Provides the generic function interestMeasure and the needed S4 method
to calculate various additional interest measures for existing sets of
itemsets or rules. Definitions and equations can be found in
Hahsler (2015).
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 differnt set of transactions than was used
for mining).
further arguments for the measure calculation.
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.
NA is returned for rules/itemsets for which a certain measure is not
defined.
For itemsets \(X\) the following measures are implemented:
Is defined on itemsets as the minimum confidence of all possible rule generated from the itemset.
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, i.e., \(\frac{min(supp(x \in X))}{max(supp(x \in X))}\). 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.
Range: \([0, 1]\)
Probability (support) of the itemset over the product of the probabilities of all items in the itemset, i.e., \(\frac{supp(X)}{\prod_{x \in X} supp(X)}\). This is a measure of dependence similar to lift for rules.
Range: \([0, \infty]\) (1 indicated independence)
Support is an estimate of \(P(X)\) a measure of generality of the itemset.
Range: \([0, 1]\)
For rules \(X \Rightarrow Y\) the following measures are implemented. In the following we use the notation \(supp(X \Rightarrow Y) = supp(X \cup Y)\) to indicate the support of the union of the itemsets \(X\) and \(Y\), i.e., the proportion of the transactions that contain both itemsets. We also use \(\overline{X}\) as the complement itemset to \(X\) with \(supp(\overline{X}) = 1 - supp(X)\), i.e., the proportion of transactions that do not contain \(X\).
Defined as \(conf(X \Rightarrow Y) - supp(Y)\)
Range: \([-.5, 1]\)
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. 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, substitutes effects can be tested using
the parameter complements = FALSE.
Range: \([0, \infty]\)
The certainty factor is a measure of variation of the probability that Y is in a transaction when only considering transactions with X. An inreasing CF means a decrease of the probability that Y is not in a transaction that X is in. Negative CFs have a similar interpretation.
Range: \([-1, 1]\) (0 indicates independence)
Collective strength (S).
Range: \([0, \infty]\)
Rule confidence is an estimate of \(P(Y|X)\) calculated as \(\frac{supp(X \Rightarrow Y)}{supp(X)}\). Confidence is a measure of validity.
Range \([0, 1]\)
Defined as \(\frac{supp(X)supp(\overline{Y})}{supp(X \cup \overline{Y})}\).
Range: \([0, \infty]\) (\(1\) indicates unrelated items)
Defined as \(\frac{supp(X \cup Y)}{\sqrt{(supp(X)supp(Y))}}\)
Range: \([0, 1]\)
Support of the left-hand-side of the rule, i.e., \(supp(X)\). A measure of to how often the rule can be applied.
Range: \([0, 1]\)
Confidence confirmed by its negative as \(conf(X \Rightarrow Y) - conf(X \Rightarrow \overline{Y})\).
Range: \([-1, 1]\)
Confidence reinforced by negatives given by \(\frac{1}{2}(conf(X \Rightarrow Y) + conf(\overline{Y} \Rightarrow \overline{X}))\).
Range: \([0, 1]\)
Support improved by negatives given by \(supp(X \cup Y) - supp(\overline{X} \cup \overline{Y})\).
Range: \([-1, 1]\)
\(\frac{supp(X \cup Y) - supp(X \cup \overline{Y})}{supp(X \cup Y)}\)
Range: \([0, 1]\)
Defined by \(supp(X \cup Y) - supp(X \cup \overline{Y})\).
Range: \([0, 1]\)
Defined as \(conf(X \Rightarrow Y) - conf(\overline{X} \Rightarrow Y)\).
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.
Note that it is equal to hyper-confidence with significance=TRUE.
Range: \([0, 1]\) (p-value scale)
Measures quadratic entropy.
Range: \([0, 1]\) (0 for independence)
Adaptation of the lift measure which is more robust for low counts. It is based on the idea that under independence the count \(c_{XY}\) of the transactions which contain all items in a rule \(X \Rightarrow Y\) follows a hypergeometric distribution (represented by the random variable \(C_{XY}\)) with the parameters given by the counts \(c_X\) and \(c_Y\).
Hyper-lift is defined as: $$\mathrm{hyperlift}(X \Rightarrow Y) = \frac{c_{XY}}{Q_{\delta}[C_{XY}]},$$
where \(Q_{\delta}[C_{XY}]\) is the
quantile of the hypergeometric distribution given by \(\delta\).
The quantile can be given
as parameter d (default: d=0.99).
Range: \([0, \infty]\) (1 indicates independence)
Confidence level for observation of too high/low counts
for rules \(X \Rightarrow Y\) using the hypergeometric model.
Since the counts are drawn from a hypergeometric distribution
(represented by the random variable \(C_{XY}\)) with
known parameters given by the counts \(c_X\) and \(c_Y\),
we can calculate a confidence interval for the observed counts
\(c_{XY}\) stemming from the distribution. Hyper-confidence
reports the confidence level
(significance level if significance=TRUE is used) for
\(1 - P[C_{XY} >= c_{XY} | c_X, c_Y]\)
\(1 - P[C_{XY} < c_{XY} | c_X, c_Y]\).
By default complementary effects are mined, substitutes can be found
by using the parameter complements = FALSE.
Range: \([0, 1]\)
IR is defined as \(\frac{|supp(X) - supp(Y)|}{supp(X) + supp(Y) - supp(X \Rightarrow Y))}\) gauges the degree of imbalance between 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.
Range: \([0, 1]\) (0 indicates a balanced rule)
Defined as \(\sqrt{N} \frac{supp(X \cup \overline{Y}) - supp(X)supp(\overline{Y})}{\sqrt{supp(X)supp(\overline{Y})}}\). Represents a variation of the Lerman similarity.
Range: \([0, 1]\) (0 means independence)
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). Defined as \(min_{X' \subset X}(conf(X \Rightarrow Y) - conf(X' \Rightarrow Y)\)
Range: \([0, 1]\)
Null-invariant measure defined as \(\frac{supp(X \cup Y)}{supp(X) + supp(Y) - supp(X \cup Y)}\)
Range: \([-1, 1]\) (0 for independence)
Measures cross entrophy.
Range: \([0, 1]\) (0 for independence)
Defined as \(\frac{supp(X \cup Y) + supp(\overline{X} \cup \overline{Y}) - supp(X)supp(Y) - supp(\overline{X})supp(\overline{Y})}{1- supp(X)supp(Y) - supp(\overline{X})supp(\overline{Y})}\)
Range: \([-1,1]\) (0 means independence)
Defined as \(\sqrt{supp(X \cup Y)} conf(X \Rightarrow Y) - supp(Y)\)
Range: \([-1, 1]\) (0 for independence)
Calculate the null-invariant Kulczynski measure with a preference for skewed patterns.
Range: \([0, 1]\)
Range: \([0, 1]\)
Estimates confidence (decreases with lower support).
Range: \([0, 1]\)
\(\frac{supp(X \cup Y) - supp(X \cup \overline{Y})}{supp(Y)}\).
Range: \([-1, 1]\)
Defined as \(\sqrt{N} \frac{supp(X \cup Y) - supp(X)supp(Y)}{\sqrt{supp(X)supp(Y)}}\)
Range: \([0, 1]\)
PS is defined as \(supp(X \Rightarrow Y) - supp(X)supp(Y)\). It 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.
Range: \([-1, 1]\) (0 indicates intependence)
Lift quantifies dependence between X and Y by \(\frac{supp(X \cup Y)}{supp(X)supp(Y)}\).
Range: \([0, \infty]\) (1 means independence)
Null-invariant measure defined as \(max(conf(X \Rightarrow Y), conf(X \Rightarrow Y))\).
Range: \([0, 1]\)
Measures the information gain for Y provided by X.
Range: \([0, 1]\) (0 for independence)
The odds of finding X in transactions which contain Y divided by the odds of finding X in transactions which do not contain Y.
Range: \([0, \infty]\) (\(1\) indicates that Y is not associated to X)
Equivalent to Pearsons Product Moment Correlation Coefficient \(\rho\).
Range: \([-1, 1]\) (0 when X and Y are independent)
Range: \([0, 1]\)
RLD 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. The code was contributed by Silvia Salini.
Range: \([0, 1]\)
Product of support and confidence. Can be seen as rule confidence weighted by support.
Range: \([0, 1]\)
Defined as \(\frac{supp(X \cup Y)}{supp(X \cup \overline{Y})}\)
Range: \([0, 1]\)
Support is an estimate of \(P(X \cup Y)\) and measures the generality of the rule.
Range: \([0, 1]\)
Defined as \(\frac{supp(X \cup Y)}{supp(X)supp(Y)}-1\). Is equivalent to \(lift(X \Rightarrow Y) -1\)
Range: \([-1, 1]\) (0 for independence)
Defined as \(\frac{\alpha-1}{\alpha+1}\) where \(\alpha\) is the odds ratio.
Range: \([-1, 1]\)
Defined as \(\frac{\sqrt{\alpha}-1}{\sqrt{\alpha}+1}\) where \(\alpha\) is the odds ratio.
Range: \([-1, 1]\)
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# 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 differnt set of transactions (I use a sample here)
## Note: reuse = TRUE (default) would just retun the stored support on the
## data set used for mining
newTrans <- sample(Income, 100)
m2 <- interestMeasure(rules, "support", transactions = newTrans, reuse = FALSE)
head(m2)
# }
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