Frequent Itemsets

Frequent Itemsets

1. Association rule discovery

1.1. Overview

• Example motivation: supermarket shelf management
• Goal:
  • Identify items that are bought together by sufficiently many customers
• Approach: Process the sales data collected with barcode scanners to find dependencies among items
• Example:
  • If someone buys diaper and milk, then he/she is likely to buy beer

1.2. The market-basket model

• Describe a common form of many-to-many relationship between two kinds of objects.
• A large set of items. e.g.: things sold in a supermarket.
• A large set of baskets
  • each of which is a small set of items. e.g.: things one customer buys on one trip to the supermarket.
• Number of baskets cannot fit into memory.
• Want to discover association rule:
  • People who bought {x,y,z} tend to buy {v,w}
  • Amazon’s recommendation

1.3. Example applications

• Baskets = sets of products someone bought in one trip to the store; Items = products
  • Real market baskets: Chain stores keep TBs of data about what customers buy together
    • Tells how typical customers navigate stores, lets them position tempting items
    • Suggests tie-in tricks, e.g., run sale on diapers and raise the price of beer
    • Need the rule to occur frequently, or no monetary benefit
• Baskets = sentences; Items = documents containing those sentences
  • Items that appear together too often could represent plagiarism
  • Notice items do not have to be in baskets
• Baskets = patients; Items = drugs and side-effects
  • Has been used to detect combinations of drugs that result in particular side-effects
  • But requires extension: Absence of an item needs to be observed as well as presence

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2. Definition of a frequent itemsets.

2.1. Definition

• A set of items that appears in many baskets is said to be frequent.
• Assume a value s: support threshold.
• If I is a set of items.
  • The support for I is the number of baskets in which I is a subset.
• I is frequent if its support is s or higher.

2.2. Example of frequent itemsets

• Items = {milk, coke, pepsi, beer, juice}
• Observed baskets:
  • B1: m,c,b
  • B2: m,p,j
  • B3: m,b
  • B4: c,j
  • B5: m,p,b
  • B6: m,c,b,j
  • B7: c,b,j
  • B8: b,c
• Support value s = 3 (items appear in at least three baskets)
• Frequent itemsets:
  • {m} appears 5 times in baskets: B1, B2, B3, B6, B6
  • {c} appeats 5 times in baskets: B1, B4, B6, B7, B8
  • {b} appears 6 times in baskets: B1, B3, B5, B6, B7, B8
  • {j} appears 4 times in baskets: B2, B4, B6, B7
  • Pair {m,b} appears 4 times in baskets: B1, B3, B5, B6
  • Pair {b,c} appears 4 times in baskets: B1, B6, B7, B8
  • Pair {c,j} appears 3 times in baskets: B4, B6, B7

2.3. Association rules

• If-then rules about the contents of baskets.
• Notation ${i_1, i_2, …, i_k} \rightarrow j$ means
  • If a basket contains all of items $i_1,…,i_k$ then it is likely to contain item $j$.
• Confidence of this association rule is the probability of having item $j$ in a basket given that basket already contained items ${i_1,…,i_k}$.
  • The fraction of the basket with ${i_1,…,i_k}$ that also contain j.
• Example:
  • Given the following baskets:
    • B1: m,c,b
    • B2: m,p,j
    • B3: m,b
    • B4: c,j
    • B5: m,p,b
    • B6: m,c,b,j
    • B7: c,b,j
    • B8: b,c
  • An association rule ${m,b} \rightarrow c$ represents the relationship between having item $c$ and having both items $m$ and $b$.
    • There are four baskets containing both $m$ and $b$: B1, B3, B5, B6
    • Out of these four baskets, two baskets also contain $c$: B1, B6
    • The confidence of this association rule is: $C = \frac{2}{4} = 0.5$

2.4. Interesting association rules

• Not all high-confidence rules are interesting
• The rule $X \rightarrow milk$ may have high confidence for many itemsets X, because milk is just purchased very often (independent of X) and the confidence will be high.
  • Not interesting!
• Interest of an association rule $I \rightarrow j$: difference between its confidence and the fraction of baskets that contain j
• Interesting rules are those with high positive or negative interest values (usually above 0.5)
• Example:
  • Given the following baskets:
    • B1: m,c,b
    • B2: m,p,j
    • B3: m,b
    • B4: c,j
    • B5: m,p,b
    • B6: m,c,b,j
    • B7: c,b,j
    • B8: b,c
  • An association rule: ${m,b} \rightarrow c$ has confidence $C = \frac{2}{4} = 0.5$
  • Since item $c$ apepars in 5 out of the total 8 baskets, fraction of baskets that contains $c$ is $\frac{5}/{8} = 0.625$.

  • Association rule interest is: $

    0.5-5/8

    = 0.125$

    • This association rule is not very interesting!

2.5. Finding association rules

• Problem: Find all association rules with support greater than s and confidence greater than c.
  • Note: Support of an association rule is the support of the set of items on the left side
• Hard part: Finding the frequent itemsets!
  • If ${i_1, i_2,…, i_k} \rightarrow j$ has high support and confidence, then both ${i_1, i_2,…, i_k}$ and ${i_1, i_2,…,i_k, j}$ will be “frequent”

2.6. Mining association rules

• Step 1: Find all frequent itemsets I
• Step 2: Rule generation

  • For every subset A of I, generate a rule $A \rightarrow I

    A$

    • Since I is frequent, A is also frequent
    • Variant 1: Single pass to compute the rule confidence $confidence(A,B \rightarrow C,D) = support(A,B,C,D) / support(A,B)$
    • Variant 2: Observation: If $A,B,C \rightarrow D$ is below confidence, so is $A,B \rightarrow C,D$
      • Can generate bigger rules from smaller ones!
  • Output the rules above the confidence threshold

2.7. Example

• Given the following baskets:
  • B1: m,c,b
  • B2: m,p,j
  • B3: m,b
  • B4: c,j
  • B5: m,p,b
  • B6: m,c,b,j
  • B7: c,b,j
  • B8: b,c
• We have the following parameters:
  • Support threshold $s = 3$: We want frequent itemsets that appear at least three times or higher.
  • Association rule confidence $c = 0.75$: We want association rules with confience threshold at least 0.75 or higher.
• Given the support threshold $s=3$, we have the following frequent itemsets:
  • {b,m}, {b,c}, {c,m}, {c,j}, {m,c,b}

• Based on these itemsets, we can generate the following rules:

  $b \rightarrow m$

• Item $b$ appears in 6 baskets: B1, B3, B5, B6, B7, B8
• Out of these baskets, 4 of them contains $m$: B1, B3, B5, B6
• Confidence $C = \frac{4}{6} = 0.6667$
• This is below confidence threshold

</details>

<summary>$m \rightarrow b$</summary>

• Item $m$ appears in 5 baskets: B1, B2, B3, B5, B6
• Out of these baskets, 4 of them contains $b$: B1, B3, B5, B6
• Confidence $C = \frac{4}{5} = 0.8$
• Basket fraction of $b$ is $\frac{6}{8} = 0.75$

• Interest = $$

  0.8-0.75

  = 0.05$ (not very interesting!)

</details>

<summary>$b \rightarrow c$</summary>

• Item $b$ appears in 6 baskets: B1, B3, B5, B6, B7, B8
• Out of these baskets, 4 of them contains $c$: B1, B6, B7, B8
• Confidence $C = \frac{4}{6} = 0.6667$
• Basket fraction of $c$ is $\frac{4}{8} = 0.5$

• Interest = $$

  0.6667-0.5

  = 0.1667$ (not very interesting!)

</details>

<summary>$c \rightarrow b$</summary>

• Item $c$ appears in 5 baskets: B1, B4, B6, B7, B8
• Out of these baskets, 4 of them contains $b$: B1, B6, B7, B8
• Confidence $C = \frac{4}{5} = 0.8$
• Basket fraction of $b$ is $\frac{6}{8} = 0.75$

• Interest = $$

  0.8-0.75

  = 0.15$ (not very interesting!)

</details>

<summary>$b \rightarrow c,m$</summary>

• Item $b$ appears in 6 baskets: B1, B3, B5, B6, B7, B8
• Out of these baskets, 2 of them contains both $c$ and $m$: B1, B6
• Confidence $C = \frac{2}{6} = 0.3333$
• Basket fraction of the pair $m$ and $c$ is $\frac{2}{8} = 0.25$

• Interest = $$

  0.3333-0.25

  = 0.08$ (not very interesting!)x

</details>

<summary>$b,c \rightarrow m$</summary>

• Pair $b$ and $c$ appears in 4 baskets: B1, B6, B7, B8
• Out of these baskets, 2 of them contains $m$: B1, B6
• Confidence $C = \frac{2}{4} = 0.5$
• Basket fraction of $m$ is $\frac{5}{8} = 0.625$

• Interest = $$

  0.5-0.625

  = 0.125$ (not very interesting!)

</details>

<summary>$b,m \rightarrow c$</summary>

• Item $b$ and $m$ appears in 4 baskets: B1, B3, B5, B6
• Out of these baskets, 2 of them contains $m$: B1, B6
• Confidence $C = \frac{2}{4} = 0.5$
• Basket fraction of $c$ is $\frac{4}{8} = 0.5$

• Interest = $$

  0.5-0.5

  = 0$ (not very interesting!)

</details>

• Exercise: Identify whether the remaining association rules of {c,m}, {c,j}, and {m,c,b} meet the confidence threshold. If they do, identify if they are interesting!

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3. Finding frequent itemsets

3.1. Computation model

• Back to finding frequent itemsets
• Typically, data is kept in flat files rather than in a database system:
  • Stored on disk
  • Stored basket-by-basket
  • Baskets are small but we have many baskets and many items
    • Expand baskets into pairs, triples, etc. as you read baskets
    • Use k nested loops to generate all sets of size k

3.2. Computational cost

• The true cost of mining disk-resident data is usually the number of disk I/Os
• In practice, association-rule algorithms read the data in passes: all baskets read in each pass.
• We measure the cost by the number of passes an algorithm makes over the data

3.3. Main-memory bottleneck

• For many frequent-itemset algorithms, main-memory is the critical resource.
• As we read baskets, we need to count something, e.g., occurrences of pairs of items
• The number of different things we can count is limited by main memory
  • Swapping counts in/out is a disaster

3.4. Finding frequent pairs

• The hardest problem often turns out to be finding the frequent pairs of items ${i_1, i_2}$.
  • Why? Freq. pairs are common, freq. triples are rare, and frequent sets with higher item counts are much rarer!
• Let’s first concentrate on pairs, then extend to larger sets
• The approach:
  • We always need to generate all the itemsets
  • But we would only like to count (keep track) of those itemsets that in the end turn out to be frequent

3.5. Naive approach

• Read file once, counting in main memory the occurrences of each pair:
  • From each basket of n items, generate its $\frac{n(n-1)}{2}$ pairs by two nested loops
• Fails if the square of number of items exceeds main memory
  • Remember: number of items can be 100K (Wal-Mart) or 10B (Web pages)
  • Suppose $10^5$ items, counts are 4-byte integers
  • Number of pairs of items: $10^{5}(10^{5}-1)/2 = 5*10^9$
  • Therefore, $2*10^{10}$, or 20 GB of memory, is needed.

3.6. Counting pairs in memory

Approach 1: Count all pairs using a matrix

• Requires 4 bytes per pair
• If n = total number items
  • Count pair of items {i, j} only if $i \leq j$
  • Keep pair counts in lexicographic order:
    • {1,2}, {1,3},…, {1,n}, {2,3}, {2,4},…,{2,n}, {3,4},…
  • Pair {i, j} is at position (i –1)(n – i/2) + j –1
  • Total number of pairs n(n –1)/2; total bytes= $2n^2$

Approach 2: Keep a table of triples [i, j, c]

• The count of the pair of items {i, j} is c.
• If integers and item ids are 4 bytes, we need approximately 12 bytes for pairs with count > 0
• Plus some additional overhead for the hashtable
• but only for pairs with count > 0
• Beats Approach 1 if less than 1/3 of possible pairs actually occur

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4. A-Priori algorithm.

4.1. Overview

• Limit the need for main memory.
• Key idea: monotonicity
  • If a set of items I appears at least s times, so does every subset J of I.
• Contrapositive: If an item i does not appear in s baskets, then no pair containing i can appear in s baskets.

4.2. A-Priori algorithm

• Pass 1: read baskets and count the item occurrences. Only keep items that appear at least s times - frequent items.
  • Requires only amount of memory proportional to number of items
• Pass 2: read baskets again and only count in main memory only those pairs whose both items were found to be frequent from Pass 1.
  • Requires memory proportional to square of frequent items only (for counts)
  • Plus a list of the frequent items (so you know what must be counted)
• Repeat the process with increasing number of items added to only sets found to be frequent.

4.3. Frequent triples

• For each k, we construct two sets of k-tuples (sets of size k):
  • $C_k$ = candidate k-tuples = those that might be frequent sets (support > s) based on information from the pass for k–1
  • $L_k$ = the set of truly frequent k-tuples

4.4. Example

• Review the sequential implementation of A-Priori
• Confirm that the output matches with the example in Section 2.2.
• Question: How to turn this sequential implementation into the mapreduce programming paradigm for Spark (See Section 6)?

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5. PCY (Park-Chen-Yu) Algorithm

5.1. Observation

• Observation: In pass 1 of A-Priori, most memory is idle
  • We store only individual item counts
  • Can we use the idle memory to reduce memory required in pass 2?
• Pass 1 of PCY: In addition to item counts, maintain a hash table with as many buckets as fit in memory
  • Keep a count for each bucket into which pairs of items are hashed
  • For each bucket just keep the count, not the actual pairs that hash to the bucket!

5.2. PCY Algorithm - First Pass

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FOR (each basket) :
    FOR (each item in the basket) :
        add 1 to item’s count;
    FOR (each pair of items) :
        hash the pair to a bucket;
        add 1 to the count for that bucket;
 

• Few things to note:
  • Pairs of items need to be generated from the input file; they are not present in the file
  • We are not just interested in the presence of a pair, but we need to see whether it is present at least s (support) times
• If a bucket contains a frequent pair, then the bucket is surely frequent
  • However, even without any frequent pair, a bucket can still be frequent
• For a bucket with total count less than s, none of its pairs can be frequent
• Pairs that hash to this bucket can be eliminated as candidates (even if the pair consists of 2 frequent items)

5.3. PCY Algorithm - Second Pass

• Before the second pass
  • Replace the buckets by a bit-vector
  • 1 means the bucket count exceeded the support s (call it a frequent bucket); 0 means it did not.
  • 4-byte integer counts are replaced by bits, so the bit-vector requires 1/32 of memory
  • Also, decide which items are frequent and list them for the second pass
• The second pass
  • Count all pairs {i,j} that meet the conditions for being a candidate pair:
    • Both i and j are frequent items
    • The pair {i,j} hashes to a bucket whose bit in the bit vector is 1 (i.e., a frequent bucket)
  • Both conditions are necessary for the pair to have a chance of being frequent

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6. SON: Savasere-Omiecinski-Navathe

6.1. Overview

• Under two passes.
• Adaptable to a distributed data model (mapreduce).
• Repeatedly read small subsets of the baskets into main memory and perform a-priori on these subsets, using a support that is equal to the main support divided by the total numbers of subsets.
• Aggregate all candidate itemsets and determine which are frequent in the entire set.

6.2. SON: MapReduce Phase 1

• Find candidate itemsets
• Map: Take the assigned subset of the baskets and find the itemsets frequent in the subset.
  • Lower the support threshold from s to ps if each Map task gets fraction $p$ ($p$ < 1) of the total input file.
  • The output is a set of key-value pairs (F, 1), where F is a frequent itemset from the sample. The value is always 1 and is irrelevant.
• Reduce: Each Reduce task is assigned a set of keys, which are itemsets. The value is ignored, and the Reduce task simply produces those keys (itemsets) that appear one or more times.
• The output of the first Reduce function is the candidate itemsets.

6.3. SON: MapReduce Phase 2

• Find true frequent itemsets
• Map: The Map tasks for the second Map function take all the output from the first Reduce Function (the candidate itemsets) and a portion of the input data file.
  • Each Map task counts the number of occurrences of each of the candidate itemsets among the baskets in the portion of the dataset that it was assigned.
  • The output is a set of key-value pairs (C, v), where C is one of the candidate sets and v is the support for that itemset among the baskets that were input to this Map task.
• Reduce: The Reduce tasks take the itemsets they are given as keys and sum the associated values.
  • The result is the total support for unique itemsets (the keys)
  • Those itemsets whose sum of values is at least s are frequent in the whole dataset, so the Reduce task outputs these itemsets.