma_{j}( \overrightarrow{v}_{j})\mid\overrightarrow{v}_{j}\subset\overrightarrow{V }^{i},\ j\in\mathbb{N}\ and\ j>1\};\end{split}  \tag{1}\]  
The value of \(i\) indicates the same dimensions for both sets. It is not a hyper-parameter of the algorithm, but the sum up of all declared functions inside any category. One example of possible \begin{table}
\begin{tabular}{c c l} \hline \hline
**Hyperparameter** & **Default Value** & **Description** \\ \hline \(N\) & 1000 & Number of triads to build \\ \(\Theta^{i}\) & \(\{\emptyset\}\) & Set of parametric functions \\ \(\Gamma^{j}\) & \(\{\emptyset\}\) & Set of non-parametric functions \\ \(\Lambda\) & \(\{\lambda_{1}\lambda_{2}\lambda_{3}\}\) & Set of proportions among the three main triads set, \\  & & being that \(\lambda_{1}+\lambda_{2}+\lambda_{3}=1\) \\ \(\lambda_{1}\) & 0.5 & Heterogeneous triads proportion \\ \(\lambda_{2}\) & 0.25 & Homogeneous non-parametric triads proportion \\ \(\lambda_{3}\) & 0.25 & Homogeneous parametric triads proportion \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ensemble Triads hyperparameters. Figure 3: Ensemble triads general concept. functions for \(\Theta\) or \(\Gamma\) can be visualized in Equation (2). In the example of Equation (2), parametric functions are Artificial Neural Networks (ANN) and Naive Bayes (NB), while non-parametric functions are Support Vector Machine (SVM) and Decision Tree (DT). The analyst may choose other parametric and/or non-parametric functions as, for instance, Linear Discriminant Analysis (LDA), as done on the case study of this work. \[ \begin{split}\Theta^{i}&=\{ANN,NB\},\ i=2\\ \Gamma^{j}&=\{SVM,DT\},\ j=2\end{split}  \tag{2}\]  
For the heterogeneous ensemble triads, the same input data are used for all inducers. Each inducer of each heterogeneous ET will receive as input the same subset of features extracted on Activities 2 and 3 of GADHA. The input vector consumed by each inducer is a subset of all features vector and is called the Input Feature Vector. Therefore, the heterogeneous ET has only one Input Feature Vector consumed by all inducers. On the other hand, for the homogeneous ensemble triads, whether the ET is parametric or not, different input vectors are used for different inducers, once the same inducer would be prone to perform the same classification when subjected to the same data. This means that each inducer of each homogeneous ET will receive as input a different subset of features extracted on Activities 2 and 3 of GADHA. The input vector consumed by each inducer is a subset of all features vector and is called the Input Feature Vector. Therefore, the homogeneous ET has three different Input Feature Vectors, one for each inducer.