5. MULTIPLE LEVELS AND CROSS LEVELS ASSOCIATION RULES UNDER CONSTRAINTS

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1 5. MULTIPLE LEVELS AND CROSS LEVELS ASSOCIATION RULES UNDER CONSTRAINTS Association rules generated from mining data at multiple levels of abstraction are called multiple level or multi level association rules. Similarly, association rules generated from mining data at cross levels of abstraction are called cross level association rules. Multi level and cross level association rules allow user to discover association rules at different levels of abstraction in the concept hierarchy. 5.1 CONCEPT HIERARCHY Concept hierarchy is a tree like structure of concepts and items stored in a database [7]. It is a useful representation scheme for organizing concepts or items in huge repository. In many business applications data is provided at multiple levels of abstractions and these applications require mining at multiple levels and cross levels of abstraction such as sales and purchase applications database where items are available at multiple levels of abstraction. Items at the lower levels are descendents and items at the higher levels contain various lower levels items as shown in Fig Concept hierarchy is traversed in a top down or bottom up manner and large number of item sets are generated. Generally, top down strategy is used, where counts are accumulated for the calculation of frequent item sets, at each concept level, starting at the concept level 1 and working downward in the concept hierarchy toward the more specific concept levels, up till no more frequent item set can be found. 92

2 Stationary Items Pen Marker Calculator Writing Tool... Ball Scientific White Board Note book Reynolds Sony JP Fig. 5.1 Concept Hierarchy Representing Stationary Items Using top down deepening method when large number of item sets are discovered at level n, large item sets are generated for level n+1. Large mitem sets at one level in the concept hierarchy are used as candidates to generate large L item sets for children at the next level. Multiple levels association rules involves concepts at different levels of abstraction. Fig. 5.1 shows a partial concept hierarchy for stationary items. This hierarchy shows that Reynolds is a type of ball pen. Concept hierarchy describes a sequence of mapping from lower level concepts to more general higher level concepts. The concepts hierarchy shown in Fig.5.1 has 5 levels referred to as level 0 to level 5, starting from level 0 at the root node MULTI LEVELS AND CROSS LEVELS CONSTRAINTS Many business database represents categorization of items such as in a departmental store, sold items reflect categorization of items. In such database users may desire to discover association rules among items only at a same level that span over different levels. This task is known as multi level association rules. For example database organized in the form of concept hierarchy given in Fig

3 allows the user to find the relationship among white board and note book which is known as mining at the multi level association rules. When user discovers relationship between white board and marker, is known as cross level association rule mining. Multi level and cross level association rules can be mined using concept hierarchies under the support and confidence framework Support and Confidence Framework There are number of variations for using support and confidence framework. Using predefined uniform minimum support and minimum confidence threshold for all levels: In this framework same value for higher level and lower level is given by the user or expert for support and confidence threshold to mine the patterns. Using predefined reduced minimum support and minimum confidence threshold at lower levels: In this scheme higher threshold value for upper levels of items and lower threshold value for downwards items of hierarchy are given for support and confidence by the expert or user to discover valuable patterns. Using predefined item or group-based minimum support and minimum confidence threshold: In this framework group wise threshold value for support and confidence is input by the user or expert. Using user specified minimum support and confidence threshold framework: In this scheme any mix of above mentioned framework is used to discover interesting patterns. Uniform minimum support strategy generates many uninteresting patterns. If high minimum support is used at upper level and lower levels also, it generates many uninteresting rules at lower levels. If low minimum support is applied at upper levels it avoids generating many interesting rules at higher levels. Therefore, 94

4 for optimization it is required to use high minimum support at higher level and reduced minimum support at lower levels. 5.3 MINING MULTI LEVELS ASSOCIATION RULE To mine multiple level association rule, one need to provide data at multiple level of abstraction and an efficient method to mine multiple level rules. With the recent development of data warehousing technology arranging data at multiple level of abstraction has been a common practice in the organizations. Therefore in the present study it is assumed that the database contains items in hierarchical order in the following manner: (i) The description of each Item in the form of [X, Description], X Item. (ii) A transaction data set T, which consists of a set of transactions [TID i, I 1, I 2.I n ], where TID n is transaction identifier. Let Item = [i 1, i 2, i 3 i m ], a set of m items. Let TID = [t 1, t 2, t 3 t n ], a transaction data base of n transactions. Each transaction is composed of an identifier TID, which is a subset of Item Coding Technique Concept hierarchy of items is numbered level 0 to level + 1. Transaction T contains the items associated to the terminal level. Level 0 is root/parent node and level + 1 is child node. Coding technique proposed in [31] is used for encoding the items of transaction database. For instance, according to Fig. 5.1 the item pen can be encoded as 1**, since it belongs to level 1, the ball pen can be encoded as11* and the Reynolds can be encoded as 111, which provide the clear identification about the location of Reynolds ball pen in the concept hierarchy. The problem of mining multi-level association rules, to discover the association rules consisting items associated to the different levels of abstraction, meeting the minimum user 95

5 specified support and confidence threshold is extended to discover multi level correlation rules to affirm the mined results with statistical techniques. In this study user specified support and confidence framework is used for mining at each level of abstraction. At initial level filtered and encoded tables are prepared as per the user specified support threshold. The codes shown in strikethrough style has less support threshold from the user specified support threshold as represented in Table 5.3, Table 5.4 and Table 5.5. At each level user specified support threshold is applied to avoid the uninteresting rules. In this study user is asked to provide following inputs to enforce the constraints on the mining process: Enter choice of strategy: Different Minimum support and confidence threshold at each level to filter items from encoded transaction table Uniform Minimum support and confidence threshold at each level To simplify the discussion a typical example which simulates the real life example related to stationary items is discussed and analyzed. List of stationary sales items is obtained through the survey of stationary shop keepers located in Meerut city. In all other chapters a real life database of farmers is used but in this chapter a transaction database related to stationary items is used due to the need of organization of data in concept hierarchy to mine multiple levels and cross levels of items. The list of stationary items is obtained after surveying the shopkeepers of stationary items. Example A shopping transaction database consists of two relations: (i) A Sales-Item description table: It consists of sets of attributes: item-code, category, brand, and price as described in Table 5.1. (ii) Sales-Transaction table: It registers each transaction, the transaction-id and set of items purchased as described in Table

6 Sample data of three days is used to discover interesting association rules. Each tuple in the table is a unique item set purchased in a day. Query is to find multi level interesting association rules and cross level interesting association rules in the database for the purchase patterns related to stationary items. Query can be expressed as standard structured query like data mining language. Interesting association rules are discovered using the algorithm shown in Fig.5.2. Table 5.1: Description of Sales Item Database Item Code Category Brand Price in Rupees 9101 Note book Capital 50/= 9102 Pen Parker 100/= Table 5.2 Sales Transaction Table of Stationary Items DAY TID Item Code 1 T1 9101, 9102,9103, 8102, T2 9101,9102,9103,8102, Algorithm to Discover Multiple Levels Association Rules Information represented in Table 5.3, Table 5.4 and Table 5.5 are used as input for algorithm shown in Fig Information given in the tables are scanned, support percentage is computed for each pattern and frequent patterns are identified by comparing the user input support thresholds with the computed support thresholds at level one, two and three. Similarly, confidence threshold is computed for each pattern. Strong rules are discovered at each level by comparing the user input level wise support threshold with the computed support thresholds. Filtered and encoded tables are prepared by removing the items of less support threshold from the user specified support threshold. In this example it is assumed 97

7 that the user specified support threshold is 1 %( 3 transaction out of 30 transaction). The items with less than 1 support per day are removed from the filtered tables. Strikethrough style is applied to represent the removed items as shown in Table 5.3, 5.4 and 5.5. Table 5.3: Filtered and encoded Transactions (Day1) Minimum Support 1% TID Items Purchased T1 111, 431,321,131,432 T2 111,431,321,131,432 T3 111,431,321 T4 111,431,321 T5 111,431,321,221 T6 221,412,222 T7 221,412,311 T8 221,412,231 T9 221,412 T10 221, ,321 Table 5.4: Filtered Encoded Transaction (Day2) Minimum Support 1% TID Items Purchased T1 111,131 T2 111,131 T3 111,431,321 T4 131,432,312 T5 131,432,221 T6 221,412 T7 221,412 T8 131,432,321 T9 221,412,112 T10 221,412 98

8 Algorithm-6: Identifying frequent patterns from concept hierarchy Input information: (i) A hierarchical-information encoded table of task related data in the form of Transaction-id(TID) and Item sets(items). Each item in the item sets contains encoded concept hierarchical information. (ii) User specified minimum support for each concept level. Begin { Find unique item set and corresponding support factor and store them in table 5.3, table 5.4 and 5.5. Items available in the table 5.3, 5.4 and 5.5 are compared with user input support and items which are below the user input thresholds are discarded. do(level 1 frequent one item sets) { Database is scanned discovering level 1, frequent one item sets (l=1 to ln{l represent level 1.}) Compute the support thresholds and stored in the respective tables item=item+1 } do (frequent two item sets) { do(level 1 to Leveln(n=number of levels present)) { Table is scanned and possible two pairs are discovered from level 1 using the codes of level 1 level 2 using the codes of level 2 level 3 using the codes of level 3 computed support >=user specified support store d in the respective tables itemi=itemi+2 } } while table is not empty do(frequent three item sets) { Find three items from table itemi, itemi+1,itemi+2 and corresponding support. If support >= user specified support store in table itemi=itemi+3 } end Output: Output is shown in various tables. Fig. 5.2 Algorithm-6 to Mine Association Rules from Concept Hierarchy 99

9 Table 5.5: Filtered Encoded Transaction (Day3) Minimum Support 1% TID Items Purchased T1 111,131,432 T2 111,321,431 T3 111,431,321 T4 131,432,311 T5 131,432 T6 221,412 T7 221,412 T8 131,432,321 T9 221,412,212 T10 221,412, Level 1 Frequent 1 Item Sets To discover frequent 1 item set at level 1, items 1**, 2**, 3** and 4** are discovered, support thresholds are computed and computed support thresholds are compared with user input support threshold. In this example user input support for level is 20%. Item sets found equal to and greater than 20% support are stored in the Table 5.6. Table 5.6: Level 1 Frequent 1 Item set (Minimum Support 20%) Items Support (%) 1** 60 2** 50 3** 36 4** 93 Total transactions in the data set are 30 and items with minimum 3 support threshold are considered for the frequent one item sets at level

10 Level 1 Frequent 2 Item Sets Level 1 frequent 2 item sets 1** and 2** are discovered from the data set, support thresholds are computed and compared with user input support. Table 5.7 indicates the level 1 frequent two item sets with minimum support threshold 20%. Table 5.7: Level 1 Frequent 2 Item Sets (Minimum Support 20%) Items Sets Support (%) 1**, 3** 33 1**, 4** 53 2**, 4** 50 3**, 4** Level 1 Frequent 3 Item Sets Table 5.8 indicates the level 1 frequent three item sets with support threshold 20%. Table 5.8: Level 1 Frequent 3 Item sets (Minimum Support 20%) Item sets Support (%) 1**,3**,4** Level 2 Frequent 1 Item Sets To discover frequent 1 item sets for level 2 minimum support thresholds is assumed as 10%. Table 5.9 indicates the level 2 frequent one item sets. Table 5.9: Level 2 Frequent 1 Item sets (Minimum Support 10%) Item Sets Support (%) 11* 40 13* 36 22* 50 32* 33 41* 43 43*

11 Level 2 Frequent 2 Item Sets To discover level 2 frequent 2 item sets, two items such as 11* and 43* are discovered, support threshold value is computed and compared with user input support threshold which is 10% for level 2. Table 5.10 indicates the level 2 frequent 2 item sets with minimum support 10%. Table 5.10: Level 2 Frequent 2 Item sets (Minimum Support 10%) Item Sets Support (%) 11*, 43* 23 11*,32* 26 22*, *,43* 30 13*,43* 20 11*,13* Level 2 Frequent 3 Item Sets To discover level 2 frequent 3 item sets, three item sets such as 11*, 32* and 43* are discovered, support threshold is computed and compared with user input support threshold which is 10% for level 2. Table 5.11 indicates the level 2 frequent three item sets with minimum support 10%. Table 5.11: Level 2 Frequent 3 Item sets (Minimum Support 10%) Item Sets Support (%) 11*, 43*,32* Level 3 Frequent 1 Item Sets To discover level 3 frequent 1 item sets, items such as 111 is discovered, support threshold is computed and compared with user input support threshold which is 6% for level 3. Table 5.12 indicates the level 3 frequent one item sets with minimum support 6%. 102

12 Table 5.12: Level 3 Frequent 1 Item sets (Minimum Support 6%) Item Sets Support (%) Level 3 Frequent 2 Item Sets To discover level 3 frequent 2 item sets, two item sets such as 111 and 431 are discovered, support threshold is computed and compared with user input support threshold which is 6% for level 3. Table 5.13 indicates the level 3 frequent two item sets with minimum support 6%. Table 5.13: Level 3 Frequent 2 Item sets (Minimum Support 6%) Item Sets Support (%) 111, , , , Level 3 Frequent 3 Item Sets To discover level 3 frequent 3 item sets, three item sets such as 111, 431 and 432 are discovered, support threshold is computed and compared with user input support threshold which is 6% for level 3. Table 5.12 indicates the level 3 frequent 3 item sets with minimum support 6%. Table 5.14: Level 3 Frequent 3 Item sets (Minimum Support 6%) Item Sets Support (%) 111,431,

13 5.3.3 Mining Multilevel Correlation Coefficient Researchers realized the demerits of confidence measure as confidence ignores the probability of body of the association rule and a new measure lift was given [80]. Lift originally called interest, measures how many times more often item X and item Y occur together than expected if they were statistically independent. The range of lift is 0 to For example there are two items, {X,Y} where X => Y has a support of 15% and a confidence of 60%. Because these values are high, therefore a specific association rule algorithm would assume this to be an interesting rule. However, if the probability to purchase item Y is 70%, then it is found that probability of purchasing Y has actually gone down, presumably because item X was purchased. In this result a negative correlation is found between buying item X, and buying item Y. The lift correlation can be defined as the below: lift (X=> Y) = 5.1) in this example : lift = (X => Y) = = If this correlation value is lower than 1, it indicates negative correlation between item X and item Y. Lift measure considers both P (X) and P (Y). If lift < 1 then item X and item Y appear less frequently together in the data than expected under the assumption of conditional independence. Item X and item Y are said to be negatively inter dependents. If lift =1 then item X and item Y appear as frequently together as expected under the assumption of conditional independence. If lift > 1 then item X and item Y appear more frequently together in the data than expected under the assumption of conditional independence. Item X and item Y are said to positively inter dependent. 104

14 The difference between confidence and lift lies in their formulation. Confidence is sensitive to the probability of item Y. Higher frequency of Y will ensure a higher confidence even if there is not true relationship between item X and item Y. If the threshold of the confidence is increased to avoid this situation, some important patterns of lower frequency will be lost. In contrast to confidence, lift value is not susceptible to the rare items problem. Lift focuses on the ratio of joint probability of item X and item Y with respect to their expected probabilities if they are independent. Item set with lower frequency can have high lift values [54]. Table 5.15 indicates the correlation obtained by lift measure between item sets at level 1. Table 5.16 shows the correlation between item sets at level 2 and table 5.17 indicates the correlation between item sets at level 3. Table 5.18 shows the correlation between cross level item sets Level 1 Correlation Coefficients of Frequently used items Correlation at level 1 between item code 1** and item code 2** lift = (a => b) = = 0.33 It indicate negative correlation between item code 1** and item code 3**. Correlation at level 1 between item code 1** and item code 3** lift = (a => b) = = 1.1 It indicate positive correlation between item code 1** and item code 3**. Correlation at level 1 between item code 1** and item code 4** lift = (a => b) = = 0.94 It indicate negative correlation between item code 1** and item code 4**. Correlation at level 1 between item code 2** and item code 4** lift = (a => b) = = 1.07 It indicate positive correlation between item code 1** and item code 4**. 105

15 Correlation at level 1 between item code 2** and item code 3** lift = (a => b) = =.36 It indicate negative correlation between item code 2** and item code 3**. Correlation at level 1 between item code 3** and item code 4** lift = (a => b) = = 1.07 It indicate positive correlation between item code 3** and item code 4**. All computed lift values are shown in the Table Table 5.15: Level 1, Correlation Coefficients of Frequently used Items Item Set Item Set Correlation 1** 2**.33 1** 3** 1.1 1** 4**.94 2** 3**.36 2** 4** ** 4** Level 2 Correlation Coefficient of Frequently used items Correlation at level 2 between item code 11* and item code 43* lift = (a => b) = = 1.15 It indicate positive correlation between item code 11* and item code 43*. Correlation at level 2 between item code 11* and item code 32* lift = (a => b) = = 1.96 It indicate positive correlation between item code 11* and item code 43*. Correlation at level 2 between item code 22* and item code 41* lift = (a => b) = = 2 It indicate positive correlation between item code 22* and item code 41*. 106

16 Correlation at level 2 between item code 32* and item code 43* lift = (a => b) = = 1.81 It indicate positive correlation between item code 32* and item code 43*. Correlation at level 2 between item code 13* and item code 43* lift = (a => b) = = 1.11 It indicates positive correlation between item code 32* and item code 43*. Correlation at level 2 between item code 11* and item code 13* lift = (a => b) = = 1.11 It indicates positive correlation between item code 32* and item code 43*. Table 5.16 shows the correlation between items at level 2. Table 5.16: Level 2, Correlation Coefficients of Frequently used Items Item Set Item Set Correlation 11* 43* * 32* * 41* * 43* * 43* * 13* Level 3 Correlation Coefficient of Frequently used Items Correlation at level 3 between item code 111 and item code 431 lift = (a => b) = = 2.5 It indicates positive correlation between item code 111 and item code 431. Correlation at level 3 between item code 111 and item code 321 lift = (a => b) = = 1.80 It indicates positive correlation between item code 111 and item code 321. Correlation at level 3 between item code 321 and item code 431 lift = (a => b) = =

17 It indicates positive correlation between item code 321 and item code 431. Correlation at level 3 between item code 221 and item code 412 lift = (a => b) = = 2.0 It indicates positive correlation between item code 221 and item code 412. Table 5.17 indicates the correlation between items at level 3. Table 5.17: Level 3, Correlation Coefficients of Frequently used items Item Set Item Set Correlation Mining Cross level Association Rules and Correlation Coefficients Cross level association rules refers association of one item to another item related to different categories and levels such as item coded as 1** (pen) and item coded as 43*(notebook) as shown in Fig Cross Level Correlation Coefficients Correlation at level 3 between item code 1** and item code 43* lift = (a => b) = = 1.6 It indicates positive correlation between item code 221 and item code 412. Table 5.18 indicates the correlation between item sets 1** and 43*. Table 5.18 shows the cross level correlation coefficient of code 1** and code 43* that is 1.6, confidence is 81% and support threshold is 43%. It conclude that 1** and 43* are frequent patterns due to the high support threshold 43% as well as strong rules due 108

18 to the high confidence threshold. Lift value is 1.6 which affirms the confidence threshold and ensures the validity of the discovered rule. Table 5.18: Cross Level Correlation Coefficient Item Set Item Set Correlation 1** 43* SUMMARY This study extended the scope of mining association rules from single level to multiple levels association rules and cross levels association rules, from uniform support framework to reduced support framework and also from mining multiple levels association rules to mining multiple levels and cross levels correlation coefficients from large transaction database. Methods of mining cross levels association rules and cross levels correlation coefficient are also studied and discussed to validate the results. Study shows that mining multiple levels and cross levels correlations validate the results discovered from mining multiple levels association rules and mining cross levels association rules. Mining multiple levels association allow to mine interesting patterns when data is available at multiple levels of abstraction, which has been a common practice nowadays and discovering correlation rules at multiple levels not only affirm the discovered patterns but also indicate the positive or negative relationships in to item sets. In this chapter for the purpose of simplicity a typical example similar to real life example is used. More concrete and generalized results can be obtained through the sample of real data. 109

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