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1 .. Cal Poly CSC 4: Knowledge Discovery from Data Alexander Dekhtyar.. Data Mining: Mining Association Rules Examples Course Enrollments Itemset. I = { CSC3, CSC3, CSC40, CSC40, CSC4, CSC44, CSC4, CSC44, CSC4, CSC47, CSC474, CSC480}. Column Course Number Course CSC3 Intro Databases CSC3 Database Design, Modeling, Implementation 3 CSC40 Software Requirements 4 CSC40 Software Construction CSC4 Autonomous Mobile Robotics CSC44 Implementation of OS 7 CSC4 Intro Computer Security 8 CSC44 Intro Networks 9 CSC4 Advanced Networks 0 CSC47 Intro Graphics CSC474 Computer Animation CSC480 Artificial Intelligence Market Baskets. The market baskets in our dataset consist of the Computer Science electives selected by individual students. Consider the list of market baskets in Figure This list can be represented as a full binary matrix as shown in Figure. Example. Consider the itemset T = {CSC3, CSC3, CSC4}. Support set of T in the dataset is Sup(T) = {s,s 8 }. Therefore, support(t) = Sup(T) = = 0..
2 s s s 3 s 4 s s s 7 s 8 s 9 s 0 s s s 3 s 4 s s s 7 s 8 s 9 s CSC3, CSC3, CSC40, CSC40, CSC44 CSC40, CSC40, CSC44, CSC4, CSC480 CSC3, CSC4, CSC44, CSC44 CSC3, CSC3, CSC47, CSC474 CSC3, CSC3, CSC4, CSC47, CSC474, CSC480 CSC40, CSC40, CSC480 CSC4, CSC44, CSC4, CSC44, CSC4, CSC480 CSC4, CSC44, CSC4 CSC47, CSC474 CSC3, CSC4, CSC44, CSC47, CSC480 CSC4, CSC4, CSC44, CSC480 CSC3, CSC3, CSC40, CSC480 CSC3, CSC40, CSC40, CSC44 CSC40, CSC47, CSC480 CSC3, CSC3, CSC4, CSC44, CSC4 CSC47, CSC474, CSC480 CSC44, CSC47 CSC3, CSC3, CSC4, CSC480 CSC40, CSC40, CSC47, CSC474 CSC44, CSC480 Figure : Student Enrollment Dataset: Market Baskets Item s s s s s s s s s s s s s s s s s s s s Count: Support: Figure : Student Enrollment Dataset: Full Binary Vectors
3 Example. Consider an association rule R = CSC40 CSC40. The support set for R is Sup(R ) = {s,s,s,s 3,s 9 }. The support of R is support(r ) = Sup(R ) = = 0.. The support set for {CSC40} is {s,s,s,s,s 3,s 4,s 9 }. The confidence of the rule R is then confidence(r ) = support(r ) support({csc40}) = = 7 = Apriori Algorithm minconf. Consider the value of minimal support, minsup = 0.. Goal. We trace the work of the Apriori algorithm in discovery of frequent itemsets with support of at least minsup (0.). Step. Itemsets of size. First, we discover frequent itemsets of size. F = {{CSC3}, {CSC3}, {CSC40}, {CSC40}, {CSC4}, {CSC44}, {CSC4}, {CSC44}, {CSC47}, {CSC474}, {CSC480}}. Note: support({csc4}) = 0. < minsup, so CSC4 is excluded from consideration. All other columns have support of 0. or higher and they are included. Step.. Itemsets of size. Join Step. pairs of items from C. On this step, we construct the list of all Note: The join step for size itemsets is trivial: it involves computing cartesian product of C. C = F F. Step.. Itemsets of size. Pruning Step. For itemsets of size, the pruning step of the cadidategen() function is trivial. Nothing is pruned, C remains intact. Step.3. Itemsets of size. Support computation. Step. generated 0 = possible pairings. We now need to prune this set, by excluding from it all pairs of courses that have low support. We can construct the following Support table for our dataset: 3
4 From the table above, the following pairs of courses exceed minsup: Itemset Baskets Frequency support {CSC3,CSC3} {s,s 4,s,s,s,s 8 } 0.3 {CSC3,CSC480} {s,s 0,s,s 3,s 8 } 0. {CSC40,CSC40} {s,s,s,s 3,s 9 } 0. {CSC40,CSC480} {s,s,s,s 3,s 4 } 0. {CSC47,CSC474} {s 4,s,s 9,s,s 9 } 0. So, F = {{CSC3,CSC3}, {CSC3,CSC480}, {CSC40,CSC40}, {CSC40,CSC480}, {CSC47,CSC474}}. Step 3.. Itemsets of size 3. Join Step. On this step, we join all pairs of sets from F trying to form candidate frequent itemsets of size 3. We are able to join the following pairs of sets: First itemset Second itemset Join ID {CSC3,CSC3} {CSC3,CSC480} {CSC3,CSC3,CSC480} c {CSC40,CSC40} {CSC40,CSC480} {CSC40,CSC40,CSC480} c {CSC3,CSC480} {CSC40,CSC480} {CSC3,CSC40,CSC480} c 3 C 3 = {{CSC3,CSC3,CSC480}, {CSC40,CSC40,CSC480}, {CSC3,CSC40,CSC480}}. Step 3.. Itemsets of size 3. Pruning Step. For {CSC3, CSC3, CSC480}: {CSC3,CSC3} F {CSC3,CSC480} F {CSC3,CSC480} F For {CSC40, CSC40, CSC480}: {CSC40,CSC40} F {CSC40,CSC480} F {CSC40,CSC480} F For {CSC3, CSC40, CSC480}: {CSC3,CSC480} F {CSC40,CSC480} F {CSC3,CSC40} F Therefore, all three elements of C 3 are not frequent itemsets and the Apriori Algoritm can stop there and return F = F F as the set of all frequent itemsets. 4
5 Takehome Problem Run Apriori Algorithm by hand with minsup = 0.. Generation of Association Rules Frequent Itemsets. In previous section, we discovered that Student Enrollment dataset has (eleven) frequent itemsets of size (all singleton sets except for {CSC4}) and five frequent itemsets of size : {{CSC3, CSC3}, {CSC3, CSC480}, {CSC40, CSC40}, {CSC40, CSC480}, {CSC47, CSC474}}. This gives rise to 0 candidate association rules with a single item on the right side. For each of them, we compute confidence. ID Rule Frequent Itemset Support Left side support Confidence 9 R CSC3 CSC3 3 = 0.7 R CSC3 CSC3 9 R 3 CSC3 CSC480 9 = 0. R 4 CSC480 CSC3 = R CSC40 CSC40 7 = 0.74 R CSC40 CSC40 7 R 7 CSC40 CSC480 7 = 0.74 R 8 CSC480 CSC40 = R 9 CSC47 CSC474 8 = 0. R 0 CSC474 CSC47 Depending on the values of minconf, we will report the following: minconf =. We report rules R,R and R minconf <. We report rules R,R and R 0 from above, plus R and R 7. minconf > 0.. In addition to the rules above, we report R,R 3 and R 9. Takehome Problem After discovering all frequent itemsets with support of at least 0., report all association rules in the dataset for minconf levels of, 0.7, 0. and 0..
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