先验算法（Apriori Algorithm）原理及python代码实现

先验算法（Apriori Algorithm）是关联规则学习的经典算法之一。先验算法的设计目的是为了处理包含交易信息内容的数据库（例如,顾客购买的商品清单，或者网页常访清单。）而其他的算法则是设计用来寻找无交易信息（如Winepi算法和Minepi算法）或无时间标记（如DNA测序）的数据之间的联系规则。关联分析的目的是从大规模数据集中寻找有趣关系的任务。这些关系可以有两种形式：频繁项集或者关联规则。频繁项集(frequent item sets)是指经常出现在一起的物品的集合，关联关系(association rules)暗示两种物品之间可能存在很强的关系。

先验算法采用广度优先搜索算法进行搜索并采用树结构来对候选项目集进行高效计数。它通过长度为 k-1的候选项目集来产生长度为k的候选项目集，然后从中删除包含不常见子模式的候选项。根据向下封闭性引理,该候选项目集包含所有长度为 k的频繁项目集。之后，就可以通过扫描交易数据库来决定候选项目集中的频繁项目集。

from __future__ import division, print_function
import numpy as np
import itertoolsclass Rule():def __init__(self, antecedent, concequent, confidence, support):self.antecedent = antecedentself.concequent = concequentself.confidence = confidenceself.support = supportclass Apriori():"""A method for determining frequent itemsets in a transactional database andalso for generating rules for those itemsets. Parameters:-----------min_sup: floatThe minimum fraction of transactions an itemets needs tooccur in to be deemed frequentmin_conf: float:The minimum fraction of times the antecedent needs to implythe concequent to justify rule"""def __init__(self, min_sup=0.3, min_conf=0.81):self.min_sup = min_supself.min_conf = min_confself.freq_itemsets = None       # List of freqeuent itemsetsself.transactions = None        # List of transactionsdef _calculate_support(self, itemset):count = 0for transaction in self.transactions:if self._transaction_contains_items(transaction, itemset):count += 1support = count / len(self.transactions)return supportdef _get_frequent_itemsets(self, candidates):""" Prunes the candidates that are not frequent => returns list with only frequent itemsets """frequent = []# Find frequent itemsfor itemset in candidates:support = self._calculate_support(itemset)if support >= self.min_sup:frequent.append(itemset)return frequentdef _has_infrequent_itemsets(self, candidate):""" True or false depending on the candidate has anysubset with size k - 1 that is not in the frequent itemset """k = len(candidate)# Find all combinations of size k-1 in candidate# E.g [1,2,3] => [[1,2],[1,3],[2,3]]subsets = list(itertools.combinations(candidate, k - 1))for t in subsets:# t - is tuple. If size == 1 get the elementsubset = list(t) if len(t) > 1 else t[0]if not subset in self.freq_itemsets[-1]:return Truereturn Falsedef _generate_candidates(self, freq_itemset):""" Joins the elements in the frequent itemset and prunesresulting sets if they contain subsets that have been determinedto be infrequent. """candidates = []for itemset1 in freq_itemset:for itemset2 in freq_itemset:# Valid if every element but the last are the same# and the last element in itemset1 is smaller than the last# in itemset2valid = Falsesingle_item = isinstance(itemset1, int)if single_item and itemset1 < itemset2:valid = Trueelif not single_item and np.array_equal(itemset1[:-1], itemset2[:-1]) and itemset1[-1] < itemset2[-1]:valid = Trueif valid:# JOIN: Add the last element in itemset2 to itemset1 to# create a new candidateif single_item:candidate = [itemset1, itemset2]else:candidate = itemset1 + [itemset2[-1]]# PRUNE: Check if any subset of candidate have been determined# to be infrequentinfrequent = self._has_infrequent_itemsets(candidate)if not infrequent:candidates.append(candidate)return candidatesdef _transaction_contains_items(self, transaction, items):""" True or false depending on each item in the itemset isin the transaction """# If items is in fact only one itemif isinstance(items, int):return items in transaction# Iterate through list of items and make sure that# all items are in the transactionfor item in items:if not item in transaction:return Falsereturn Truedef find_frequent_itemsets(self, transactions):""" Returns the set of frequent itemsets in the list of transactions """self.transactions = transactions# Get all unique items in the transactionsunique_items = set(item for transaction in self.transactions for item in transaction)# Get the frequent itemsself.freq_itemsets = [self._get_frequent_itemsets(unique_items)]while(True):# Generate new candidates from last added frequent itemsetscandidates = self._generate_candidates(self.freq_itemsets[-1])# Get the frequent itemsets among those candidatesfrequent_itemsets = self._get_frequent_itemsets(candidates)# If there are no frequent itemsets we're doneif not frequent_itemsets:break# Add them to the total list of frequent itemsets and start overself.freq_itemsets.append(frequent_itemsets)# Flatten the array and return every frequent itemsetfrequent_itemsets = [itemset for sublist in self.freq_itemsets for itemset in sublist]return frequent_itemsetsdef _rules_from_itemset(self, initial_itemset, itemset):""" Recursive function which returns the rules where confidence >= min_confidenceStarts with large itemset and recursively explores rules for subsets """rules = []k = len(itemset)# Get all combinations of sub-itemsets of size k - 1 from itemset# E.g [1,2,3] => [[1,2],[1,3],[2,3]]subsets = list(itertools.combinations(itemset, k - 1))support = self._calculate_support(initial_itemset)for antecedent in subsets:# itertools.combinations returns tuples => convert to listantecedent = list(antecedent)antecedent_support = self._calculate_support(antecedent)# Calculate the confidence as sup(A and B) / sup(B), if antecedent# is B in an itemset of A and Bconfidence = float("{0:.2f}".format(support / antecedent_support))if confidence >= self.min_conf:# The concequent is the initial_itemset except for antecedentconcequent = [itemset for itemset in initial_itemset if not itemset in antecedent]# If single item => get itemif len(antecedent) == 1:antecedent = antecedent[0]if len(concequent) == 1:concequent = concequent[0]# Create new rulerule = Rule(antecedent=antecedent,concequent=concequent,confidence=confidence,support=support)rules.append(rule)# If there are subsets that could result in rules# recursively add rules from subsetsif k - 1 > 1:rules += self._rules_from_itemset(initial_itemset, antecedent)return rulesdef generate_rules(self, transactions):self.transactions = transactionsfrequent_itemsets = self.find_frequent_itemsets(transactions)# Only consider itemsets of size >= 2 itemsfrequent_itemsets = [itemset for itemset in frequent_itemsets if not isinstance(itemset, int)]rules = []for itemset in frequent_itemsets:rules += self._rules_from_itemset(itemset, itemset)# Remove empty valuesreturn rules

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