在本案例中，我们结合商场的购物记录，实现经典的关联规则挖掘算法APRIORI算法和FPGrowth算法。具体来看，我们将实现频繁K项集的查找，条件模式基的查找，FP树的构建。最后，我们将从购物记录中寻找关系密切的商品，从而能给商场的商品销售提供建议支持。
import pandas as pd
import numpy as np
由于数据集文件以txt文件存储，我们选择Pandas中的read_table()函数读入数据。
data = pd.read_table('./input/shopping-record.txt', header=None)
data.head()
| 0 | |
|---|---|
| 0 | citrus fruit,semi-finished bread,margarine,rea... |
| 1 | tropical fruit,yogurt,coffee |
| 2 | whole milk |
| 3 | pip fruit,yogurt,cream cheese,meat spreads |
| 4 | other vegetables,whole milk,condensed milk,lon... |
从结果上来看，虽然每行数据记录的项之间以逗号隔开，但是数据集并没有严格的格式规范，各行数据记录所包含的项数量并不相同，无法在读入函数中指定统一的分隔符。因此，我们选择使用Python的标准函数，直接读入，将数据集以嵌套列表的形式存储。
# 数据转换
data = []
with open('./input/shopping-record.txt') as f:
for line in f.readlines():
# 去除各种符号
line = line.strip()
#去除空行
if line:
row_list = line.split(',')
data.append(row_list)
## 输出数据集的前10条记录
data[:10]
[['citrus fruit', 'semi-finished bread', 'margarine', 'ready soups'], ['tropical fruit', 'yogurt', 'coffee'], ['whole milk'], ['pip fruit', 'yogurt', 'cream cheese', 'meat spreads'], ['other vegetables', 'whole milk', 'condensed milk', 'long life bakery product'], ['whole milk', 'butter', 'yogurt', 'rice', 'abrasive cleaner'], ['rolls/buns'], ['other vegetables', 'UHT-milk', 'rolls/buns', 'bottled beer', 'liquor (appetizer)'], ['potted plants'], ['whole milk', 'cereals']]
数据集一共有9835条数据记录。
len(data)
9835
from collections import defaultdict
min_support = 100
frequent_1 = defaultdict(int)
for item in data:
for record in item:
frequent_1[frozenset([record])] += 1
frequent_1 = {v:frequent_1[v] for v in frequent_1 if frequent_1[v] >= min_support}
print u'频繁1项集的个数：', len(frequent_1)
## 输出前10个元素
frequent_1_list = frequent_1.items()
for index in range(10):
print frequent_1_list[index]
频繁1项集的个数： 88 (frozenset(['beef']), 516) (frozenset(['rolls/buns']), 1809) (frozenset(['cling film/bags']), 112) (frozenset(['spread cheese']), 110) (frozenset(['liquor']), 109) (frozenset(['flower (seeds)']), 102) (frozenset(['white bread']), 414) (frozenset(['dessert']), 365) (frozenset(['pork']), 567) (frozenset(['whipped/sour cream']), 705)
我们使用getFrequentItemSetWithSupport()函数完成APRIORI算法中的连接步和剪枝步。
def getSubset(item, k):
import itertools as its
return [frozenset(item) for item in its.combinations(item, k)]
# k >= 2时
def getFrequentItemSetWithSupport(frequent_k, min_support, k, item_list):
from collections import defaultdict
items = frequent_k.keys()
#k项候选集
candidate_items = []
current_k = defaultdict(int)
### 连接步，生成候选集
candidate_items = set([m.union(n) for m in items for n in items if m != n and len(m.union(n)) == k])
### 剪枝步，剔除含有非频繁项子集的项集
final_candidate = set()
for candidate in candidate_items:
sub_items = getSubset(candidate, (k - 1))
if any([item < candidate for item in sub_items]):
final_candidate.add(candidate)
else:
continue
# 遍历数据集data，对final_candidate中的元素进行统计，保留支持度大于最小阈值的频繁项
for item in final_candidate:
for record in item_list:
#数据记录record中含有频繁项item
if item.issubset(record):
current_k[item] += 1
else:
continue
#print 'current_candidate: ', current_k
return {v: current_k[v] for v in current_k if current_k[v] >= min_support}
在本案例中，我们选择k值的上限为5, 反复调用上述函数，来生成频繁k项集。
upper_bound = 5
final_itemsets= []
final_itemsets.append(frequent_1)
min_conf = 0.4
frequent_k_minus_1 = frequent_1
k = 2
while k <= upper_bound:
#print frequent_k_minus_1
#print ""
frequent_k = getFrequentItemSetWithSupport(frequent_k_minus_1, min_support, k, data)
final_itemsets.append(frequent_k)
frequent_k_minus_1 = frequent_k
k += 1
## 输出频繁2项集中的前10个元素
final_itemsets_2_list = final_itemsets[1].items()
for index in range(10):
print final_itemsets_2_list[index]
(frozenset(['yogurt', 'whipped/sour cream']), 204) (frozenset(['tropical fruit', 'sausage']), 137) (frozenset(['pork', 'other vegetables']), 213) (frozenset(['other vegetables', 'frankfurter']), 162) (frozenset(['whole milk', 'whipped/sour cream']), 317) (frozenset(['other vegetables', 'napkins']), 142) (frozenset(['whole milk', 'frozen vegetables']), 201) (frozenset(['shopping bags', 'bottled water']), 108) (frozenset(['brown bread', 'soda']), 124) (frozenset(['yogurt', 'frankfurter']), 110)
从k=4开始，频繁项集集合为空,也就是说，在给定的最小支持度水平下，数据集至多存在频繁3项集。
生成频繁项集之后，我们可以直接得到同时满足最小支持度和最小置信度的强关联规则。例如，对于频繁项集{'A','B'}来说，关联规则 $A \rightarrow B$的置信度如下所示：
\begin{equation}
\text{conf}(A \rightarrow B) = \dfrac{\text{support}(\{A,B\})}{\text{support}(\{A\})}
\end{equation}
def getAllSubsets(item):
subsets = []
for i in xrange(len(item) - 1):
subsets.extend(getSubset(item, i + 1))
return subsets
association_rules = []
for item_set in final_itemsets[1:]:
if len(item_set) == 0:
continue
for value in item_set:
print 'frequent item set: ', value
#形如 if condition, then conclusion
for condition in getAllSubsets(value):
print ' condition: ', condition,
conclusion_items = frozenset(x for x in value if x not in condition)
print ' conclusion: ', conclusion_items,
if len(conclusion_items) > 0:
confidence = float(final_itemsets[len(value) - 1][value])/final_itemsets[len(condition) - 1][condition]
print confidence
if confidence > min_conf:
association_rules.append([[condition, conclusion_items],
confidence])
print ''
frequent item set: frozenset(['yogurt', 'whipped/sour cream'])
condition: frozenset(['yogurt']) conclusion: frozenset(['whipped/sour cream']) 0.148688046647
condition: frozenset(['whipped/sour cream']) conclusion: frozenset(['yogurt']) 0.289361702128
frequent item set: frozenset(['tropical fruit', 'sausage'])
condition: frozenset(['tropical fruit']) conclusion: frozenset(['sausage']) 0.132751937984
condition: frozenset(['sausage']) conclusion: frozenset(['tropical fruit']) 0.148268398268
frequent item set: frozenset(['pork', 'other vegetables'])
condition: frozenset(['pork']) conclusion: frozenset(['other vegetables']) 0.375661375661
condition: frozenset(['other vegetables']) conclusion: frozenset(['pork']) 0.111928533894
frequent item set: frozenset(['other vegetables', 'frankfurter'])
condition: frozenset(['other vegetables']) conclusion: frozenset(['frankfurter']) 0.0851287440883
condition: frozenset(['frankfurter']) conclusion: frozenset(['other vegetables']) 0.279310344828
frequent item set: frozenset(['whole milk', 'whipped/sour cream'])
condition: frozenset(['whole milk']) conclusion: frozenset(['whipped/sour cream']) 0.126144050935
condition: frozenset(['whipped/sour cream']) conclusion: frozenset(['whole milk']) 0.449645390071
frequent item set: frozenset(['other vegetables', 'napkins'])
condition: frozenset(['other vegetables']) conclusion: frozenset(['napkins']) 0.0746190225959
condition: frozenset(['napkins']) conclusion: frozenset(['other vegetables']) 0.27572815534
frequent item set: frozenset(['whole milk', 'frozen vegetables'])
condition: frozenset(['whole milk']) conclusion: frozenset(['frozen vegetables']) 0.0799840827696
condition: frozenset(['frozen vegetables']) conclusion: frozenset(['whole milk']) 0.424947145877
frequent item set: frozenset(['shopping bags', 'bottled water'])
condition: frozenset(['shopping bags']) conclusion: frozenset(['bottled water']) 0.111455108359
condition: frozenset(['bottled water']) conclusion: frozenset(['shopping bags']) 0.099356025759
frequent item set: frozenset(['brown bread', 'soda'])
condition: frozenset(['brown bread']) conclusion: frozenset(['soda']) 0.194357366771
condition: frozenset(['soda']) conclusion: frozenset(['brown bread']) 0.0723032069971
frequent item set: frozenset(['yogurt', 'frankfurter'])
condition: frozenset(['yogurt']) conclusion: frozenset(['frankfurter']) 0.0801749271137
condition: frozenset(['frankfurter']) conclusion: frozenset(['yogurt']) 0.189655172414
frequent item set: frozenset(['rolls/buns', 'soda'])
condition: frozenset(['rolls/buns']) conclusion: frozenset(['soda']) 0.208402432283
condition: frozenset(['soda']) conclusion: frozenset(['rolls/buns']) 0.219825072886
frequent item set: frozenset(['yogurt', 'frozen vegetables'])
condition: frozenset(['yogurt']) conclusion: frozenset(['frozen vegetables']) 0.0889212827988
condition: frozenset(['frozen vegetables']) conclusion: frozenset(['yogurt']) 0.257928118393
frequent item set: frozenset(['whole milk', 'dessert'])
condition: frozenset(['whole milk']) conclusion: frozenset(['dessert']) 0.0537206526064
condition: frozenset(['dessert']) conclusion: frozenset(['whole milk']) 0.369863013699
frequent item set: frozenset(['whole milk', 'bottled beer'])
condition: frozenset(['whole milk']) conclusion: frozenset(['bottled beer']) 0.0799840827696
condition: frozenset(['bottled beer']) conclusion: frozenset(['whole milk']) 0.253787878788
frequent item set: frozenset(['rolls/buns', 'pastry'])
condition: frozenset(['rolls/buns']) conclusion: frozenset(['pastry']) 0.113875069099
condition: frozenset(['pastry']) conclusion: frozenset(['rolls/buns']) 0.235428571429
frequent item set: frozenset(['margarine', 'rolls/buns'])
condition: frozenset(['margarine']) conclusion: frozenset(['rolls/buns']) 0.251736111111
condition: frozenset(['rolls/buns']) conclusion: frozenset(['margarine']) 0.0801547816473
frequent item set: frozenset(['tropical fruit', 'domestic eggs'])
condition: frozenset(['tropical fruit']) conclusion: frozenset(['domestic eggs']) 0.108527131783
condition: frozenset(['domestic eggs']) conclusion: frozenset(['tropical fruit']) 0.179487179487
frequent item set: frozenset(['whole milk', 'curd'])
condition: frozenset(['whole milk']) conclusion: frozenset(['curd']) 0.102268205332
condition: frozenset(['curd']) conclusion: frozenset(['whole milk']) 0.490458015267
frequent item set: frozenset(['sausage', 'pastry'])
condition: frozenset(['sausage']) conclusion: frozenset(['pastry']) 0.133116883117
condition: frozenset(['pastry']) conclusion: frozenset(['sausage']) 0.140571428571
frequent item set: frozenset(['rolls/buns', 'shopping bags'])
condition: frozenset(['rolls/buns']) conclusion: frozenset(['shopping bags']) 0.106135986733
condition: frozenset(['shopping bags']) conclusion: frozenset(['rolls/buns']) 0.198142414861
frequent item set: frozenset(['other vegetables', 'root vegetables'])
condition: frozenset(['other vegetables']) conclusion: frozenset(['root vegetables']) 0.244876510772
condition: frozenset(['root vegetables']) conclusion: frozenset(['other vegetables']) 0.434701492537
frequent item set: frozenset(['yogurt', 'curd'])
condition: frozenset(['yogurt']) conclusion: frozenset(['curd']) 0.123906705539
condition: frozenset(['curd']) conclusion: frozenset(['yogurt']) 0.324427480916
frequent item set: frozenset(['whole milk', 'onions'])
condition: frozenset(['whole milk']) conclusion: frozenset(['onions']) 0.0473537604457
condition: frozenset(['onions']) conclusion: frozenset(['whole milk']) 0.390163934426
frequent item set: frozenset(['whole milk', 'coffee'])
condition: frozenset(['whole milk']) conclusion: frozenset(['coffee']) 0.0732192598488
condition: frozenset(['coffee']) conclusion: frozenset(['whole milk']) 0.322241681261
frequent item set: frozenset(['whole milk', 'hamburger meat'])
condition: frozenset(['whole milk']) conclusion: frozenset(['hamburger meat']) 0.0576999602069
condition: frozenset(['hamburger meat']) conclusion: frozenset(['whole milk']) 0.443425076453
frequent item set: frozenset(['tropical fruit', 'whipped/sour cream'])
condition: frozenset(['tropical fruit']) conclusion: frozenset(['whipped/sour cream']) 0.131782945736
condition: frozenset(['whipped/sour cream']) conclusion: frozenset(['tropical fruit']) 0.192907801418
frequent item set: frozenset(['canned beer', 'rolls/buns'])
condition: frozenset(['canned beer']) conclusion: frozenset(['rolls/buns']) 0.145287958115
condition: frozenset(['rolls/buns']) conclusion: frozenset(['canned beer']) 0.06135986733
frequent item set: frozenset(['root vegetables', 'soda'])
condition: frozenset(['root vegetables']) conclusion: frozenset(['soda']) 0.170708955224
condition: frozenset(['soda']) conclusion: frozenset(['root vegetables']) 0.106705539359
frequent item set: frozenset(['whole milk', 'beef'])
condition: frozenset(['whole milk']) conclusion: frozenset(['beef']) 0.08316752885
condition: frozenset(['beef']) conclusion: frozenset(['whole milk']) 0.40503875969
frequent item set: frozenset(['hamburger meat', 'other vegetables'])
condition: frozenset(['hamburger meat']) conclusion: frozenset(['other vegetables']) 0.415902140673
condition: frozenset(['other vegetables']) conclusion: frozenset(['hamburger meat']) 0.0714661061482
frequent item set: frozenset(['coffee', 'other vegetables'])
condition: frozenset(['coffee']) conclusion: frozenset(['other vegetables']) 0.231173380035
condition: frozenset(['other vegetables']) conclusion: frozenset(['coffee']) 0.0693641618497
frequent item set: frozenset(['white bread', 'soda'])
condition: frozenset(['white bread']) conclusion: frozenset(['soda']) 0.243961352657
condition: frozenset(['soda']) conclusion: frozenset(['white bread']) 0.0588921282799
frequent item set: frozenset(['yogurt', 'beef'])
condition: frozenset(['yogurt']) conclusion: frozenset(['beef']) 0.0838192419825
condition: frozenset(['beef']) conclusion: frozenset(['yogurt']) 0.222868217054
frequent item set: frozenset(['whole milk', 'butter milk'])
condition: frozenset(['whole milk']) conclusion: frozenset(['butter milk']) 0.0453641066454
condition: frozenset(['butter milk']) conclusion: frozenset(['whole milk']) 0.414545454545
frequent item set: frozenset(['yogurt', 'margarine'])
condition: frozenset(['yogurt']) conclusion: frozenset(['margarine']) 0.102040816327
condition: frozenset(['margarine']) conclusion: frozenset(['yogurt']) 0.243055555556
frequent item set: frozenset(['other vegetables', 'shopping bags'])
condition: frozenset(['other vegetables']) conclusion: frozenset(['shopping bags']) 0.119810825013
condition: frozenset(['shopping bags']) conclusion: frozenset(['other vegetables']) 0.235294117647
frequent item set: frozenset(['whole milk', 'domestic eggs'])
condition: frozenset(['whole milk']) conclusion: frozenset(['domestic eggs']) 0.117389574214
condition: frozenset(['domestic eggs']) conclusion: frozenset(['whole milk']) 0.472756410256
frequent item set: frozenset(['yogurt', 'pastry'])
condition: frozenset(['yogurt']) conclusion: frozenset(['pastry']) 0.126822157434
condition: frozenset(['pastry']) conclusion: frozenset(['yogurt']) 0.198857142857
frequent item set: frozenset(['newspapers', 'soda'])
condition: frozenset(['newspapers']) conclusion: frozenset(['soda']) 0.183439490446
condition: frozenset(['soda']) conclusion: frozenset(['newspapers']) 0.0839650145773
frequent item set: frozenset(['margarine', 'other vegetables'])
condition: frozenset(['margarine']) conclusion: frozenset(['other vegetables']) 0.336805555556
condition: frozenset(['other vegetables']) conclusion: frozenset(['margarine']) 0.101944298476
frequent item set: frozenset(['tropical fruit', 'newspapers'])
condition: frozenset(['tropical fruit']) conclusion: frozenset(['newspapers']) 0.112403100775
condition: frozenset(['newspapers']) conclusion: frozenset(['tropical fruit']) 0.147770700637
frequent item set: frozenset(['other vegetables', 'pastry'])
condition: frozenset(['other vegetables']) conclusion: frozenset(['pastry']) 0.116657908565
condition: frozenset(['pastry']) conclusion: frozenset(['other vegetables']) 0.253714285714
frequent item set: frozenset(['chicken', 'other vegetables'])
condition: frozenset(['chicken']) conclusion: frozenset(['other vegetables']) 0.417061611374
condition: frozenset(['other vegetables']) conclusion: frozenset(['chicken']) 0.0924855491329
frequent item set: frozenset(['whole milk', 'waffles'])
condition: frozenset(['whole milk']) conclusion: frozenset(['waffles']) 0.049741345006
condition: frozenset(['waffles']) conclusion: frozenset(['whole milk']) 0.330687830688
frequent item set: frozenset(['canned beer', 'shopping bags'])
condition: frozenset(['canned beer']) conclusion: frozenset(['shopping bags']) 0.146596858639
condition: frozenset(['shopping bags']) conclusion: frozenset(['canned beer']) 0.115583075335
frequent item set: frozenset(['yogurt', 'root vegetables'])
condition: frozenset(['yogurt']) conclusion: frozenset(['root vegetables']) 0.185131195335
condition: frozenset(['root vegetables']) conclusion: frozenset(['yogurt']) 0.236940298507
frequent item set: frozenset(['margarine', 'soda'])
condition: frozenset(['margarine']) conclusion: frozenset(['soda']) 0.173611111111
condition: frozenset(['soda']) conclusion: frozenset(['margarine']) 0.0583090379009
frequent item set: frozenset(['whole milk', 'oil'])
condition: frozenset(['whole milk']) conclusion: frozenset(['oil']) 0.0441703143653
condition: frozenset(['oil']) conclusion: frozenset(['whole milk']) 0.402173913043
frequent item set: frozenset(['other vegetables', 'frozen vegetables'])
condition: frozenset(['other vegetables']) conclusion: frozenset(['frozen vegetables']) 0.0919600630583
condition: frozenset(['frozen vegetables']) conclusion: frozenset(['other vegetables']) 0.369978858351
frequent item set: frozenset(['whole milk', 'napkins'])
condition: frozenset(['whole milk']) conclusion: frozenset(['napkins']) 0.0771985674493
condition: frozenset(['napkins']) conclusion: frozenset(['whole milk']) 0.376699029126
frequent item set: frozenset(['sausage', 'soda'])
condition: frozenset(['sausage']) conclusion: frozenset(['soda']) 0.258658008658
condition: frozenset(['soda']) conclusion: frozenset(['sausage']) 0.139358600583
frequent item set: frozenset(['whole milk', 'pork'])
condition: frozenset(['whole milk']) conclusion: frozenset(['pork']) 0.0867489056904
condition: frozenset(['pork']) conclusion: frozenset(['whole milk']) 0.384479717813
frequent item set: frozenset(['citrus fruit', 'fruit/vegetable juice'])
condition: frozenset(['citrus fruit']) conclusion: frozenset(['fruit/vegetable juice']) 0.125307125307
condition: frozenset(['fruit/vegetable juice']) conclusion: frozenset(['citrus fruit']) 0.143459915612
frequent item set: frozenset(['tropical fruit', 'brown bread'])
condition: frozenset(['tropical fruit']) conclusion: frozenset(['brown bread']) 0.101744186047
condition: frozenset(['brown bread']) conclusion: frozenset(['tropical fruit']) 0.164576802508
frequent item set: frozenset(['whole milk', 'white bread'])
condition: frozenset(['whole milk']) conclusion: frozenset(['white bread']) 0.066852367688
condition: frozenset(['white bread']) conclusion: frozenset(['whole milk']) 0.405797101449
frequent item set: frozenset(['whole milk', 'sausage'])
condition: frozenset(['whole milk']) conclusion: frozenset(['sausage']) 0.116991643454
condition: frozenset(['sausage']) conclusion: frozenset(['whole milk']) 0.318181818182
frequent item set: frozenset(['pastry', 'soda'])
condition: frozenset(['pastry']) conclusion: frozenset(['soda']) 0.236571428571
condition: frozenset(['soda']) conclusion: frozenset(['pastry']) 0.120699708455
frequent item set: frozenset(['domestic eggs', 'soda'])
condition: frozenset(['domestic eggs']) conclusion: frozenset(['soda']) 0.195512820513
condition: frozenset(['soda']) conclusion: frozenset(['domestic eggs']) 0.0711370262391
frequent item set: frozenset(['other vegetables', 'fruit/vegetable juice'])
condition: frozenset(['other vegetables']) conclusion: frozenset(['fruit/vegetable juice']) 0.108775617446
condition: frozenset(['fruit/vegetable juice']) conclusion: frozenset(['other vegetables']) 0.291139240506
frequent item set: frozenset(['sausage', 'bottled water'])
condition: frozenset(['sausage']) conclusion: frozenset(['bottled water']) 0.127705627706
condition: frozenset(['bottled water']) conclusion: frozenset(['sausage']) 0.108555657774
frequent item set: frozenset(['whole milk', 'cream cheese'])
condition: frozenset(['whole milk']) conclusion: frozenset(['cream cheese']) 0.0644647831277
condition: frozenset(['cream cheese']) conclusion: frozenset(['whole milk']) 0.415384615385
frequent item set: frozenset(['citrus fruit', 'other vegetables'])
condition: frozenset(['citrus fruit']) conclusion: frozenset(['other vegetables']) 0.348894348894
condition: frozenset(['other vegetables']) conclusion: frozenset(['citrus fruit']) 0.149238045192
frequent item set: frozenset(['shopping bags', 'soda'])
condition: frozenset(['shopping bags']) conclusion: frozenset(['soda']) 0.249742002064
condition: frozenset(['soda']) conclusion: frozenset(['shopping bags']) 0.14110787172
frequent item set: frozenset(['tropical fruit', 'pip fruit'])
condition: frozenset(['tropical fruit']) conclusion: frozenset(['pip fruit']) 0.19476744186
condition: frozenset(['pip fruit']) conclusion: frozenset(['tropical fruit']) 0.270161290323
frequent item set: frozenset(['yogurt', 'fruit/vegetable juice'])
condition: frozenset(['yogurt']) conclusion: frozenset(['fruit/vegetable juice']) 0.134110787172
condition: frozenset(['fruit/vegetable juice']) conclusion: frozenset(['yogurt']) 0.258790436006
frequent item set: frozenset(['soda', 'whipped/sour cream'])
condition: frozenset(['soda']) conclusion: frozenset(['whipped/sour cream']) 0.066472303207
condition: frozenset(['whipped/sour cream']) conclusion: frozenset(['soda']) 0.16170212766
frequent item set: frozenset(['other vegetables', 'rolls/buns'])
condition: frozenset(['other vegetables']) conclusion: frozenset(['rolls/buns']) 0.220178665265
condition: frozenset(['rolls/buns']) conclusion: frozenset(['other vegetables']) 0.231619679381
frequent item set: frozenset(['yogurt', 'sausage'])
condition: frozenset(['yogurt']) conclusion: frozenset(['sausage']) 0.140670553936
condition: frozenset(['sausage']) conclusion: frozenset(['yogurt']) 0.208874458874
frequent item set: frozenset(['yogurt', 'berries'])
condition: frozenset(['yogurt']) conclusion: frozenset(['berries']) 0.0758017492711
condition: frozenset(['berries']) conclusion: frozenset(['yogurt']) 0.318042813456
frequent item set: frozenset(['citrus fruit', 'bottled water'])
condition: frozenset(['citrus fruit']) conclusion: frozenset(['bottled water']) 0.163390663391
condition: frozenset(['bottled water']) conclusion: frozenset(['citrus fruit']) 0.122355105796
frequent item set: frozenset(['rolls/buns', 'whipped/sour cream'])
condition: frozenset(['rolls/buns']) conclusion: frozenset(['whipped/sour cream']) 0.0796019900498
condition: frozenset(['whipped/sour cream']) conclusion: frozenset(['rolls/buns']) 0.204255319149
frequent item set: frozenset(['whole milk', 'pastry'])
condition: frozenset(['whole milk']) conclusion: frozenset(['pastry']) 0.130123358536
condition: frozenset(['pastry']) conclusion: frozenset(['whole milk']) 0.373714285714
frequent item set: frozenset(['yogurt', 'rolls/buns'])
condition: frozenset(['yogurt']) conclusion: frozenset(['rolls/buns']) 0.246355685131
condition: frozenset(['rolls/buns']) conclusion: frozenset(['yogurt']) 0.186843559978
frequent item set: frozenset(['sausage', 'other vegetables'])
condition: frozenset(['sausage']) conclusion: frozenset(['other vegetables']) 0.286796536797
condition: frozenset(['other vegetables']) conclusion: frozenset(['sausage']) 0.139253809774
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condition: frozenset(['yogurt', 'other vegetables']) conclusion: frozenset(['root vegetables']) 0.297423887588
condition: frozenset(['yogurt', 'root vegetables']) conclusion: frozenset(['other vegetables']) 0.5
condition: frozenset(['other vegetables', 'root vegetables']) conclusion: frozenset(['yogurt']) 0.272532188841
frequent item set: frozenset(['whole milk', 'pork', 'other vegetables'])
condition: frozenset(['whole milk']) conclusion: frozenset(['pork', 'other vegetables']) 0.0397930760048
condition: frozenset(['pork']) conclusion: frozenset(['whole milk', 'other vegetables']) 0.176366843034
condition: frozenset(['other vegetables']) conclusion: frozenset(['whole milk', 'pork']) 0.0525486074619
condition: frozenset(['whole milk', 'pork']) conclusion: frozenset(['other vegetables']) 0.45871559633
condition: frozenset(['whole milk', 'other vegetables']) conclusion: frozenset(['pork']) 0.135869565217
condition: frozenset(['pork', 'other vegetables']) conclusion: frozenset(['whole milk']) 0.469483568075
frequent item set: frozenset(['yogurt', 'whole milk', 'other vegetables'])
condition: frozenset(['yogurt']) conclusion: frozenset(['whole milk', 'other vegetables']) 0.159620991254
condition: frozenset(['whole milk']) conclusion: frozenset(['yogurt', 'other vegetables']) 0.0871468364505
condition: frozenset(['other vegetables']) conclusion: frozenset(['yogurt', 'whole milk']) 0.115081450342
condition: frozenset(['yogurt', 'whole milk']) conclusion: frozenset(['other vegetables']) 0.397459165154
condition: frozenset(['yogurt', 'other vegetables']) conclusion: frozenset(['whole milk']) 0.512880562061
condition: frozenset(['whole milk', 'other vegetables']) conclusion: frozenset(['yogurt']) 0.297554347826
frequent item set: frozenset(['whole milk', 'other vegetables', 'whipped/sour cream'])
condition: frozenset(['whole milk']) conclusion: frozenset(['other vegetables', 'whipped/sour cream']) 0.0573020294469
condition: frozenset(['other vegetables']) conclusion: frozenset(['whole milk', 'whipped/sour cream']) 0.0756699947451
condition: frozenset(['whipped/sour cream']) conclusion: frozenset(['whole milk', 'other vegetables']) 0.204255319149
condition: frozenset(['whole milk', 'other vegetables']) conclusion: frozenset(['whipped/sour cream']) 0.195652173913
condition: frozenset(['whole milk', 'whipped/sour cream']) conclusion: frozenset(['other vegetables']) 0.454258675079
condition: frozenset(['other vegetables', 'whipped/sour cream']) conclusion: frozenset(['whole milk']) 0.507042253521
frequent item set: frozenset(['yogurt', 'whole milk', 'rolls/buns'])
condition: frozenset(['yogurt']) conclusion: frozenset(['whole milk', 'rolls/buns']) 0.111516034985
condition: frozenset(['whole milk']) conclusion: frozenset(['yogurt', 'rolls/buns']) 0.0608834062873
condition: frozenset(['rolls/buns']) conclusion: frozenset(['yogurt', 'whole milk']) 0.0845771144279
condition: frozenset(['yogurt', 'whole milk']) conclusion: frozenset(['rolls/buns']) 0.277676950998
condition: frozenset(['yogurt', 'rolls/buns']) conclusion: frozenset(['whole milk']) 0.452662721893
condition: frozenset(['whole milk', 'rolls/buns']) conclusion: frozenset(['yogurt']) 0.274685816876
## 输出前10条关联规则
for index in range(10):
print association_rules[index]
[[frozenset(['whipped/sour cream']), frozenset(['whole milk'])], 0.44964539007092197] [[frozenset(['frozen vegetables']), frozenset(['whole milk'])], 0.4249471458773784] [[frozenset(['curd']), frozenset(['whole milk'])], 0.4904580152671756] [[frozenset(['root vegetables']), frozenset(['other vegetables'])], 0.43470149253731344] [[frozenset(['hamburger meat']), frozenset(['whole milk'])], 0.4434250764525994] [[frozenset(['beef']), frozenset(['whole milk'])], 0.4050387596899225] [[frozenset(['hamburger meat']), frozenset(['other vegetables'])], 0.41590214067278286] [[frozenset(['butter milk']), frozenset(['whole milk'])], 0.41454545454545455] [[frozenset(['domestic eggs']), frozenset(['whole milk'])], 0.47275641025641024] [[frozenset(['chicken']), frozenset(['other vegetables'])], 0.41706161137440756]
首先，我们构造创建FP树所需的类：Node和FPTree。
'''
Node类,包含以下变量：
name： 当前节点代表的频繁项
count：当前节点出现的频数
link：FP树的其他路径中代表相同频繁项的节点
parent：当前节点的父节点
children：当前节点的子节点
包含以下方法：
__init__()： 构造方法
increment()：增加当前节点代表的频繁项的支持度
display()：可视化节点，以及对应的频繁项支持度
'''
class Node:
'''
输入参数：name：节点代表的频繁项
count：频繁项的支持度
parent：节点的父节点
'''
def __init__(self, name, count, parent):
self.name = name
self.count = count
self.link = None
self.parent = parent
self.children = {}
'''
输入参数： num：整数，代表频繁项支持度增加的数值
'''
def increment(self, num):
self.count += num
'''
输入参数：lens：节点的初始支持度
'''
def display(self, lens = 1):
print ' '*lens, self.name, ' ', self.count
for child in self.children.values():
child.display(lens + 1)
'''
FPTree类,包含以下变量：
flag：表示构建FPTree还是条件FPTree，取决于输入数据
root： FP树的根节点
data：FP树代表的数据集
data_type：数据集存储的原始类型（字典或者嵌套列表）
min_support：最小支持度（绝对）
frequent：频繁1项集集合
headerTable：FP树对应的频繁项表
包含以下方法：
__init__()： 构造方法
find_frequent_items()：根据数据集生成频繁1项集集合
build_tree()：将数据集中的每条数据记录插入FP树种，也就是建树过程
get_tree()：返回树的根节点
get_frequent_items()：返回频繁1项集集合
get_headertable()：返回频繁项表
get_data()：返回数据集
show()：可视化FP树的结构
'''
class FPTree:
'''
输入参数：transactions：数据集
min_support：最小支持度
root_value：根节点代表的频繁项
count：根节点代表的频繁项的支持度
'''
def __init__(self, transactions, min_support, root_value, count, flag):
self.flag = flag
self.root = Node(root_value, count, None)
self.data = transactions
self.data_type = type(self.data)
self.min_support = min_support
self.frequent = self.find_frequent_items()
self.headerTable = {v:[self.frequent[v], None] for v in self.frequent}
'''
返回参数：以字典格式存储的频繁1项集集合
'''
def find_frequent_items(self):
from collections import defaultdict
freq1 = defaultdict(int)
#统计列表中各个元素的出现频次
if self.data_type == list and self.flag == 'fptree':
flatten_list = [element for item in self.data for element in item]
for value in flatten_list:
freq1[frozenset([value])] += 1
# 条件FP树
elif self.data_type == dict and self.flag == 'cfptree':
for item in self.data:
for element in item:
if type(element) == frozenset:
freq1[element] += self.data[item]
else:
freq1[frozenset([element])] += self.data[item]
# 过滤掉非频繁项
return {v:freq1[v] for v in freq1 if freq1[v] >= self.min_support}
def build_tree(self):
root = Node('null', 1, None)
sorted_headertable = [v[0] for v in sorted(self.frequent.items(), key=lambda kv: (-kv[1], list(kv[0])[0]))]
# 将业务数据库中的每条记录插入树中，更新频繁项表
for record in self.data:
# 找到业务数据纪录中的频繁项
sorted_items = [item for item in sorted_headertable if list(item)[0] in record]
#print u'将排序后的频繁项集放入FP树中: '
#print sorted_items
#print ''
node = self.root
while len(sorted_items) > 0:
first_value = sorted_items[0]
#print u' 开始插入节点 ', first_value
if first_value in node.children:
#print u' 当前节点的子节点中包含节点 ', first_value
if self.data_type == list:
node.children[first_value].increment(1)
elif self.data_type == dict:
node.children[first_value].increment(self.data[record])
else:
# 创建新的子节点
#print u' 创建新节点 ', first_value
if self.data_type == list:
node.children[first_value] = Node(list(first_value)[0], 1, node)
elif self.data_type == dict:
node.children[first_value] = Node(first_value, self.data[record], node)
# 更新频繁项表
if self.headerTable[first_value][1] == None:
#print u' 在频繁项表中，添加对应链表的第一个节点', first_value
self.headerTable[first_value][1] = node.children[first_value]
else:
#print u' 添加到频繁项表对应链表的末尾'
currentNode = self.headerTable[first_value][1]
# 遍历连表，确保在表的尾部插入新的节点指针
while currentNode.link != None:
currentNode = currentNode.link
currentNode.link = node.children[first_value]
#print '***********'
sorted_items.pop(0)
node = node.children[first_value]
#print ''
def get_tree(self):
return self.root
def get_frequent_items(self):
return self.frequent
def get_headertable(self):
return self.headerTable
def get_data(self):
return self.data
def show(self, depth = 1):
print ' '*depth, self.root.name, ' ', self.root.count
for node in self.root.children.values():
node.display(depth + 1)
#最小支持度
min_support = 100
#创建实例
fptree = FPTree(data, min_support, 'null', 1, 'fptree')
然后，我们可以返回数据集的频繁1项集集合。
freq1 = fptree.get_frequent_items()
## 输出频繁1项集的前10个元素
freq1_list = freq1.items()
for index in range(10):
print freq1_list[index]
(frozenset(['beef']), 516) (frozenset(['rolls/buns']), 1809) (frozenset(['cling film/bags']), 112) (frozenset(['spread cheese']), 110) (frozenset(['liquor']), 109) (frozenset(['flower (seeds)']), 102) (frozenset(['white bread']), 414) (frozenset(['dessert']), 365) (frozenset(['pork']), 567) (frozenset(['whipped/sour cream']), 705)
fptree.build_tree()
FP树创建好之后，我们构造mine_tree()函数，来实现FP树的挖掘过程。
find_prefix()函数返回在FP书中以当前节点为后缀的所有路径，构造条件模式库mine_tree()函数递归挖掘以条件模式库为基础构造的条件FP树'''
输入参数：Node节点
返回参数：以输入节点为后缀的条件模式库
'''
def find_prefix(node):
#条件模式库
cpb = {}
suffix_list = []
# 判断条件，是否已经达到链表的尾部
while node != None:
suffix_list.append(node)
node = node.link
for item in suffix_list:
path = []
num = item.count
# 判断条件，是否已经回溯到树的顶点
while item.parent != None:
path.append(item.name)
item = item.parent
# 除去当前节点的部分作为条件模式基，并以当前节点的支持度作为条件模式库的支持度
cpb[frozenset(path[1:])] = num
return cpb
'''
输入参数：frequent_items：数据集的频繁1项集集合
headerTable：数据集的频繁项表
frequent：每次迭代过程生成的频繁项
item_list：存储所有迭代过程生成的的频繁项集
'''
def mine_tree(frequent_items, headerTable, min_support, frequent, item_list):
# 频繁项表中的元素降序排列
candidates = [v[0] for v in sorted(frequent_items.items(), key=lambda kv: (-kv[1], list(kv[0])[0]))]
#print candidates
for item in candidates[::-1]:
#从以下的元素开始
#print 'from the node: ', item
# 针对每个条件模式库，生成频繁项集
freq_set = frequent.copy()
freq_set.add(item)
item_list.append(freq_set)
# 生成条件模式库
cpbs = find_prefix(headerTable[item][1])
#print 'its cpbs: ', cpbs
#创建条件FP树
cTree = FPTree(cpbs, min_support, 'root', 1, 'cfptree')
cTree.build_tree()
#print '-----headerTable: ', cTree.get_headertable()
#判断条件：频繁项表为空
if len(cTree.get_headertable()) != 0:
#print 'condtional tree for: ', freq_set
#cTree.show(1)
mine_tree(cTree.get_frequent_items(), cTree.get_headertable(), min_support, freq_set, item_list)
根据原始数据集构建FP树之后，可以得到FP树的节点root_node，频繁项表headertable，以及频繁1项集集合freq_items。
将这些变量作为输入参数，带入到函数mine_tree()函数中，生成的频繁项集翻入到列表变量frequent_item中。
frequent_item = []
frequent = set()
root_node = fptree.get_tree()
headertable = fptree.get_headertable()
freq_items = fptree.get_frequent_items()
mine_tree(freq_items, headertable, min_support, frequent, frequent_item)
打印requent_item的内容。从中我们可以发现，商场售卖的受客户欢迎的商品，比如yorgurt，soda等；当然，我们还发现顾客通常会一起购买whole milk和yourgurt等。更多的信息，可以继续分析frequent_item的内容。
frequent_item
[{frozenset({'roll products'})},
{frozenset({'condensed milk'})},
{frozenset({'flower (seeds)'})},
{frozenset({'dish cleaner'})},
{frozenset({'salt'})},
{frozenset({'frozen dessert'})},
{frozenset({'canned vegetables'})},
{frozenset({'liquor'})},
{frozenset({'spread cheese'})},
{frozenset({'cling film/bags'})},
{frozenset({'frozen fish'})},
{frozenset({'mustard'})},
{frozenset({'packaged fruit/vegetables'})},
{frozenset({'cake bar'})},
{frozenset({'seasonal products'})},
{frozenset({'pasta'})},
{frozenset({'canned fish'})},
{frozenset({'herbs'})},
{frozenset({'processed cheese'})},
{frozenset({'soft cheese'})},
{frozenset({'potted plants'})},
{frozenset({'flour'})},
{frozenset({'dishes'})},
{frozenset({'semi-finished bread'})},
{frozenset({'baking powder'})},
{frozenset({'pickled vegetables'})},
{frozenset({'white wine'})},
{frozenset({'red/blush wine'})},
{frozenset({'detergent'})},
{frozenset({'chewing gum'})},
{frozenset({'grapes'})},
{frozenset({'cat food'})},
{frozenset({'sliced cheese'})},
{frozenset({'sliced cheese'}), frozenset({'whole milk'})},
{frozenset({'hard cheese'})},
{frozenset({'ice cream'})},
{frozenset({'meat'})},
{frozenset({'ham'})},
{frozenset({'ham'}), frozenset({'whole milk'})},
{frozenset({'beverages'})},
{frozenset({'specialty bar'})},
{frozenset({'butter milk'})},
{frozenset({'butter milk'}), frozenset({'other vegetables'})},
{frozenset({'butter milk'}), frozenset({'whole milk'})},
{frozenset({'oil'})},
{frozenset({'oil'}), frozenset({'whole milk'})},
{frozenset({'misc. beverages'})},
{frozenset({'frozen meals'})},
{frozenset({'candy'})},
{frozenset({'specialty chocolate'})},
{frozenset({'onions'})},
{frozenset({'onions'}), frozenset({'whole milk'})},
{frozenset({'onions'}), frozenset({'other vegetables'})},
{frozenset({'hygiene articles'})},
{frozenset({'hygiene articles'}), frozenset({'whole milk'})},
{frozenset({'hamburger meat'})},
{frozenset({'other vegetables'}), frozenset({'hamburger meat'})},
{frozenset({'whole milk'}), frozenset({'hamburger meat'})},
{frozenset({'berries'})},
{frozenset({'other vegetables'}), frozenset({'berries'})},
{frozenset({'berries'}), frozenset({'yogurt'})},
{frozenset({'berries'}), frozenset({'whole milk'})},
{frozenset({'UHT-milk'})},
{frozenset({'sugar'})},
{frozenset({'sugar'}), frozenset({'other vegetables'})},
{frozenset({'sugar'}), frozenset({'whole milk'})},
{frozenset({'dessert'})},
{frozenset({'other vegetables'}), frozenset({'dessert'})},
{frozenset({'whole milk'}), frozenset({'dessert'})},
{frozenset({'long life bakery product'})},
{frozenset({'other vegetables'}), frozenset({'long life bakery product'})},
{frozenset({'whole milk'}), frozenset({'long life bakery product'})},
{frozenset({'salty snack'})},
{frozenset({'other vegetables'}), frozenset({'salty snack'})},
{frozenset({'whole milk'}), frozenset({'salty snack'})},
{frozenset({'waffles'})},
{frozenset({'waffles'}), frozenset({'whole milk'})},
{frozenset({'cream cheese'})},
{frozenset({'yogurt'}), frozenset({'cream cheese'})},
{frozenset({'other vegetables'}), frozenset({'cream cheese'})},
{frozenset({'whole milk'}), frozenset({'cream cheese'})},
{frozenset({'white bread'})},
{frozenset({'soda'}), frozenset({'white bread'})},
{frozenset({'other vegetables'}), frozenset({'white bread'})},
{frozenset({'white bread'}), frozenset({'whole milk'})},
{frozenset({'chicken'})},
{frozenset({'root vegetables'}), frozenset({'chicken'})},
{frozenset({'chicken'}), frozenset({'whole milk'})},
{frozenset({'other vegetables'}), frozenset({'chicken'})},
{frozenset({'frozen vegetables'})},
{frozenset({'frozen vegetables'}), frozenset({'rolls/buns'})},
{frozenset({'root vegetables'}), frozenset({'frozen vegetables'})},
{frozenset({'frozen vegetables'}), frozenset({'yogurt'})},
{frozenset({'other vegetables'}), frozenset({'frozen vegetables'})},
{frozenset({'frozen vegetables'}), frozenset({'whole milk'})},
{frozenset({'chocolate'})},
{frozenset({'chocolate'}), frozenset({'rolls/buns'})},
{frozenset({'chocolate'}), frozenset({'other vegetables'})},
{frozenset({'chocolate'}), frozenset({'soda'})},
{frozenset({'chocolate'}), frozenset({'whole milk'})},
{frozenset({'napkins'})},
{frozenset({'napkins'}), frozenset({'rolls/buns'})},
{frozenset({'soda'}), frozenset({'napkins'})},
{frozenset({'napkins'}), frozenset({'yogurt'})},
{frozenset({'other vegetables'}), frozenset({'napkins'})},
{frozenset({'napkins'}), frozenset({'whole milk'})},
{frozenset({'beef'})},
{frozenset({'beef'}), frozenset({'yogurt'})},
{frozenset({'beef'}), frozenset({'rolls/buns'})},
{frozenset({'beef'}), frozenset({'root vegetables'})},
{frozenset({'beef'}), frozenset({'other vegetables'})},
{frozenset({'beef'}), frozenset({'whole milk'})},
{frozenset({'curd'})},
{frozenset({'curd'}), frozenset({'tropical fruit'})},
{frozenset({'whipped/sour cream'}), frozenset({'curd'})},
{frozenset({'root vegetables'}), frozenset({'curd'})},
{frozenset({'other vegetables'}), frozenset({'curd'})},
{frozenset({'curd'}), frozenset({'yogurt'})},
{frozenset({'curd'}), frozenset({'whole milk'})},
{frozenset({'butter'})},
{frozenset({'whipped/sour cream'}), frozenset({'butter'})},
{frozenset({'root vegetables'}), frozenset({'butter'})},
{frozenset({'rolls/buns'}), frozenset({'butter'})},
{frozenset({'yogurt'}), frozenset({'butter'})},
{frozenset({'other vegetables'}), frozenset({'butter'})},
{frozenset({'other vegetables'}),
frozenset({'whole milk'}),
frozenset({'butter'})},
{frozenset({'whole milk'}), frozenset({'butter'})},
{frozenset({'pork'})},
{frozenset({'pork'}), frozenset({'rolls/buns'})},
{frozenset({'soda'}), frozenset({'pork'})},
{frozenset({'root vegetables'}), frozenset({'pork'})},
{frozenset({'other vegetables'}), frozenset({'pork'})},
{frozenset({'pork'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'pork'}), frozenset({'whole milk'})},
{frozenset({'coffee'})},
{frozenset({'coffee'}), frozenset({'rolls/buns'})},
{frozenset({'other vegetables'}), frozenset({'coffee'})},
{frozenset({'coffee'}), frozenset({'whole milk'})},
{frozenset({'margarine'})},
{frozenset({'soda'}), frozenset({'margarine'})},
{frozenset({'bottled water'}), frozenset({'margarine'})},
{frozenset({'root vegetables'}), frozenset({'margarine'})},
{frozenset({'yogurt'}), frozenset({'margarine'})},
{frozenset({'rolls/buns'}), frozenset({'margarine'})},
{frozenset({'other vegetables'}), frozenset({'margarine'})},
{frozenset({'whole milk'}), frozenset({'margarine'})},
{frozenset({'frankfurter'})},
{frozenset({'frankfurter'}), frozenset({'root vegetables'})},
{frozenset({'frankfurter'}), frozenset({'yogurt'})},
{frozenset({'frankfurter'}), frozenset({'soda'})},
{frozenset({'frankfurter'}), frozenset({'other vegetables'})},
{frozenset({'frankfurter'}), frozenset({'rolls/buns'})},
{frozenset({'frankfurter'}), frozenset({'whole milk'})},
{frozenset({'domestic eggs'})},
{frozenset({'domestic eggs'}), frozenset({'citrus fruit'})},
{frozenset({'domestic eggs'}), frozenset({'tropical fruit'})},
{frozenset({'domestic eggs'}), frozenset({'soda'})},
{frozenset({'domestic eggs'}), frozenset({'yogurt'})},
{frozenset({'domestic eggs'}), frozenset({'root vegetables'})},
{frozenset({'domestic eggs'}), frozenset({'rolls/buns'})},
{frozenset({'domestic eggs'}), frozenset({'other vegetables'})},
{frozenset({'domestic eggs'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'domestic eggs'}), frozenset({'whole milk'})},
{frozenset({'brown bread'})},
{frozenset({'root vegetables'}), frozenset({'brown bread'})},
{frozenset({'brown bread'}), frozenset({'tropical fruit'})},
{frozenset({'brown bread'}), frozenset({'sausage'})},
{frozenset({'soda'}), frozenset({'brown bread'})},
{frozenset({'brown bread'}), frozenset({'rolls/buns'})},
{frozenset({'brown bread'}), frozenset({'yogurt'})},
{frozenset({'brown bread'}), frozenset({'other vegetables'})},
{frozenset({'brown bread'}), frozenset({'whole milk'})},
{frozenset({'whipped/sour cream'})},
{frozenset({'whipped/sour cream'}), frozenset({'citrus fruit'})},
{frozenset({'whipped/sour cream'}), frozenset({'soda'})},
{frozenset({'whipped/sour cream'}), frozenset({'tropical fruit'})},
{frozenset({'whipped/sour cream'}), frozenset({'rolls/buns'})},
{frozenset({'whipped/sour cream'}), frozenset({'root vegetables'})},
{frozenset({'whipped/sour cream'}), frozenset({'yogurt'})},
{frozenset({'whipped/sour cream'}),
frozenset({'other vegetables'}),
frozenset({'yogurt'})},
{frozenset({'whipped/sour cream'}),
frozenset({'whole milk'}),
frozenset({'yogurt'})},
{frozenset({'whipped/sour cream'}), frozenset({'other vegetables'})},
{frozenset({'whipped/sour cream'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'whipped/sour cream'}), frozenset({'whole milk'})},
{frozenset({'fruit/vegetable juice'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'citrus fruit'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'shopping bags'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'root vegetables'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'tropical fruit'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'bottled water'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'rolls/buns'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'soda'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'yogurt'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'other vegetables'})},
{frozenset({'fruit/vegetable juice'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'fruit/vegetable juice'}), frozenset({'whole milk'})},
{frozenset({'pip fruit'})},
{frozenset({'bottled water'}), frozenset({'pip fruit'})},
{frozenset({'pastry'}), frozenset({'pip fruit'})},
{frozenset({'sausage'}), frozenset({'pip fruit'})},
{frozenset({'soda'}), frozenset({'pip fruit'})},
{frozenset({'citrus fruit'}), frozenset({'pip fruit'})},
{frozenset({'rolls/buns'}), frozenset({'pip fruit'})},
{frozenset({'root vegetables'}), frozenset({'pip fruit'})},
{frozenset({'yogurt'}), frozenset({'pip fruit'})},
{frozenset({'tropical fruit'}), frozenset({'pip fruit'})},
{frozenset({'other vegetables'}), frozenset({'pip fruit'})},
{frozenset({'other vegetables'}),
frozenset({'whole milk'}),
frozenset({'pip fruit'})},
{frozenset({'whole milk'}), frozenset({'pip fruit'})},
{frozenset({'canned beer'})},
{frozenset({'rolls/buns'}), frozenset({'canned beer'})},
{frozenset({'shopping bags'}), frozenset({'canned beer'})},
{frozenset({'soda'}), frozenset({'canned beer'})},
{frozenset({'newspapers'})},
{frozenset({'bottled water'}), frozenset({'newspapers'})},
{frozenset({'root vegetables'}), frozenset({'newspapers'})},
{frozenset({'newspapers'}), frozenset({'tropical fruit'})},
{frozenset({'soda'}), frozenset({'newspapers'})},
{frozenset({'newspapers'}), frozenset({'yogurt'})},
{frozenset({'other vegetables'}), frozenset({'newspapers'})},
{frozenset({'newspapers'}), frozenset({'rolls/buns'})},
{frozenset({'newspapers'}), frozenset({'whole milk'})},
{frozenset({'bottled beer'})},
{frozenset({'bottled beer'}), frozenset({'rolls/buns'})},
{frozenset({'bottled beer'}), frozenset({'bottled water'})},
{frozenset({'bottled beer'}), frozenset({'other vegetables'})},
{frozenset({'bottled beer'}), frozenset({'soda'})},
{frozenset({'bottled beer'}), frozenset({'whole milk'})},
{frozenset({'citrus fruit'})},
{frozenset({'sausage'}), frozenset({'citrus fruit'})},
{frozenset({'soda'}), frozenset({'citrus fruit'})},
{frozenset({'bottled water'}), frozenset({'citrus fruit'})},
{frozenset({'rolls/buns'}), frozenset({'citrus fruit'})},
{frozenset({'root vegetables'}), frozenset({'citrus fruit'})},
{frozenset({'root vegetables'}),
frozenset({'other vegetables'}),
frozenset({'citrus fruit'})},
{frozenset({'tropical fruit'}), frozenset({'citrus fruit'})},
{frozenset({'yogurt'}), frozenset({'citrus fruit'})},
{frozenset({'whole milk'}),
frozenset({'citrus fruit'}),
frozenset({'yogurt'})},
{frozenset({'other vegetables'}), frozenset({'citrus fruit'})},
{frozenset({'other vegetables'}),
frozenset({'whole milk'}),
frozenset({'citrus fruit'})},
{frozenset({'whole milk'}), frozenset({'citrus fruit'})},
{frozenset({'pastry'})},
{frozenset({'root vegetables'}), frozenset({'pastry'})},
{frozenset({'shopping bags'}), frozenset({'pastry'})},
{frozenset({'pastry'}), frozenset({'sausage'})},
{frozenset({'pastry'}), frozenset({'tropical fruit'})},
{frozenset({'pastry'}), frozenset({'yogurt'})},
{frozenset({'pastry'}), frozenset({'rolls/buns'})},
{frozenset({'soda'}), frozenset({'pastry'})},
{frozenset({'pastry'}), frozenset({'other vegetables'})},
{frozenset({'pastry'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'pastry'}), frozenset({'whole milk'})},
{frozenset({'sausage'})},
{frozenset({'bottled water'}), frozenset({'sausage'})},
{frozenset({'sausage'}), frozenset({'tropical fruit'})},
{frozenset({'root vegetables'}), frozenset({'sausage'})},
{frozenset({'shopping bags'}), frozenset({'sausage'})},
{frozenset({'sausage'}), frozenset({'yogurt'})},
{frozenset({'soda'}), frozenset({'sausage'})},
{frozenset({'other vegetables'}), frozenset({'sausage'})},
{frozenset({'sausage'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'sausage'}), frozenset({'whole milk'})},
{frozenset({'sausage'}), frozenset({'rolls/buns'})},
{frozenset({'shopping bags'})},
{frozenset({'shopping bags'}), frozenset({'bottled water'})},
{frozenset({'root vegetables'}), frozenset({'shopping bags'})},
{frozenset({'shopping bags'}), frozenset({'tropical fruit'})},
{frozenset({'shopping bags'}), frozenset({'yogurt'})},
{frozenset({'shopping bags'}), frozenset({'rolls/buns'})},
{frozenset({'shopping bags'}), frozenset({'other vegetables'})},
{frozenset({'shopping bags'}), frozenset({'whole milk'})},
{frozenset({'soda'}), frozenset({'shopping bags'})},
{frozenset({'tropical fruit'})},
{frozenset({'bottled water'}), frozenset({'tropical fruit'})},
{frozenset({'soda'}), frozenset({'tropical fruit'})},
{frozenset({'root vegetables'}), frozenset({'tropical fruit'})},
{frozenset({'root vegetables'}),
frozenset({'whole milk'}),
frozenset({'tropical fruit'})},
{frozenset({'root vegetables'}),
frozenset({'other vegetables'}),
frozenset({'tropical fruit'})},
{frozenset({'rolls/buns'}), frozenset({'tropical fruit'})},
{frozenset({'whole milk'}),
frozenset({'rolls/buns'}),
frozenset({'tropical fruit'})},
{frozenset({'yogurt'}), frozenset({'tropical fruit'})},
{frozenset({'other vegetables'}),
frozenset({'tropical fruit'}),
frozenset({'yogurt'})},
{frozenset({'whole milk'}),
frozenset({'tropical fruit'}),
frozenset({'yogurt'})},
{frozenset({'other vegetables'}), frozenset({'tropical fruit'})},
{frozenset({'other vegetables'}),
frozenset({'whole milk'}),
frozenset({'tropical fruit'})},
{frozenset({'whole milk'}), frozenset({'tropical fruit'})},
{frozenset({'root vegetables'})},
{frozenset({'root vegetables'}), frozenset({'bottled water'})},
{frozenset({'root vegetables'}), frozenset({'soda'})},
{frozenset({'root vegetables'}), frozenset({'rolls/buns'})},
{frozenset({'root vegetables'}),
frozenset({'other vegetables'}),
frozenset({'rolls/buns'})},
{frozenset({'root vegetables'}),
frozenset({'rolls/buns'}),
frozenset({'whole milk'})},
{frozenset({'root vegetables'}), frozenset({'yogurt'})},
{frozenset({'root vegetables'}),
frozenset({'other vegetables'}),
frozenset({'yogurt'})},
{frozenset({'root vegetables'}),
frozenset({'whole milk'}),
frozenset({'yogurt'})},
{frozenset({'root vegetables'}), frozenset({'other vegetables'})},
{frozenset({'root vegetables'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'root vegetables'}), frozenset({'whole milk'})},
{frozenset({'bottled water'})},
{frozenset({'bottled water'}), frozenset({'yogurt'})},
{frozenset({'bottled water'}), frozenset({'rolls/buns'})},
{frozenset({'bottled water'}), frozenset({'other vegetables'})},
{frozenset({'bottled water'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'soda'}), frozenset({'bottled water'})},
{frozenset({'bottled water'}), frozenset({'whole milk'})},
{frozenset({'yogurt'})},
{frozenset({'soda'}), frozenset({'yogurt'})},
{frozenset({'soda'}), frozenset({'whole milk'}), frozenset({'yogurt'})},
{frozenset({'rolls/buns'}), frozenset({'yogurt'})},
{frozenset({'other vegetables'}),
frozenset({'rolls/buns'}),
frozenset({'yogurt'})},
{frozenset({'whole milk'}), frozenset({'rolls/buns'}), frozenset({'yogurt'})},
{frozenset({'other vegetables'}), frozenset({'yogurt'})},
{frozenset({'other vegetables'}),
frozenset({'whole milk'}),
frozenset({'yogurt'})},
{frozenset({'whole milk'}), frozenset({'yogurt'})},
{frozenset({'soda'})},
{frozenset({'soda'}), frozenset({'other vegetables'})},
{frozenset({'soda'}),
frozenset({'other vegetables'}),
frozenset({'whole milk'})},
{frozenset({'soda'}), frozenset({'rolls/buns'})},
{frozenset({'soda'}), frozenset({'whole milk'})},
{frozenset({'rolls/buns'})},
{frozenset({'other vegetables'}), frozenset({'rolls/buns'})},
{frozenset({'other vegetables'}),
frozenset({'rolls/buns'}),
frozenset({'whole milk'})},
{frozenset({'rolls/buns'}), frozenset({'whole milk'})},
{frozenset({'other vegetables'})},
{frozenset({'other vegetables'}), frozenset({'whole milk'})},
{frozenset({'whole milk'})}]