COMP7103 Topic 2 Association Rules

作者: Arthur | 848 字, 4 分钟 | 2021-02-25 | 分类: Notes

comp7103, data mining, hku

翻译: EN

COMP7103 Data Mining

Topic 2 Association Rules

Market-Basket Model

A general many-many mapping (association) between two kinds of things

  • 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 the items, e.g., the things one customer buys on one day

Frequent Itemsets

Support

Support for itemset I (s(I)) = the number of baskets containing all items in I

Given a support threshold s, sets of items that appear in at least s baskets are called frequent itemsets

frequent_itemsets

Monotonicity

For any sets of items I and any set of items J, it holds that

association_rules_monotonicity

Applications

  • given that many people buy beer and diapers together
    • Run a sale on diapers; raise price of beer
    • Only useful if many buy diapers & beer
  • Items that appear together too often could represent plagiarism
  • Unusual words appearing together in a large number of documents

Association Rules

If-then rules I → j about the contents of baskets, I is a set of items and j is an item

  • i → j means
    • if a basket contains all the items in I then it is likely to contain j

Confidence

The probability of j given I

association_rules_confidence

association_rules_confidence_example

Finding Association Rules

find all association rules with support ≥ s and confidence ≥ c

Computation Model

  • Data is kept in raw files rather than in a database system
    • Stored on disk
    • Stored basket-by-basket
  • The true cost of mining disk-resident data is usually the number of disk I/O’s
  • In practice, association-rule algorithms read data in passes – all baskets read in turn
  • we measure the cost by the number of passes an algorithm takes

Association Rules Algorithms

Naïve Algorithm

  • Read file once, counting in main memory the occurrences of each pair
    • From each basket of n items, generate its n (n -1)/2 pairs by two nested loops
  • Fails if (#items)^2 exceeds main memory

A-Priori Algorithm

  • A two-pass approach called a-priori limits the need for main memory
  • Key idea: monotonicity
    • If a set of items appears at least s times, so does every subset
    • For pairs: if item i does not appear in s baskets, then no pair including i can appear in s baskets
  • Process
    • Pass 1
      • Read baskets and count in main memory the occurrences of each item (Requires only memory proportional to #items)
        • Items that appear at least s times are the frequent items
    • Pass 2
      • Read baskets again and count in main memory only those pairs both of which were found in pass 1 to be frequent
        • To count number of item pairs use a hash function
        • Requires memory proportional to square of frequent items only, plus a list of the frequent items, plus space for hashing

APriori_Algorithm

  • One pass for each k
  • Needs room in main memory to count each candidate k -set
  • For typical market-basket data and reasonable support (e.g., 1%), k = 2 requires the most memory

PCY Algorithm

  • Main observation: during pass 1 of A-priori, most memory is idle
  • Use that memory to keep additional info to improve storage during pass 2 of A-priori
  • Passes > 2 are the same as in A-Priori
  • Process
    • Pass 1
      • Use a hash function which bucketizes item pairs, that is, maps them to integers in [1,k]
      • Each bucket i in [1,k] is associated with a counter ci
      • During pass 1, as we examine a basket (e.g. {m,b,d,o})
        • update counters of single items
        • Generate all item pairs for that basket, hash each of them and add 1 to the corr. counter
    • Pass 2
      • 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 frequent bucket
      • Ignore all pairs belonging to non-frequent buckets (do not use a counter for them)

Simple Algorithm

  • Take a random sample of the market baskets
    • give a full pass on the data and keep a basket in main memory with probability p
    • A random sample is the best representative of a dataset
    • Keeping only the first baskets might not contain iPhones for example
    • If we cannot have a sample large enough then
      • Remove false positives with one more pass
  • Run A-priori or one of its improvements in main memory, so you don’t pay for disk I/O each time you give a pass on the data
    • Be sure you leave enough space for counts
  • Adjust the support threshold s accordingly

SON Algorithm

  • Two passes
  • No false positives or false negatives
  • Divide the dataset into chunks, where each chunk contains a subset of baskets
  • Process
    • Pass 1
      • Divide the dataset into chunks, where each chunk contains a subset of baskets
      • Let pi such that the ith chunk contains a fraction pi of the dataset
      • For each chunk i compute all frequent itemsets with support p i x s and store them on disk. This is the set of candidates for next pass
    • Pass 2
      • Read all frequent itemsets found in the previous pass (candidates)
      • For each of them count the number of occurrences and output only those with support at least s

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Arthur

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