Association rule induction [Agrawal et al. 1993] is a powerful method for so-called market basket analysis, which aims at finding regularities in the shopping behavior of customers of supermarkets, mail-order companies and the like. With the induction of association rules one tries to find sets of products that are frequently bought together, so that from the presence of certain products in a shopping cart one can infer (with a high probability) that certain other products are present. Such information, expressed in the form of rules, can often be used to increase the number of items sold, for instance, by appropriately arranging the products in the shelves of a supermarket (they may, for example, be placed adjacent to each other in order to invite even more customers to buy them together) or by directly suggesting items to a customer, which may be of interest for him/her.
An association rule is a rule like "If a customer buys wine and bread, he often buys cheese, too." It expresses an association between (sets of) items, which may be products of a supermarket or a mail-order company, special equipment options of a car, optional services offered by telecommunication companies etc. An association rule states that if we pick a customer at random and find out that he selected certain items (bought certain products, chose certain options etc.), we can be confident, quantified by a percentage, that he also selected certain other items (bought certain other products, chose certain other options etc.).
Of course, we do not want just any association rules, we want "good" rules, rules that are "expressive" and "reliable". The standard measures to assess association rules are the support and the confidence of a rule, both of which are computed from the support of certain item sets. These notions are discussed here in more detail. However, these standard criteria are often not sufficient to restrict the set of rules to the interesting ones. Therefore some additional rule evaluation measures are considered here.
The main problem of association rule induction is that there are so many possible rules. For example, for the product range of a supermarket, which may consist of several thousand different products, there are billions of possible association rules. It is obvious that such a vast amount of rules cannot be processed by inspecting each one in turn. Therefore efficient algorithms are needed that restrict the search space and check only a subset of all rules, but, if possible, without missing important rules. One such algorithm is the apriori algorithm, which was developed by [Agrawal et al. 1994] and which is implemented in a specific way in my apriori program. A brief description of some implementation aspects can be found in these papers:
By the way: Earlier versions of my apriori program are incorporated in the well-known data mining tool Clementine (apriori version 1.8 in Clementine version 5.0, apriori version 2.7 in Clementine version 7.0), available from SPSS. Newer versions of Clementine still use my program, but I am not completely sure about the version number of the underlying apriori program.
Enjoy,
Christian Borgelt
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Let T be the set of all transactions under consideration, e.g., let T be the set of all "baskets" or "carts" of products bought by the customers of a supermarket - on a given day if you like. The support of an item set S is the percentage of those transactions in T which contain S. In the supermarket example this is the number of "baskets" that contain a given set S of products, for example S = { bread, wine, cheese }. If U is the set of all transactions that contain all items in S, then
support(S) = (|U| / |T|) *100%,
where |U| and |T| are the number of elements in U and T, respectively. For example, if a customer buys the set X = { milk, bread, apples, wine, sausages, cheese, onions, potatoes } then S is obviously a subset of X, hence S is in U. If there are 318 customers and 242 of them buy such a set U or a similar one that contains S, then support(S) = 76.1%.
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This is the measure used by [Agrawal et al. 1993], the inventors of the apriori algorithm, to evaluate association rules. The confidence of a rule R = "A and B -> C" is the support of the set of all items that appear in the rule divided by the support of the antecedent of the rule, i.e.
confidence(R) = (support({A, B, C}) / support({A, B})) *100%.
More intuitively, the confidence of a rule is the number of cases in which the rule is correct relative to the number of cases in which it is applicable. For example, let R = "wine and bread -> cheese". If a customer buys wine and bread, then the rule is applicable and it says that he/she can be expected to buy cheese. If he/she does not buy wine or does not buy bread or buys neither, than the rule is not applicable and thus (obviously) does not say anything about this customer.
If the rule is applicable, it says that the customer can be expected to buy cheese. But he/she may or may not buy cheese, that is, the rule may or may not be correct. Of course, we are interested in how good the rule is, i.e., how often its prediction that the customer buys cheese is correct. The rule confidence measures this: It states the percentage of cases in which the rule is correct. It computes the percentage relative to the number of cases in which the antecedent holds, since these are the cases in which the rule makes a prediction that can be true or false. If the antecedent does not hold, then the rule does not make a prediction, so these cases are excluded.
With this measure a rule is selected if its confidence exceeds or is equal to a given lower limit. That is, we look for rules that have a high probability of being true, i.e., we look for "good" rules, which make correct (or very often correct) predictions. My apriori program always uses this measure to select association rules. The default value for the confidence limit is 80%. It can be changed with the option -c.
In addition to the rule confidence my apriori program lets you select several other rule evaluation measures, which are explained below, but it will also use rule confidence. If you want to rely entirely on some other measure, you can do so by setting the minimal rule confidence to zero. (Attention: If you have a large number of items, setting the minimal rule confidence to zero can result in very high memory consumption.)
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The support of rules may cause some confusion, because I use this term in a different way than [Agrawal et al. 1993] do. For them, the support of a rule "A and B -> C" is the support of the set {A, B, C}. This is fine if rule confidence is the only rule evaluation measure, but it causes problems if some other measure is used. For these other measures it is often much more appropriate to call the support of the antecedent of the rule, i.e. the support of {A, B} in the example above, the support of the rule.
The difference can also be stated in the following way: For [Agrawal et al. 1993], the support of the rule is the (relative) number of cases in which the rule is correct (i.e., in which the presence of the item C follows from the presence of the items A and B), whereas for me (and thus my apriori program) the support of a rule is the (relative) number of cases in which it is applicable (i.e., in which the antecedent of the rule holds), although in some of these cases it may be false (because only the items A and B are present, but the item C is missing).
One reason for this, as already mentioned, is that the definition of [Agrawal et al. 1993] does not work well for evaluation measures other than rule confidence. This is explained in more detail below. Another reason is that I prefer the support of a rule to say something about the "statistical" support of a rule and its confidence, i.e., from how many cases the confidence is computed in order to express how well founded the assertion about the confidence is.
Maybe an example will make this clearer. Suppose you have a die which you suspect to be biased. To test this hypothesis, you throw the die, say, a thousand times. 307 times the 6 turns up. Hence you assume that the die is actually biased, since the relative frequency is about 30% although for an unbiased die it should be around 17%. Now, what is the "statistical" support of this assertion, i.e., on how many experiments does it rest? Obviously it rests on all 1000 experiments and not only on the 307 experiments in which the 6 turned up. This is so, simply because you had to do 1000 experiments to find out that the relative frequency is around 30% and not only the 307 in which a 6 turned up.
Or suppose you are doing an opinion poll to find out about the acceptance of a certain political party, maybe with the usual question "If an election were held next Sunday ...?" You ask 2000 persons, of which 857 say that they would vote for the party you are interested in. What is the support of the assertion that this party would get around 43% of all votes? It is the size of your sample, i.e., all 2000 persons, and not only the 857 that answered in the positive. Again you had to ask all 2000 people to find out about the percentage of 43%. Of course, you could have asked fewer people, say, 100, of which, say, 43 said that they would vote for the party, but then your assertion would be less reliable, because it is less "supported". The number of votes for the party could also be 40% or 50%, because of some random influences. Such deviations are much less likely, if you asked 2000 persons, since then the random influences can be expected to cancel out.
The rule support can be used to select association rules by stating a lower bound for the support of a rule. This is equivalent to saying that you are interested only in such rules that have a large enough statistical basis (since my apriori program uses the term "support" in my interpretation and not in the one used by [Agrawal et al. 1993]). The default value for the support limit is 10%. It can be changed with the option -s. If the number given is negative, it is interpreted as an absolute number (number of transactions) rather than a percentage. (Note that in this case the option -a reverses its meaning: if it is not given only the absolute support is printed, if it is added, the relative supoort is printed, too.) The lower bound for the rule support is combined with the lower bound for the rule confidence, i.e., my apriori program generates only rules the confidence of which is greater than or equal to the confidence limit and the support of which is greater than or equal to the support limit.
Despite the above arguments in favor of my definition of the support of an association rule, a rule support compatibility mode is available. With the option -o the original rule support definition can be selected. In this case the support of an association rule is the support of the set with the items in the antecedent and the consequent of the rule, i.e. the support of a rule as defined in [Agrawal et al. 1993].
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The target type, which can be selected via the option -t, is either association rules (option -tr, default), frequent item sets (option -ts), closed item sets (option -tc), maximal item sets (option -tm), or association hyperedges (option -th).
By default my apriori program produces association rules with a single item in the consequent. The restriction to single item consequents is due to the following considerations: In the first place, association rule mining usually produces too many rules even if one confines oneself to rules with only one item in the consequent. So why should one make the situation worse by allowing more than one item in the consequent? (It merely blows up the output size.)
Secondly, I do not know any application in which rules with more than one item in the consequent are of any real use. The reason, in my opinion, is that such more complex rules add almost nothing to the insights about the data set. To understand this, consider the simpler rules that correspond to a rule with multiple items in the consequent, that is, rules having the same antecedent and consequents with only single items from the consequent of the complex rule. All of these rules must necessarily be in the output, because neither their support nor their confidence can be less than that of the more complex rule. That is, if you have a rule c d <- a b, you will necessarily also have the rules c <- a b and d <- a b in the output. Of course, these latter two rules together do not say the same as the more complex rule. However, what do you gain from the additional information the more complex rule gives you? How can you use it? And is this little extra information worth having to analyze a much bigger rule set?
Sometimes one may not want to find association rules, but only the frequent item sets underlying them. That is, one wants to find all item sets with a support exceeding a certain threshold. My apriori program supports this search, too: If the option -ts is given, only frequent item sets are determined.
A frequent item set is called closed if no superset has the same support. If the option -tc is given, the found frequent item sets are subsequently filtered and only the closed item sets are kept.
A frequent item set is called maximal if no superset is frequent, i.e., has a support exceeding the minimal support. If the option -tm is given, the found frequent item sets are subsequently filtered and only the maximal item sets are kept.
My apriori program can also find association hyperedges, i.e., sets of items that are strongly predictive w.r.t. each other. In this mode no rules are generated, only item sets are selected. The selection criterion is as follows: Given an item set with enough support (option -s), all rules are checked which can be formed using this set with all items appearing in the rule. For example, for the item set {a b c}, the rules c <- a b, b <- a c, a <- b c would be considered. The confidences of these rules are computed and averaged. If the resulting average confidence is greater than the minimal confidence (option -c), the item set is selected. (I am grateful to Bastien Duclaux for requesting the possibility to generate association hyperedges.)
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If rules are selected using the rule confidence, the following problem arises: "Good" rules (rules that are often true) are not always "interesting" rules (rules that reveal something about the interdependence of the items). You certainly know the examples that are usually given to illustrate this fact. For instance, it is easy to find out in a medical database that the rule "pregnant -> female" is true with a confidence of 100%. Hence it is a perfect rule, it never fails, but, of course, this is not very surprising. Although the measures explained below cannot deal with this problem (which is semantical), they may be able to improve on the results in a related case.
Let us look at the supermarket example again and let us assume that 60% of all customers buy some kind of bread. Consider the rule "cheese -> bread", which holds with a confidence of, say, 62%. Is this an important rule? Obviously not, since the fact that the customer buys cheese does not have a significant influence on him/her buying bread: The percentages are almost the same. But if you had set a confidence limit of 60%, you would get both rules "-> bread" (confidence 60%) and "cheese -> bread" (confidence 62%), although the first would suffice (the first, since it is the simpler of the two). The idea of all measures that can be used in addition or instead of rule confidence is to handle such situations and to suppress the second rule.
In addition, consider the following case: Assume that the confidence of the rule "cheese -> bread" is not 62% but 35%. With a confidence limit of 60% it would not be selected, but it may be very important to know about this rule! Together with cheese bread is bought much less frequent than it is bought at all. Is cheese some kind of substitute for bread, so that one does not need any bread, if one has cheese? Ok, maybe this is not a very good example. However, what can be seen is that a rule with low confidence can be very interesting, since it may capture an important influence. Furthermore, this is a way to express negation (though only in the consequent of a rule), since "cheese -> bread" with confidence 35% is obviously equivalent to "cheese -> no bread" with confidence 65%. This also makes clear why the support of the item set that contains all items in the body and the head of the rule is not appropriate for this measure. An important rule may have confidence 0 and thus a support (in the interpretation of [Agrawal et al. 1993]) of 0. Hence it is not reasonable to set a lower bound for this kind of support.
I hope that the intention underlying all this is already clear: Potentially interesting rules differ significantly in their confidence from the confidence of rules with the same consequent, but a simpler antecedent. Adding an item to the antecedent is informative only if it significantly changes the confidence of the rule. Otherwise the simpler rule suffices.
Unfortunately the measures other than rule confidence do not solve the rule selection problem in the very general form in which it was stated above. It is not that easy to deal with all rules that have a simpler antecedent, to keep track of which of these rules were selected (this obviously influences the selection of more complicated rules), to deal with the special type of Poincare paradox that can occur, etc. Hence the measures always compare the confidence of a rule with the confidence of the rule with empty antecedent, i.e. with the relative frequency of the consequent.
I call the confidence of a rule with empty antecedent the prior confidence, since it is the confidence that the item in the consequent of the rule will be present in an item set prior to any information about other items that are present. The confidence of a rule with non-empty antecedent (and the same consequent) I call the posterior confidence, since it is the confidence that the item in the consequent of the rule will be present after it gets known that the items in the antecedent of the rule are present.
All measures that can be used in addition to rule confidence are computed from these two values: the prior confidence and the posterior confidence. Only those rules are selected for which the value of the chosen additional evaluation measure exceeds or is equal to a certain limit. The measures are chosen with the option -e, the limit is passed to the program via the option -d. The default value for the limit is 10%.
All additional rule evaluation measures are combined with the limits for rule confidence and rule support. I.e., my apriori program selects only those rules the confidence of which is greater than or equal to the confidence limit, the support of which is greater than or equal to the support limit, and for which the additional evaluation value is greater than or equal to the limit for this measure. The default is to use no additional evaluation measure, i.e., to rely only on rule confidence and rule support. Of course you can remove the restriction that the rule confidence must exceed a certain limit by simply setting this limit to zero. In this case rules are selected using only the limits for the rule support and the additional evaluation measure. (Attention: If you have a large number of items, setting the minimal rule confidence to zero can result in very high memory consumption.)
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The simplest way to compare the two confidences is to compute the absolute value of their difference. I.e., if "-> bread" has a confidence of 60% and "cheese -> bread" has a confidence of 62%, then the value of this measure is 2%. The parameter given via the option -d to the program states a lower bound for this difference. It follows that this measure selects rules the confidence of which differs more than a given threshold from the corresponding prior confidence. For example, with the option -d20 (and, of course, the option -ed to select the measure) for the item "bread" only rules with a confidence less than 40% or greater than 80% would be selected. Of course, for other items, with a different prior confidence, the upper and lower bounds are different, too.
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An equally simple way to compare the two confidences is to compute their quotient. Since either the prior or the posterior confidence can be greater (which was handled by computing the absolute value for the previous measure), this quotient or its reciprocal, whichever is smaller, is then compared to one. A quotient of one says that the rule is not interesting, since the prior and the posterior confidence are identical. The more the quotient differs from one, the more "interesting" the rule is. Hence, just as above, a lower bound for this difference is given via the option -d. For the bread example, with the option -d20 rules with a confidence less than or equal to (1 -20%) *60% = 0.8 *60% = 48% or a confidence greater than or equal to 60% / (1 -20%) = 60% / 0.8 = 75% are selected. The difference between this measure and the absolute confidence difference to the prior is that the deviation that is considered to be significant depends on the prior confidence. If it is high, then the deviation of the posterior confidence must also be high, and if it is low, then the deviation need only be low. For example, if "-> bread" had a confidence of only 30%, then the option -d20 (just as above) would select rules the confidence of which is less than 0.8 *30% = 24% or greater than 30% /0.8 = 37.5%. As you can see, for a prior confidence of 60% the deviation has to be at least 12%/15%, for a prior confidence of 30% it has to be only 6%/7.5% in order to make a rule eligible. The idea is that an increment of the confidence from 30% to 40% is more important than an increment from 60% to 70%, since the relative change is greater.
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This measure is very similar to the preceding one. Actually, if the confidence of a rule is smaller than the prior confidence, then it coincides with it. The improvement value is simply the posterior confidence divided by the prior confidence. It is greater than one if the confidence increases due to the antecedent, and it is smaller than one if the confidence decreases due to the antecedent. By computing the absolute value of the difference to one, the improvement value can easily be made a rule selection measure. The advantage of this measure over the preceding one is that it is symmetric w.r.t. changes of the confidence due to the antecedent of a rule. For the bread example, with the option -d20 rules with a confidence less than or equal to (1 -20%) *60% = 0.8 *60% = 48% or a confidence greater than or equal to (1 +20%) *60% = 1.2 * 60% = 72% are selected. (Note the difference of 72% compared to 75% for the preceding measure.) Similarly, for the second bread example discussed above, the numbers are 0.8 *30% = 24% and 1.2 *30% = 36%. Note that this is the only measure for which a value greater than 100 may be specified with the -d option, since it can exceed 100% if the posterior confidence of a rule exceeds twice the prior confidence. (I am grateful to Roland Jonscher, who pointed out this measure to me.)
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This measure is simply the information gain criterion that can be used in decision tree learners like C4.5 to select the split attributes. Its idea is as follows: Without any further information about other items in the set, we have a certain probability (or, to be exact, a relative frequency) distribution for, say "bread" and "no bread". Let us assume it is 60% : 40% (prior confidence of the item "bread", just as above). This distribution has a certain entropy
H = - P(bread) log2 P(bread) - P(no bread) log2 P(no bread),
where P(bread) is equivalent to the support of "bread", which in turn is equivalent to the prior confidence of "bread". The entropy of a probability distribution is, intuitively, a lower bound on the number of yes-no-questions you have to ask in order to determine the actual value. This cannot be understood very well with only two possible values, but it can be made to work for this case too. I skip the details here, they are not that important.
After we get the information that the items in the antecedent of the rule are present (say, cheese), we have a different probability distribution, say 35% : 65%. I.e., P(bread|cheese) = 0.35 and P(no bread|cheese) = 0.65. If we also know the support of the item "cheese" (let it be P(cheese) = 0.4 and P(no cheese) = 0.6), then we can also compute the probabilities P(bread|no cheese) = 0.77 and P(no bread|no cheese) = 0.23. Hence we have two posterior probability distributions. The question now is: How much information do we receive from observing the antecedent of the rule? Information is measured as a reduction of entropy. Hence the entropies of the two conditional probability distributions (for "cheese" and "no cheese") are computed and summed weighted with the probability of their occurrence (i.e., the relative frequency of "cheese" and "no cheese", respectively). This gives the expected value of the posterior or conditional entropy. The difference of this value to the prior entropy (see above) is the gain in information from the antecedent of the rule or, as I called it, the information difference to the prior.
The value that can be given via the -d option is a lower bound for the information gain, measured in hundreds of a bit. Since all items can only be present or absent, the information gain can be at most one bit. Therefore a percent value is still reasonable.
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The chi2 measure is well known from statistics. It is often used to measure the difference between a supposed independent distribution of two discrete variables and the actual joint distribution in order to determine how strongly two variables depend on each other. This measure (as it is defined in statistics) contains the number of cases it is computed from as a factor. This is not very appropriate if one wants to evaluate rules that can have varying support. Hence this factor is removed by simply dividing the measure by the number of items sets (the total number, i.e. with the names used above, the number of sets in X). With this normalization, the chi2 measure can assume values between 0 (no dependence) and 1 (very strong dependence). The value that can be given via the -d option is a lower bound for the strength of the dependence of the head on the body in percent (0 - no dependence, 100 - perfect dependence). Only those rules are selected, in which the head depends on the body with a higher degree of dependence.
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In the directory apriori/doc you can find a Gnuplot script named arem.gp (arem stands for additional rule evaluation measures) which visualizes the behavior of the additional rule evaluation measures. This script draws eight 3d graphs, one for the absolute confidence difference, one for the difference of the confidence quotient to one, three for the information difference to the prior confidence and three for the normalized chi2 measure. All graphs show the value of an additional rule evaluation measure over a plane defined by the prior and the posterior confidence of a rule. The latter two measures need three graphs, since they depend on the antecedent support of a rule as a third parameter. Setting a minimal value for an additional rule evaluation measure is like flooding the corresponding graph landscape up to a certain level (given as a percentage, since all considered measures assume values between 0 and 1). Only those rules are selected that sit on dry land.
The first graph shows the behavior of the absolute confidence difference. For the diagonal, i.e. the line where the prior and the posterior confidence are identical, its value is zero (as expected). The more the two confidences differ, the higher the value of this measure gets, but in a linear way.
The second graph shows the behavior of the confidence quotient to one. Again its value is zero for the diagonal (as the value of all measures is) and becomes greater the more the prior and the posterior confidence differ. But it is much steeper for a small prior confidence than for a large one and it is non-linear.
The third to fifth graph show the information difference to the prior confidence for an antecedent support (which is identical to the rule support in my interpretation, see above) of 0.2 (20%), 0.3 (30%) and 0.4 (40%). The regions at the margins, where the measure is zero, correspond to impossible combinations of prior and posterior confidence and antecedent support. As you can see, the valley gets narrower with increasing antecedent support. I.e., with the same minimal value for this measure, rules with low antecedent support need a higher confidence difference to be selected than rules with a high antecedent support. This nicely models the statistical significance of confidence changes. If you only have a few cases to support your rule, even a large deviation from the prior confidence can be explained by random fluctuations, since only a few transactions need to be different to produce a considerable change. However, if the antecedent support is large, even a small deviation (in percent) has to be considered significant (non random), since it takes a lot of changes to transactions to produce even a small change in the percentage. This dependence on the antecedent support of the rule and that the valley is not pointed at the diagonal (which means that even a low minimal value can exclude a lot of rules) is the main difference between the information gain and the normalized chi2 measure on the one hand and the absolute confidence difference and difference of the confidence quotient to one on the other.
The sixth to eighth graph show the normalized chi2 measure for an antecedent support of 0.2, 0.3, and 0.4. The valleys are very similar to those for the information difference to the prior confidence, they only have slightly steeper flanks, especially for low antecedent support. So in practice there is no big difference between the information difference and the normalized chi2 measure.
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My apriori program provides a simple way to restrict the rules to generate w.r.t. the items that shall appear in them. It accepts a third optional input file, in which item appearances can be given. For each item it can be stated whether it may appear in the body (antecedent) of a rule, in the head (consequent), or in both. A description of the format of this additional input file, including examples, can be found here. If no item appearances file is given, the rule selection is not restricted. (I am grateful to the people at Integral Solutions Ltd., who developed the well-known data mining tool Clementine, but are now part of SPSS, for requesting the possibility to restrict item appearances.)
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Since version 4.20 there are extended selection possibilities for frequent item sets, too. (These were added due to a coopertion with Sonja Gruen, FU Berlin.)
An expected value for the support of an item set is computed from the support values of the individual items, assuming independence. Then the binary logarithm of the quotient of actual support and expected support is computed. A minimum value for this measure can be set with the option -d. In this case only frequent item sets for which this measure exceeds the given threshold are kept.
As with the preceding measure the quotient of actual and expected support of an item set is computed and compared to 1 (a value of 1 signifies independence of the items). A minimum value for this measure can be set with the option -d. In this case only frequent item sets for which this measure exceeds the given threshold are kept.
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The counting process can be sped up by organizing the transactions into a prefix tree. That is, the items in each transaction are sorted and then transactions with the same prefix are grouped together and are counted, as one may say, in parallel. This way of organizing the transactions was added in version 4.03 and is the default behavior now. If you prefer that the transactions are treated individually (i.e., the transactions are stored in a simple list and only one transaction is counted at a time), use the option -h.
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My apriori program is invoked as follows:
apriori [options] infile outfile [appfile]
The normal arguments are:
| infile | file to read transactions from | |
| outfile | file to write association rules / hyperedges to | |
| appfile | file stating item appearances (optional) |
The possible options are:
| -t# | target type (default: association rules) | |
| (s: itemsets, c: closed itemsets, m: maximal itemsets, (r: association rules, h: association hyperedges) | ||
| -m# | minimal number of items per set/rule/hyperedge (default: 1) | |
| -n# | maximal number of items per set/rule/hyperedge (default: 5) | |
| -s# | minimal support of a set/rule/hyperedge (default: 10%) | |
| -S# | minimal support of a set/rule/hyperedge (default: 100%) | |
| -c# | minimal confidence of a rule/hyperedge (default: 80%) | |
| -o | use original definition of the support of a rule (body & head) | |
| -k# | item separator for output (default: " ") | |
| -p# | output format for support/confidence (default: "%.1f%%") | |
| -x | extended support output (print both rule support types) | |
| -a | print absolute support (number of transactions) | |
| -y | print lift value (confidence divided by prior) | |
| -e# | additional rule evaluation measure (default: none) | |
| -! | print a list of additional rule evaluation measures | |
| -d# | minimal value of additional evaluation measure (default: 10%) | |
| -v | print value of additional rule evaluation measure | |
| -g | write output in scanable form (quote certain characters) | |
| -l | do not load transactions into memory (work on input file) | |
| -q# | sort items w.r.t. their frequency (default: 1) | |
| (1: ascending, -1: descending, 0: do not sort, | ||
| (2: ascending, -2: descending w.r.t. transaction size sum) | ||
| -u# | filter unused items from transactions (default: 0.5) | |
| (0: do not filter items w.r.t. usage in item sets, <0: fraction of removed items for filtering, >0: take execution times ratio into account) | ||
| -h | do not organize transactions as a prefix tree | |
| -j | use quicksort to sort the transactions (default: heapsort) | |
| -z | minimize memory usage (default: maximize speed) | |
| -i# | ignore records starting with characters in the given string | |
| -b/f/r# | blank characters, field and record separators | |
| (default: " \t\r", " \t", "\n") |
(# always means a number, a letter, or a string that specifies the parameter of the option.)
Note that the effect of the option -z can depend heavily on how the items are sorted (option -q). Highest savings in memory usually result if items are sorted with descending frequency (-q-1). However, this often worsens the processing time considerably.
A note on the option -j: Constructing the prefix tree for the transactions requires sorting the transactions. Since version 4.17 heap sort is the default sorting method for the transactions, because it turned out that in conjunction with the item sorting (and especially for artificial datasets like T10I4D100K) quicksort can lead to very bad processing times (almost worst case behavior, i.e., O(n2) run time for the sorting). However, sometimes this is not a problem and then quicksort is slightly faster, which can be activated with the option -j.
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A text file structured by field and record separators and blanks. Record separators, not surprisingly, separate records, usually lines, field separators fields (or columns), usually words. Blanks are used to fill fields (columns), e.g. to align them. In the transactions file each record must contain one transaction, i.e. a list of item identifiers, which are separated by field separators. An empty record is interpreted as an empty transaction.
Examples can be found in the directory apriori/ex in the source package. Refer to the file apriori/ex/readme, which explains how to process the different example files in the directory apriori/ex in the source package.
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A text file structured by field and record separators and blanks. (Note: For this file the same field and record separators and blanks are used as for the transactions file.)
The first record, which must have one field, contains the default appearance to be used with all items not mentioned in the appearances file. Other records state the appearance of specific items. The first field states the item, the second the appearance indicator. If no appearance indicator is given, the item will be ignored (i.e. may appear neither in the body (antecedent) nor in the head (consequent) of a rule). Empty records are ignored.
The following appearance indicators are recognized:
Example 1: Generate only rules with item "x" in the consequent.
in
x out
Example 2: Item "x" may appear only in a rule head (consequent), item "y" only in a rule body (antecedent); appearance of all other items is not restricted.
both
x head
y body
Providing no item appearances file is equivalent to an item appearances file containing only an indicator like "both", which does not restrict the appearance of any items.
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Each line of the output file contains one association rule in the format
c <- a b ... (x%, y%)
where a, b, and c are item identifiers, and
| x | the percentage of transactions that contain all items appearing in the rule body (antecedent), that is, in the example above, a and b. (support of the rule, i.e., the support in my interpretation) | |
| y | the confidence of the rule, which is computed as the quotient of the percentage of transactions that contain all items appearing in the rule body (antecedent) and the rule head (consequent) - that is, in the example above, a, b, and c - and the above percentage x. |
If the option -o is used, x is replaced by the rule support in the original definition (i.e., the one used by [Agrawal et al. 1993]), namely the percentage of transactions that contain all items appearing in the rule (antecedent) and the rule head (consequent), that is, in the example above, a, b, and c. The value of y, however, is still computed from the value of x as described above.
If the option -x is given, both types of rule support (support of all items in the rule and support of the items in the body/antecedent of the rule) will be printed. The confidence of a rule (see above) is the quotient of the two support values (* 100%), i.e., a rule will be printed as
c <- a b ... (x1%, x2%, y%)
where x1 is the support of the set of all items in the rule, x2 is the support of the set of items in the body (antecedent) of the rule, and y = x1/x2 * 100% is the confidence of the rule.
If the option -a is given, the support percentage x is supplemented by the absolute number of transactions underlying it:
c <- a b ... (x%/s, y%)
where s is the absolute number of transactions. If the option -x is given, the absolute support is printed for both types of rule support.
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Each line of the output file contains one item set in the format
a b c ... (x%)
where a, b, and c are item identifiers and x is the percentage of transactions that contain this item set (item set support).
If the option -a is given, this percentage is supplemented by the absolute number of transactions underlying it:
a b c ... (x%/s)
where s is the absolute number of transactions.
If the option -x is given, the percentage of transactions that are identical to the item set is printed, too (whereas the normal support is the percentage of transactions that are a superset of the item set):
a b c ... (x%, %y)
where x is the normal item set support and y is the percentage of transactions identical to the item set. (This output option was added in response to a request by Laura Maruster.) If the option -a is also given, both percentages are supplemented by the absolute number of transactions underlying these percentages.
Note that for frequent item sets the option -x cannot be combined with the option -y. That is, in order to compute the second support measure for item sets, the transactions have to be loaded into memory.
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Each line of the output file contains one hyperedge the format
a b c ... (x%, y%)
where a, b, and c are item identifiers, and
| x | the percentage of transactions that contain all items appearing in the hyperedge, that is, in the example above, a, b, and c. | |
| y | the average confidence of all rules that can be formed using the items in the hyperedge with all items appearing in the rule (see above), i.e., for the example above, the average confidence of the rules c <- a b, b <- a c, and a <- b c. |
If the option -a is given, the support percentage x is supplemented by the absolute number of transactions underlying it:
a b c ... (x%/s, y%)
where s is the absolute number of transactions.
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The program can be compiled with two additional compilation options (see makefile), namely -DBENCH and -DARCH64.
Compiling the program with -DBENCH produces a version that prints some benchmark information on termination, in particular about the memory used during the item set tree construction (number of nodes, counters, necessary counters, child pointers, necessary child pointers). Collecting the memory usage information slightly, but negligibly increases the execution time.
Compiling the program with -DARCH64 produces a version for 64 bit machines (architecture model: pointers are 64 bits, integers are 32 bits wide), by removing some alignment issues in the transaction and item set tree representations, which would otherwise lead to bus errors. These adaptations slightly, but negligibly increase memory consumption. (I am grateful to Anthony Casaletto, SPSS Inc., for helping me a lot to identify these alignment problems, by compiling and testing the program on a 64 bit machine, since I do not have access to one.)
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apriori -
find association rules/hyperedges with apriori algorithm
copyright © 1996-2003 Christian Borgelt
This program is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser (Library) General Public License as published by the Free Software Foundation.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser (Library) General Public License for more details.
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Download page with most recent version.
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| Snail mail: |
Christian Borgelt Working Group Neural Networks and Fuzzy Systems Department of Knowledge Processing and Language Engineering School of Computer Science Otto-von-Guericke-University of Magdeburg Universitätsplatz 2 D-39106 Magdeburg Germany | |
| E-mail: |
christian.borgelt@cs.uni-magdeburg.de borgelt@iws.cs.uni-magdeburg.de | |
| Phone: | +49 391 67 12700 | |
| Fax: | +49 391 67 12018 | |
| Office: | 29.015 |
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