## 3 Results
### Assessing the Strength of Association Rules with 7-Score
Figure 3: Three different exemplary datasets (**a**–**c**) with the rules of the same confidence of 100%. The rule: if light green, then dark green. The next issue is assessing the rules of low confidence. Please observe the two examples illustrated in Figure 5.  
Figure 4: Two different exemplary datasets (**a**,**b**) with the rules of the same confidence of 100% and support of 33.3%. The rule: if light green, then dark green. Figure 5: Two different exemplary datasets (**a**,**b**) with the rules of the same confidence of 20% and support of 16.7%, and lift of 1. The rule: if light green, then dark green. In case (a), the heads are multi-colored and the rule--if light green, then dark green--seems meaningless. In case (b), where the head is dichotomous, it seems that finding the opposite rule (if light green, then blue) brings a better result (conf = 80%, sup = 66.7%, lift = 1). The same result will be achieved in case (a) if the rule will be stated as follows: if light green, then not dark green. It can be concluded that the rules of low confidence are meaningless. To assess the strength of rules, the following aim function is created:  
\[\text{strength of rule}=\text{lift}+\text{N}^{2}\times\text{sup}+\text{N}^{2} \times\text{conf} \tag{6}\]  
where N is for the total number of cases in a database. Equation (6) considers the following assumptions. Assumption 1:  
\[\text{I}(\text{sup})>\text{I}(\text{lift}) \tag{7}\]  
where I is a function of the importance of the rule. Equation (7) is achieved by making the sum component of support equal or higher than lift, as follows:  
\[\text{sup}\times\text{N}=\text{n}_{bh} \tag{8}\]  
where \(n_{bh}\) is the number of cases meeting the rule and, as the maximum lift is N and the minimum support is \(\frac{1}{N}\), the following Equation is met:  
\[\text{N}^{2}\times\text{sup}\geq\text{lift} \tag{9}\]  
Meeting assumption 2 presented in Equation (10), using Equation (6),  
\[I(conf)>I(sup) \tag{10}\]  
is achieved by multiplying the confidence by the same number as the support, i.e., by \(N^{2}\), as the confidence is greater than the support for each rule (as the number of bodies meeting the rule is always lower than \(N\)). Equation (6) for the strength of rule introduces possible cases where the joint impact of lift and support is greater than the impact of confidence on the strength of the rule. These kinds of cases are partially limited by excluding from the analysis the cases of low confidence (below 50%). To observe how the rules are assessed, the exemplary database is created of 10 bodies and 10 heads. The number of bodies meeting the rule \(n_{b}\) changes from 1 to 9, and the number of heads meeting the rule \(n_{h}\) also changes from 1 to 9. The number of cases meeting the rule \(n_{bh}\) changes from 1 to a number defined as \(min(n_{b},\ n_{h})\). All possible combinations are assumed, and all rules are found in the created cases. Confidence, support, lift, and the strengths of the rules are calculated. From the full set of rules, regardless of the rules of confidence lower than 0.5, the cases with lift lower than 1 are excluded too. When the lift is lower than 1, this means that, when predicting the head, the better result can be achieved by applying the probability of appearance of a specific head, rather than basing it on a specific body appearance. The remaining data and results (scores) are presented in Appendix A Table A1. As it is difficult to present a 4-dimentional chart in a 2D figure, Figure 6 is prepared. Support and confidence are on the horizontal axes and 7-Score values are on the vertical axis. It can be observed that, for several pairs of identical conf and sup, there are several values of 7-Score. This is because of the influence of lift--which is also considered in 7-Score and in Figure 3:0 Lift differentiates 7-Score for the cases of the same support and confidence, as was assumed while the formula for 7-Score was created.