|
| 1 | +# Interest Measures |
| 2 | + |
| 3 | +## Support |
| 4 | + |
| 5 | +Support is defined on an itemset as the proportion of transactions that contain the attribute $`X`$. |
| 6 | + |
| 7 | +```math |
| 8 | +supp(X) = \frac{n_{X}}{|D|}, |
| 9 | +``` |
| 10 | + |
| 11 | +where $`|D|`$ is the number of records in the transactional database. |
| 12 | + |
| 13 | +For an association rule, support is defined as the support of all the attributes in the rule. |
| 14 | + |
| 15 | +```math |
| 16 | +supp(X \implies Y) = \frac{n_{XY}}{|D|} |
| 17 | +``` |
| 18 | + |
| 19 | +**Range:** $`[0, 1]`$ |
| 20 | + |
| 21 | +**Reference:** Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, |
| 22 | +2015, URL: https://mhahsler.github.io/arules/docs/measures |
| 23 | + |
| 24 | +## Confidence |
| 25 | + |
| 26 | +Confidence of the rule, defined as the proportion of transactions that contain |
| 27 | +the consequent in the set of transactions that contain the antecedent. This proportion is an estimate |
| 28 | +of the probability of seeing the consequent, if the antecedent is present in the transaction. |
| 29 | + |
| 30 | +```math |
| 31 | +conf(X \implies Y) = \frac{supp(X \implies Y)}{supp(X)} |
| 32 | +``` |
| 33 | + |
| 34 | +**Range:** $`[0, 1]`$ |
| 35 | + |
| 36 | +**Reference:** Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, |
| 37 | +2015, URL: https://mhahsler.github.io/arules/docs/measures |
| 38 | + |
| 39 | +## Lift |
| 40 | + |
| 41 | +Lift measures how many times more often the antecedent and the consequent Y |
| 42 | +occur together than expected if they were statistically independent. |
| 43 | + |
| 44 | +```math |
| 45 | +lift(X \implies Y) = \frac{conf(X \implies Y)}{supp(Y)} |
| 46 | +``` |
| 47 | + |
| 48 | +**Range:** $`[0, \infty]`$ (1 means independence) |
| 49 | + |
| 50 | +**Reference:** Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, |
| 51 | +2015, URL: https://mhahsler.github.io/arules/docs/measures |
| 52 | + |
| 53 | +## Coverage |
| 54 | + |
| 55 | +Coverage, also known as antecedent support, is an estimate of the probability that |
| 56 | +the rule applies to a randomly selected transaction. It is the proportion of transactions |
| 57 | +that contain the antecedent. |
| 58 | + |
| 59 | +```math |
| 60 | +cover(X \implies Y) = supp(X) |
| 61 | +``` |
| 62 | + |
| 63 | +**Range:** $`[0, 1]`$ |
| 64 | + |
| 65 | +**Reference:** Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, |
| 66 | +2015, URL: https://mhahsler.github.io/arules/docs/measures |
| 67 | + |
| 68 | +## RHS Support |
| 69 | + |
| 70 | +Support of the consequent. |
| 71 | + |
| 72 | +```math |
| 73 | +RHSsupp(X \implies Y) = supp(Y) |
| 74 | +``` |
| 75 | + |
| 76 | +**Range:** $`[0, 1]`$ |
| 77 | + |
| 78 | +**Reference:** Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, |
| 79 | +2015, URL: https://mhahsler.github.io/arules/docs/measures |
| 80 | + |
| 81 | +## Conviction |
| 82 | + |
| 83 | +Conviction can be interpreted as the ratio of the expected frequency that the antecedent occurs without |
| 84 | +the consequent. |
| 85 | + |
| 86 | +```math |
| 87 | +conv(X \implies Y) = \frac{1 - supp(Y)}{1 - conf(X \implies Y)} |
| 88 | +``` |
| 89 | + |
| 90 | +**Range:** $`[0, \infty]`$ (1 means independence, $`\infty`$ means the rule always holds) |
| 91 | + |
| 92 | +**Reference:** Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, |
| 93 | +2015, URL: https://mhahsler.github.io/arules/docs/measures |
| 94 | + |
| 95 | +## Inclusion |
| 96 | + |
| 97 | +Inclusion is defined as the ratio between the number of attributes of the rule |
| 98 | +and all attributes in the database. |
| 99 | + |
| 100 | +```math |
| 101 | +inclusion(X \implies Y) = \frac{|X \cup Y|}{m}, |
| 102 | +``` |
| 103 | + |
| 104 | +where $`m`$ is the total number of attributes in the transactional database. |
| 105 | + |
| 106 | + |
| 107 | +**Range:** $`[0, 1]`$ |
| 108 | + |
| 109 | +**Reference:** I. Fister Jr., V. Podgorelec, I. Fister. Improved Nature-Inspired Algorithms for Numeric Association |
| 110 | +Rule Mining. In: Vasant P., Zelinka I., Weber GW. (eds) Intelligent Computing and Optimization. ICO 2020. Advances in |
| 111 | +Intelligent Systems and Computing, vol 1324. Springer, Cham. |
| 112 | + |
| 113 | +## Amplitude |
| 114 | + |
| 115 | +Amplitude measures the quality of a rule, preferring attributes with smaller intervals. |
| 116 | + |
| 117 | +```math |
| 118 | +ampl(X \implies Y) = 1 - \frac{1}{n}\sum_{k = 1}^{n}{\frac{Ub_k - Lb_k}{max(o_k) - min(o_k)}}, |
| 119 | +``` |
| 120 | + |
| 121 | +where $`n`$ is the total number of attributes in the rule, $`Ub_k`$ and $`Lb_k`$ are upper and lower |
| 122 | +bounds of the selected attribute, and $`max(o_k)`$ and $`min(o_k)`$ are the maximum and minimum |
| 123 | +feasible values of the attribute $`o_k`$ in the transactional database. |
| 124 | + |
| 125 | +**Range:** $`[0, 1]`$ |
| 126 | + |
| 127 | +**Reference:** I. Fister Jr., I. Fister A brief overview of swarm intelligence-based algorithms for numerical |
| 128 | +association rule mining. arXiv preprint arXiv:2010.15524 (2020). |
| 129 | + |
| 130 | +## Interestingness |
| 131 | + |
| 132 | +Interestingness of the rule, defined as: |
| 133 | + |
| 134 | +```math |
| 135 | +interest(X \implies Y) = \frac{supp(X \implies Y)}{supp(X)} \cdot \frac{supp(X \implies Y)}{supp(Y)} |
| 136 | +\cdot (1 - \frac{supp(X \implies Y)}{|D|}) |
| 137 | +``` |
| 138 | + |
| 139 | +Here, the first part gives us the probability of generating the rule based on the antecedent, the second part |
| 140 | +gives us the probability of generating the rule based on the consequent and the third part is the probability |
| 141 | +that the rule won't be generated. Thus, rules with very high support will be deemed uninteresting. |
| 142 | + |
| 143 | +**Range:** $`[0, 1]`$ |
| 144 | + |
| 145 | +**Reference:** I. Fister Jr., I. Fister A brief overview of swarm intelligence-based algorithms for numerical |
| 146 | +association rule mining. arXiv preprint arXiv:2010.15524 (2020). |
| 147 | + |
| 148 | +## Comprehensibility |
| 149 | + |
| 150 | +Comprehensibility of the rule. Rules with fewer attributes in the consequent are more |
| 151 | +comprehensible. |
| 152 | + |
| 153 | +```math |
| 154 | +comp(X \implies Y) = \frac{log(1 + |Y|)}{log(1 + |X \cup Y|)} |
| 155 | +``` |
| 156 | + |
| 157 | +**Range:** $`[0, 1]`$ |
| 158 | + |
| 159 | +**Reference:** I. Fister Jr., I. Fister A brief overview of swarm intelligence-based algorithms for numerical |
| 160 | +association rule mining. arXiv preprint arXiv:2010.15524 (2020). |
| 161 | + |
| 162 | +## Netconf |
| 163 | + |
| 164 | +The netconf metric evaluates the interestingness of |
| 165 | +association rules depending on the support of the rule and the |
| 166 | +support of the antecedent and consequent of the rule. |
| 167 | + |
| 168 | +```math |
| 169 | +netconf(X \implies Y) = \frac{supp(X \implies Y) - supp(X)supp(Y)}{supp(X)(1 - supp(X))} |
| 170 | +``` |
| 171 | + |
| 172 | +**Range:** $`[-1, 1]`$ (Negative values represent negative dependence, positive values represent positive |
| 173 | +dependence and 0 represents independence) |
| 174 | + |
| 175 | +**Reference:** E. V. Altay and B. Alatas, "Sensitivity Analysis of MODENAR Method for Mining of Numeric Association |
| 176 | +Rules," 2019 1st International Informatics and Software Engineering Conference (UBMYK), 2019, pp. 1-6, |
| 177 | +doi: 10.1109/UBMYK48245.2019.8965539. |
| 178 | + |
| 179 | +## Yule's Q |
| 180 | + |
| 181 | +The Yule's Q metric represents the correlation between two possibly related dichotomous events. |
| 182 | + |
| 183 | +```math |
| 184 | +yulesq(X \implies Y) = |
| 185 | +\frac{supp(X \implies Y)supp(\neg X \implies \neg Y) - supp(X \implies \neg Y)supp(\neg X \implies Y)} |
| 186 | +{supp(X \implies Y)supp(\neg X \implies \neg Y) + supp(X \implies \neg Y)supp(\neg X \implies Y)} |
| 187 | +``` |
| 188 | + |
| 189 | +**Range:** $`[-1, 1]`$ (-1 reflects total negative association, 1 reflects perfect positive association |
| 190 | +and 0 reflects independence) |
| 191 | + |
| 192 | +**Reference:** E. V. Altay and B. Alatas, "Sensitivity Analysis of MODENAR Method for Mining of Numeric Association |
| 193 | +Rules," 2019 1st International Informatics and Software Engineering Conference (UBMYK), 2019, pp. 1-6, |
| 194 | +doi: 10.1109/UBMYK48245.2019.8965539. |
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