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Basic association rules

Before you begin…

Download and install Titus. This article was tested with Titus 0.7.1; newer versions should work with no modification. Python >= 2.6 and < 3.0 is required.

Launch a Python prompt and import the following:

Python 2.7.6
Type "help", "copyright", "credits" or "license" for more information.
>>> import random
>>> import json
>>> from collections import Counter
>>> import titus.prettypfa
>>> from titus.genpy import PFAEngine

The basic form

PFA has no dedicated library for association rules because the scoring procedure is easy to express in terms of functional primitives. If you’re building an association rule model with many rules, be sure to make the antecedent a map-lookup (constant time) rather than a loop through an array of rules (linear time).

Map keys in PFA are always strings (a restriction inherited from Avro and JSON), so you’ll need a way to project your sets of antecedent items onto a string. Use a.sort to remove dependency on item order and s.join to make one string with a DELIMITER that is not in your item’s alphabet. Then it simply becomes a matter of looking up a value in a map.


pfaDocument = titus.prettypfa.jsonNode('''
  Item = string;
  Rule = record(Rule,
                consequent: array(Item),
                confidence: double)

input: array(Item)
output: Rule
  rules(map(Rule)) = {}
  ifnotnull(rule: try rules[s.join(a.sort(input), <<DELIMITER>>)])
    json(Rule, {consequent: [], confidence: 0.0})
''', DELIMITER = {"string": DELIMITER})

The French quotes (<< and >>) are a mechanism for inserting simple values into a PrettyPFA document. The values you insert have to be raw PFA (hence {"string": " - "} rather than just " - ").

Producing a model

Like most models, association rules are much easier to score than to produce. You can use any tool to produce them as long as you make a JSON structure adhering to your chosen Rule declaration above. For the sake of providing a complete example, here’s a simple association rule producer in Python.

items = ["milk", "bread", "butter", "beer", "diapers"]
transactions = [random.sample(items, random.randint(1, len(items))) for x in xrange(10000)]

def pairsOfSubsets(transaction):
    for binary in xrange(2**0, 2**len(transaction) - 1):
        yield [transaction[i] for i in xrange(len(transaction)) if binary & 2**i != 0], \
              [transaction[i] for i in xrange(len(transaction)) if binary & 2**i == 0]

def makeHashable(set):
    return DELIMITER.join(sorted(set))

supports = Counter()
for transaction in transactions:
    for antecedent, consequent in pairsOfSubsets(transaction):
        supports[makeHashable(antecedent)] += 1
        supports[makeHashable(consequent)] += 1
    supports[makeHashable(transaction)] += 1

rules = {}
for transaction in transactions:
    numer = float(supports[makeHashable(transaction)])
    for antecedent, consequent in pairsOfSubsets(transaction):
        confidence = numer / supports[makeHashable(consequent)]
        rules[makeHashable(antecedent)] = {"consequent": consequent, "confidence": confidence}

Insert the model into PFA

These rules already have the right structure for our Rule type, so we can just insert them into the PFA. We’ll find out if it was the right structure when we try to make an engine out of it.

pfaDocument["cells"]["rules"]["init"] = rules

Test it!

engine, = PFAEngine.fromJson(pfaDocument)

for transaction in transactions:
    for antecedent, consequent in pairsOfSubsets(transaction):
        print "{0:50s} -> {1}".format(json.dumps(antecedent), json.dumps(engine.action(antecedent)))