Homework 5

Answer the following questions: (10 point each)

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1- Consider the data set shown in Table 5.1.

Table 5.1. Example of market basket transactions.

Customer ID

Transaction ID

Items Bought

1

0001

{a, d, e}

2

0024

{a, b, c, e}

2

0012

{a, b, d, e}

2

0031

{a, c, d, e}

3

0015

{b, c, e}

3

0022

{b, d, e}

3

0029

{c, d}

4

0040

{a, b, c}

5

0033

{a, d, e}

(a) Compute the support for item sets {e}, {b, d}, and {b, d, e} by treating each transaction ID as a market basket.

S({c}) =

S({e, d}) =

S({a, b, d}) =

S({c, d}) =

(b) Use the results in part (a) to compute the confidence for the association

rules {b, d} {e} and {e} {b, d}. Is confidence a symmetric measure?

c(bd e) =

c(e bd) =

2. Consider the market basket transactions shown in Table 5.2.

Table 5.2. Market basket transactions

Transaction ID

Items Bought

1

{Milk, Beer, Diapers}

2

{Bread, Butter, Milk}

3

{Milk, Diapers, Cookies}

4

{Bread, Butter, Cookies}

5

{Beer, Cookies, Diapers}

6

{Milk, Diapers, Bread, Butter, Cheese}

7

{Bread, Butter, Diapers}

8

{Beer, Diapers}

9

{Milk, Diapers, Bread, Butter}

10

{Beer, Cookies}

(a) What is the maximum number of association rules that can be extracted from this data (including rules that have zero support)?

Answer:

(b) What is the maximum size of frequent item sets that can be extracted (assuming minusup > 0)?

Answer:

(c) Write an expression for the maximum number of size-3 itemsets that can be derived from this data set.

Answer:

(d) Find an itemset (of size 2 or larger) that has the largest support.

Answer:

(e) Find a pair of items, a and b, such that the rules {a} → {b} and {b} → {a} have the same confidence.

Answer:

3. The Apriori algorithm uses a generate-and-count strategy for deriving frequent itemsets. Candidate itemsets of size k+1 are created by joining a pair of frequent itemsets of size k (this is known as the candidate generation step). A candidate is discarded if any one of its subsets is found to be infrequent during the candidate pruning step. Suppose the Apriori algorithm is applied to the data set shown in Table 5.3 with minsup = 30%, i.e., any itemset occurring in less than 3 transactions is considered to be infrequent.

Table 5.3. Example of market basket transactions.

Transaction ID

Items Bought

1

{a, b, c}

2

{b, c, d}

3

{a, b, d}

4

{a, c, d}

5

{b, c}

6

{c, d}

7

{c}

8

{a, b, c}

9

{a, d}

10

{b, d}

(a) Draw an itemset lattice representing the data set given in Table 5.3.

Label each node in the lattice with the following letter(s):

• N: If the itemset is not considered to be a candidate itemset by the Apriori algorithm. There are two reasons for an itemset not to be considered as a candidate itemset: (1) it is not generated at all during the candidate generation step, or (2) it is generated during the candidate generation step but is subsequently removed during the candidate pruning step because one of its subsets is found to be infrequent.

• F: If the candidate itemset is found to be frequent by the Apriori algorithm.

• I: If the candidate itemset is found to be infrequent after support counting.

Answer:

(b) What is the percentage of frequent itemsets (with respect to all itemsets in the lattice)?

Answer:

(c) What is the pruning ratio of the Apriori algorithm on this data set? (Pruning ratio is defined as the percentage of itemsets not considered to be a candidate because (1) they are not generated during candidate generation or (2) they are pruned during the candidate pruning step.)

Answer:

(d) What is the false alarm rate (i.e, percentage of candidate itemsets that are found to be infrequent after performing support counting)?

Answer:

1

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