To overcome the fundamental problem of CAH, methods related to the temporal complexity of distance calculations, we initially use a representative sample of the total set of individuals. This reduces the size of the data and therefore the number of distance calculations between pairs of individuals at the initial stage.

In addition, for the aggregation of classes, to allow qualitative data to be considered, we use the notion of “close resilience boundary” rather than “geometric distance” as is the case in many methods, such as the CURE algorithm [12] . Thus, two classes are aggregated and if at iteration t, these two classes have the largest number of pairs of individuals with close resilience boundaries (i.e. the global resilience boundary).

The second step of our classification method concerns the characterization of the classes obtained. We use the decision tree technique here because of their ability to provide simple and readable visualization as well as an easy interpretation of results.

We define the concept of resilience boundary below.

Definition 1: The elementary resilience threshold ( ${\beta }_{ij}^t$) of an individual (i) at time (t) relative to a given capacity (j) represents the minimum threshold below which the individual is vulnerable to this capacity.

Definition 2: The overall resilience threshold ( ${\beta }_i^t$) of an individual (i) at time (t) represents the minimum threshold, below which the individual is vulnerable to all abilities. It is obtained by averaging the elementary resilience thresholds of all capacities.

${\beta }_i^t=\frac{{{\displaystyle \sum }}_{j=1}^{card\left(D\right)}{\beta }_{ij}^t}{card\left(D\right)}$ (1)

Where $card\left(D\right)$ is the number of dimensions considered.

Note

$\Omega =\left\{x_1,x_2,\cdots ,x_n\right\}$, the set of individuals who suffered the traumatic shock to be classified.

$P^m=\left\{C_1,C_2,\cdots ,C_k\right\}$, the set of individual partitions at the iteration m.

$C_i$, the i^th class of individuals.

${\beta }_i^t$, the resilience boundary of individual i, at time t;

$\left|\Delta {\beta }_i^j\right|$, the positive difference in individual resilience thresholds $x_i$and $x_j$;

$\theta$, the threshold of difference.

$N_{Br}\left(C_i,C_j\right)$, the number of pairs of individuals belonging to classes $C_i$ and having close resilience boundaries.

Two individuals $x_i$ and $x_j$ have close resiliency thresholds, if the positive difference in their resilience thresholds is less than or equal to the threshold set $\theta$.

Thus, for $x_1,x_2$ and $x_3$, individuals belonging to different classes:

$\left|\Delta {\beta }_1^3\right|\le \theta \Rightarrow {\beta }_1\simeq {\beta }_3$(2)

$\left|\Delta {\beta }_1^2\right|>\theta \Rightarrow {\beta }_1>{\beta }_2$ or ${\beta }_1<{\beta }_2$(3)

The classes $C_a$ and $C_b$ are grouped, if they have the largest number of pairs of individuals with close resiliency boundaries. In this case, all individuals of the class $C_b$ are added to the class $C_a$:

$If{N}_{Br}\left(C_a,C_b\right)\leftarrow \underset{\forall C_i,C_j\in P^m}{Max}{N}_{Br}\left(C_i,C_j\right)$ then $C_a\leftarrow x_i,\forall x_i\in C_b$ (4)

Decision trees [13] are a supervised learning method used for data exploration and prediction. The technique consists in iteratively con-structing homogeneous classes of individuals by asking a succession of binary questions on the different attributes. It allows to predict the value of a qualitative variable from decryptable variables qualitative and/or quantitative. One of the main advantages of decision trees is their graphical representation. It provides a simple and readable visualization of anything that facilitates the interpretation of results. This graphical representation is in the form of a tree consisting of terminal sheets (the classes of individuals) and nodes corresponding to binary questions on a variable of the data set.

Tree construction process:

The effectiveness of a decision tree depends heavily on the purity of the leaves generated by the different nodes. The objective is to create for each node, two sheets that are more homogeneous than the preceding node. This requires that the questions chosen for the construction of the tree be as discriminating as possible. Two mathematical tools are generally used to test the purity of a leaf: the Gini index [14] and probabilistic entropy [15] . The first one is used by the CART algorithm while the C4.5 algorithm uses probabilistic entropy.

If we designate by Xi the ith discrete variable describing an individual, $x_{i1},\cdots ,x_{in}$, the n modalities that can take this variable and $p_{i1},\cdots ,p_{in}$, the respective probabilities of obtaining these modalities, the entropy H(Xi) of the variable Xi is calculated in base b as follows :

$H_b\left(X_i\right)=-{\displaystyle \underset{j=1}{\overset{n}{\sum }}{p}_j{\log }_b\left(p_j\right)}$ (5)

Entropy, like the Gini index, is also described as a measure of disorder. More precisely, entropy is used to measure the disorder generated by the choice of the variable Xi as the cut-off variable of the node α and the $x_{ik}$ modality as the criteria for the distribution of individuals in the different classes.

Entropy gives a value of zero when the variable Xi is deterministic and returns only one value xo and it is maximum (positive) when Xi is uniformly distributed, i.e., $\forall j$, $p_j=1/n$.

In the process of constructing the tree, it happens that one ends up in a situation of over-learning or over-adjustment, characterized by an «exaggerated» description of the data set. This situation results in too many terminal sheets in the decision tree, which becomes de facto long and illegible. This is a dimensioning problem which can be solved by the so-called tree pruning technique. It allows to keep a correct level of generality by proceeding by cross validation. In other words, the dataset is divided into sub-sets, each serving as a learning sample and then as a validation sample.

Principle and stopping criteria:

Like all learning algorithms, the decision tree algorithm has the following stopping criteria:

The decision tree is constructed according to the following principle:

a) Check whether the present level meets one of the stop criteria listed above.

b) If yes, stop the process.

c) If not,

The classification algorithm we propose here has the advantage of providing both a categorization of individuals victims of a traumatic shock and a characterization of the partitions generated using a decision tree. It is therefore a hybrid method between unsupervised and supervised learning.

Note by mad(.) the characterization of classes by the decision tree technique as used in the CART algorithm. The decision tree will be represented by $\Lambda$.

The hierarchical algorithm for classification of latent traumatic states is as follows:

We present here the application of our hybrid hierarchical classification model to the data collected in the survey. The execution of our algorithm allows to obtain the different significant classes in which are distributed the individuals composing the target population but also their characterization. We refer here to rclass.cah, the function implementing our hierarchical classification algorithm and res.class_cah, the object containing the results of the classification. The main command line of our algorithm is as follows:

Res.class_ag<-rclass.cah(mbase.stat,dist="Simsymb",agglom="borneResilience",dendo=TRUE, list.rep=TRUE, arbre.d=TRUE)

Figure 1 below shows the dendrogram corresponding to the hierarchical classification of 50 people from our samples. As a reminder, a dendrogram is a

hierarchical grouping diagram, enabling data to be organized in a tree structure according to their similarities.

As can be seen, Figure 1 shows the existence of four (4) classes among the individuals presented. Each class groups similar individuals according to a common characteristic. The dendogram is therefore limited to showing the number of groupings that can logically emerge from the sample, without saying more.

The characterization of these classes using the decision tree technique will therefore provide more information on these groupings, for eventual decision-making.

After the identification of classes, the next step in our algorithm concerns their characterization or the identification of discriminant variables by the decision tree technique. For this, we use the initial database (base.stat). Generally, for a better exploration of a data set, there is much interest in the characterization of certain variables. The aim is to understand the structuring of the data set by highlighting the existing relationships between certain variables and to find the most discriminating variables.

The execution of our basic function rclass.cah, allows to obtain the following characterization based on the psychosocial state of individuals belonging to different classes ( Figure 2 ).

Based on the information collected, reading the tree above (see the leaves of the tree) allows us to distinguish 4 categories or classes of people, according to their level of resilience of which 3 groups are resilient and the last is not resilient. This result of the decision tree is therefore in phase with the number of classes identified in the first part of our algorithm. The categories identified are:

Moreover, it can be seen that the degree of consistency of individuals (CONSC) was considered by the algorithm as the most discriminating variable among the different descriptors used.

To validate our approach, we will compare its performance with that of some methods in the literature. However, it is important to remember that the performance of a classification method depends heavily on the field of application of its measure of similarity or dissimilarity and the type of data used. Based on this assumption, it would be more appropriate to compare our approach with classification models adapted to the analysis of social resilience processes. Thus, only those of (Coulibaly et al., 2017) [8] meet this requirement.

In the field, the performance of machine learning algorithms is usually evaluated using the classical precision, recall and F-measure indices. These are calculated from the four (4) response possibilities when a classification query is made (see Table 1 below).

In our context, relative to a given class:

$Precisio{n}_{C_i}=\frac{T{P}_{C_i}}{T{P}_{C_i}+F{P}_{C_i}}$(6)

$Recal{l}_{C_i}=\frac{T{P}_{C_i}}{T{P}_{C_i}+F{N}_{C_i}}$(7)

$F-Measur{e}_{C_i}=2*\frac{Precisio{n}_{C_i}*Recal{l}_{C_i}}{Precisio{n}_{C_i}+Recal{l}_{C_i}}$(8)

In this section, to calculate these three indices, we will use the results from the implementation of the two classification approaches: [Coulibaly et al., 2017) [8] and our present approach], with the data mentioned in section 5.

The results from the calculation of indices are given in Table 2 below. The values reported are the averages obtained from 8 runs of each approach under the same initial conditions.

The following Figures 3-5 respectively show the results of Table 2 above according to Precision, Recall and F-Measure.

The analysis of these results allows us to make the following observations: