Editing Association rule learning (section)

=== Confidence ===
Confidence is the percentage of all transactions satisfying {{mvar|X}} that also satisfy {{mvar|Y}}.<ref>{{Cite web|last=Wong|first=Pak|date=1999|title=Visualizing Association Rules for Text Mining|url=https://neuro.bstu.by/ai/Data-mining/Stock-market/InfoVis1999Association.pdf|url-status=live|website=BSTU Laboratory of Artificial Neural Networks|archive-url=https://web.archive.org/web/20211129082512/https://neuro.bstu.by/ai/Data-mining/Stock-market/InfoVis1999Association.pdf |archive-date=2021-11-29 }}</ref>

With respect to {{mvar|T}}, the confidence value of an association rule, often denoted as <math>X \Rightarrow Y</math>, is the ratio of transactions containing both {{mvar|X}} and {{mvar|Y}} to the total amount of {{mvar|X}} values present, where {{mvar|X}} is the antecedent and {{mvar|Y}} is the consequent.

Confidence can also be interpreted as an estimate of the [[conditional probability]] <math>P(E_Y | E_X)</math>, the probability of finding the RHS of the rule in transactions under the condition that these transactions also contain the LHS.<ref name=":0" /><ref name="hipp">{{Cite journal | last1 = Hipp | first1 = J. | last2 = Güntzer | first2 = U. | last3 = Nakhaeizadeh | first3 = G. | title = Algorithms for association rule mining --- a general survey and comparison | doi = 10.1145/360402.360421 | journal = ACM SIGKDD Explorations Newsletter | volume = 2 | pages = 58–64 | year = 2000 | citeseerx = 10.1.1.38.5305 | s2cid = 9248096 }}</ref>

It is commonly depicted as:

:<math>\mathrm{conf}(X \Rightarrow Y) = P(Y | X) = \frac{\mathrm{supp}(X \cap Y)}{ \mathrm{supp}(X) }=\frac{\text{number of transactions containing }X\text{ and }Y}{\text{number of transactions containing }X}</math>

The equation illustrates that confidence can be computed by calculating the co-occurrence of transactions {{mvar|X}} and {{mvar|Y}} within the dataset in ratio to transactions containing only {{mvar|X}}. This means that the number of transactions in both  {{mvar|X}} and {{mvar|Y}}  is divided by those just in {{mvar|X}} .

For example, Table 2 shows the rule <math>\{\mathrm{butter, bread}\} \Rightarrow \{\mathrm{milk}\}</math> which has a confidence of <math>\frac{1/5}{1/5}=\frac{0.2}{0.2}=1.0</math> in the dataset, which denotes that every time a customer buys butter and bread, they also buy milk. This particular example demonstrates the rule being correct 100% of the time for transactions containing both butter and bread. The rule <math>\{\mathrm{fruit}\} \Rightarrow \{\mathrm{eggs}\}</math>, however, has a confidence of <math>\frac{2/5}{3/5}=\frac{0.4}{0.6}=0.67</math>. This suggests that eggs are bought 67% of the times that fruit is brought. Within this particular dataset, fruit is purchased a total of 3 times, with two of those times consisting of egg purchases.

For larger datasets, a minimum threshold, or a percentage cutoff, for the confidence can be useful for determining item relationships. When applying this method to some of the data in Table 2, information that does not meet the requirements are removed. Table 4 shows association rule examples where the minimum threshold for confidence is 0.5 (50%). Any data that does not have a confidence of at least 0.5 is omitted. Generating thresholds allow for the association between items to become stronger as the data is further researched by emphasizing those that co-occur the most. The table uses the confidence information from Table 3 to implement the Support &times; Confidence column, where the relationship between items via their both confidence and support, instead of just one concept, is highlighted. Ranking the rules by Support &times; Confidence multiples the confidence of a particular rule to its support and is often implemented for a more in-depth understanding of the relationship between the items.

  {| class="wikitable sortable"
|+Table 4. Example of Confidence and Support &times; Confidence
!if Antecedent then Consequent
!Confidence 
!Support &times; Confidence
|-
|if buy milk, then buy bread
|{{frac|2|2}} = 1.0
|0.4&times;1.0= 0.4
|-
|if buy milk, then buy eggs
|{{1/2}} = 0.5
|0.2&times;0.5= 0.1
|-
|if buy bread, then buy fruit
|{{2/3}} &asymp; 0.66
|0.4&times;0.66= 0.264
|-
|if buy fruit, then buy eggs
|{{2/3}} &asymp; 0.66
|0.4&times;0.66= 0.264
|-
|if buy milk and bread, then buy fruit
|{{frac|2|2}} = 1.0
|0.4&times;1.0= 0.4
|}

Overall, using confidence in association rule mining is great way to bring awareness to data relations. Its greatest benefit is highlighting the relationship between particular items to one another within the set, as it compares co-occurrences of items to the total occurrence of the antecedent in the specific rule. However, confidence is not the optimal method for every concept in association rule mining. The disadvantage of using it is that it does not offer multiple difference outlooks on the associations. Unlike support, for instance, confidence does not provide the perspective of relationships between certain items in comparison to the entire dataset, so while milk and bread, for example, may occur 100% of the time for confidence, it only has a support of 0.4 (40%). This is why it is important to look at other viewpoints, such as Support &times; Confidence, instead of solely relying on one concept incessantly to define the relationships.