ASSOCIATION RULES
Chapter 6:
Association Analysis
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Objectives
Introduction to association analysis
Measuring importance of derived association rules
Building association rules
Building sequential association rules
SAS EM examples
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Introduction
What is association analysis?
A process to discover the frequency of occurrence, jointly or sequentially, between sets of items from historical data.
Some data mining professions consider that this as a type of unsupervised techniques, but some consider this as a supervised technique.
A commonly used technique for market basket analysis (MBA).
MBA refers to a set of business problems related to understanding point-of-transaction data.
What is the average number of unique items per order?
Did customer accept or decline a particular cross-sell offer?
What are the most popular items?
What items are frequently purchased together?
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Introduction
Applications
Items purchased on a credit card, give insight into the next product that customers are likely to purchase.
Optional telecommunications services purchases by customers help determine how to bundle these services together to maximize revenue.
Banking services used by customers identify customers likely to want other services.
Unusual combinations of insurance claims can be a sign of fraud and can spark further investigation.
Medical patient histories can give indications of complications based on certain combinations of treatments.
Discover common features among a group of virus infected patients.
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Introduction
The results of a typical association analysis are expressed in the form of a set of rules which are formed by the set of items that are jointly occurred in the data set.
Structure of an association rule:
{Body} {Head}
Body and Head can be referred as a set of items
For practical reason, the head often contain only 1 item.
Example: {Coke} {Orange Juice}
This rule says Coke and Orange Juice have been purchased together. If a customer purchases Coke, then the customer also purchases Orange Juice.
How to measure the importance of these rules?
How to build association rules?
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Measuring Importance of Rules
Rule importance is measured by confidence and support.
Consider a simplified transaction
Co-occurrence table (2-dimensional)
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Measuring Importance of Rules
Rule support percentage
Support percentage measures how often the items in a rule {X} {Y} occur together in the transactions.
It is the percentage of transactions in the database that contains X and Y.
i.e. {X} {Y} support percentage = Number of transactions containing both X and Y / Total number of transactions x 100%
Support is symmetric.
Example: Consider the rule {Coke} {OJ} (or {OJ} {Coke}).
In all 5 transactions, 2 contains both coke and OJ.
The rule support percentage is 40%.
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Measuring Importance of Rules
Rule confidence
Confidence measures the certainty of a rule {X} {Y}.
It is the percentage of transactions containing X and Y in transactions containing X.
i.e. {X} {Y} confidence = support count of the rule / support count of X x 100% = support % of the rule / support % of X x 100%
Example: The rule {Coke} {OJ} has a confidence of 100%.
{Coke} {OJ} confidence = 2 / 2 x100% = 100%.
Example: What is the confidence of the rule {OJ} {Coke}?
{OJ} {Coke} confidence = 2 / 4 x 100% = 50%.
Confidence is not symmetric.
The complete meaning of the rule {Coke} {OJ} is: When a customer buys coke, there is a 100% of chance that the customer will also buy OJ. This happens in 40% of all purchases.
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Measuring Importance of Rules
Is the rule a useful predictor for the Head?
Consider the following table of support percentage of items:
Which one of these three rules is a useful rule?
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Measuring Importance of Rules
Is the rule a useful predictor for the Head?
A rule with high confidence may not be a useful rule.
E.g.
Chance of A occurs randomly:
The rule {B, C} {A} actually worse than saying A randomly occurs in the transactions.
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Is the rule a useful predictor for the Head?
Useful rules have improvement (lift) greater than 1.
Lift = Confidence of the rule
Support % of rule head only
E.g. Consider the rule B & C A:
The rule is not an useful rule.
E.g. Consider the rule A B:
A predicts B 32% better than chance.
Measuring Importance of Rules
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Measuring Importance of Rules
Is the rule a useful predictor for the Head?
Actionable rules: easily understood, suggests possible causes and possible interventions.
Toy shop customers who purchase remote control cars have a 60% likelihood of also purchasing a chocolate bar.
Trivial rules: Known by anyone familiar with the business.
Toy shop customers who purchase remote control cars have a 98% likelihood of also purchasing batteries.
Inexplicable rules: Seem to have no explanation and do not suggest a course of action
When a new hardware store opens, one of the most commonly sold items is toilet bowl cleaner.
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Building Association Rules
Data preparation
Often use transaction data over a period of time.
Minimum data requirement:
Basket or transaction ID;
Items purchased in each transaction;
Quantity is often not considered.
Additional columns:
Customer ID and timestamp are not needed but more powerful analysis can be applied if available.
Different software may require the data to be prepared in different format.
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Building Association Rules
Common approach involves two parts:
Find all frequent k-itemsets, k=1, 2, 3, …
The set of items in a rule is referred as an itemset.
A rule contains k items forms a k-itemset.
The occurrence frequency of an k-itemset is the number of transactions that contain all k items in the itemset.
An itemset satisfies a minimum support percentage (or count) is called a frequent itemset.
Generate strong association rules of k items from a frequent k-itemsets.
A rule that satisfies both specified minimum support threshold and minimum confidence threshold is called a strong rule.
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Building Association Rules
Finding frequent itemsets generally is straight forward.
Naïve approach: count all possible k-itemsets that appear in any transaction, with k = 1, 2, 3, …
Given a set of m items, there are 2m-1 possible subsets
When m=5, there are potentially 31 possible itemsets; when m=30, this becomes 1073741823.
Such a direct search is very costly.
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Building Association Rules
Downward closed principle:
If a k-itemset is a frequent itemset, then any subset of the set must also be frequent itemset.
E.g. If {A,B,C} is a frequent 3-itemset, both {A,B}, {B,C}, {A,C} must also be frequent 2-itemsets.
This means that if we know that an itemset doesn’t satisfy minimum support threshold, none of its supersets will either.
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Building Association Rules
Apriori algorithm
K-itemset is generated through repeating the following two operations.
k-itemset candidates generation
Assume items in each transaction are arranged in lexicographic order.
A pair of frequent (k-1)-itemsets will be merged to form a candidate k-itemset only if their first k-2 items are identical.
Forming frequent k-itemset by candidate pruning.
Eliminate a candidate k-itemset if its support is lower than a specified minimum support.
Those survived k-itemsets become frequent k-itemsets.
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Building Association Rules
Apriori algorithm
Step 1:
Arrange items in a transaction in lexicographic order.
Form a set of candidate 1-itemset, denote as C1.
Count the support of each candidate in C1.
Given support threshold s, Only 1-itemsets with support >= s will be selected as frequent 1-itemsets, denoted as L1.
Example: Suppose s =3 (support count)
Database
C1
L1
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Building Association Rules
Apriori algorithm
Step 2
Pairs of items in L1 become the set of candidate 2-itemset, denoted as C2.
The items in C2 whose count reaches s are the frequent 2-itemsets, denoted as L2.
Example
C2
L2
L1
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Building Association Rules
Apriori algorithm
Step k
To form Ck, the candidate generation procedure merges a pair of frequent (k-1)-itemsets only if their first k-2 items are identical.
The items in Ck whose count reaches s are the frequent k-itemsets, denoted as Lk.
Example, k = 3
L2
C3
L3
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Building Association Rules
Apriori algorithm
Rule Generation
Strong rules can be generate from each frequent k-itemset.
Partitioning the itemset into two non-empty and non-overlapped subsets such that the joining of the two subsets equal to the itemset.
Each pair of subset form an association rule.
Compute the confidence for each formed rule.
A rule is a strong rule if its minimum confidence threshold is satisfied.
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Building Association Rules
Apriori algorithm
Rule Generation
Example
Rules from Frequent 3-itemset {I1, I2, I3}.
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Building Association Rules
Apriori algorithm
Computational complexity
Support threshold
Size of frequent itemsets tends to increase with lower support thresholds, hence higher computation cost.
Number of items
The number of frequent itemsets grows with the number of items, computation cost will increase.
Number of transactions
Run time increases with a larger number of transactions.
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Building Association Rules (Self-Study)
Other considerations
Choosing the set of items at the right level of detail
What constitutes a particular item depends on the business need.
A frozen pizza might be considered an item for analysis in a grocery store.
The pizza toppings that are ordered are item for analysis in a pizza restaurant.
Different levels of detail may be needed at different times.
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Building Association Rules (Self-Study)
Other considerations
Choosing the set of items at the right level of detail.
Brands, sizes
Strawberry ice-cream, …
Garden peas, mint peas,…
Chicken curry, lamb curry, …
Ice-cream, apple pie, …
Peas, carrots, mixed,…
Curry, meat pie, steak, …
Desserts,
vegetables, dinners
Frozen
foods
Product hierarchy
Number
of items
More
Less
Number of strong rules
Less
More
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Building Association Rules (Self-Study)
Other considerations
Choosing the set of items at the right level of detail.
Generalized items (higher product hierarchy) help to find rules with sufficient support.
May begin analysis with generalized items first, then focus on more specific items.
Not all items need to be analyzed at the same level of product hierarchy.
Rare items can be rolled up to increase the support value while more common items can stay at lower level.
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Building Association Rules (Self-Study)
Other considerations
Complexity of rules
The more items in the transactions, the longer it takes to generate rules of a given complexity.
Look for rules with lower complexity when there are less number of items purchased together in one transaction.
Example: Express check-out in supermarkets.
Look for rules with higher complexity when there are more number of items purchased together in one transaction.
Example: Normal check-out in supermarkets.
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Sequential Association Analysis
Transaction data often contains temporal information about when an item was purchased by customers.
Such information can be used to form sequence of transactions made by a customer over a period of time.
It allows the prediction of future occurrences of certain events.
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Sequential Association Analysis
Problem formulation
Data must have two additional features:
A time stamp or sequencing information to determine when transactions occurred relative to each other.
Identity information, such as account number or id number, that identifies different transactions as belonging to the same object.
Each rows of data records the item associated with a particular object at a given time.
Records with identical time stamp (or sequence number) and identity indicate the set of items occurred in the same transaction by the same object.
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Sequential Association Analysis
Problem formulation
A sequence of an object (s) is a list of elements (transactions) (l ) arranged in temporal order.
Denoted as s = <{l1} {l 2} … {ln}>
Each element contains one or more items (or events).
Denoted as {lj } = {e1, e2, …, ek}
We ignore the quantity of the same item.
For example, the sequence for object A:
sA = <{1,2,4} {2,3} {5}>
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Sequential Association Analysis
Problem formulation
A data sequence refers to an ordered list of elements associated with a single object.
Example: 5 data sequences are contained in the example data set.
A sequence can be characterized by the number of elements and the number of items.
l-element sequence (size): corresponds to the number of elements presents in a sequence.
E.g.: Customer A’s sequence is a 3-element sequence
k-item sequence (length): a sequence that contains k items.
E.g.: Customer A’s sequence is a 6-item sequence
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Sequential Association Analysis
Problem formulation
A sequence <{t1} {t2} … {tn}> is a subsequence of another sequence <{s1} {s2} … {sm}> with m ≥ n if there exists integers i1 < i2 < … < in such that t1 ⊆ si1, t2 ⊆ si2, … , tn ⊆ sin .
We can also say t is supported by s.
Examples:
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Sequential Association Analysis
Problem formulation
Support percentage of a sequence s is the percentage of all data sequences that contain s.
Example: Consider the following customer purchase data sequences:
The support % of sequence sA = 20%.
The support % of a sequence t: <{1,2} {2,3}> = 60%.
Sequential Association Analysis
Problem formulation
If the support (percentage or count) for s is greater than or equal to a user-specified threshold, then s is declared to be a frequent sequence.
The task of sequential pattern discovery is to find all frequent sequences.
The number of candidate sequences will grow exponentially by a brute-force approach.
Downward closed principle still holds for sequential data because any k-item frequent sequence must contains all of its (k-1)-item frequent subsequences.
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Sequential Association Analysis
Apriori-like algorithm
Step 1
Arrange all items of each element in lexicographic order.
Form a set of candidate 1-item subsequences, denoted as C1.
Compute the support of each candidate in C1.
Given a support threshold d (% or count) , only those 1-item subsequences with support >= d will be selected as 1-item frequent subsequences, denote the list as L1.
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Sequential Association Analysis
Apriori-like algorithm
Step 1
Example: Suppose the support threshold count is 3.
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Sequential Association Analysis
Apriori-like algorithm
Step 2
Form a set of candidate 2-item subsequences from L1, denoted as C2.
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Sequential Association Analysis
Apriori-like algorithm
Steps k>2
A (k-1)-item sequence s is merged with another (k-1)-item sequence t only if the subsequence obtained by dropping the first item in s is identical to the subsequence obtained by dropping the last item in t .
The resulting candidate is the sequence s concatenated with the last item from t in two possible ways:
If the last two items in t belong to the same element, then the last item in t is added to the last element in s .
For example, s = <{1,2}> and t = <{2,3}>. Merged sequence is <{1,2,3}>.
If the last two items in t belong to different elements, then the last item in t is appended to the end of s as a separate element.
For example, s= <{1,2}> and t = <{2} {2}>. Merged sequence is <{1,2} {2}>.
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Sequential Association Analysis
Apriori-like algorithm
Step 3:
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Sequential Association Analysis
Apriori-like algorithm
Step 4:
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Sequential Association Analysis
Apriori-like algorithm
Rules generation
Sequential rules are generated from each frequent subsequence list Lk with at least 2 elements.
To avoid duplicated rules formation, a k-item frequent subsequence in list Lk only forms rules which contain these k-item exactly with chain length equals to the number of elements in the subsequence.
Examples:
2-item subsequence <{2,3}>: no rule
2-item subsequence <{2} {3}>: <{2}> <{3}>
3-item subsequence <{2} {3,4}>: <{2}> <{3,4}>
3-item subsequence <{2} {3} {5}>: <{2}> <{3}> <{5}>
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Sequential Association Analysis (SS)
Apriori-like algorithm
Rules generation
There is no universally agreed way in calculating the confidence and lift values of a sequential rule.
Confidence of a sequential rule with k-chain can be considered as the percentage of data sequences containing the k-chain in the data sequences containing the first (k-1)-chain (body) .
Lift of a sequential rule with k-chain = confidence of the rule / support % of the last element (the head) of the rule
Example: Consider the rule <{1}> <{2}> <{4}> generated from the following data sequences:
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Support % = 20%
Confidence = 1 / 4 x 100% = 25%
Lift = 25% / 100% = 0.25
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Sequential Association Analysis
Timing constraints
Consolidate time (Window size)
Items within an element do not have to occur at the same time.
E.g. Two transactions made by the same customer in the same day might be considered to be a single transaction.
A consolidate time can be defined to specify the maximum allowed time difference between the earliest and latest occurrences of items in any element.
A consolidate time of 0 means all items in the same element must occur simultaneously.
If consolidate time is changed, the content of a data sequence will also be changed.
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Sequential Association Analysis
Timing constraints
Consolidate time
Example: Consider the data sequence of a customer as follows:
Consolidate time = 0, the sequence is
<{1,2,4} {2,3} {5} {6}>.
Consolidate time =1, the sequence becomes <{1,2,3,4} {5,6}>.
Consolidate time =2, the sequence becomes <{1,2,3,4,5} {6}>.
Consolidate time =3, the sequence becomes <{1,2,3,4,5,6}>.
Sequential Association Analysis
Timing constraints
Maximum transaction duration (Maximum span )
A sequence of elements that covers a very long time period may contains associations which are already obsolete.
One may specify the maximum transaction duration between the earliest and the latest occurrences of elements in the entire sequence to reduce the number of obsolete associations.
E.g. Two elements more than 3 months apart do not constitute a sequence.
A maximum transaction duration = means a sequence may contains elements occurred at any time without any restrictions.
Changing the maximum transaction duration changes the forming of a subsequence.
Value of maximum transaction duration must be greater than the consolidate time value.
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Sequential Association Analysis
Timing constraints
Maximum transaction duration
Example: Consider the sequence of a customer as follows:
Maximum transaction duration = 3, <{4} {6}> is supported.
Maximum transaction duration = 2, <{4} {6}> is not supported, but <{4} {5}> is supported.
Maximum transaction duration =1, <{4} {5}> is not supported, but <{4} {3}> is supported.
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SAS Association Node
To perform association discovery and sequence discovery.
Data must contain as least two columns: identity and item involved. It must be in the format of 1 record for each identity value and each item.
Association node allows user to specify:
The minimum support.
The minimum confidence level.
The maximum number of items in an association.
Option to generate association rules with or without sequence (require an addition of time sequence column).
Option to extract rules from the generated rules for k-itemset, or rules contain specific items.
SAS Association Node
Example 6.1
Import Association_Sample1.sas7bdat to a new project.
Set the Role of variable Object to ID.
Set the Role of variable Product to Target and Level to Nominal.
Set the Role of variable Time to Sequence and Level to Interval (or Ordinal).
The role of the data set must be Transaction.
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SAS Association Node
Example 6.1
Name a new diagram Sample1.
Drag the Association_Sample1 Data Sources into the diagram and connect it to an Association node.
Set the following properties of the Association node:
Set USE of variable Time in the Variable table to No.
So that only association discovery is performed.
Set Maximum Items to 5.
So that a rule contains at most 5 items.
Set Minimum Confidence Level to 50.
Set Support Type to Count and Support Count to 3.
Support percentage refers to the proportion of the largest single item frequency, not the number of transactions.
Set Numbers to Keep to the largest value so that all found rules can be kept in the result.
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SAS Association Node
Example 6.1
Results: Rules Table
Select View | Rules | Rule Table to view the set of generated rules and the respective statistics.
Click the Relations column header to sort the table in the ascending order of the number of items in each relation.
Or click any column header to sort the table in ascending or descending order of the selected column.
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SAS Association Node
Example 6.1
Link Graph
To view associations among selected items, in the Results, click View | Link Graph.
Larger nodes have greater counts than smaller nodes.
The thicker the links are, the higher confidence the rules have.
The default link graph shows too many nodes and links. It can be simplified.
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SAS Association Node
Example 6.1
Link Graph
To view how selected items are associated with other items, right click in the link graph and select Graph Properties.
Select Constellation – Links, uncheck Show all links in Link visibility, and then uncheck Show links into Selection and Show links among selection; Select Show Directed Links. Click Apply button.
Select Constellation – Nodes, from the drop list of Auto Layout, select Spring Force. Click Apply button and then OK button.
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SAS Association Node
Example 6.1
Link Graph
In Link Graph, select a node, say node of item 3.
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SAS Association Node
Example 6.1
Select subset of rules
In Rules property of the Association node, open Rules Selector window:
Select Rules tab and click Subset to open the RuleSelectorWhereBuilder window.
Select rules equal to item 3 on either left hand side or right hand side of the rule.
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SAS Association Node
Example 6.2
Import Grocery1.sas7bdat into Project Association.
Set the Role of CustomerID to ID.
Set the Role of Product to Target.
Set the Role of the data set to Transaction.
Create a new diagram named Grocery1 and drag the Grocery1 data into the diagram.
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SAS Association Node
Example 6.2
Preliminary analysis:
How many transactions?
How many products?
How frequent these products are purchased by the customers?
Change the Train Role of the data to Raw. Then connect the Data Sources node to a StatExplore node.
SAS Association Node
Example 6.2
Drag the Grocery1 from Data Sources into the diagram and connect the data source node to an Association node.
Set the following properties of the Association node:
Maximum Items: 5
Minimum Confidence: 50
Support Type: Percent
Support Percentage: 20. This is the percentage of the largest 1-itemset. That is 0.2×256 = 51.2 or 52.
Set the value of Number of Rules to keep as 10000.
Run the node.
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SAS Association Node
Example 6.2
Which two products can be bundled together?
From the Rules table,
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SAS Association Node
Example 6.2
How to promote the sales of toothpaste?
From the Rules table
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SAS Association Node
Example 6.3
Refer to the data Association_Sample1 in Project Association.
The data set contains the following data sequence:
A: <{1,2,4}, {2,3}, {5}>
B: <{1,2}, {2,3,4}>
C: <{1,2}, {2,3,4}, {2,4,5}>
D: <{3,4}, {4,5}>
E: <{1,3}, {2,4,5}>
Drag the data node into the diagram Sample1 and connect it to a new Association node. Name this node Sequence1.
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SAS Association Node
Example 6.3
Set the following properties of Sequence1:
Set Use of variable for Time in the Variable table to Yes, so that sequential discovery will be performed.
The Association and Sequence property settings are interrelated.
Set Association Maximum item to 20.
This defines the maximum number of items of each element in each derived rule.
Set Association Support Type to Count, and Association Support Count to 1.
This ensures all items will be considered for sequential association.
Only those 1-itemsets with sequence support count in data > specified Association Support Count value will be considered for forming the sequential rules.
Value of Minimum Confidence is not important.
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SAS Association Node
Example 6.3
Set the following properties of Sequence1:
Set Sequence Chain Count (chain length) to 10, so that each derived rule contains at most 10 elements.
Set Sequence Support type to Count, and Sequence Support count to 1.
Set Number of Rules to keep as 10000.
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SAS Association Node
Example 6.3
Results: Rules Table
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SAS Association Node
Example 6.3
Copy Sequence1 and paste it into the diagram. Name it as Sequence2. Connect Sample1 data source node to it.
Set Consolidate Time property to 1.
The data sequences become:
A: <{1,2,3,4}, {5}>
B: <{1,2,3,4}>
C: <{1,2,3,4}, {2,4,5}>
D: <{3,4,5}>
E: <{1,2,3,4,5}>
Set Maximum Transaction Duration to an arbitrary large value, say 10.
Maximum Transaction Duration > Consolidate Time.
Run the node.
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SAS Association Node
Example 6.3
Copy Sequence1 and paste it into the diagram. Name it as Sequence3. Connect Sample1 data source node to it.
Set Consolidate Time property to 0, and Maximum Transaction Duration property to 1.
The supported subsequences are:
A: <{1,2,4}, {2,3}> , <{2,3}, {5}>
B: <{1,2}, {2,3,4}>
C: <{1,2}, {2,3,4}>, <{2,3,4}, {2,4,5}>
D: <{3,4}, {4,5}>
E: <{1,3}, {2,4,5}>
Run the node.
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SAS Association Node
Example 6.4
A market analyst for a grocery chain wants to identify likely product purchase sequences.
Import data Grocery_seq.sas7bdat into project Association.
Set Role of Customer to ID.
Set Role of Product to Target.
Set Role of Time to Sequence and Level to Interval.
Set Role of the data to Transaction.
Create a new diagram named Grocery_Seq. Drag the data into a new diagram.
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SAS Association Node
Example 6.4
Preliminary analysis
How many customers?
How many products?
What is the maximum transaction span in the data?
Connect the Data Sources node to a StatExplore node. Change the Train Role of the data to Raw and Role of Time of Input.
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SAS Association Node
Example 6.4
Drag the Grocery_seq data source node into the diagram. Connect it to an Association node and set the following properties:
Association:
20 for Maximum Items.
Count for Support Type and 1 for Support count.
Sequence:
10 for Chain Count.
Count for Support Type and 1 for Support count.
Rules:
10000 for numbers to keep.
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SAS Association Node
Example 6.4
Results:
Only the first 10,000 (according to the specified Sort Criterion) are kept.
E.g. Search for all rules that have a chain length of 3 and equals to heineken in first element and ham in the last element.
In Rules
CustomerItems
1orange juice, coke
2milk, orange juice, window cleaner
3orange juice, detergent
4orange juice, detergent, coke
5window cleaner
OJWi ClMilkCokeDetergent
OJ41122
Wi Cl12100
Milk11100
Coke20021
Detergent10002
Sheet1
Customer Items
1 orange juice, coke
2 milk, orange juice, window cleaner
3 orange juice, detergent
4 orange juice, detergent, coke
5 window cleaner
Sheet2
Sheet3
OJWi ClMilkCokeDetergent
OJ41121
Wi Cl12100
Milk11100
Coke21020
Detergent10002
Sheet1
Customer Items OJ Wi Cl Milk Coke Detergent
1 orange juice, coke OJ 4 1 1 2 1
2 milk, orange juice, window cleaner Wi Cl 1 2 1 0 0
3 orange juice, detergent Milk 1 1 1 0 0
4 orange juice, detergent, coke Coke 2 1 0 2 0
5 window cleaner Detergent 1 0 0 0 2
Sheet2
Sheet3
Sheet1
OJ Wi Cl Milk Coke Detergent
OJ 4 1 1 2 2
Wi Cl 1 2 1 0 0
Milk 1 1 1 0 0
Coke 2 0 0 2 1
Detergent 1 0 0 0 2
Sheet2
Sheet3
MBD000DC5A0.xls
IDItems
100I1,I2,I5
200I2,I4,I5
300I2,I3
400I1,I2,I4
500I1,I3
600I1,I2,I3
700I1,I3
800I1,I2,I3,I5
900I1,I2,I3
ItemsetSupport Cnt
I17
I27
I36
I42
I53
ItemsetSupport Cnt
I17
I27
I36
I53
ItemsetSupport Cnt
I1,I25
I1,I35
I1,I52
I2,I34
I2,I53
I3,I51
ItemsetSupport Cnt
I1,I25
I1,I35
I2,I34
I2,I53
ItemsetSupport Cnt
I17
I27
I36
I53
ItemsetSupport Cnt
I1,I2,I33
I2,I3,I51
ItemsetSupport Cnt
I1,I2,I33
BodyHeadConfidence
I1I2, I342.86%
I2I1,I342.86%
I3I1,I250.00%
I1,I2I360.00%
I1,I3I260.00%
I2,I3I175.00%
ObjectTimestampEvents
A11,2,4
A22,3
A35
B11,2
B22,3,4
C11,2
C22,3,4
C32,4,5
D12
D23,4
D34,5
E11,3
E22,4,5
Sequence sSequence tIs t a subsequence of s?
<{2,4} {3,5,6} {8}> <{2} {3,6} {8}> Yes
<{2,4} {3,5,6} {8}> <{5} {8}> Yes
<{2,4} {3,5,6} {8}> <{8}> Yes
<{2,4} {3,5,6} {8}> <{2} {4}> No
<{2,4} {3,5,6} {8}> <{3,5} {2,4}> No
<{2,4} {3,5,6} {8}> <{2} {3,7}> No
s
A
= <{1,2,4} {2,3} {5}>
s
B
= <{1,2} {2,3,4}>
s
C
= <{1,2} {2,3,4} {2,4,5}>
s
D
= <{2} {3,4} {4,5}>
s
E
= <{1,3} {2,4,5}>
SequenceCount
<{1}>4
<{2}>5
<{3}>5
<{4}>5
<{5}>4
C1
SequenceCount
<{1}>4
<{2}>5
<{3}>5
<{4}>5
<{5}>4
L1
SequenceCountSequenceCountSequenceCountSequenceCount
<{1} {1}>0<{2,3}>3<{4} {1}>0<{5} {1}>0
<{1} {2}>4<{2,4}>4<{4} {2}>2<{5} {2}>0
<{1} {3}>3<{2,5}>3<{4} {3}>1<{5} {3}>0
<{1} {4}>3<{3} {1}>0<{4} {4}>1<{5} {4}>0
<{1} {5}>3<{3} {2}>2<{4} {5}>3<{5} {5}>0
<{1,2}>3<{3} {3}>0<{4,5}>3
<{1,3}>1<{3} {4}>3
<{1,4}>1<{3} {5}>4
<{1,5}>0<{3,4}>3
<{2} {1}>0<{3,5}>0
<{2} {2}>3
<{2} {3}>4
<{2} {4}>3
<{2} {5}>3
C2
SequenceCountSequenceCountSequenceCountSequenceCount
<{1} {2}>4<{2} {3}>4<{3} {4}>3<{4} {5}>3
<{2} {1}>0<{3} {2}>2<{4} {3}>1<{5} {4}>0
<{1,2}>3<{2,3}>3<{3,4}>3<{4,5}>3
<{1} {3}>3<{2} {4}>3<{3} {5}>4
<{3} {1}>0<{4} {2}>2<{5} {3}>0
<{1,3}>1<{2,4}>4<{3,5}>0
<{1} {4}>3<{2} {5}>3
<{4} {1}>0<{5} {2}>0
<{1,4}>1<{2,5}>3
<{1} {5}>3
<{5} {1}>0
<{1,5}>0
C2
SequenceCountSequenceCount
<{1} {2}>4<{2,3}>3
<{1} {3}>3<{2,4}>4
<{1} {4}>3<{2,5}>3
<{1} {5}>3<{3} {4}>3
<{1,2}>3<{3} {5}>4
<{2} {2}>3<{3,4}>3
<{2} {3}>4<{4} {5}>3
<{2} {4}>3<{4,5}>3
<{2} {5}>3
L2
SequenceCountSequenceCountSequenceCountSequenceCount
<{1} {2}>4<{2} {3}>4<{3} {4}>3<{4} {5}>3
<{2} {1}>0<{3} {2}>2<{4} {3}>1<{5} {4}>0
<{1,2}>3<{2,3}>3<{3,4}>3<{4,5}>3
<{1} {3}>3<{2} {4}>3<{3} {5}>4
<{3} {1}>0<{4} {2}>2<{5} {3}>0
<{1,3}>1<{2,4}>4<{3,5}>0
<{1} {4}>3<{2} {5}>3
<{4} {1}>0<{5} {2}>0
<{1,4}>1<{2,5}>3
<{1} {5}>3
<{5} {1}>0
<{1,5}>0
C2
SequenceCountSequenceCountSequenceCount
<{1} {2}>4<{2} {4}>3<{4} {5}>3
<{1,2}>3<{2,4}>4<{4,5}>3
<{1} {3}>3<{2} {5}>3
<{1} {4}>3<{2,5}>3
<{1} {5}>3<{3} {4}>3
<{2} {3}>4<{3,4}>3
<{2,3}>3<{3} {5}>4
L2
SequenceCount
<{1} {2,3}>3
<{1} {2,4}>3
<{1,2} {2}>3
<{1,2} {3}>3
<{2} {2,3}>3
<{2} {3} {5}>3
<{2} {3,4}>3
<{3} {4,5}>3
L3
ObjectTimestampEvents
A11,2,4
A22,3
A35
B11,2
B22,3,4
C11,2
C22,3,4
C32,4,5
D12
D23,4
D34,5
E11,3
E22,4,5
SequenceCountSequenceCountSequenceCount
<{1} {2} {3}>0<{1} {3} {4}>1<{2} {4} {5}>2
<{1} {2,3}>3<{1} {3,4}>2<{2} {4,5}>2
<{1} {2} {4}>1<{1} {3} {5}>2<{2,4} {5}>2
<{1} {2,4}>3<{1} {4} {5}>1<{2,4,5}>2
<{1} {2} {5}>2<{1} {4,5}>2<{3} {4} {5}>0
<{1} {2,5}>2<{2} {3} {4}>2<{3} {4,5}>3
<{1,2} {3}>3<{2} {3,4}>3<{3,4} {5}>2
<{1,2,3}>0<{2} {3} {5}>3<{3,4,5}>0
<{1,2} {4}>2<{2,3} {4}>1
<{1,2,4}>1<{2,3,4}>2
<{1,2} {5}>2<{2,3} {5}>2
<{1,2,5}>0
C3
SequenceCountSequenceCountSequenceCount
<{1} {2} {2}>1<{1,2} {2}>3<{2} {2,5}>1
<{1} {2} {3}>0<{1,2} {3}>3<{2} {3} {4}>2
<{1} {2} {4}>1<{1,2} {4}>2<{2} {3} {5}>3
<{1} {2} {5}>2<{1,2} {5}>2<{2} {3,4}>3
<{1} {2,3}>3<{1,2,3}>0<{2} {4} {5}>2
<{1} {2,4}>3<{1,2,4}>1<{2} {4,5}>2
<{1} {2,5}>2<{1,2,5}>0<{3} {4} {5}>0
<{1} {3} {4}>1<{2} {2} {3}>0<{3} {4,5}>3
<{1} {3} {5}>2<{2} {2} {4}>1<{3,4} {5}>2
<{1} {3,4}>2<{2} {2} {5}>2<{3,4,5}>0
<{1} {4} {5}>1<{2} {2,3}>3
<{1} {4,5}>1<{2} {2,4}>2
C3
ObjectTimestampEvents
A11,2,4
A22,3
A35
B11,2
B22,3,4
C11,2
C22,3,4
C32,4,5
D12
D23,4
D34,5
E11,3
E22,4,5
SequenceCount
<{1} {2,3}>3
<{1} {2,4}>3
<{1,2} {3}>3
<{2} {3,4}>3
<{2} {3} {5}>3
<{3} {4,5}>3
L3
SequenceCount
<{2} {2,5}>1
<{2} {3} {4}>2
<{2} {3} {5}>3
<{2} {3,4}>3
<{2} {4} {5}>2
<{2} {4,5}>2
<{2,3} {4}>1
<{2,3} {5}>1
<{2,3,4}>1
<{2,4} {5}>2
<{2,4,5}>1
<{3} {4} {5}>0
<{3} {4,5}>3
<{3,4} {5}>2
<{3,4,5}>0
SequenceCount
<{1,2} {2,3}>3
<{1,2} {3} {5}>2
<{1,2} {3,4}>2
C4
ObjectTimestampEvents
A11,2,4
A22,3
A35
B11,2
B22,3,4
C11,2
C22,3,4
C32,4,5
D12
D23,4
D34,5
E11,3
E22,4,5
SequenceCount
<{1} {2,3}>3
<{1} {2,4}>3
<{1,2} {3}>3
<{2} {3,4}>3
<{2} {3} {5}>3
<{3} {4,5}>3
L3
SequenceCount
<{1,2} {2,3}>3
L4
ObjectTimestampEvents
A11,2,4
A22,3
A35
B11,2
B22,3,4
C11,2
C22,3,4
C32,4,5
D12
D23,4
D34,5
E11,3
E22,4,5
Timestamp
Elements
4
{2,3}{5}{6}><{1,2,4}
123
A:{1,2,3,4,5}
B:{1,2,3,4}
C:{1,2,3,4,5}
D:{3,4,5}
E:{1,2,3,4,5}
Transaction Records
/docProps/thumbnail.jpeg