interestingness_measures.md

Interestingness measures

Support

Support is defined on an itemset as the proportion of transactions that contain the attribute $X$.

$$supp(X) = \frac{n_{X}}{|D|},$$

where $|D|$ is the number of records in the transactional database.

For an association rule, support is defined as the support of all the attributes in the rule.

$$supp(X \implies Y) = \frac{n_{XY}}{|D|}$$

Range: $[0, 1]$

Reference: Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, 2015, URL: https://mhahsler.github.io/arules/docs/measures

Confidence

Confidence of the rule, defined as the proportion of transactions that contain the consequent in the set of transactions that contain the antecedent. This proportion is an estimate of the probability of seeing the consequent, if the antecedent is present in the transaction.

$$conf(X \implies Y) = \frac{supp(X \implies Y)}{supp(X)}$$

Range: $[0, 1]$

Reference: Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, 2015, URL: https://mhahsler.github.io/arules/docs/measures

Lift

Lift measures how many times more often the antecedent and the consequent Y occur together than expected if they were statistically independent.

$$lift(X \implies Y) = \frac{conf(X \implies Y)}{supp(Y)}$$

Range: $[0, \infty]$ (1 means independence)

Reference: Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, 2015, URL: https://mhahsler.github.io/arules/docs/measures

Coverage

Coverage, also known as antecedent support, is an estimate of the probability that the rule applies to a randomly selected transaction. It is the proportion of transactions that contain the antecedent.

$$cover(X \implies Y) = supp(X)$$

Range: $[0, 1]$

Reference: Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, 2015, URL: https://mhahsler.github.io/arules/docs/measures

RHS Support

Support of the consequent.

$$RHSsupp(X \implies Y) = supp(Y)$$

Range: $[0, 1]$

Reference: Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, 2015, URL: https://mhahsler.github.io/arules/docs/measures

Conviction

Conviction can be interpreted as the ratio of the expected frequency that the antecedent occurs without the consequent.

$$conv(X \implies Y) = \frac{1 - supp(Y)}{1 - conf(X \implies Y)}$$

Range: $[0, \infty]$ (1 means independence, $\infty$ means the rule always holds)

Reference: Michael Hahsler, A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, 2015, URL: https://mhahsler.github.io/arules/docs/measures

Inclusion

Inclusion is defined as the ratio between the number of attributes of the rule and all attributes in the database.

$$inclusion(X \implies Y) = \frac{|X \cup Y|}{m},$$

where $m$ is the total number of attributes in the transactional database.

Range: $[0, 1]$

Reference: I. Fister Jr., V. Podgorelec, I. Fister. Improved Nature-Inspired Algorithms for Numeric Association Rule Mining. In: Vasant P., Zelinka I., Weber GW. (eds) Intelligent Computing and Optimization. ICO 2020. Advances in Intelligent Systems and Computing, vol 1324. Springer, Cham.

Amplitude

Amplitude measures the quality of a rule, preferring attributes with smaller intervals.

$$ampl(X \implies Y) = 1 - \frac{1}{n}\sum_{k = 1}^{n}{\frac{Ub_k - Lb_k}{max(o_k) - min(o_k)}},$$

where $n$ is the total number of attributes in the rule, $Ub_k$ and $Lb_k$ are upper and lower bounds of the selected attribute, and $max(o_k)$ and $min(o_k)$ are the maximum and minimum feasible values of the attribute $o_k$ in the transactional database.

Range: $[0, 1]$

Reference: I. Fister Jr., I. Fister A brief overview of swarm intelligence-based algorithms for numerical association rule mining. arXiv preprint arXiv:2010.15524 (2020).

Interestingness

Interestingness of the rule, defined as:

$$interest(X \implies Y) = \frac{supp(X \implies Y)}{supp(X)} \cdot \frac{supp(X \implies Y)}{supp(Y)} \cdot (1 - \frac{supp(X \implies Y)}{|D|})$$

Here, the first part gives us the probability of generating the rule based on the antecedent, the second part gives us the probability of generating the rule based on the consequent and the third part is the probability that the rule won't be generated. Thus, rules with very high support will be deemed uninteresting.

Range: $[0, 1]$

Reference: I. Fister Jr., I. Fister A brief overview of swarm intelligence-based algorithms for numerical association rule mining. arXiv preprint arXiv:2010.15524 (2020).

Comprehensibility

Comprehensibility of the rule. Rules with fewer attributes in the consequent are more comprehensible.

$$comp(X \implies Y) = \frac{log(1 + |Y|)}{log(1 + |X \cup Y|)}$$

Range: $[0, 1]$

Reference: I. Fister Jr., I. Fister A brief overview of swarm intelligence-based algorithms for numerical association rule mining. arXiv preprint arXiv:2010.15524 (2020).

Netconf

The netconf metric evaluates the interestingness of association rules depending on the support of the rule and the support of the antecedent and consequent of the rule.

$$netconf(X \implies Y) = \frac{supp(X \implies Y) - supp(X)supp(Y)}{supp(X)(1 - supp(X))}$$

Range: $[-1, 1]$ (Negative values represent negative dependence, positive values represent positive dependence and 0 represents independence)

Reference: E. V. Altay and B. Alatas, "Sensitivity Analysis of MODENAR Method for Mining of Numeric Association Rules," 2019 1st International Informatics and Software Engineering Conference (UBMYK), 2019, pp. 1-6, doi: 10.1109/UBMYK48245.2019.8965539.

Yule's Q

The Yule's Q metric represents the correlation between two possibly related dichotomous events.

$$yulesq(X \implies Y) = \frac{supp(X \implies Y)supp(\neg X \implies \neg Y) - supp(X \implies \neg Y)supp(\neg X \implies Y)} {supp(X \implies Y)supp(\neg X \implies \neg Y) + supp(X \implies \neg Y)supp(\neg X \implies Y)}$$

Range: $[-1, 1]$ (-1 reflects total negative association, 1 reflects perfect positive association and 0 reflects independence)

Reference: E. V. Altay and B. Alatas, "Sensitivity Analysis of MODENAR Method for Mining of Numeric Association Rules," 2019 1st International Informatics and Software Engineering Conference (UBMYK), 2019, pp. 1-6, doi: 10.1109/UBMYK48245.2019.8965539.

Zhang's Metric

Zheng's metric measures the strength of association (positive or negative) between the antecedent and consequent, taking into account both their co-occurrence and non-co-occurrence.

$$zhang(X \implies Y) = \frac{conf(X \implies Y) - conf(\neg X \implies Y)}{max\{conf(X \implies Y), conf(\neg X \implies Y)\}}$$

Range: $[-1, 1]$ (-1 reflects total negative association, 1 reflects perfect positive association and 0 reflects independence)

Reference: T. Zhang, “Association Rules,” in Knowledge Discovery and Data Mining. Current Issues and New Applications, 2000, pp. 245–256. doi: 10.1007/3-540-45571-X_31.

Leverage

Leverage metric is difference between the frequency of antecedent and the consequent appearing together and the expected frequency of them appearing separately based on their individual support

$$leverage(X \implies Y) = support(X \implies Y) - (support(X) \times support(Y))$$

Range: [-1, 1] (-1 reflects total negative association, 1 reflects perfect positive association and 0 reflects independence)

Reference: Gregory Piatetsky-Shapiro. 1991. Discovery, Analysis, and Presentation of Strong Rules. In Knowledge Discovery in Databases, Gregory Piatetsky-Shapiro and William J. Frawley (Eds.). AAAI/MIT Press, 229–248.