Provides the generic function interestMeasure and the needed S4 method
to calculate various additional interest measures for existing sets of
itemsets or rules. A searchable list of definitions, equations and references for all available interest measures can be found here:
https://mhahsler.github.io/arules/docs/measures
interestMeasure(x, measure, transactions = NULL, reuse = TRUE, ...)a set of itemsets or rules.
name or vector of names of the desired interest measures (see details for available measures). If measure is missing then all available measures are calculated.
the transaction data set used to mine
the associations or a set of different transactions to calculate
interest measures from (Note: you need to set reuse=FALSE in the
later case).
logical indicating if information in quality slot should
be reuse for calculating the measures. This speeds up the process
significantly since only very little (or no) transaction counting
is necessary if support, confidence and lift are already available.
Use reuse=FALSE to force counting (might be very slow but
is necessary if you use a different set of transactions than was used
for mining).
further arguments for the measure calculation. Many measures are based on contingency table counts
and zero counts can produce NaN values (division by zero). This issue can be resolved by using the
additional parameter smoothCounts which performs additive smoothing by adds a "pseudo count" of
smoothCounts to each count in the contingency table. Use smoothCounts = 1 or larger values for
Laplace smoothing. Use smoothCounts = .5 for Haldane-Anscombe correction often used for chi-squared,
phi correlation and related measures.
If only one measure is used, the function returns a numeric vector
containing the values of the interest measure for each association
in the set of associations x.
If more than one measures are specified, the result is a data.frame containing the different measures for each association as columns.
NA is returned for rules/itemsets for which a certain measure is not
defined.
The following measures are implemented for itemsets \(X\):
Is defined on itemsets as the minimum confidence of all possible rule generated from the itemset.
See details: https://mhahsler.github.io/arules/docs/measures#all-confidence
Range: \([0, 1]\)
Defined on itemsets as the ratio of the support of the least frequent item to the support of the most frequent item. Cross-support patterns have a ratio smaller than a set threshold. Normally many found patterns are cross-support patterns which contain frequent as well as rare items. Such patterns often tend to be spurious.
See details: https://mhahsler.github.io/arules/docs/measures#cross-support-ratio
Range: \([0, 1]\)
Lift is typically only defined for rules. In a similar way, we use the probability (support) of the itemset over the product of the probabilities of all items in the itemset, i.e., \(\frac{supp(X)}{\prod_{x \in X} supp(X)}\).
Range: \([0, \infty]\) (1 indicated independence)
Support is an estimate of \(P(X)\), a measure of generality of the itemset. It is estimated by the number of transactions that contain the itemset over the total number of transactions in the data set.
See details: https://mhahsler.github.io/arules/docs/measures#support
Range: \([0, 1]\)
Absolute support count of the itemset, i.e., the number of transactions that contain the itemset.
See details: https://mhahsler.github.io/arules/docs/measures#support
Range: \([0, \infty]\)
The following measures are implemented for rules of the form \(X \Rightarrow Y\):
Defined as the rule confidence minus the rule's support.
See details: https://mhahsler.github.io/arules/docs/measures#added-value
Range: \([-.5, 1]\)
Confidence boost is the ratio of the confidence of a rule to the confidence of any more general rule (i.e., a rule with the same consequent but one or more items removed in the LHS). Values larger than 1 mean the new rule boosts the confidence compared to the best, more general rule. The measure is related to the improvement measure.
See details: https://mhahsler.github.io/arules/docs/measures#confidence-boost
Range: \([0, \infty]\)
The chi-squared statistic to test for independence between the lhs and rhs of the rule. The critical value of the chi-squared distribution with \(1\) degree of freedom (2x2 contingency table) at \(\alpha=0.05\) is \(3.84\); higher chi-squared values indicate that the lhs and the rhs are not independent.
See details: https://mhahsler.github.io/arules/docs/measures#chi-squared
Note that the contingency table is likely to have cells with low expected values and that thus Fisher's Exact Test might be more appropriate (see below).
Called with significance = TRUE, the p-value of the test for
independence is returned instead of the chi-squared statistic.
For p-values, substitution effects (the ocurrence of one item makes the ocurrance of another item less likely) can be tested using
the parameter complements = FALSE.
Correction for multiple comparisons can be done using p.adjust.
Range: \([0, \infty]\) or p-value scale
The certainty factor is a measure of variation of the probability that Y is in a transaction when only considering transactions with X. An increasing CF means a decrease of the probability that Y is not in a transaction that X is in. Negative CFs have a similar interpretation.
See details: https://mhahsler.github.io/arules/docs/measures#certainty-factor
Range: \([-1, 1]\) (0 indicates independence)
Collective strength gives 0 for perfectly negative correlated items, infinity for perfectly positive correlated items, and 1 if the items co-occur as expected under independence.
See details: https://mhahsler.github.io/arules/docs/measures#collective-strength
Range: \([0, \infty]\)
Confidence is a measure of rule validity. Rule confidence is an estimate of \(P(Y|X)\).
See details: https://mhahsler.github.io/arules/docs/measures#confidence
Range: \([0, 1]\)
Conviction was developed as an alternative to lift that also incorporates the direction of the rule.
See details: https://mhahsler.github.io/arules/docs/measures#conviction
Range: \([0, \infty]\) (\(1\) indicates unrelated items)
A measure if correlation between the items in X and Y.
See details: https://mhahsler.github.io/arules/docs/measures#cosine
Range: \([0, 1]\)(\(.5\) indicates no correlation)
Absolute support count of the rule, i.e., the number of transactions that contain all items in the rule.
See details: https://mhahsler.github.io/arules/docs/measures#support
Range: \([0, \infty]\)
It measures the probability that a rule applies to a randomly selected transaction. It is estimated by the proportion of transactions that contain the antecedent (LHS) of the rule. Therefore, coverage is sometimes called antecedent support or LHS support.
See details: https://mhahsler.github.io/arules/docs/measures#coverage
Range: \([0, 1]\)
How much higher is the confidence of a rule compared to the confidence of the rule \(X \Rightarrow \overline{Y}\).
See details: https://mhahsler.github.io/arules/docs/measures#descriptive-confirmed-confidence
Range: \([-1, 1]\)
Confidence reinforced by the confidence of the rule \(\overline{X} \Rightarrow \overline{Y}\).
See details: https://mhahsler.github.io/arules/docs/measures#casual-confidence
Range: \([0, 1]\)
Support reinforced by the support of the rule \(\overline{X} \Rightarrow \overline{Y}\).
See details: https://mhahsler.github.io/arules/docs/measures#casual-support Range: \([-1, 1]\)
Rate of the examples minus the rate of counter examples (i.e., \(X \Rightarrow \overline{Y}\)).
See details: https://mhahsler.github.io/arules/docs/measures#example-and-counter-example-rate
Range: \([0, 1]\)
Defined as the difference in confidence of the rule and the rule \(\overline{X} \Rightarrow Y\)
See details: https://mhahsler.github.io/arules/docs/measures#difference-of-confidence Range: \([-1, 1]\)
p-value of Fisher's exact test used in the analysis of contingency tables
where sample sizes are small.
By default complementary effects are mined, substitutes can be found
by using the parameter complements = FALSE.
See details: https://mhahsler.github.io/arules/docs/measures#fishers-exact-test
Note that it is equal to hyper-confidence with significance=TRUE.
Correction for multiple comparisons can be done using p.adjust.
Range: \([0, 1]\) (p-value scale)
Measures quadratic entropy of a rule.
See details: https://mhahsler.github.io/arules/docs/measures#gini-index
Range: \([0, 1]\) (0 means the rule provides no information for the data set)
Confidence level that the observed co-occurrence count of the LHS and RHS is too high given the expected count using the hypergeometric model.
See details: https://mhahsler.github.io/arules/docs/measures#hyper-confidence
Hyper-confidence reports the confidence level by default and the
significance level if significance=TRUE is used.
By default complementary effects are mined, substitutes (too low co-occurrence counts) can be found
by using the parameter complements = FALSE.
Range: \([0, 1]\)
Adaptation of the lift measure which evaluates the deviation from independence using a quantile of the hypergeometric distribution defined by the counts of the LHS and RHS. HyperLift can be used to calculate confidence intervals for the lift measure.
The used quantile can be given
as parameter level (default: level = 0.99).
See details: https://mhahsler.github.io/arules/docs/measures#hyper-lift
Range: \([0, \infty]\) (1 indicates independence)
IR measures the degree of imbalance between the two events that the lhs and the rhs are contained in a transaction. The ratio is close to 0 if the conditional probabilities are similar (i.e., very balanced) and close to 1 if they are very different. See also: https://mhahsler.github.io/arules/docs/measures#imbalance-ratio
Range: \([0, 1]\) (0 indicates a balanced rule)
A variation of the Lerman similarity.
See details: https://mhahsler.github.io/arules/docs/measures#implication-index
Range: \([0, 1]\) (0 means independence)
Log likelihood of the right-hand side of the rule, given the left-hand side of the rule using Laplace corrected confidence.
See details: https://mhahsler.github.io/arules/docs/measures#importance
Range: \([-Inf, Inf]\)
The improvement of a rule is the minimum difference between its confidence and the confidence of any more general rule (i.e., a rule with the same consequent but one or more items removed in the LHS).
Special case: We define improvement for a rules with an empty LHS as its confidence.
The idea of improvement can be generalized to other measures than confidence. Other measures like
lift can be specified with the extra parameter improvementMeasure.
See details: https://mhahsler.github.io/arules/docs/measures#improvement
Range: \([0, 1]\)
Null-invariant measure of dependence defined as the Jaccard similarity between the two sets of transactions that contain the items in X and Y, respectively.
See details: https://mhahsler.github.io/arules/docs/measures#jaccard-coefficient
Range: \([0, 1]\)
A scaled measures of cross entropy to measure the information content of a rule.
See details: https://mhahsler.github.io/arules/docs/measures#j-measure
Range: \([0, 1]\) (0 indicates X does not provide information for Y)
Cohen's Kappa of the rule (seen as a classifier) given as the rule's observed rule accuracy (i.e., confidence) corrected by the expected accuracy (of a random classifier).
See details: https://mhahsler.github.io/arules/docs/measures#kappa
Range: \([-1,1]\) (0 means the rule is not better than a random classifier)
Defined as \(\sqrt{supp(X \cup Y)} conf(X \Rightarrow Y) - supp(Y)\)
See details: https://mhahsler.github.io/arules/docs/measures#klosgen
Range: \([-1, 1]\) (0 for independence)
Calculate the null-invariant Kulczynski measure with a preference for skewed patterns.
See details: https://mhahsler.github.io/arules/docs/measures#kulczynski
Range: \([0, 1]\)
Goodman and Kruskal's lambda to assess the association between the LHS and RHS of the rule.
See details: https://mhahsler.github.io/arules/docs/measures#lambda
Range: \([0, 1]\)
Estimates confidence by increasing each count by 1. Parameter k can be used
to specify the number of classes (default is 2).
Prevents counts of 0 and L decreases with lower support.
See details: https://mhahsler.github.io/arules/docs/measures#laplace-corrected-confidence
Range: \([0, 1]\)
Probability of finding a matching transaction minus the probability of finding a contradicting transaction normalized by the probability of finding a transaction containing Y.
See details: https://mhahsler.github.io/arules/docs/measures#least-contradiction
Range: \([-1, 1]\)
Defined as \(\sqrt{N} \frac{supp(X \cup Y) - supp(X)supp(Y)}{\sqrt{supp(X)supp(Y)}}\)
See details: https://mhahsler.github.io/arules/docs/measures#lerman-similarity
Range: \([0, 1]\)
PS measures the difference of X and Y appearing together in the data set and what would be expected if X and Y where statistically dependent. It can be interpreted as the gap to independence.
See details: https://mhahsler.github.io/arules/docs/measures#leverage
Range: \([-1, 1]\) (0 indicates independence)
Lift quantifies dependence between X and Y by comparing the probability that X and Y are contained in a transaction to the expected probability under independence (i.e., the product of the probabilities that X is contained in a transaction times the probability that Y is contained in a transaction).
See details: https://mhahsler.github.io/arules/docs/measures#lift
Range: \([0, \infty]\) (1 means independence between LHS and RHS)
Null-invariant symmetric measure defined as the larger of the confidence of the rule or the rule with X and Y exchanged.
See details: https://mhahsler.github.io/arules/docs/measures#maxconfidence Range: \([0, 1]\)
Measures the information gain for Y provided by X.
See details: https://mhahsler.github.io/arules/docs/measures#mutual-information
Range: \([0, 1]\) (0 means that X does not provide information for Y)
The odds of finding X in transactions which contain Y divided by the odds of finding X in transactions which do not contain Y. For zero counts, Haldane-Anscombe correction (adding .5 to all zells) is applied.
See details: https://mhahsler.github.io/arules/docs/measures#odds_ratio
Range: \([0, \infty]\) (\(1\) indicates that Y is not associated to X)
Calculates the lower and upper bounds of the confidence interval around the odds ratio (using a normal approximation).
The used confidence level defaults to 0.95, but can be adjusted with the additional
parameter confidenceLevel.
See details: https://mhahsler.github.io/arules/docs/measures#odds-ratio
Range: \([0, \infty]\)
Correlation coefficient between the transactions containing X and Y represented as two binary vectors. Phi correlation is equivalent to Pearson's Product Moment Correlation Coefficient \(\rho\) with 0-1 values.
See details: https://mhahsler.github.io/arules/docs/measures#phi-correlation-coefficient Range: \([-1, 1]\) (0 when X and Y are independent)
The measure is defined as the probability that a transaction contains X but not Y. A smaller value is better.
See details: https://mhahsler.github.io/arules/docs/measures#ralambondrainy
Range: \([0, 1]\)
Range: \([0, 1]\)
RLD is an association measure motivated by indices used in population genetics. It evaluates the deviation of the support of the whole rule from the support expected under independence given the supports of the LHS and the RHS.
See details: https://mhahsler.github.io/arules/docs/measures#relative-linkage-disequilibrium
The code was contributed by Silvia Salini.
Range: \([0, 1]\)
Product of support and confidence. Can be seen as rule confidence weighted by support.
See details: https://mhahsler.github.io/arules/docs/measures#rule-power-factor
Range: \([0, 1]\)
Confidence of a rule divided by the confidence of the rule \(X \Rightarrow \overline{Y}\).
See details: https://mhahsler.github.io/arules/docs/measures#sebag-schoenauer
Range: \([0, 1]\)
Standardized lift uses the minimum and maximum lift can reach for each rule to standardize lift between 0 and 1. By default, the measure is corrected for minimum support and minimum confidence. Correction can be disabled by using the argument correct = FALSE.
See details: https://mhahsler.github.io/arules/docs/measures#standardized-lift
Range: \([0, 1]\)
Support is an estimate of \(P(X \cup Y)\) and measures the generality of the rule.
See details: https://mhahsler.github.io/arules/docs/measures#support
Range: \([0, 1]\)
Returns the counts for the contingency table. The values are labeled \(n_{XY}\)
where \(X\) and \(Y\) represent the presence (1) or absence (0) of the LHS and RHS of the rule, respectively.
If several measures are specified, then the counts have the prefix table.
Range: counts
Defined as the lift of a rule minus 1 so 0 represents independence.
See details: https://mhahsler.github.io/arules/docs/measures#Varying-Rates-Liaison
Range: \([-1, \infty]\) (0 for independence)
Defined as \(\frac{\alpha-1}{\alpha+1}\) where \(\alpha\) is the odds ratio.
See details: https://mhahsler.github.io/arules/docs/measures#yules-q-and-yules-y
Range: \([-1, 1]\)
Defined as \(\frac{\sqrt{\alpha}-1}{\sqrt{\alpha}+1}\) where \(\alpha\) is the odds ratio.
See details: https://mhahsler.github.io/arules/docs/measures#yules-q-and-yules-y
Range: \([-1, 1]\)
A complete list of references for each individual measure is available in the following document:
Hahsler, Michael (2015). A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules, 2015, URL: https://mhahsler.github.io/arules/docs/measures.
# NOT RUN {
data("Income")
rules <- apriori(Income)
## calculate a single measure and add it to the quality slot
quality(rules) <- cbind(quality(rules),
hyperConfidence = interestMeasure(rules, measure = "hyperConfidence",
transactions = Income))
inspect(head(rules, by = "hyperConfidence"))
## calculate several measures
m <- interestMeasure(rules, c("confidence", "oddsRatio", "leverage"),
transactions = Income)
inspect(head(rules))
head(m)
## calculate all available measures for the first 5 rules and show them as a
## table with the measures as rows
t(interestMeasure(head(rules, 5), transactions = Income))
## calculate measures on a different set of transactions (I use a sample here)
## Note: reuse = TRUE (default) would just return the stored support on the
## data set used for mining
newTrans <- sample(Income, 100)
m2 <- interestMeasure(rules, "support", transactions = newTrans, reuse = FALSE)
head(m2)
## calculate all available measures for the 5 frequent itemsets with highest support
its <- apriori(Income, parameter = list(target = "frequent itemsets"))
its <- head(its, 5, by = "support")
inspect(its)
interestMeasure(its, transactions = Income)
# }
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