Data 624: Extra Credit
Nakesha Fray
2025-04-30
Extra Credit:
Imagine 10000 receipts sitting on your table. Each receipt represents a transaction with items that were purchased. The receipt is a representation of stuff that went into a customer’s basket - and therefore ‘Market Basket Analysis’. That is exactly what the Groceries Data Set contains: a collection of receipts with each line representing 1 receipt and the items purchased. Each line is called a transaction and each column in a row represents an item. The data set is attached. Your assignment is to use R to mine the data for association rules. You should report support, confidence and lift and your top 10 rules by lift. Extra credit: do a simple cluster analysis on the data as well. Use whichever packages you like.
# Read in dataset
grocery <- read.csv("C:/Users/Kesha/Desktop/Spring 2025/DATA 624/GroceryDataSet.csv")I loaded the CSV file in as a transaction for the arules package.
# Load the file as transactions
grocery_trans <- read.transactions("C:/Users/Kesha/Desktop/Spring 2025/DATA 624/GroceryDataSet.csv",
format = "basket", sep = ",")Here I took a quick look at the data with summary and inspect. We can see that there are 9,835 transactions and 169 unique items. The density was 2.6% which means most transactions contained only a few items. We can also see that the most frequent items were whole milk, other vegetables, rolls/buns, soda, and yogurt.
# View data
summary(grocery_trans)## transactions as itemMatrix in sparse format with
## 9835 rows (elements/itemsets/transactions) and
## 169 columns (items) and a density of 0.02609146
##
## most frequent items:
## whole milk other vegetables rolls/buns soda
## 2513 1903 1809 1715
## yogurt (Other)
## 1372 34055
##
## element (itemset/transaction) length distribution:
## sizes
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
## 2159 1643 1299 1005 855 645 545 438 350 246 182 117 78 77 55 46
## 17 18 19 20 21 22 23 24 26 27 28 29 32
## 29 14 14 9 11 4 6 1 1 1 1 3 1
##
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 2.000 3.000 4.409 6.000 32.000
##
## includes extended item information - examples:
## labels
## 1 abrasive cleaner
## 2 artif. sweetener
## 3 baby cosmeticsinspect(grocery_trans[1:15])## items
## [1] {citrus fruit,
## margarine,
## ready soups,
## semi-finished bread}
## [2] {coffee,
## tropical fruit,
## yogurt}
## [3] {whole milk}
## [4] {cream cheese,
## meat spreads,
## pip fruit,
## yogurt}
## [5] {condensed milk,
## long life bakery product,
## other vegetables,
## whole milk}
## [6] {abrasive cleaner,
## butter,
## rice,
## whole milk,
## yogurt}
## [7] {rolls/buns}
## [8] {bottled beer,
## liquor (appetizer),
## other vegetables,
## rolls/buns,
## UHT-milk}
## [9] {pot plants}
## [10] {cereals,
## whole milk}
## [11] {bottled water,
## chocolate,
## other vegetables,
## tropical fruit,
## white bread}
## [12] {bottled water,
## butter,
## citrus fruit,
## curd,
## dishes,
## flour,
## tropical fruit,
## whole milk,
## yogurt}
## [13] {beef}
## [14] {frankfurter,
## rolls/buns,
## soda}
## [15] {chicken,
## tropical fruit}Next, I generated the association rules using the apriori() function with a minimum support of 0.01 and a minimum confidence of 0.5. This resulted in 15 strong association rules, each involving three items. The support values ranged from 1.01% to 2.23%, confidence from 0.50 to 0.59, and lift values from 1.98 to 3.03, indicating positive associations and varying strength of association between items. The rule with the highest lift involved citrus fruit and root vegetables leading to the purchase of other vegetables, suggesting that customers who buy citrus fruit and root vegetables are about three times more likely to also buy other vegetables. Additionally, the rules revealed strong links between root vegetables, tropical fruit and other vegetables.
rules <- apriori(grocery_trans, parameter = list(supp = 0.01, conf = 0.5))## Apriori
##
## Parameter specification:
## confidence minval smax arem aval originalSupport maxtime support minlen
## 0.5 0.1 1 none FALSE TRUE 5 0.01 1
## maxlen target ext
## 10 rules TRUE
##
## Algorithmic control:
## filter tree heap memopt load sort verbose
## 0.1 TRUE TRUE FALSE TRUE 2 TRUE
##
## Absolute minimum support count: 98
##
## set item appearances ...[0 item(s)] done [0.00s].
## set transactions ...[169 item(s), 9835 transaction(s)] done [0.00s].
## sorting and recoding items ... [88 item(s)] done [0.00s].
## creating transaction tree ... done [0.00s].
## checking subsets of size 1 2 3 4 done [0.00s].
## writing ... [15 rule(s)] done [0.00s].
## creating S4 object ... done [0.00s].inspect(rules)## lhs rhs support
## [1] {curd, yogurt} => {whole milk} 0.01006609
## [2] {butter, other vegetables} => {whole milk} 0.01148958
## [3] {domestic eggs, other vegetables} => {whole milk} 0.01230300
## [4] {whipped/sour cream, yogurt} => {whole milk} 0.01087951
## [5] {other vegetables, whipped/sour cream} => {whole milk} 0.01464159
## [6] {other vegetables, pip fruit} => {whole milk} 0.01352313
## [7] {citrus fruit, root vegetables} => {other vegetables} 0.01037112
## [8] {root vegetables, tropical fruit} => {other vegetables} 0.01230300
## [9] {root vegetables, tropical fruit} => {whole milk} 0.01199797
## [10] {tropical fruit, yogurt} => {whole milk} 0.01514997
## [11] {root vegetables, yogurt} => {other vegetables} 0.01291307
## [12] {root vegetables, yogurt} => {whole milk} 0.01453991
## [13] {rolls/buns, root vegetables} => {other vegetables} 0.01220132
## [14] {rolls/buns, root vegetables} => {whole milk} 0.01270971
## [15] {other vegetables, yogurt} => {whole milk} 0.02226741
## confidence coverage lift count
## [1] 0.5823529 0.01728521 2.279125 99
## [2] 0.5736041 0.02003050 2.244885 113
## [3] 0.5525114 0.02226741 2.162336 121
## [4] 0.5245098 0.02074225 2.052747 107
## [5] 0.5070423 0.02887646 1.984385 144
## [6] 0.5175097 0.02613116 2.025351 133
## [7] 0.5862069 0.01769192 3.029608 102
## [8] 0.5845411 0.02104728 3.020999 121
## [9] 0.5700483 0.02104728 2.230969 118
## [10] 0.5173611 0.02928317 2.024770 149
## [11] 0.5000000 0.02582613 2.584078 127
## [12] 0.5629921 0.02582613 2.203354 143
## [13] 0.5020921 0.02430097 2.594890 120
## [14] 0.5230126 0.02430097 2.046888 125
## [15] 0.5128806 0.04341637 2.007235 219summary(rules)## set of 15 rules
##
## rule length distribution (lhs + rhs):sizes
## 3
## 15
##
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 3 3 3 3 3 3
##
## summary of quality measures:
## support confidence coverage lift
## Min. :0.01007 Min. :0.5000 Min. :0.01729 Min. :1.984
## 1st Qu.:0.01174 1st Qu.:0.5151 1st Qu.:0.02089 1st Qu.:2.036
## Median :0.01230 Median :0.5245 Median :0.02430 Median :2.203
## Mean :0.01316 Mean :0.5411 Mean :0.02454 Mean :2.299
## 3rd Qu.:0.01403 3rd Qu.:0.5718 3rd Qu.:0.02598 3rd Qu.:2.432
## Max. :0.02227 Max. :0.5862 Max. :0.04342 Max. :3.030
## count
## Min. : 99.0
## 1st Qu.:115.5
## Median :121.0
## Mean :129.4
## 3rd Qu.:138.0
## Max. :219.0
##
## mining info:
## data ntransactions support confidence
## grocery_trans 9835 0.01 0.5
## call
## apriori(data = grocery_trans, parameter = list(supp = 0.01, conf = 0.5))Theses are the top 10 association rules, sorted by lift. The output showed somewhat strong co-purchasing patterns. The highest lift value of 3.0 appeared in two rules: {citrus fruit, root vegetables} => {other vegetables} and {root vegetables, tropical fruit} => {other vegetables}, indicating that customers who purchase these combinations are three times more likely to also buy other vegetables. These rules suggest strong associations within fruits and vegetables. Other notable rules include {rolls/buns, root vegetables} => {other vegetables} with a lift of 2.6 and {curd, yogurt} => {whole milk} with a lift of 2.3, confidence of 58%, and support of 1.0%. Furthermore, rules predicting the purchase of whole milk were also commonly paired with items like butter, yogurt, and root vegetables, with lift values ranging from 2.1 to 2.3.
rules_sorted <- sort(rules, by = "lift", decreasing = TRUE)
top10_rules <- head(rules_sorted, 10)
inspect(top10_rules)## lhs rhs support
## [1] {citrus fruit, root vegetables} => {other vegetables} 0.01037112
## [2] {root vegetables, tropical fruit} => {other vegetables} 0.01230300
## [3] {rolls/buns, root vegetables} => {other vegetables} 0.01220132
## [4] {root vegetables, yogurt} => {other vegetables} 0.01291307
## [5] {curd, yogurt} => {whole milk} 0.01006609
## [6] {butter, other vegetables} => {whole milk} 0.01148958
## [7] {root vegetables, tropical fruit} => {whole milk} 0.01199797
## [8] {root vegetables, yogurt} => {whole milk} 0.01453991
## [9] {domestic eggs, other vegetables} => {whole milk} 0.01230300
## [10] {whipped/sour cream, yogurt} => {whole milk} 0.01087951
## confidence coverage lift count
## [1] 0.5862069 0.01769192 3.029608 102
## [2] 0.5845411 0.02104728 3.020999 121
## [3] 0.5020921 0.02430097 2.594890 120
## [4] 0.5000000 0.02582613 2.584078 127
## [5] 0.5823529 0.01728521 2.279125 99
## [6] 0.5736041 0.02003050 2.244885 113
## [7] 0.5700483 0.02104728 2.230969 118
## [8] 0.5629921 0.02582613 2.203354 143
## [9] 0.5525114 0.02226741 2.162336 121
## [10] 0.5245098 0.02074225 2.052747 107The plot shows that whole milk is the most frequently purchased item, being in approximately 25% of customer transactions, followed by other vegetables (~19%), rolls/buns (~18%), and soda (~17.5%). These stand out due to their higher purchase frequencies compared to the rest of the items which suggest that they are commonly found in customer baskets.
itemFrequencyPlot(grocery_trans, topN = 20)
For the simple cluster analysis, I performed k-means clustering on the grocery data. First, I converted the grocery_trans transaction data into a binary matrix, where a value of 1 indicates a purchased item and 0 indicates a non-purchased item. Then, I performed k-means clustering with 3 centers, grouping the transactions into 3 distinct clusters.
The cluster plot showed three groups, all of which overlap. The blue cluster, which is similar in size to the green cluster, contains the majority of data points from all three clusters (red, blue, and green). The green cluster overlaps about half of its area with the blue cluster and the other half with the red cluster. Most of its data points are concentrated in the overlap between the green, blue, and red clusters, with fewer data points in the green/red-only overlap. Lastly, the red cluster, the largest in size, overlaps with both the green and blue clusters, with about one-third of its area in each. The majority of its data points are found in the green/blue overlap, and far fewer are located in the red-only section.
The overlapping suggests that there are no clearly distinct groups, which aligns with earlier findings from the association rules, further confirming the relationships between purchased items. The blue cluster, which contains the majority of the data points, likely captures many commonly purchased patterns shared across transactions. The red and green clusters, while somewhat distinct, still overlap heavily with the blue cluster, indicating that these clusters represent a mix of both common and less common purchasing patterns across transactions.
groceries_matrix <- as(grocery_trans, "matrix")
set.seed(11233)
kmeans <- kmeans(groceries_matrix, centers = 3, nstart = 10)
kmeans$cluster## [1] 3 3 2 3 1 2 3 1 3 2 1 2 3 3 3 3 3 3 3 3 1 3 2 3 1 3 3 3 1 3 3 1 1 1 3 3 3
## [38] 3 1 3 3 2 3 3 3 3 3 3 3 1 3 3 1 1 2 2 3 3 3 3 1 3 2 3 3 2 3 3 2 2 3 2 2 2
## [75] 3 1 2 3 3 3 3 2 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 2 3 3 2 3 2 3 3 3 2 3 3 1
## [112] 3 2 3 3 2 2 1 2 1 3 3 3 1 3 2 3 3 3 3 3 2 3 3 3 1 3 3 3 3 2 1 2 2 3 3 1 3
## [149] 3 3 1 3 3 1 1 3 3 2 1 3 2 3 3 3 1 3 2 2 3 1 1 1 3 3 3 2 2 3 3 1 2 1 1 3 3
## [186] 1 3 3 2 1 3 3 3 3 3 2 3 3 3 3 3 3 3 3 2 3 1 3 3 3 3 3 1 2 2 3 3 3 3 3 3 3
## [223] 1 3 3 3 3 3 3 3 1 1 2 3 3 1 3 1 2 2 3 1 2 3 2 3 3 3 1 2 3 2 3 3 1 1 3 3 1
## [260] 2 1 3 3 3 3 3 3 1 3 3 2 3 3 2 1 1 1 2 1 3 3 2 3 3 2 2 3 3 3 1 3 1 1 1 2 3
## [297] 3 3 3 1 3 3 3 3 3 3 3 3 2 3 2 3 3 3 1 3 3 3 3 3 3 3 2 3 1 1 3 3 1 3 3 3 3
## [334] 3 3 3 3 3 3 1 2 3 3 3 3 1 3 2 2 2 3 3 3 3 3 2 3 3 3 2 2 2 3 3 2 3 1 3 3 3
## [371] 1 3 3 1 3 2 2 3 3 3 3 3 3 3 3 1 3 3 3 2 3 3 3 3 3 2 3 3 2 1 3 1 2 1 1 3 3
## [408] 3 3 3 3 1 1 2 3 1 3 3 3 1 3 3 2 3 3 3 3 3 3 2 3 3 1 3 3 3 3 3 1 3 1 3 1 3
## [445] 3 3 1 3 1 1 2 3 2 1 3 1 3 3 3 3 3 3 3 2 3 3 1 3 2 3 1 3 2 3 1 2 1 3 2 3 3
## [482] 2 3 3 3 3 1 3 1 3 3 3 3 2 3 3 3 1 1 1 3 2 3 3 3 3 3 2 1 3 3 3 3 3 3 2 3 2
## [519] 1 3 3 3 1 1 3 1 3 2 3 3 3 3 3 2 3 3 3 2 2 3 3 3 3 1 3 2 2 2 2 1 3 3 3 3 3
## [556] 3 3 1 3 3 3 1 2 2 3 3 3 3 2 3 3 1 1 3 3 3 3 3 3 2 3 3 3 3 3 2 2 2 3 3 3 3
## [593] 3 3 1 1 2 2 3 3 3 3 3 2 1 3 3 2 1 3 3 3 1 3 1 2 3 3 3 2 3 3 3 3 2 2 3 1 2
## [630] 2 1 2 2 1 3 1 3 1 3 3 3 1 1 3 3 3 3 3 1 3 3 2 3 1 1 3 3 3 1 2 3 1 3 3 3 2
## [667] 3 2 1 3 3 3 2 3 3 2 3 2 2 2 1 2 3 3 1 2 3 3 3 3 3 3 3 3 2 3 1 3 2 3 2 3 3
## [704] 3 2 3 1 2 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 1 2 1 3 3 3 3 3 1
## [741] 2 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 1 3 1 2 1 3 1 3 3 2 2 3 1 3 3 1 2 1 1
## [778] 1 3 1 3 2 2 3 2 3 3 3 3 1 3 2 2 3 3 2 1 2 3 3 2 1 3 2 3 3 3 3 2 2 2 2 3 3
## [815] 3 3 2 1 3 3 1 2 2 1 3 2 3 1 3 2 2 1 2 3 3 2 3 3 3 2 2 3 2 1 3 2 1 1 1 1 2
## [852] 3 1 3 1 2 3 3 3 1 1 1 3 3 2 3 3 3 2 3 3 2 3 2 2 3 3 3 3 3 3 3 3 3 3 2 3 3
## [889] 3 1 3 3 1 2 1 2 2 3 3 2 3 1 2 3 1 3 3 2 3 2 3 3 3 3 3 3 3 3 3 3 2 2 2 1 3
## [926] 3 3 2 3 3 2 2 3 3 3 3 2 3 3 3 3 3 2 3 3 1 2 3 1 1 1 3 2 3 3 3 3 2 2 3 3 1
## [963] 1 3 3 3 2 3 1 3 3 3 3 3 1 3 3 3 1 1 1 3 3 3 2 2 3 3 1 2 1 3 3 3 3 1 1 3 3
## [1000] 1 3 3 3 1 3 3 3 3 2 2 3 1 2 1 3 3 1 3 2 2 3 3 2 3 3 3 1 3 3 3 2 3 3 3 3 2
## [1037] 3 3 3 2 3 1 1 3 1 1 1 3 3 2 3 2 2 3 3 3 2 3 2 3 3 3 2 1 3 3 1 3 3 3 3 3 3
## [1074] 1 1 3 1 3 3 3 3 2 3 1 2 3 3 3 3 1 1 1 2 3 3 3 3 3 3 3 3 1 1 3 2 1 3 3 1 3
## [1111] 3 3 3 3 1 3 3 3 3 2 3 1 1 1 3 3 1 3 3 3 3 3 3 1 1 3 3 1 3 1 3 3 3 3 1 3 3
## [1148] 1 1 1 3 3 1 3 3 3 3 3 3 3 3 3 3 3 3 1 2 1 1 1 1 3 3 3 1 3 3 3 3 3 3 3 1 3
## [1185] 1 3 3 2 2 2 3 3 3 3 3 3 2 1 2 3 1 3 3 3 1 2 2 3 2 1 2 1 2 3 3 2 1 1 1 1 3
## [1222] 1 3 3 3 3 1 1 1 2 3 3 3 3 1 2 2 3 3 3 3 3 1 1 1 3 3 2 3 3 3 3 2 3 2 1 3 3
## [1259] 3 3 2 3 1 3 3 3 1 2 3 2 3 1 3 3 1 3 3 3 2 3 3 1 3 1 1 3 3 3 3 2 2 3 3 1 3
## [1296] 3 3 3 3 3 1 1 2 3 1 3 2 1 1 3 1 3 1 3 1 3 3 3 1 3 3 3 1 1 2 3 3 3 3 3 2 1
## [1333] 2 3 2 3 3 2 1 1 3 3 3 2 3 1 2 2 3 3 3 1 3 1 3 3 3 1 2 3 1 3 3 2 3 1 3 3 3
## [1370] 3 3 1 3 2 1 3 3 3 3 3 3 2 3 3 2 1 3 1 3 3 3 2 1 3 3 3 3 3 3 2 3 3 3 3 3 1
## [1407] 2 3 3 2 2 3 3 1 3 3 3 1 3 2 3 3 3 3 3 2 2 3 3 3 2 2 3 2 3 2 3 3 3 3 3 3 2
## [1444] 2 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 1 3 3 2 3 1 3 3 2 3 1 3
## [1481] 2 3 2 3 3 3 1 3 1 1 3 3 2 3 2 3 1 3 3 3 2 3 3 3 3 3 1 1 2 3 3 1 3 1 2 3 1
## [1518] 3 2 3 3 3 2 3 3 1 3 3 3 3 3 3 3 1 2 2 3 2 1 3 1 1 1 1 3 3 3 3 2 3 1 2 3 3
## [1555] 1 3 3 1 1 3 3 2 3 3 3 2 3 3 3 3 3 3 3 3 3 1 3 2 3 3 3 3 3 3 3 3 3 3 3 1 3
## [1592] 1 3 1 3 3 3 3 3 1 3 3 1 3 3 3 1 2 3 3 3 3 2 3 3 3 2 3 3 3 1 1 1 3 3 3 3 3
## [1629] 2 3 3 3 3 3 3 3 3 3 1 3 2 3 3 3 3 3 3 3 3 2 1 2 3 1 3 3 3 1 2 3 1 1 2 2 2
## [1666] 1 2 3 3 3 2 2 3 3 3 1 3 1 3 1 3 1 3 3 3 3 2 2 3 3 3 3 2 3 1 1 3 1 3 3 3 3
## [1703] 3 1 2 1 1 3 3 2 1 3 2 3 3 3 1 3 1 3 2 1 3 3 2 2 2 3 3 3 3 3 3 3 3 1 3 3 2
## [1740] 2 3 3 1 3 3 3 3 3 3 3 2 3 3 1 3 1 3 1 3 3 3 3 2 3 2 2 1 3 3 1 3 3 2 1 1 3
## [1777] 3 3 3 1 2 3 3 3 3 2 2 3 3 2 2 3 3 2 2 3 2 2 3 3 3 3 1 3 3 1 3 3 3 1 3 3 3
## [1814] 3 2 3 3 3 2 3 3 3 3 2 3 2 1 3 3 2 2 3 3 3 3 2 1 3 3 3 1 3 1 3 3 1 2 3 3 2
## [1851] 3 3 2 1 3 3 2 3 3 3 3 3 3 1 3 3 3 2 2 3 2 3 2 2 1 3 3 3 1 3 3 1 1 3 1 3 2
## [1888] 3 1 3 3 2 1 3 3 1 3 2 2 3 3 3 1 2 2 1 3 3 3 3 3 3 1 1 3 2 3 1 2 2 1 1 1 1
## [1925] 3 3 2 3 3 3 3 3 1 2 3 2 2 1 2 3 1 2 3 2 3 1 3 2 1 2 3 2 2 3 3 3 3 3 3 3 3
## [1962] 3 3 3 3 1 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 2 3 1 3 3 3 2 3 2 3
## [1999] 3 3 3 1 1 3 1 3 1 1 2 3 1 2 3 1 1 1 3 1 3 3 1 2 3 3 2 3 1 3 3 3 3 3 3 3 1
## [2036] 2 3 3 3 2 2 3 2 3 3 3 2 2 3 3 3 1 3 1 1 1 3 3 3 1 2 3 3 1 3 3 3 2 3 3 2 2
## [2073] 3 3 3 3 2 3 3 3 2 3 2 3 3 2 3 2 1 1 3 3 3 2 1 2 3 3 1 2 2 3 3 3 3 1 3 1 3
## [2110] 3 3 3 3 3 3 3 3 2 3 3 1 1 3 1 3 3 3 3 3 3 2 3 1 2 1 1 3 1 1 3 3 3 3 2 3 3
## [2147] 3 2 2 3 3 3 3 3 3 3 2 2 3 3 1 1 3 3 3 1 3 3 1 1 1 2 2 3 3 3 1 1 3 3 3 3 3
## [2184] 1 3 1 3 1 2 3 3 1 3 3 2 3 2 3 1 3 3 3 3 3 3 3 2 3 2 3 3 1 1 1 3 2 2 1 3 3
## [2221] 3 3 1 2 3 2 2 2 1 3 2 3 1 2 3 3 3 3 3 3 2 3 1 3 3 2 3 3 3 3 3 3 2 1 3 1 2
## [2258] 3 2 3 3 1 3 2 3 3 3 3 3 3 1 3 3 1 3 3 1 3 3 2 3 3 3 3 3 3 3 3 1 2 3 1 3 3
## [2295] 3 3 3 3 3 2 1 3 2 3 3 2 2 2 3 1 3 3 2 1 2 1 3 3 3 1 1 3 1 2 3 3 3 3 2 2 2
## [2332] 3 2 3 1 2 1 3 3 3 3 3 3 2 2 1 3 3 1 3 3 3 3 1 3 3 1 3 3 3 3 3 3 3 1 2 3 3
## [2369] 3 3 2 2 1 3 3 3 1 3 3 3 3 3 3 3 3 2 3 1 3 2 1 3 3 2 3 3 3 3 3 3 3 2 1 3 3
## [2406] 3 3 3 1 2 3 2 3 3 1 3 2 1 3 3 3 3 3 3 3 3 3 1 3 3 2 1 2 3 1 1 3 3 3 3 3 3
## [2443] 2 3 1 3 3 3 3 2 3 3 2 3 1 2 3 3 3 3 3 2 2 3 3 3 2 1 3 1 3 3 1 2 3 3 3 2 1
## [2480] 2 3 3 3 3 3 3 1 3 3 2 3 3 1 3 3 3 1 3 1 3 3 3 3 3 2 3 3 1 3 3 3 3 3 2 3 3
## [2517] 1 3 3 1 3 2 3 3 3 2 1 2 1 3 2 3 1 3 3 1 3 1 3 2 3 3 1 1 2 1 3 3 3 1 2 3 3
## [2554] 1 2 2 3 3 3 1 2 3 2 3 3 2 3 2 3 3 3 3 3 1 3 3 2 2 3 3 1 3 2 3 1 3 3 1 1 1
## [2591] 3 3 3 3 2 3 1 3 1 2 3 3 3 2 3 3 3 3 2 3 2 1 3 3 2 3 3 3 3 2 3 2 3 1 2 3 3
## [2628] 3 1 1 3 1 2 3 2 2 1 2 3 3 3 1 3 3 2 3 2 3 3 3 3 1 3 3 3 3 3 1 3 3 3 2 1 3
## [2665] 2 2 3 1 3 3 2 3 3 1 2 3 3 3 3 2 3 1 3 1 3 3 3 3 1 3 1 3 3 3 1 3 3 3 3 1 3
## [2702] 3 1 3 1 3 3 1 3 1 3 3 3 3 2 2 2 3 3 3 3 3 2 2 3 2 3 1 3 3 1 3 1 1 3 3 1 3
## [2739] 2 3 2 2 3 1 1 3 3 3 3 1 3 3 1 3 3 1 3 3 3 3 3 2 1 1 3 3 3 3 3 3 3 3 2 3 3
## [2776] 1 3 1 2 3 2 1 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3
## [2813] 3 3 2 1 3 3 3 2 3 3 3 3 3 1 3 3 3 3 2 3 3 3 3 2 2 3 2 3 3 3 1 3 3 3 3 2 2
## [2850] 3 2 3 3 3 3 3 1 3 3 3 3 3 1 2 3 2 3 3 2 3 3 3 3 3 2 3 3 3 3 3 1 2 2 3 2 2
## [2887] 2 2 3 3 3 3 1 3 3 3 3 3 1 1 3 3 3 2 3 3 3 3 2 1 1 3 3 3 3 3 3 2 3 3 3 3 3
## [2924] 3 3 3 3 1 3 3 2 3 3 3 3 3 3 2 1 3 3 3 1 2 3 3 3 1 3 3 1 3 2 1 1 3 2 3 3 3
## [2961] 1 2 1 3 2 3 3 2 1 3 1 1 2 1 2 3 3 3 3 3 3 1 1 3 3 3 3 1 3 1 3 1 3 3 3 1 2
## [2998] 3 2 2 3 3 1 3 1 2 3 3 1 1 2 3 1 3 3 1 1 1 1 3 2 2 1 3 3 3 1 1 3 3 2 1 1 3
## [3035] 3 1 3 3 2 3 1 1 3 2 3 1 3 3 1 3 3 1 2 1 2 1 1 3 1 3 3 3 3 2 3 2 1 3 3 2 3
## [3072] 2 3 2 3 3 3 3 2 1 3 1 3 1 3 3 1 3 3 3 1 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3
## [3109] 3 3 3 3 1 3 3 3 3 3 1 3 3 3 3 1 3 3 3 3 3 3 3 3 1 1 2 1 3 3 3 3 3 3 3 3 3
## [3146] 2 1 2 2 1 1 3 1 1 3 3 1 3 2 3 3 1 3 3 3 3 3 1 2 1 3 3 1 2 3 2 3 1 2 3 3 3
## [3183] 2 3 3 2 3 3 3 1 3 3 1 3 1 2 1 2 1 3 3 3 3 1 1 3 1 3 1 1 3 3 3 1 1 1 2 1 3
## [3220] 1 1 1 2 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 2 1 2 1 3 2 3 3 1 3 3 3 3 3 3 2
## [3257] 2 3 3 1 1 3 1 3 3 1 3 3 1 3 2 3 3 3 1 3 3 3 3 3 3 2 2 3 1 1 3 3 2 3 3 1 3
## [3294] 3 2 1 3 1 3 2 2 3 2 3 3 1 3 2 3 3 3 3 3 3 3 2 3 1 1 3 3 3 3 3 3 3 3 3 3 1
## [3331] 1 2 3 3 1 1 3 1 1 2 3 3 2 3 1 3 3 3 3 1 3 3 3 3 2 3 3 1 3 3 3 3 3 2 3 3 3
## [3368] 3 3 3 3 2 3 3 2 3 3 3 3 3 1 2 1 1 3 3 3 3 3 1 2 2 3 3 3 1 3 3 3 3 3 3 3 2
## [3405] 2 2 3 3 3 2 2 3 3 3 2 3 3 1 3 1 3 2 3 1 3 3 3 1 2 3 3 3 3 3 2 3 2 3 3 2 3
## [3442] 3 2 3 3 3 3 3 2 3 1 3 3 2 3 3 3 3 3 1 3 3 3 3 3 1 3 2 2 2 3 3 3 3 1 2 3 3
## [3479] 3 3 3 3 3 3 3 3 2 3 3 2 2 3 1 3 2 2 3 3 3 3 1 3 1 3 3 2 3 2 2 3 3 3 3 3 1
## [3516] 3 3 3 2 3 3 3 3 3 3 3 1 3 3 3 3 2 3 3 3 3 3 1 3 3 2 3 3 2 3 2 2 2 3 3 1 3
## [3553] 3 3 3 1 2 2 3 1 3 1 3 3 3 2 3 2 3 3 3 3 3 3 3 2 1 3 3 3 3 3 3 3 3 2 3 3 3
## [3590] 2 3 3 2 1 3 2 3 3 3 3 2 3 3 3 3 3 3 1 1 3 2 3 3 2 2 3 3 3 3 3 3 1 3 3 1 3
## [3627] 3 1 3 1 1 1 3 2 3 3 3 3 3 2 2 2 3 2 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 1 3
## [3664] 3 3 1 3 3 1 3 3 2 1 1 3 1 3 1 1 3 3 3 3 3 2 3 1 1 3 3 1 2 2 2 3 1 3 3 3 3
## [3701] 3 1 3 3 3 2 3 1 3 2 3 3 3 3 3 1 3 1 3 1 3 3 3 3 1 3 2 3 3 2 3 3 3 3 2 3 3
## [3738] 2 2 3 2 3 3 2 3 3 3 3 3 3 2 1 3 3 1 3 3 3 1 3 2 1 3 3 3 3 3 2 3 3 3 3 3 3
## [3775] 3 3 3 1 3 3 1 3 3 3 3 3 3 3 3 2 1 3 3 3 3 3 2 3 3 2 1 3 3 3 3 3 3 3 3 3 2
## [3812] 3 3 1 1 3 3 1 3 3 1 2 3 3 3 3 3 1 3 3 3 3 1 2 3 3 3 2 2 2 1 1 3 3 2 1 3 3
## [3849] 1 1 3 1 3 1 1 2 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 2 3 2 3 3 3 1 3 2 1
## [3886] 3 2 2 3 3 1 1 3 3 1 2 3 3 2 3 3 3 1 2 1 3 1 1 3 3 1 2 3 2 2 2 3 2 3 3 3 3
## [3923] 3 1 3 3 3 3 3 1 3 3 3 1 2 2 3 1 3 2 3 3 3 3 3 3 3 1 2 1 2 1 1 1 1 1 3 3 3
## [3960] 3 3 3 2 2 3 3 3 3 3 1 1 1 2 1 3 3 1 1 2 3 3 3 3 2 1 3 3 3 2 3 3 1 3 1 3 3
## [3997] 1 3 3 3 1 3 3 3 2 3 3 3 3 2 2 2 3 2 1 2 2 3 3 1 3 3 3 3 3 3 3 3 2 3 3 1 1
## [4034] 2 3 1 2 1 3 3 3 1 3 3 1 3 2 2 1 2 3 1 1 3 3 2 3 2 3 1 2 3 3 1 3 1 3 1 3 3
## [4071] 3 2 2 1 3 3 2 3 2 1 1 3 3 2 3 3 3 1 2 1 2 3 3 3 3 2 2 3 2 3 2 2 2 2 1 3 1
## [4108] 1 3 3 3 2 2 3 3 3 3 3 3 2 3 2 3 2 3 1 1 2 3 1 3 2 2 3 1 3 1 3 3 2 3 3 3 3
## [4145] 1 3 3 3 3 3 1 3 3 3 3 3 1 3 1 1 2 3 1 1 2 1 2 1 3 3 3 3 3 2 2 3 3 3 3 1 3
## [4182] 3 1 1 3 1 3 3 3 3 2 3 1 3 3 3 3 3 3 3 3 3 3 3 1 1 3 3 1 3 2 3 3 1 3 3 2 3
## [4219] 1 3 3 1 1 3 3 2 2 3 1 1 1 1 1 2 1 2 3 2 3 3 3 3 3 3 2 3 2 2 3 1 3 2 3 3 3
## [4256] 1 2 1 3 3 2 3 3 3 3 3 3 3 3 2 3 1 3 2 2 3 2 3 3 1 3 3 3 3 3 3 2 2 1 2 2 3
## [4293] 3 2 1 3 2 3 3 3 2 1 2 3 1 3 1 3 3 3 2 2 2 1 2 3 1 2 1 3 2 3 3 1 2 3 2 1 1
## [4330] 1 3 1 2 3 3 2 2 2 1 2 3 3 1 3 1 1 2 3 2 3 3 3 3 1 1 3 1 1 3 1 2 3 1 3 3 3
## [4367] 1 2 1 1 3 1 3 2 1 1 3 1 2 3 1 2 2 3 3 1 3 3 1 3 2 3 2 3 1 1 1 3 1 2 3 3 3
## [4404] 3 1 2 3 3 3 1 3 1 3 3 1 1 1 3 2 1 1 3 2 3 3 3 2 3 1 3 2 2 2 3 3 3 3 3 3 2
## [4441] 3 2 3 1 3 1 3 1 2 1 1 3 3 3 2 3 1 3 1 1 1 3 1 3 3 3 3 2 3 3 3 1 3 3 3 1 3
## [4478] 3 2 1 3 3 2 3 2 3 2 1 3 3 3 1 3 1 3 3 3 3 1 2 3 3 3 1 1 3 3 3 3 3 2 3 2 1
## [4515] 3 1 2 3 1 3 3 1 2 3 3 2 3 3 1 3 1 3 3 3 3 3 2 3 2 1 3 3 1 1 3 1 3 3 1 1 2
## [4552] 3 3 1 2 1 3 1 3 3 3 3 3 3 2 1 3 3 2 3 1 3 3 1 1 3 2 3 2 3 3 3 3 3 3 2 3 2
## [4589] 3 3 3 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 1 2 3 2 3 3 3 3 3 2 2 3 3
## [4626] 1 1 3 1 3 3 3 3 3 2 2 3 3 2 3 3 3 1 3 3 3 2 3 3 3 3 2 3 3 3 2 3 3 3 3 3 3
## [4663] 3 2 1 1 3 3 3 3 3 3 1 2 3 3 1 2 3 1 2 1 3 3 3 3 1 3 3 3 3 3 2 3 1 2 3 3 3
## [4700] 2 3 3 3 2 3 2 1 3 3 3 3 1 3 1 3 2 3 2 2 3 1 1 3 3 1 2 3 3 2 3 1 1 3 3 3 3
## [4737] 3 3 2 2 1 1 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2 1 2 3 3 3 3 3 2 3 3 3 3 3 1
## [4774] 1 3 3 3 3 3 2 1 2 3 1 3 3 2 3 3 3 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3
## [4811] 3 3 3 3 3 2 3 3 3 3 1 3 3 3 3 3 3 2 2 3 3 3 3 2 3 3 2 3 1 3 1 1 2 3 3 3 1
## [4848] 1 2 3 3 3 3 3 2 2 3 3 3 3 3 3 2 3 3 1 3 1 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3
## [4885] 2 3 3 2 3 3 2 3 1 3 3 3 1 3 2 3 2 3 3 2 3 3 3 3 3 3 3 3 2 3 2 2 1 3 3 3 3
## [4922] 3 2 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 1 2 3 3 1 1 3 3 2 1 2 2 3 3 3 3 3 2 3
## [4959] 2 2 3 1 3 3 3 3 3 1 3 1 3 3 3 3 3 1 3 3 2 3 3 2 3 1 3 3 3 3 3 1 2 2 3 3 3
## [4996] 1 1 3 3 3 3 1 2 3 1 3 2 3 2 3 3 2 2 3 3 3 3 3 1 3 3 3 3 3 2 1 3 1 2 3 2 3
## [5033] 3 1 3 3 3 3 3 3 3 3 3 3 2 3 3 2 1 1 2 3 3 3 3 3 3 3 3 3 2 3 2 3 3 2 1 3 2
## [5070] 1 2 2 3 2 3 3 3 1 3 3 2 3 2 3 3 3 3 2 3 1 3 3 3 3 3 3 3 2 3 3 2 3 2 3 3 3
## [5107] 1 3 2 3 1 3 3 3 1 3 3 3 1 3 3 3 1 3 3 3 3 1 3 2 2 3 1 1 3 3 3 3 3 3 3 3 3
## [5144] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 2 1 1 1 3 1 3 1 3 3 1 1 3 1 3 3 2 3 3
## [5181] 3 3 3 3 3 3 3 3 1 3 3 2 3 2 1 1 3 3 2 1 3 3 3 3 1 1 3 3 2 3 2 3 3 3 2 1 3
## [5218] 3 3 1 3 3 3 3 3 2 3 3 1 3 3 3 3 3 3 3 1 3 1 3 2 1 3 1 1 1 3 3 3 3 3 3 3 1
## [5255] 3 3 2 2 3 3 3 2 3 1 3 3 3 3 2 1 2 3 3 2 3 3 3 3 3 3 2 3 3 2 1 3 1 3 3 3 3
## [5292] 2 3 2 1 1 2 1 3 2 3 3 3 3 3 1 3 3 2 3 2 2 1 3 3 1 3 3 3 3 3 2 2 3 2 3 1 3
## [5329] 3 2 2 3 3 3 3 1 3 1 2 2 3 3 2 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 1 3 3 3 2 1
## [5366] 2 3 3 3 3 3 3 3 1 2 3 3 3 2 3 3 3 3 1 2 2 3 3 3 3 3 1 3 3 3 3 1 2 3 3 3 2
## [5403] 3 3 3 1 3 1 3 3 3 3 3 1 3 3 3 1 2 3 3 3 1 1 1 3 1 3 3 2 3 3 3 2 2 2 3 3 3
## [5440] 1 2 3 3 2 2 3 2 3 2 3 2 2 3 3 3 1 3 3 3 2 3 1 3 1 3 3 2 2 3 1 3 2 3 1 3 3
## [5477] 1 2 1 3 3 1 1 1 3 1 3 3 2 3 1 2 3 1 3 3 3 3 2 2 1 3 3 3 1 3 3 3 2 2 2 2 2
## [5514] 3 1 3 1 1 3 2 3 3 3 3 1 3 1 1 1 1 3 3 1 3 1 3 2 1 3 2 3 1 3 3 3 3 3 1 1 1
## [5551] 1 1 1 3 2 3 3 2 1 3 3 3 3 3 3 3 3 3 2 2 3 2 2 3 3 3 1 3 2 3 3 3 1 3 1 3 3
## [5588] 3 3 3 3 2 3 2 1 1 3 2 3 3 3 2 3 3 2 3 3 3 3 3 2 2 2 1 1 3 3 3 2 2 1 1 1 1
## [5625] 1 3 2 3 3 3 2 3 3 1 1 2 2 3 2 2 1 2 2 1 1 1 1 1 3 1 1 3 3 3 3 3 1 3 3 1 1
## [5662] 3 1 3 2 3 3 1 1 1 2 1 1 3 2 3 1 2 2 3 3 3 1 1 1 2 3 2 1 3 2 1 3 1 3 3 2 2
## [5699] 1 3 3 1 3 1 1 3 2 3 1 2 2 3 3 1 3 3 3 1 2 2 1 3 1 2 1 3 2 2 3 2 3 1 2 1 2
## [5736] 1 1 3 1 2 3 3 1 3 2 2 3 2 3 1 3 1 3 1 3 3 2 3 3 3 3 3 2 1 3 1 3 1 3 3 1 3
## [5773] 3 1 1 3 3 3 3 1 2 1 3 2 1 1 1 1 1 3 3 3 3 3 2 1 3 2 1 3 3 1 1 1 2 1 1 1 2
## [5810] 1 1 3 3 3 3 2 1 1 3 3 3 1 3 3 1 3 3 3 3 3 2 2 3 1 3 3 3 3 3 3 3 3 3 3 3 3
## [5847] 3 3 2 3 3 3 3 1 3 3 1 3 3 1 3 3 2 2 3 2 3 1 3 1 3 3 3 2 3 3 3 1 3 3 3 3 3
## [5884] 3 3 1 2 1 3 3 1 1 3 3 3 1 1 1 3 2 3 1 2 1 1 3 3 2 2 3 1 1 1 3 3 1 3 3 3 3
## [5921] 1 1 1 2 3 3 2 3 3 3 2 1 3 3 2 3 3 3 2 1 3 3 3 3 3 2 3 3 3 3 1 2 2 1 3 3 1
## [5958] 2 3 3 3 1 3 3 3 2 1 2 2 1 2 1 3 3 1 2 3 2 2 1 3 2 2 3 3 3 1 3 3 3 3 3 3 3
## [5995] 2 3 1 3 3 3 2 3 3 3 2 3 2 2 3 3 3 2 3 3 2 3 1 3 2 3 3 3 3 3 3 3 1 3 3 3 3
## [6032] 3 3 3 3 1 1 2 3 3 3 3 3 2 3 3 3 3 3 1 2 2 3 3 2 3 3 3 3 3 3 2 2 3 3 3 3 1
## [6069] 3 2 1 1 3 3 3 3 3 2 3 3 3 3 2 3 3 3 3 3 2 3 2 3 1 3 2 3 1 3 3 1 3 3 3 2 3
## [6106] 3 1 3 3 2 2 3 3 3 3 3 3 3 3 2 1 3 2 3 3 3 3 3 3 3 3 2 3 3 1 3 3 3 2 3 3 3
## [6143] 3 3 1 3 3 3 3 2 1 3 1 3 3 3 3 2 3 3 1 1 3 3 3 3 3 2 3 2 3 3 3 3 3 1 3 1 3
## [6180] 3 3 3 3 3 3 3 3 3 3 1 2 3 3 3 3 3 2 3 2 1 1 1 3 3 3 3 3 3 1 3 2 3 1 3 2 2
## [6217] 3 3 3 3 2 3 3 3 1 3 3 1 3 3 3 3 3 2 3 1 3 3 2 3 1 3 3 2 3 3 2 3 3 3 2 3 2
## [6254] 3 2 1 3 2 2 3 1 2 1 3 1 1 3 3 3 3 2 3 3 1 3 2 3 3 3 1 3 2 3 3 2 3 3 1 2 1
## [6291] 3 3 3 3 3 3 3 3 1 3 3 2 3 3 3 3 2 3 3 2 1 1 2 3 2 3 3 3 3 1 3 3 3 1 1 3 2
## [6328] 1 1 3 3 3 3 3 1 2 1 3 1 1 3 3 3 1 3 3 3 1 3 3 3 3 1 3 3 2 3 3 3 3 1 3 3 3
## [6365] 3 3 3 3 1 1 3 1 3 3 1 3 3 3 3 3 3 3 3 3 3 1 2 2 3 3 3 1 3 3 1 2 1 3 1 3 3
## [6402] 3 2 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 1 1 3 3 1 2 3
## [6439] 1 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 2 3 1 3 3 1 1
## [6476] 3 3 3 1 3 3 2 1 3 3 3 3 3 2 1 3 2 3 3 3 1 3 3 3 1 3 3 3 3 3 1 3 3 2 2 1 2
## [6513] 3 1 3 1 3 3 3 3 3 3 3 1 3 3 3 2 3 2 2 1 3 3 3 3 3 3 3 3 2 2 3 3 3 3 1 1 3
## [6550] 3 3 2 2 3 3 3 1 1 2 3 3 3 2 1 3 3 1 3 3 3 3 2 2 2 3 3 3 3 1 2 3 3 3 3 3 3
## [6587] 2 3 3 3 1 3 3 3 2 3 3 3 2 3 2 3 1 3 3 3 3 1 3 3 3 3 1 3 3 3 3 1 3 1 3 3 1
## [6624] 3 1 3 3 3 1 2 3 3 3 1 3 1 3 3 3 3 1 3 3 2 3 3 3 3 3 3 3 2 3 3 2 3 3 3 3 1
## [6661] 3 3 3 3 3 1 3 3 3 3 1 3 2 3 3 3 3 2 2 2 1 2 3 3 2 3 2 2 2 3 3 3 2 2 1 3 3
## [6698] 2 2 1 3 3 2 1 2 2 2 2 1 3 1 2 2 3 3 1 1 2 3 3 2 2 2 3 3 3 3 3 3 3 1 2 3 2
## [6735] 3 3 3 3 1 1 1 3 3 2 2 2 1 1 3 3 3 1 1 1 3 2 2 2 3 2 2 3 1 1 3 3 3 2 2 1 3
## [6772] 2 3 1 3 1 3 3 1 2 1 2 3 1 1 3 2 1 1 3 1 2 3 2 2 3 3 3 1 3 3 1 1 2 2 3 2 3
## [6809] 1 1 1 2 3 3 3 3 1 3 1 3 1 2 1 3 1 1 2 1 1 1 3 3 1 3 2 3 2 3 3 3 1 3 3 3 1
## [6846] 3 3 3 3 1 3 1 3 3 1 2 3 3 1 3 2 3 1 1 1 3 3 3 3 3 1 3 3 3 3 3 3 1 3 3 3 1
## [6883] 3 1 3 3 1 3 3 3 2 1 3 3 3 3 1 2 3 3 3 3 1 3 2 3 3 3 3 3 3 3 3 3 3 3 1 3 3
## [6920] 3 1 3 2 3 2 2 3 2 3 3 3 3 3 1 3 3 1 2 3 3 3 1 3 3 3 2 1 3 3 1 3 2 3 2 3 3
## [6957] 3 3 3 3 3 1 3 1 3 3 3 3 3 3 2 3 2 3 2 3 2 2 3 3 3 3 3 3 2 3 3 3 3 1 3 3 3
## [6994] 3 2 1 1 3 3 3 3 3 3 3 3 3 1 3 3 3 2 3 1 1 3 3 2 3 3 1 3 2 3 2 3 2 3 2 3 2
## [7031] 3 1 3 3 3 1 3 3 1 3 3 3 3 3 3 2 3 3 3 3 3 2 3 3 2 3 3 3 3 2 1 3 3 3 3 2 3
## [7068] 1 1 3 3 3 3 3 3 3 2 3 3 3 3 2 2 2 1 2 3 3 3 3 3 3 3 2 1 3 3 1 3 3 3 3 2 3
## [7105] 3 3 1 3 2 3 3 3 3 1 3 1 2 2 2 3 2 3 1 3 3 3 3 3 3 2 3 2 1 1 1 3 2 2 3 2 3
## [7142] 3 2 1 1 1 3 2 3 3 3 3 3 3 2 3 3 3 1 1 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1
## [7179] 3 3 2 2 1 2 2 3 3 2 3 1 3 3 3 2 3 2 3 3 3 3 3 2 2 3 3 2 3 3 3 1 1 3 1 3 3
## [7216] 3 2 3 1 3 2 3 3 3 3 3 3 2 3 3 3 1 3 3 3 1 3 3 3 3 3 2 3 3 3 3 3 3 3 1 3 2
## [7253] 1 3 3 2 2 1 2 2 1 1 3 3 3 3 2 3 3 1 3 3 2 3 3 3 3 3 2 3 3 3 1 2 1 3 2 2 3
## [7290] 2 2 3 3 2 3 3 1 3 1 3 1 3 1 1 3 1 3 3 3 3 3 3 2 3 3 3 3 1 1 3 3 3 3 3 3 3
## [7327] 3 3 3 3 3 3 2 3 1 3 3 2 2 3 3 2 2 3 3 3 3 3 3 3 1 3 3 2 2 3 2 3 3 3 3 2 3
## [7364] 2 3 1 1 2 3 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 2 3 3 3 3 1 3 3 2 3 3 1 2 3 3 3
## [7401] 3 3 3 3 3 3 3 1 3 3 3 3 3 2 3 3 3 1 2 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 1 3 2
## [7438] 3 2 3 3 3 2 3 3 3 3 2 2 3 3 2 1 3 1 2 2 2 3 3 3 1 2 2 2 3 3 3 1 1 2 2 1 1
## [7475] 3 3 3 3 3 3 3 2 2 1 3 3 2 3 3 3 3 1 2 3 1 3 3 3 3 1 1 3 3 3 3 3 3 2 2 2 3
## [7512] 3 3 1 3 1 1 3 2 2 3 3 3 1 3 3 2 3 3 3 3 2 3 1 3 3 2 3 2 3 1 1 1 3 1 3 3 1
## [7549] 3 1 3 2 2 1 1 3 3 1 1 3 3 3 3 3 3 1 3 1 3 3 1 2 3 1 1 2 3 3 3 3 3 1 3 3 2
## [7586] 3 3 3 3 3 3 2 1 3 1 3 2 1 1 3 3 3 3 2 3 1 1 2 3 3 2 3 1 3 3 3 3 3 3 3 3 3
## [7623] 3 3 3 2 3 3 3 3 3 3 3 2 3 3 3 3 3 1 1 3 2 3 3 1 3 2 3 3 3 2 2 3 3 2 3 3 3
## [7660] 1 3 1 3 3 3 2 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 1 1 3 3 3 2 1 3 2 3 3 3 3 3 3
## [7697] 2 1 2 3 2 1 3 3 2 3 3 3 3 2 3 2 3 3 3 3 2 1 1 3 3 2 3 2 3 1 1 3 2 2 3 3 2
## [7734] 3 3 3 3 3 2 3 3 1 1 3 2 3 1 2 3 3 3 1 3 2 1 3 1 3 2 1 3 3 2 1 1 1 3 1 3 2
## [7771] 2 2 3 3 3 2 1 3 3 3 2 2 1 3 3 1 3 3 3 3 3 2 3 3 3 3 2 2 3 3 2 3 3 1 3 1 1
## [7808] 1 3 1 2 1 2 3 3 3 2 3 3 3 3 3 3 2 2 1 2 1 3 2 3 2 2 3 1 3 2 1 3 3 3 2 1 3
## [7845] 3 2 1 3 3 1 3 3 3 2 2 2 1 3 1 3 3 3 3 1 2 1 3 3 2 2 3 3 3 1 3 2 3 2 2 1 3
## [7882] 3 2 3 1 3 3 3 2 3 2 3 2 3 3 3 3 2 2 3 3 3 2 3 2 3 3 3 3 3 3 3 3 3 3 2 3 2
## [7919] 3 2 3 3 2 1 2 3 3 3 3 2 3 2 3 3 3 1 3 1 3 3 3 3 3 2 3 1 3 3 2 3 2 3 3 3 3
## [7956] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 1 3 1 3 3 3 3 3 3 1 2 1 3 3
## [7993] 3 3 3 3 1 3 3 2 3 3 3 2 3 3 3 3 1 3 3 3 3 3 2 1 3 1 1 3 2 3 3 3 3 2 2 2 3
## [8030] 3 3 3 3 3 3 3 1 3 3 3 1 3 3 1 3 3 2 2 3 3 2 2 3 3 1 1 2 3 3 3 3 3 2 3 3 1
## [8067] 2 1 3 3 2 3 3 3 3 3 3 3 2 3 3 3 3 3 2 3 3 3 2 3 2 3 1 3 3 3 2 2 3 3 3 2 3
## [8104] 1 3 3 3 2 3 3 3 3 3 3 1 3 1 3 1 3 3 3 3 3 3 1 3 3 1 3 3 3 3 3 3 3 3 3 1 3
## [8141] 2 3 3 1 2 3 1 3 1 3 3 3 3 3 3 3 3 1 1 3 3 1 2 3 3 3 3 3 3 2 3 3 3 1 3 1 3
## [8178] 3 3 3 3 2 3 1 2 1 2 1 3 3 3 3 3 3 3 2 2 3 3 3 3 3 2 3 3 2 3 3 3 1 3 3 3 3
## [8215] 3 3 2 3 3 3 3 3 1 3 3 3 3 3 3 2 1 2 1 1 1 2 3 3 1 3 3 3 1 3 3 3 3 2 2 3 2
## [8252] 3 3 3 2 3 2 3 3 3 3 3 3 2 2 3 3 1 3 3 3 3 1 3 3 3 3 1 3 3 1 3 3 3 2 2 1 1
## [8289] 2 2 3 3 3 3 1 3 1 3 3 3 3 3 3 3 2 1 3 3 2 2 3 3 3 1 3 3 3 2 3 2 3 3 3 2 3
## [8326] 3 1 3 3 2 3 1 3 3 3 3 3 3 3 2 3 3 3 3 1 3 3 2 2 3 3 3 1 2 3 2 3 1 3 2 3 2
## [8363] 3 3 3 2 3 2 3 3 2 3 3 3 3 3 3 3 3 1 3 1 3 3 2 3 2 1 3 3 1 3 3 3 1 3 3 1 2
## [8400] 3 3 1 3 3 2 3 2 3 2 3 3 3 1 3 1 3 3 3 3 3 3 2 3 3 2 1 3 3 2 3 3 1 3 3 3 3
## [8437] 1 3 3 3 3 3 3 1 2 3 2 3 3 3 2 3 2 2 3 3 3 3 3 1 3 3 3 3 3 1 3 2 3 3 1 3 1
## [8474] 1 3 3 3 2 2 3 3 2 3 3 3 3 3 3 3 2 3 3 3 3 3 3 2 1 3 3 3 3 3 3 3 2 3 2 3 3
## [8511] 3 3 3 2 1 3 3 3 2 3 3 3 3 3 3 3 2 2 2 2 3 1 1 3 1 3 2 1 3 2 3 3 1 3 2 3 3
## [8548] 3 1 3 3 3 3 3 3 3 3 3 3 2 2 3 3 1 1 3 2 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3
## [8585] 1 2 3 3 3 2 2 2 2 3 3 3 2 2 3 1 3 3 1 1 3 3 3 3 3 1 1 3 3 3 3 1 3 3 3 3 3
## [8622] 3 3 3 3 3 2 2 1 3 3 3 3 3 3 3 3 1 3 2 3 2 1 3 1 3 3 3 2 3 3 3 3 3 3 1 3 3
## [8659] 3 1 3 3 2 3 2 1 3 2 1 3 3 3 3 3 3 1 1 2 1 3 2 2 3 3 1 3 1 2 3 3 3 1 1 1 3
## [8696] 1 3 3 3 3 3 1 3 1 3 2 3 3 2 2 2 3 2 2 1 2 3 2 2 3 1 3 3 3 3 1 3 3 3 1 3 3
## [8733] 3 3 3 3 3 3 3 3 3 2 1 3 2 3 3 1 2 3 3 2 1 3 1 3 1 3 1 3 3 3 2 3 1 2 3 3 2
## [8770] 3 3 2 3 2 3 3 1 3 3 3 3 3 3 3 1 3 2 3 1 3 3 3 1 3 3 3 3 3 3 2 3 3 3 3 3 2
## [8807] 3 3 1 3 3 3 3 2 3 1 3 2 3 3 3 3 1 3 2 3 3 2 1 1 3 2 3 1 3 1 3 3 2 3 3 3 3
## [8844] 1 2 3 1 2 3 2 2 1 2 3 2 1 3 1 3 2 1 1 1 1 2 1 3 3 3 2 3 2 3 3 1 3 3 3 3 3
## [8881] 3 3 1 1 3 2 3 1 3 3 2 1 3 3 1 1 3 1 3 1 3 2 1 3 3 3 2 3 3 3 3 3 1 2 3 3 3
## [8918] 3 2 1 3 3 3 3 1 2 3 1 3 3 1 3 1 1 3 3 3 2 1 3 2 1 2 2 1 3 3 2 3 1 1 3 3 3
## [8955] 3 1 3 1 3 1 3 3 3 3 1 2 1 3 1 3 3 1 3 1 3 1 2 1 2 3 3 2 2 3 2 3 1 3 3 3 2
## [8992] 3 3 1 3 3 1 2 1 3 2 2 3 2 3 3 3 1 3 1 3 2 1 3 1 3 3 1 3 1 1 3 3 3 2 2 2 3
## [9029] 1 1 3 1 2 2 3 3 1 3 1 1 3 3 1 1 1 3 1 3 1 3 3 3 3 3 1 1 3 3 3 3 3 3 1 3 1
## [9066] 1 1 3 3 2 2 2 1 2 1 1 3 3 3 2 3 1 3 3 1 1 1 3 1 3 1 1 1 3 2 1 3 3 1 1 3 3
## [9103] 3 3 1 1 3 3 3 1 1 3 2 1 3 2 3 1 2 3 1 3 1 1 2 1 1 3 2 3 2 3 1 3 1 3 1 3 1
## [9140] 3 1 3 3 3 3 1 1 2 3 3 3 3 3 2 3 3 1 3 3 3 3 3 3 1 3 3 3 1 1 3 1 3 2 1 2 1
## [9177] 2 3 3 1 1 1 1 1 3 2 1 1 3 1 3 1 3 3 1 3 3 3 3 1 3 2 3 3 3 1 1 1 3 3 3 1 3
## [9214] 1 3 1 1 3 3 3 1 3 2 3 3 1 3 3 3 1 3 1 3 3 1 1 3 3 3 3 3 3 3 3 1 1 3 3 3 1
## [9251] 3 1 1 3 3 3 3 3 2 1 2 2 3 1 3 3 3 3 3 3 3 2 1 3 1 3 1 1 3 3 3 3 1 3 3 2 2
## [9288] 3 1 3 3 1 3 3 3 2 3 3 3 2 3 3 1 3 3 1 3 3 1 1 3 3 3 3 2 2 3 1 1 3 3 3 3 3
## [9325] 3 2 3 2 1 3 2 1 3 1 1 3 3 2 3 3 3 1 3 3 3 3 1 1 3 3 3 3 3 2 1 3 3 3 3 3 3
## [9362] 3 2 3 3 1 1 3 3 3 1 3 2 3 2 3 2 3 2 3 3 3 3 2 3 3 3 3 1 3 2 3 3 3 3 3 3 3
## [9399] 2 3 3 1 1 3 3 3 2 2 2 3 3 1 1 3 1 3 3 2 3 3 3 3 1 1 3 3 2 1 1 2 3 2 3 3 1
## [9436] 2 3 2 3 2 3 3 3 2 3 3 2 2 3 3 1 2 1 3 1 1 3 2 1 1 1 3 3 1 3 3 3 1 3 3 3 3
## [9473] 2 1 3 1 3 3 3 3 1 2 3 3 2 2 2 3 3 3 3 2 3 3 3 3 3 3 3 1 3 2 3 3 3 3 3 3 2
## [9510] 1 2 3 1 3 3 3 3 3 3 1 3 1 2 1 3 3 1 3 2 3 3 3 3 2 3 3 3 1 3 3 3 1 3 2 3 3
## [9547] 2 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 2 3 2 3 3 3 2 3 3 1 1 2 3 1 3 2 2 2 3 2 3
## [9584] 3 3 3 3 1 3 3 3 3 3 1 1 2 3 3 3 3 2 3 3 3 1 3 3 2 1 3 3 2 3 2 2 3 1 3 2 3
## [9621] 3 3 2 3 1 3 3 1 3 3 3 3 3 3 3 1 1 2 2 2 3 3 3 3 2 2 3 3 3 3 3 3 1 1 2 3 1
## [9658] 3 2 3 1 1 1 3 2 1 3 3 2 3 3 3 3 1 2 2 3 2 1 3 3 1 3 3 3 3 3 3 1 3 3 3 3 3
## [9695] 3 3 3 2 2 3 2 2 3 2 1 3 3 2 3 3 3 2 3 3 2 3 3 1 2 3 3 3 3 1 3 1 2 3 3 1 3
## [9732] 3 2 3 2 3 3 3 3 3 2 3 1 1 1 3 2 3 3 3 3 3 2 2 3 3 3 2 3 2 2 3 3 3 3 2 1 1
## [9769] 3 3 3 3 3 1 3 3 1 2 2 3 2 2 3 2 2 3 3 3 2 3 3 1 1 3 2 3 3 3 3 1 1 3 2 1 3
## [9806] 3 2 3 3 1 3 1 3 2 2 3 3 1 3 2 3 1 3 3 3 1 1 1 3 1 2 3 1 3 1kmeans$centers## abrasive cleaner artif. sweetener baby cosmetics baby food bags
## 1 0.008363826 0.005227392 0.0005227392 0.0005227392 0.0000000000
## 2 0.005064716 0.003376477 0.0011254924 0.0000000000 0.0005627462
## 3 0.001627339 0.002603743 0.0004882018 0.0000000000 0.0004882018
## baking powder bathroom cleaner beef berries beverages bottled beer
## 1 0.038159958 0.005227392 0.10297961 0.05384213 0.02665970 0.08416100
## 2 0.027011818 0.001688239 0.06640405 0.03770400 0.02870006 0.07090602
## 3 0.008624898 0.002278275 0.03270952 0.02554923 0.02506103 0.08218063
## bottled water brandy brown bread butter butter milk cake bar
## 1 0.12859383 0.003136435 0.09670674 0.10507057 0.05331939 0.019341349
## 2 0.13055712 0.002250985 0.08778841 0.08891390 0.03826674 0.019696117
## 3 0.09910496 0.005044752 0.04833198 0.03026851 0.01708706 0.009438568
## candles candy canned beer canned fish canned fruit canned vegetables
## 1 0.012023001 0.03659174 0.04652378 0.02665970 0.005750131 0.024568740
## 2 0.012380416 0.02926280 0.03320203 0.01631964 0.005064716 0.010129432
## 3 0.006997559 0.02799024 0.10024410 0.01106591 0.001952807 0.006672091
## cat food cereals chewing gum chicken chocolate
## 1 0.03345531 0.010454783 0.02352326 0.09357031 0.06638787
## 2 0.03207653 0.012943163 0.01913337 0.05064716 0.06190208
## 3 0.01757526 0.002115541 0.02082994 0.02489829 0.04084622
## chocolate marshmallow citrus fruit cleaner cling film/bags cocoa drinks
## 1 0.010977522 0.15107162 0.008363826 0.016727653 0.002613696
## 2 0.013505909 0.09679235 0.007315701 0.015756894 0.005627462
## 3 0.007160293 0.05744508 0.003417413 0.008462164 0.001139138
## coffee condensed milk cooking chocolate cookware cream
## 1 0.06952431 0.013068479 0.004181913 0.0031364349 0.0036591741
## 2 0.06809229 0.009566685 0.003939223 0.0005627462 0.0005627462
## 3 0.05158666 0.009601302 0.001627339 0.0032546786 0.0008136697
## cream cheese curd curd cheese decalcifier dental care dessert
## 1 0.07056979 0.08991113 0.010977522 0.0026136958 0.010454783 0.06011500
## 2 0.05402364 0.09003939 0.006190208 0.0022509848 0.005064716 0.04839617
## 3 0.02587469 0.03124491 0.002929211 0.0009764036 0.004556550 0.02668836
## detergent dish cleaner dishes dog food domestic eggs
## 1 0.03345531 0.01254574 0.03136435 0.011500261 0.11657083
## 2 0.02982555 0.01012943 0.01463140 0.010129432 0.09791784
## 3 0.01171684 0.00992677 0.01415785 0.007160293 0.03694060
## female sanitary products finished products fish flour
## 1 0.007318348 0.010454783 0.003659174 0.03345531
## 2 0.006190208 0.003376477 0.003939223 0.02588633
## 3 0.005695688 0.006183889 0.002441009 0.00992677
## flower (seeds) flower soil/fertilizer frankfurter frozen chicken
## 1 0.019341349 0.001045478 0.08572922 0.0000000000
## 2 0.012943163 0.001125492 0.07146877 0.0011254924
## 3 0.006834825 0.002441009 0.04703011 0.0006509357
## frozen dessert frozen fish frozen fruits frozen meals frozen potato products
## 1 0.01881861 0.024046001 0.0041819132 0.03920544 0.013591218
## 2 0.01069218 0.012380416 0.0000000000 0.03432752 0.009003939
## 3 0.00829943 0.007648495 0.0006509357 0.02327095 0.006672091
## frozen vegetables fruit/vegetable juice grapes hair spray ham
## 1 0.09357031 0.11029796 0.04652378 0.001045478 0.04704652
## 2 0.05965110 0.08947665 0.01913337 0.001125492 0.03714125
## 3 0.03059398 0.05549227 0.01578519 0.001139138 0.01627339
## hamburger meat hard cheese herbs honey house keeping products
## 1 0.07161526 0.04966022 0.039728176 0.0015682175 0.014113957
## 2 0.04670793 0.03151379 0.020258863 0.0050647158 0.011817670
## 3 0.01741253 0.01464605 0.007811229 0.0004882018 0.005532954
## hygiene articles ice cream instant coffee Instant food products jam
## 1 0.04966022 0.02613696 0.009932044 0.014636696 0.009932044
## 2 0.04220597 0.01969612 0.006752954 0.008441193 0.009003939
## 3 0.02506103 0.02620016 0.006834825 0.005858421 0.002929211
## ketchup kitchen towels kitchen utensil light bulbs liqueur
## 1 0.008363826 0.011500261 0.0005227392 0.006795609 0.0005227392
## 2 0.005064716 0.009566685 0.0011254924 0.002813731 0.0011254924
## 3 0.002766477 0.003254679 0.0001627339 0.003742880 0.0009764036
## liquor liquor (appetizer) liver loaf long life bakery product
## 1 0.006795609 0.007318348 0.007841087 0.05488761
## 2 0.002250985 0.006190208 0.008441193 0.04333146
## 3 0.014971522 0.008624898 0.003254679 0.03026851
## make up remover male cosmetics margarine mayonnaise meat meat spreads
## 1 0.0005227392 0.004181913 0.10297961 0.019341349 0.05227392 0.004704652
## 2 0.0005627462 0.003376477 0.08272369 0.009566685 0.03207653 0.003939223
## 3 0.0009764036 0.005044752 0.03775427 0.005858421 0.01578519 0.004231082
## misc. beverages mustard napkins newspapers nut snack nuts/prunes
## 1 0.02875065 0.016727653 0.07579718 0.09932044 0.004181913 0.004704652
## 2 0.02532358 0.020258863 0.07146877 0.10523354 0.001688239 0.003939223
## 3 0.02912937 0.008136697 0.03954434 0.06639544 0.003254679 0.002766477
## oil onions organic products organic sausage other vegetables
## 1 0.05279665 0.07370622 0.003136435 0.002613696 0.9947726
## 2 0.03432752 0.03038829 0.001688239 0.003939223 0.0000000
## 3 0.01855167 0.01790073 0.001139138 0.001627339 0.0000000
## packaged fruit/vegetables pasta pastry pet care photo/film
## 1 0.01725039 0.02247778 0.11709357 0.009932044 0.005750131
## 2 0.01631964 0.02194710 0.12549240 0.009566685 0.009566685
## 3 0.01074044 0.01074044 0.06965012 0.009275834 0.010252238
## pickled vegetables pip fruit popcorn pork pot plants
## 1 0.03345531 0.13695766 0.008363826 0.11395714 0.02247778
## 2 0.02813731 0.09172763 0.011254924 0.06640405 0.02476083
## 3 0.01008950 0.05191212 0.005695688 0.03759154 0.01350692
## potato products preservation products processed cheese prosecco
## 1 0.004181913 0.0005227392 0.02822791 0.001568217
## 2 0.004501970 0.0005627462 0.02476083 0.002250985
## 3 0.001952807 0.0000000000 0.01057771 0.002115541
## pudding powder ready soups red/blush wine rice roll products
## 1 0.0041819132 0.003136435 0.02561422 0.020386827 0.024568740
## 2 0.0050647158 0.002813731 0.01181767 0.011254924 0.012943163
## 3 0.0009764036 0.001139138 0.01936534 0.002603743 0.005044752
## rolls/buns root vegetables rubbing alcohol rum salad dressing
## 1 0.2200732 0.24725562 0.0020909566 0.007841087 0.0026136958
## 2 0.2144063 0.14237479 0.0016882386 0.006752954 0.0000000000
## 3 0.1638731 0.05630594 0.0004882018 0.002766477 0.0004882018
## salt salty snack sauces sausage seasonal products
## 1 0.019341349 0.05645583 0.007841087 0.14009409 0.01881861
## 2 0.013505909 0.03489026 0.008441193 0.10917276 0.01012943
## 3 0.007323027 0.03287225 0.003905614 0.07518308 0.01399512
## semi-finished bread shopping bags skin care sliced cheese snack products
## 1 0.02718244 0.12023001 0.006272870 0.04652378 0.004704652
## 2 0.02476083 0.09341587 0.004501970 0.03432752 0.001688239
## 3 0.01269325 0.09324654 0.002441009 0.01480879 0.002929211
## soap soda soft cheese softener sound storage medium
## 1 0.002613696 0.1693675 0.037114480 0.008363826 0.0000000000
## 2 0.004501970 0.1446258 0.022509848 0.008441193 0.0000000000
## 3 0.002115541 0.1845403 0.009275834 0.003742880 0.0001627339
## soups sparkling wine specialty bar specialty cheese specialty chocolate
## 1 0.016204914 0.007841087 0.02875065 0.021955044 0.03136435
## 2 0.006190208 0.003376477 0.02250985 0.008441193 0.03263928
## 3 0.004068348 0.005532954 0.02831570 0.004393816 0.02945484
## specialty fat specialty vegetables spices spread cheese sugar
## 1 0.006272870 0.003659174 0.010977522 0.016204914 0.05541035
## 2 0.003376477 0.001125492 0.005064716 0.010692178 0.04839617
## 3 0.002929211 0.001301871 0.003417413 0.009764036 0.02294548
## sweet spreads syrup tea tidbits toilet cleaner
## 1 0.013068479 0.005750131 0.007841087 0.002090957 0.0010454783
## 2 0.014631401 0.002813731 0.005064716 0.003376477 0.0005627462
## 3 0.006183889 0.002603743 0.002278275 0.002115541 0.0006509357
## tropical fruit turkey UHT-milk vinegar waffles
## 1 0.18557240 0.020386827 0.04286461 0.013068479 0.05331939
## 2 0.13956106 0.008441193 0.01406866 0.007315701 0.04445695
## 3 0.06981286 0.004231082 0.03612693 0.004231082 0.03205858
## whipped/sour cream whisky white bread white wine whole milk yogurt
## 1 0.15211709 0.0010454783 0.07109252 0.01150026 0.384736 0.22791427
## 2 0.09735509 0.0000000000 0.06190208 0.01069218 1.000000 0.18683174
## 3 0.03921888 0.0009764036 0.02733930 0.02375915 0.000000 0.09829129
## zwieback
## 1 0.009409305
## 2 0.006190208
## 3 0.006346623fviz_cluster(kmeans, data = groceries_matrix)