## 2 Measuring System, Materials and Methods
### Methodology
#### 2.1.4 Extraction of Associations Rules  
Investigating the relationship between the physical parameters of the structure and the parameters of the waveform obtained as a result of the structure testing is a process that is difficult to standardize. Typically, the investigation begins with measurement analysis using various kinds of charts. The next step often involves determining the coordinates and values of the extremes and inflection points, followed by their description using selected statistics. A boxplot can be used for this purpose (an example of a boxplot is presented in Figure 6).  
One of the most valuable tools for uncovering the relationships between physical parameters and the measurement waveform is using algorithms designed explicitly for association rule mining, such as the Apriori algorithm. While more attributes may not be advantageous in the identification process, it facilitates a more comprehensive examination of the association rules and their statistical characterization. Ultimately, this deeper analysis contributes to an enhanced understanding of the underlying processes and aids in developing a more effective identification model [73; 74; 75; 76; 77].  
To investigate association rules, it is necessary to create a database containing comparisons of records in which structures differ by only one parameter (with the smallest possible change). The detected dependencies can belong to three classes: increase (\(\uparrow\)), decrease (\(\downarrow\)), or no change (-). In general terms, a rule is expressed as follows (1):  
\[\text{If (BODY) Then (HEAD) [_support_, _confidence_]} \tag{1}\]  
BODY (A) refers to the change in a specific parameter of the structure, while _HEAD_ (B) denotes the change in a specific parameter of the waveform. The quality of the rule is characterized by two percentage indicators: _support_ and _confidence_. _Support_ is defined as the ratio of the number of records in which the rule occurs to the total number of records in the database (D), which is expressed by Equation (2):  
\[\text{supp (A)}=\frac{\#\{\text{d}_{i}\in\text{D}:\text{A}\subset\text{d}_{i}\}}{ \#\text{D}}\approx\text{P(A)} \tag{2}\]  
Figure 6: Description of the boxplot.  
The term _confidence_ quantifies the conditional probability that the entire rule will be observed, contingent upon the presence of the BODY (A). This parameter can be mathematically represented as follows (3):  
\[\text{conf}(\text{A}\rightarrow\text{B})=\frac{\text{supp }(\text{A}\cup\text{B})}{\text{supp }(\text{A})}\approx\text{P}(\text{B}|\text{A}) \tag{3}\]  
The method employed for discovering association rules resembles the classical Apriori algorithm. It is predicated on the observation that interesting rules are derived exclusively from frequent item sets, defined as those for which the support exceeds the minimum support threshold established by the researcher. However, three significant modifications were introduced compared to the classical algorithm:  
* Set A encompassed only the physical parameters of the structure, while Set B consisted solely of attributes that described the examined waveform.
* The length of the BODY was constrained to a single element from set A.
* Depending on the architecture of the database, it was permissible to omit support as a limiting criterion.  
The implementation of these modifications resulted in the exclusion of a substantial majority of generated rules, thereby retaining only those that elucidated the relationships between variations in specific structural parameters and the resultant changes they induced [73; 74; 75; 76; 77]. The algorithm utilized for deriving association rules is presented in Figure 7.  
The modifications implemented facilitated a substantially reduced number of potential combinations, decreasing it from 2\({}^{\text{X}-1}\) to #A\(\times\)2\({}^{\text{\#B}-1}\), where X = #A + #B represents the total number of parameters. This reduction enhances the algorithm's efficiency by limiting the search space for association rules.