Association Rules - Life in L.A. <a name=top></a>¶
In this notebook, you will apply association rules mining to the Life in L.A. dataset.
The dataset consists of 16 observations, each item having 8 variables. Our goal here is to find the combination of items that occur with certain frequency, and find item-sets that provide insights to the structure of data
TUTORIAL OUTLINE¶
1. LOAD AND EXPLORE DATA <a name=load></a>¶
The very fisrt step in programming with R is to import data. Then we perform a few routines to explore the structure of data.
In [1]:
life.in.LA <- read.csv("Data/turner_life_in_LA.csv", head=TRUE)
head(life.in.LA) # look at data (first 6 rows)
str(life.in.LA) # investigate types of variables; other than lon & lat, features seem to be "categorical"
summary(life.in.LA) # provides 6-point summary for each variable
| lon | lat | affluence | unemployment | urban.stress | prop.white | lon.class | lat.class |
|---|---|---|---|---|---|---|---|
| 183 | 104 | 2 | 2 | 1 | 1 | West | North |
| 329 | 155 | 2 | 2 | 1 | 1 | Central | North |
| 87 | 183 | 1 | 2 | 1 | 1 | West | North |
| 154 | 184 | 2 | 2 | 1 | 1 | West | North |
| 224 | 225 | 2 | 2 | 1 | 1 | Central | North |
| 139 | 266 | 1 | 1 | 1 | 1 | West | Central |
'data.frame': 16 obs. of 8 variables: $ lon : int 183 329 87 154 224 139 156 222 352 433 ... $ lat : int 104 155 183 184 225 266 335 312 304 320 ... $ affluence : int 2 2 1 2 2 1 1 1 3 3 ... $ unemployment: int 2 2 2 2 2 1 1 1 2 2 ... $ urban.stress: int 1 1 1 1 1 1 1 2 2 2 ... $ prop.white : int 1 1 1 1 1 1 1 1 1 1 ... $ lon.class : Factor w/ 3 levels "Central","East",..: 3 1 3 3 1 3 3 1 2 2 ... $ lat.class : Factor w/ 3 levels "Central","North",..: 2 2 2 2 2 1 1 1 1 1 ...
lon lat affluence unemployment urban.stress Min. : 87.0 Min. :104.0 Min. :1.000 Min. :1.000 Min. :1.0 1st Qu.:176.2 1st Qu.:214.8 1st Qu.:1.750 1st Qu.:1.750 1st Qu.:1.0 Median :248.5 Median :316.0 Median :2.000 Median :2.000 Median :1.0 Mean :264.0 Mean :316.8 Mean :2.125 Mean :1.938 Mean :1.5 3rd Qu.:352.8 3rd Qu.:382.0 3rd Qu.:3.000 3rd Qu.:2.000 3rd Qu.:2.0 Max. :433.0 Max. :634.0 Max. :3.000 Max. :3.000 Max. :3.0 prop.white lon.class lat.class Min. :1.000 Central:6 Central:6 1st Qu.:1.000 East :5 North :5 Median :1.000 West :5 South :5 Mean :1.188 3rd Qu.:1.000 Max. :3.000
2. EXPLORING DATA WITH arules<a name=arules></a>¶
In previous section, we learned that variables $affluence$, $unemployment$, $urban.stress$, and $prop.white$, are technically categorical variables. We will convert these integer values to categories by function called lapply. Then we use a function called apriori, which returns the all possible $rules$ ($itemset$).<br><br>
By default, apriori creates rules with minimum support of $0.1$, minimum confidence of $0.8$, and maximum of $10$ items.
<br>
$\textbf{NOTE:}$ Since apriori is a function within a package called arules, we must load the library beforehand.
In [2]:
library(arules)
(life.in.LA[,c(3,4,5,6)] <- lapply(life.in.LA[,c(3,4,5,6)] , factor)) # Categorizes variables 3 to 6
# find association rules with default settings
rules.LA <- apriori(life.in.LA[,3:8])
Loading required package: Matrix
Attaching package: ‘arules’
The following objects are masked from ‘package:base’:
abbreviate, write
- $affluence
- 2
- 2
- 1
- 2
- 2
- 1
- 1
- 1
- 3
- 3
- 2
- 3
- 3
- 3
- 2
- 3
- $unemployment
- 2
- 2
- 2
- 2
- 2
- 1
- 1
- 1
- 2
- 2
- 3
- 1
- 3
- 3
- 2
- 2
- $urban.stress
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 2
- 2
- 2
- 1
- 2
- 3
- 2
- 1
- 2
- $prop.white
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 2
- 3
- 1
- 1
Apriori
Parameter specification:
confidence minval smax arem aval originalSupport maxtime support minlen
0.8 0.1 1 none FALSE TRUE 5 0.1 1
maxlen target ext
10 rules FALSE
Algorithmic control:
filter tree heap memopt load sort verbose
0.1 TRUE TRUE FALSE TRUE 2 TRUE
Absolute minimum support count: 1
set item appearances ...[0 item(s)] done [0.00s].
set transactions ...[18 item(s), 16 transaction(s)] done [0.00s].
sorting and recoding items ... [15 item(s)] done [0.00s].
creating transaction tree ... done [0.00s].
checking subsets of size 1 2 3 4 5 6 done [0.00s].
writing ... [444 rule(s)] done [0.00s].
creating S4 object ... done [0.00s].
For a rule $X \to Y$, arules computes the following metrics:
- $\text{Support}(X\to Y) = \frac{\text{Freq}(X,Y)}{N}$
- $\text{Confidence}(X\to Y) = \frac{\text{Freq}(X,Y)}{\text{Freq}(X)}$
- $\text{Lift}(X\to Y) = \frac{N^2 \times \text{Support}(X\to Y)}{\text{Freq}(X) \times \text{Freq}(Y)}$
The higher these values, the better.
The following function inspect returns a list of all rules created by apriori. Using the default setting, the list consists of $444$ rules (which is quite lengthy... it looks like the default setting is very lenient).
In [3]:
inspect(rules.LA)
lhs rhs support confidence lift
[1] {} => {prop.white=1} 0.8750 0.8750000 1.0000000
[2] {unemployment=3} => {lat.class=South} 0.1875 1.0000000 3.2000000
[3] {affluence=1} => {prop.white=1} 0.2500 1.0000000 1.1428571
[4] {unemployment=1} => {lat.class=Central} 0.2500 1.0000000 2.6666667
[5] {unemployment=1} => {prop.white=1} 0.2500 1.0000000 1.1428571
[6] {lon.class=East} => {affluence=3} 0.3125 1.0000000 2.6666667
[7] {affluence=3} => {lon.class=East} 0.3125 0.8333333 2.6666667
[8] {lon.class=East} => {urban.stress=2} 0.2500 0.8000000 2.1333333
[9] {lon.class=West} => {urban.stress=1} 0.3125 1.0000000 1.7777778
[10] {lon.class=West} => {prop.white=1} 0.3125 1.0000000 1.1428571
[11] {lat.class=North} => {affluence=2} 0.2500 0.8000000 2.1333333
[12] {lat.class=North} => {urban.stress=1} 0.3125 1.0000000 1.7777778
[13] {lat.class=North} => {unemployment=2} 0.3125 1.0000000 1.7777778
[14] {lat.class=North} => {prop.white=1} 0.3125 1.0000000 1.1428571
[15] {affluence=2} => {urban.stress=1} 0.3750 1.0000000 1.7777778
[16] {affluence=2} => {unemployment=2} 0.3125 0.8333333 1.4814815
[17] {affluence=2} => {prop.white=1} 0.3750 1.0000000 1.1428571
[18] {affluence=3} => {urban.stress=2} 0.3125 0.8333333 2.2222222
[19] {urban.stress=2} => {affluence=3} 0.3125 0.8333333 2.2222222
[20] {urban.stress=2} => {prop.white=1} 0.3125 0.8333333 0.9523810
[21] {lat.class=Central} => {prop.white=1} 0.3750 1.0000000 1.1428571
[22] {lon.class=Central} => {prop.white=1} 0.3750 1.0000000 1.1428571
[23] {urban.stress=1} => {prop.white=1} 0.5625 1.0000000 1.1428571
[24] {unemployment=2} => {prop.white=1} 0.5625 1.0000000 1.1428571
[25] {unemployment=3,
lon.class=East} => {lat.class=South} 0.1250 1.0000000 3.2000000
[26] {affluence=3,
unemployment=3} => {lat.class=South} 0.1250 1.0000000 3.2000000
[27] {unemployment=3,
lon.class=East} => {affluence=3} 0.1250 1.0000000 2.6666667
[28] {affluence=3,
unemployment=3} => {lon.class=East} 0.1250 1.0000000 3.2000000
[29] {unemployment=1,
lon.class=West} => {affluence=1} 0.1250 1.0000000 4.0000000
[30] {affluence=1,
unemployment=1} => {lat.class=Central} 0.1875 1.0000000 2.6666667
[31] {affluence=1,
lat.class=Central} => {unemployment=1} 0.1875 1.0000000 4.0000000
[32] {unemployment=1,
urban.stress=1} => {affluence=1} 0.1250 1.0000000 4.0000000
[33] {affluence=1,
unemployment=1} => {prop.white=1} 0.1875 1.0000000 1.1428571
[34] {lon.class=West,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[35] {affluence=1,
lon.class=West} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[36] {affluence=1,
urban.stress=1} => {lon.class=West} 0.1875 1.0000000 3.2000000
[37] {affluence=1,
lon.class=West} => {prop.white=1} 0.1875 1.0000000 1.1428571
[38] {urban.stress=1,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[39] {affluence=1,
lat.class=Central} => {prop.white=1} 0.1875 1.0000000 1.1428571
[40] {affluence=1,
urban.stress=1} => {prop.white=1} 0.1875 1.0000000 1.1428571
[41] {unemployment=1,
lon.class=West} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[42] {lon.class=West,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[43] {unemployment=1,
lon.class=West} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[44] {unemployment=1,
urban.stress=1} => {lon.class=West} 0.1250 1.0000000 3.2000000
[45] {unemployment=1,
lon.class=West} => {prop.white=1} 0.1250 1.0000000 1.1428571
[46] {unemployment=1,
urban.stress=2} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[47] {unemployment=1,
urban.stress=2} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[48] {unemployment=1,
lon.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[49] {urban.stress=2,
lon.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[50] {unemployment=1,
urban.stress=2} => {prop.white=1} 0.1250 1.0000000 1.1428571
[51] {unemployment=1,
lon.class=Central} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[52] {lon.class=Central,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[53] {unemployment=1,
urban.stress=1} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[54] {urban.stress=1,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[55] {unemployment=1,
lat.class=Central} => {prop.white=1} 0.2500 1.0000000 1.1428571
[56] {unemployment=1,
prop.white=1} => {lat.class=Central} 0.2500 1.0000000 2.6666667
[57] {unemployment=1,
lon.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[58] {unemployment=1,
urban.stress=1} => {prop.white=1} 0.1250 1.0000000 1.1428571
[59] {lon.class=East,
lat.class=South} => {affluence=3} 0.1875 1.0000000 2.6666667
[60] {affluence=3,
lat.class=South} => {lon.class=East} 0.1875 1.0000000 3.2000000
[61] {urban.stress=2,
lat.class=South} => {lon.class=East} 0.1250 1.0000000 3.2000000
[62] {affluence=2,
lat.class=South} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[63] {lon.class=Central,
lat.class=South} => {affluence=2} 0.1250 1.0000000 2.6666667
[64] {affluence=2,
lat.class=South} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[65] {urban.stress=1,
lat.class=South} => {affluence=2} 0.1250 1.0000000 2.6666667
[66] {affluence=2,
lat.class=South} => {prop.white=1} 0.1250 1.0000000 1.1428571
[67] {urban.stress=2,
lat.class=South} => {affluence=3} 0.1250 1.0000000 2.6666667
[68] {lon.class=Central,
lat.class=South} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[69] {urban.stress=1,
lat.class=South} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[70] {lon.class=Central,
lat.class=South} => {prop.white=1} 0.1250 1.0000000 1.1428571
[71] {urban.stress=1,
lat.class=South} => {prop.white=1} 0.1250 1.0000000 1.1428571
[72] {unemployment=2,
lat.class=South} => {prop.white=1} 0.1250 1.0000000 1.1428571
[73] {affluence=3,
lon.class=East} => {urban.stress=2} 0.2500 0.8000000 2.1333333
[74] {urban.stress=2,
lon.class=East} => {affluence=3} 0.2500 1.0000000 2.6666667
[75] {affluence=3,
urban.stress=2} => {lon.class=East} 0.2500 0.8000000 2.5600000
[76] {lon.class=East,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[77] {unemployment=2,
lon.class=East} => {affluence=3} 0.1875 1.0000000 2.6666667
[78] {affluence=3,
unemployment=2} => {lon.class=East} 0.1875 1.0000000 3.2000000
[79] {prop.white=1,
lon.class=East} => {affluence=3} 0.1875 1.0000000 2.6666667
[80] {lon.class=East,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[81] {unemployment=2,
lon.class=East} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[82] {unemployment=2,
urban.stress=2} => {lon.class=East} 0.1875 1.0000000 3.2000000
[83] {prop.white=1,
lon.class=East} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[84] {lon.class=East,
lat.class=Central} => {unemployment=2} 0.1250 1.0000000 1.7777778
[85] {unemployment=2,
lat.class=Central} => {lon.class=East} 0.1250 1.0000000 3.2000000
[86] {lon.class=East,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[87] {unemployment=2,
lon.class=East} => {prop.white=1} 0.1875 1.0000000 1.1428571
[88] {prop.white=1,
lon.class=East} => {unemployment=2} 0.1875 1.0000000 1.7777778
[89] {affluence=2,
lon.class=West} => {lat.class=North} 0.1250 1.0000000 3.2000000
[90] {lon.class=West,
lat.class=North} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[91] {lon.class=West,
lat.class=North} => {unemployment=2} 0.1875 1.0000000 1.7777778
[92] {unemployment=2,
lon.class=West} => {lat.class=North} 0.1875 1.0000000 3.2000000
[93] {lon.class=West,
lat.class=North} => {prop.white=1} 0.1875 1.0000000 1.1428571
[94] {affluence=2,
lon.class=West} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[95] {affluence=2,
lon.class=West} => {unemployment=2} 0.1250 1.0000000 1.7777778
[96] {affluence=2,
lon.class=West} => {prop.white=1} 0.1250 1.0000000 1.1428571
[97] {lon.class=West,
lat.class=Central} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[98] {urban.stress=1,
lat.class=Central} => {lon.class=West} 0.1250 1.0000000 3.2000000
[99] {lon.class=West,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[100] {unemployment=2,
lon.class=West} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[101] {urban.stress=1,
lon.class=West} => {prop.white=1} 0.3125 1.0000000 1.1428571
[102] {prop.white=1,
lon.class=West} => {urban.stress=1} 0.3125 1.0000000 1.7777778
[103] {unemployment=2,
lon.class=West} => {prop.white=1} 0.1875 1.0000000 1.1428571
[104] {lon.class=Central,
lat.class=North} => {affluence=2} 0.1250 1.0000000 2.6666667
[105] {affluence=2,
lat.class=North} => {urban.stress=1} 0.2500 1.0000000 1.7777778
[106] {urban.stress=1,
lat.class=North} => {affluence=2} 0.2500 0.8000000 2.1333333
[107] {affluence=2,
lat.class=North} => {unemployment=2} 0.2500 1.0000000 1.7777778
[108] {unemployment=2,
lat.class=North} => {affluence=2} 0.2500 0.8000000 2.1333333
[109] {affluence=2,
unemployment=2} => {lat.class=North} 0.2500 0.8000000 2.5600000
[110] {affluence=2,
lat.class=North} => {prop.white=1} 0.2500 1.0000000 1.1428571
[111] {prop.white=1,
lat.class=North} => {affluence=2} 0.2500 0.8000000 2.1333333
[112] {lon.class=Central,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[113] {lon.class=Central,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[114] {lon.class=Central,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[115] {urban.stress=1,
lat.class=North} => {unemployment=2} 0.3125 1.0000000 1.7777778
[116] {unemployment=2,
lat.class=North} => {urban.stress=1} 0.3125 1.0000000 1.7777778
[117] {unemployment=2,
urban.stress=1} => {lat.class=North} 0.3125 0.8333333 2.6666667
[118] {urban.stress=1,
lat.class=North} => {prop.white=1} 0.3125 1.0000000 1.1428571
[119] {prop.white=1,
lat.class=North} => {urban.stress=1} 0.3125 1.0000000 1.7777778
[120] {unemployment=2,
lat.class=North} => {prop.white=1} 0.3125 1.0000000 1.1428571
[121] {prop.white=1,
lat.class=North} => {unemployment=2} 0.3125 1.0000000 1.7777778
[122] {affluence=2,
lon.class=Central} => {urban.stress=1} 0.2500 1.0000000 1.7777778
[123] {urban.stress=1,
lon.class=Central} => {affluence=2} 0.2500 1.0000000 2.6666667
[124] {unemployment=2,
lon.class=Central} => {affluence=2} 0.1875 1.0000000 2.6666667
[125] {affluence=2,
lon.class=Central} => {prop.white=1} 0.2500 1.0000000 1.1428571
[126] {affluence=2,
urban.stress=1} => {unemployment=2} 0.3125 0.8333333 1.4814815
[127] {affluence=2,
unemployment=2} => {urban.stress=1} 0.3125 1.0000000 1.7777778
[128] {unemployment=2,
urban.stress=1} => {affluence=2} 0.3125 0.8333333 2.2222222
[129] {affluence=2,
urban.stress=1} => {prop.white=1} 0.3750 1.0000000 1.1428571
[130] {affluence=2,
prop.white=1} => {urban.stress=1} 0.3750 1.0000000 1.7777778
[131] {affluence=2,
unemployment=2} => {prop.white=1} 0.3125 1.0000000 1.1428571
[132] {affluence=2,
prop.white=1} => {unemployment=2} 0.3125 0.8333333 1.4814815
[133] {affluence=3,
lat.class=Central} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[134] {affluence=3,
unemployment=2} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[135] {unemployment=2,
urban.stress=2} => {affluence=3} 0.1875 1.0000000 2.6666667
[136] {affluence=3,
urban.stress=2} => {prop.white=1} 0.2500 0.8000000 0.9142857
[137] {affluence=3,
prop.white=1} => {urban.stress=2} 0.2500 1.0000000 2.6666667
[138] {urban.stress=2,
prop.white=1} => {affluence=3} 0.2500 0.8000000 2.1333333
[139] {unemployment=2,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[140] {affluence=3,
lat.class=Central} => {prop.white=1} 0.1875 1.0000000 1.1428571
[141] {affluence=3,
unemployment=2} => {prop.white=1} 0.1875 1.0000000 1.1428571
[142] {urban.stress=2,
lon.class=Central} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[143] {lon.class=Central,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[144] {unemployment=2,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[145] {urban.stress=2,
lat.class=Central} => {prop.white=1} 0.2500 1.0000000 1.1428571
[146] {urban.stress=2,
prop.white=1} => {lat.class=Central} 0.2500 0.8000000 2.1333333
[147] {urban.stress=2,
lon.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[148] {unemployment=2,
urban.stress=2} => {prop.white=1} 0.1875 1.0000000 1.1428571
[149] {lon.class=Central,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[150] {urban.stress=1,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[151] {unemployment=2,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[152] {unemployment=2,
lon.class=Central} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[153] {urban.stress=1,
lon.class=Central} => {prop.white=1} 0.2500 1.0000000 1.1428571
[154] {unemployment=2,
lon.class=Central} => {prop.white=1} 0.1875 1.0000000 1.1428571
[155] {unemployment=2,
urban.stress=1} => {prop.white=1} 0.3750 1.0000000 1.1428571
[156] {unemployment=3,
lon.class=East,
lat.class=South} => {affluence=3} 0.1250 1.0000000 2.6666667
[157] {affluence=3,
unemployment=3,
lat.class=South} => {lon.class=East} 0.1250 1.0000000 3.2000000
[158] {affluence=3,
unemployment=3,
lon.class=East} => {lat.class=South} 0.1250 1.0000000 3.2000000
[159] {affluence=1,
unemployment=1,
lon.class=West} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[160] {affluence=1,
lon.class=West,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[161] {unemployment=1,
lon.class=West,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[162] {affluence=1,
unemployment=1,
lon.class=West} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[163] {affluence=1,
unemployment=1,
urban.stress=1} => {lon.class=West} 0.1250 1.0000000 3.2000000
[164] {unemployment=1,
urban.stress=1,
lon.class=West} => {affluence=1} 0.1250 1.0000000 4.0000000
[165] {affluence=1,
unemployment=1,
lon.class=West} => {prop.white=1} 0.1250 1.0000000 1.1428571
[166] {unemployment=1,
prop.white=1,
lon.class=West} => {affluence=1} 0.1250 1.0000000 4.0000000
[167] {affluence=1,
unemployment=1,
urban.stress=1} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[168] {affluence=1,
urban.stress=1,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[169] {unemployment=1,
urban.stress=1,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[170] {affluence=1,
unemployment=1,
lat.class=Central} => {prop.white=1} 0.1875 1.0000000 1.1428571
[171] {affluence=1,
unemployment=1,
prop.white=1} => {lat.class=Central} 0.1875 1.0000000 2.6666667
[172] {affluence=1,
prop.white=1,
lat.class=Central} => {unemployment=1} 0.1875 1.0000000 4.0000000
[173] {affluence=1,
unemployment=1,
urban.stress=1} => {prop.white=1} 0.1250 1.0000000 1.1428571
[174] {unemployment=1,
urban.stress=1,
prop.white=1} => {affluence=1} 0.1250 1.0000000 4.0000000
[175] {affluence=1,
lon.class=West,
lat.class=Central} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[176] {affluence=1,
urban.stress=1,
lat.class=Central} => {lon.class=West} 0.1250 1.0000000 3.2000000
[177] {urban.stress=1,
lon.class=West,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[178] {affluence=1,
lon.class=West,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[179] {prop.white=1,
lon.class=West,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[180] {affluence=1,
urban.stress=1,
lon.class=West} => {prop.white=1} 0.1875 1.0000000 1.1428571
[181] {affluence=1,
prop.white=1,
lon.class=West} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[182] {affluence=1,
urban.stress=1,
prop.white=1} => {lon.class=West} 0.1875 1.0000000 3.2000000
[183] {affluence=1,
urban.stress=1,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[184] {urban.stress=1,
prop.white=1,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[185] {unemployment=1,
lon.class=West,
lat.class=Central} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[186] {unemployment=1,
urban.stress=1,
lon.class=West} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[187] {unemployment=1,
urban.stress=1,
lat.class=Central} => {lon.class=West} 0.1250 1.0000000 3.2000000
[188] {urban.stress=1,
lon.class=West,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[189] {unemployment=1,
lon.class=West,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[190] {unemployment=1,
prop.white=1,
lon.class=West} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[191] {prop.white=1,
lon.class=West,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[192] {unemployment=1,
urban.stress=1,
lon.class=West} => {prop.white=1} 0.1250 1.0000000 1.1428571
[193] {unemployment=1,
prop.white=1,
lon.class=West} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[194] {unemployment=1,
urban.stress=1,
prop.white=1} => {lon.class=West} 0.1250 1.0000000 3.2000000
[195] {unemployment=1,
urban.stress=2,
lat.class=Central} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[196] {unemployment=1,
urban.stress=2,
lon.class=Central} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[197] {unemployment=1,
lon.class=Central,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[198] {urban.stress=2,
lon.class=Central,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[199] {unemployment=1,
urban.stress=2,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[200] {unemployment=1,
urban.stress=2,
prop.white=1} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[201] {unemployment=1,
urban.stress=2,
lon.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[202] {unemployment=1,
urban.stress=2,
prop.white=1} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[203] {unemployment=1,
prop.white=1,
lon.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[204] {urban.stress=2,
prop.white=1,
lon.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[205] {unemployment=1,
lon.class=Central,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[206] {unemployment=1,
prop.white=1,
lon.class=Central} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[207] {prop.white=1,
lon.class=Central,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[208] {unemployment=1,
urban.stress=1,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[209] {unemployment=1,
urban.stress=1,
prop.white=1} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[210] {urban.stress=1,
prop.white=1,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[211] {urban.stress=2,
lon.class=East,
lat.class=South} => {affluence=3} 0.1250 1.0000000 2.6666667
[212] {affluence=3,
urban.stress=2,
lat.class=South} => {lon.class=East} 0.1250 1.0000000 3.2000000
[213] {affluence=2,
lon.class=Central,
lat.class=South} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[214] {affluence=2,
urban.stress=1,
lat.class=South} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[215] {urban.stress=1,
lon.class=Central,
lat.class=South} => {affluence=2} 0.1250 1.0000000 2.6666667
[216] {affluence=2,
lon.class=Central,
lat.class=South} => {prop.white=1} 0.1250 1.0000000 1.1428571
[217] {affluence=2,
prop.white=1,
lat.class=South} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[218] {prop.white=1,
lon.class=Central,
lat.class=South} => {affluence=2} 0.1250 1.0000000 2.6666667
[219] {affluence=2,
urban.stress=1,
lat.class=South} => {prop.white=1} 0.1250 1.0000000 1.1428571
[220] {affluence=2,
prop.white=1,
lat.class=South} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[221] {urban.stress=1,
prop.white=1,
lat.class=South} => {affluence=2} 0.1250 1.0000000 2.6666667
[222] {urban.stress=1,
lon.class=Central,
lat.class=South} => {prop.white=1} 0.1250 1.0000000 1.1428571
[223] {prop.white=1,
lon.class=Central,
lat.class=South} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[224] {urban.stress=1,
prop.white=1,
lat.class=South} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[225] {affluence=3,
lon.class=East,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[226] {urban.stress=2,
lon.class=East,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[227] {affluence=3,
unemployment=2,
lon.class=East} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[228] {unemployment=2,
urban.stress=2,
lon.class=East} => {affluence=3} 0.1875 1.0000000 2.6666667
[229] {affluence=3,
unemployment=2,
urban.stress=2} => {lon.class=East} 0.1875 1.0000000 3.2000000
[230] {affluence=3,
prop.white=1,
lon.class=East} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[231] {urban.stress=2,
prop.white=1,
lon.class=East} => {affluence=3} 0.1875 1.0000000 2.6666667
[232] {affluence=3,
lon.class=East,
lat.class=Central} => {unemployment=2} 0.1250 1.0000000 1.7777778
[233] {unemployment=2,
lon.class=East,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[234] {affluence=3,
unemployment=2,
lat.class=Central} => {lon.class=East} 0.1250 1.0000000 3.2000000
[235] {affluence=3,
lon.class=East,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[236] {prop.white=1,
lon.class=East,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[237] {affluence=3,
unemployment=2,
lon.class=East} => {prop.white=1} 0.1875 1.0000000 1.1428571
[238] {affluence=3,
prop.white=1,
lon.class=East} => {unemployment=2} 0.1875 1.0000000 1.7777778
[239] {unemployment=2,
prop.white=1,
lon.class=East} => {affluence=3} 0.1875 1.0000000 2.6666667
[240] {affluence=3,
unemployment=2,
prop.white=1} => {lon.class=East} 0.1875 1.0000000 3.2000000
[241] {urban.stress=2,
lon.class=East,
lat.class=Central} => {unemployment=2} 0.1250 1.0000000 1.7777778
[242] {unemployment=2,
lon.class=East,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[243] {unemployment=2,
urban.stress=2,
lat.class=Central} => {lon.class=East} 0.1250 1.0000000 3.2000000
[244] {urban.stress=2,
lon.class=East,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[245] {prop.white=1,
lon.class=East,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[246] {unemployment=2,
urban.stress=2,
lon.class=East} => {prop.white=1} 0.1875 1.0000000 1.1428571
[247] {urban.stress=2,
prop.white=1,
lon.class=East} => {unemployment=2} 0.1875 1.0000000 1.7777778
[248] {unemployment=2,
prop.white=1,
lon.class=East} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[249] {unemployment=2,
urban.stress=2,
prop.white=1} => {lon.class=East} 0.1875 1.0000000 3.2000000
[250] {unemployment=2,
lon.class=East,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[251] {prop.white=1,
lon.class=East,
lat.class=Central} => {unemployment=2} 0.1250 1.0000000 1.7777778
[252] {unemployment=2,
prop.white=1,
lat.class=Central} => {lon.class=East} 0.1250 1.0000000 3.2000000
[253] {affluence=2,
lon.class=West,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[254] {affluence=2,
urban.stress=1,
lon.class=West} => {lat.class=North} 0.1250 1.0000000 3.2000000
[255] {affluence=2,
lon.class=West,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[256] {affluence=2,
unemployment=2,
lon.class=West} => {lat.class=North} 0.1250 1.0000000 3.2000000
[257] {affluence=2,
lon.class=West,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[258] {affluence=2,
prop.white=1,
lon.class=West} => {lat.class=North} 0.1250 1.0000000 3.2000000
[259] {urban.stress=1,
lon.class=West,
lat.class=North} => {unemployment=2} 0.1875 1.0000000 1.7777778
[260] {unemployment=2,
lon.class=West,
lat.class=North} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[261] {unemployment=2,
urban.stress=1,
lon.class=West} => {lat.class=North} 0.1875 1.0000000 3.2000000
[262] {urban.stress=1,
lon.class=West,
lat.class=North} => {prop.white=1} 0.1875 1.0000000 1.1428571
[263] {prop.white=1,
lon.class=West,
lat.class=North} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[264] {unemployment=2,
lon.class=West,
lat.class=North} => {prop.white=1} 0.1875 1.0000000 1.1428571
[265] {prop.white=1,
lon.class=West,
lat.class=North} => {unemployment=2} 0.1875 1.0000000 1.7777778
[266] {unemployment=2,
prop.white=1,
lon.class=West} => {lat.class=North} 0.1875 1.0000000 3.2000000
[267] {affluence=2,
urban.stress=1,
lon.class=West} => {unemployment=2} 0.1250 1.0000000 1.7777778
[268] {affluence=2,
unemployment=2,
lon.class=West} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[269] {affluence=2,
urban.stress=1,
lon.class=West} => {prop.white=1} 0.1250 1.0000000 1.1428571
[270] {affluence=2,
prop.white=1,
lon.class=West} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[271] {affluence=2,
unemployment=2,
lon.class=West} => {prop.white=1} 0.1250 1.0000000 1.1428571
[272] {affluence=2,
prop.white=1,
lon.class=West} => {unemployment=2} 0.1250 1.0000000 1.7777778
[273] {urban.stress=1,
lon.class=West,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[274] {prop.white=1,
lon.class=West,
lat.class=Central} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[275] {urban.stress=1,
prop.white=1,
lat.class=Central} => {lon.class=West} 0.1250 1.0000000 3.2000000
[276] {unemployment=2,
urban.stress=1,
lon.class=West} => {prop.white=1} 0.1875 1.0000000 1.1428571
[277] {unemployment=2,
prop.white=1,
lon.class=West} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[278] {affluence=2,
lon.class=Central,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[279] {urban.stress=1,
lon.class=Central,
lat.class=North} => {affluence=2} 0.1250 1.0000000 2.6666667
[280] {affluence=2,
lon.class=Central,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[281] {unemployment=2,
lon.class=Central,
lat.class=North} => {affluence=2} 0.1250 1.0000000 2.6666667
[282] {affluence=2,
lon.class=Central,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[283] {prop.white=1,
lon.class=Central,
lat.class=North} => {affluence=2} 0.1250 1.0000000 2.6666667
[284] {affluence=2,
urban.stress=1,
lat.class=North} => {unemployment=2} 0.2500 1.0000000 1.7777778
[285] {affluence=2,
unemployment=2,
lat.class=North} => {urban.stress=1} 0.2500 1.0000000 1.7777778
[286] {unemployment=2,
urban.stress=1,
lat.class=North} => {affluence=2} 0.2500 0.8000000 2.1333333
[287] {affluence=2,
unemployment=2,
urban.stress=1} => {lat.class=North} 0.2500 0.8000000 2.5600000
[288] {affluence=2,
urban.stress=1,
lat.class=North} => {prop.white=1} 0.2500 1.0000000 1.1428571
[289] {affluence=2,
prop.white=1,
lat.class=North} => {urban.stress=1} 0.2500 1.0000000 1.7777778
[290] {urban.stress=1,
prop.white=1,
lat.class=North} => {affluence=2} 0.2500 0.8000000 2.1333333
[291] {affluence=2,
unemployment=2,
lat.class=North} => {prop.white=1} 0.2500 1.0000000 1.1428571
[292] {affluence=2,
prop.white=1,
lat.class=North} => {unemployment=2} 0.2500 1.0000000 1.7777778
[293] {unemployment=2,
prop.white=1,
lat.class=North} => {affluence=2} 0.2500 0.8000000 2.1333333
[294] {affluence=2,
unemployment=2,
prop.white=1} => {lat.class=North} 0.2500 0.8000000 2.5600000
[295] {urban.stress=1,
lon.class=Central,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[296] {unemployment=2,
lon.class=Central,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[297] {urban.stress=1,
lon.class=Central,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[298] {prop.white=1,
lon.class=Central,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[299] {unemployment=2,
lon.class=Central,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[300] {prop.white=1,
lon.class=Central,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[301] {unemployment=2,
urban.stress=1,
lat.class=North} => {prop.white=1} 0.3125 1.0000000 1.1428571
[302] {urban.stress=1,
prop.white=1,
lat.class=North} => {unemployment=2} 0.3125 1.0000000 1.7777778
[303] {unemployment=2,
prop.white=1,
lat.class=North} => {urban.stress=1} 0.3125 1.0000000 1.7777778
[304] {unemployment=2,
urban.stress=1,
prop.white=1} => {lat.class=North} 0.3125 0.8333333 2.6666667
[305] {affluence=2,
unemployment=2,
lon.class=Central} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[306] {unemployment=2,
urban.stress=1,
lon.class=Central} => {affluence=2} 0.1875 1.0000000 2.6666667
[307] {affluence=2,
urban.stress=1,
lon.class=Central} => {prop.white=1} 0.2500 1.0000000 1.1428571
[308] {affluence=2,
prop.white=1,
lon.class=Central} => {urban.stress=1} 0.2500 1.0000000 1.7777778
[309] {urban.stress=1,
prop.white=1,
lon.class=Central} => {affluence=2} 0.2500 1.0000000 2.6666667
[310] {affluence=2,
unemployment=2,
lon.class=Central} => {prop.white=1} 0.1875 1.0000000 1.1428571
[311] {unemployment=2,
prop.white=1,
lon.class=Central} => {affluence=2} 0.1875 1.0000000 2.6666667
[312] {affluence=2,
unemployment=2,
urban.stress=1} => {prop.white=1} 0.3125 1.0000000 1.1428571
[313] {affluence=2,
urban.stress=1,
prop.white=1} => {unemployment=2} 0.3125 0.8333333 1.4814815
[314] {affluence=2,
unemployment=2,
prop.white=1} => {urban.stress=1} 0.3125 1.0000000 1.7777778
[315] {unemployment=2,
urban.stress=1,
prop.white=1} => {affluence=2} 0.3125 0.8333333 2.2222222
[316] {affluence=3,
unemployment=2,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[317] {unemployment=2,
urban.stress=2,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[318] {affluence=3,
urban.stress=2,
lat.class=Central} => {prop.white=1} 0.1875 1.0000000 1.1428571
[319] {affluence=3,
prop.white=1,
lat.class=Central} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[320] {affluence=3,
unemployment=2,
urban.stress=2} => {prop.white=1} 0.1875 1.0000000 1.1428571
[321] {affluence=3,
unemployment=2,
prop.white=1} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[322] {unemployment=2,
urban.stress=2,
prop.white=1} => {affluence=3} 0.1875 1.0000000 2.6666667
[323] {affluence=3,
unemployment=2,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[324] {unemployment=2,
prop.white=1,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[325] {urban.stress=2,
lon.class=Central,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[326] {urban.stress=2,
prop.white=1,
lon.class=Central} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[327] {prop.white=1,
lon.class=Central,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[328] {unemployment=2,
urban.stress=2,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[329] {unemployment=2,
prop.white=1,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[330] {unemployment=2,
urban.stress=1,
lon.class=Central} => {prop.white=1} 0.1875 1.0000000 1.1428571
[331] {unemployment=2,
prop.white=1,
lon.class=Central} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[332] {affluence=1,
unemployment=1,
lon.class=West,
lat.class=Central} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[333] {affluence=1,
unemployment=1,
urban.stress=1,
lon.class=West} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[334] {affluence=1,
unemployment=1,
urban.stress=1,
lat.class=Central} => {lon.class=West} 0.1250 1.0000000 3.2000000
[335] {affluence=1,
urban.stress=1,
lon.class=West,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[336] {unemployment=1,
urban.stress=1,
lon.class=West,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[337] {affluence=1,
unemployment=1,
lon.class=West,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[338] {affluence=1,
unemployment=1,
prop.white=1,
lon.class=West} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[339] {affluence=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[340] {unemployment=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[341] {affluence=1,
unemployment=1,
urban.stress=1,
lon.class=West} => {prop.white=1} 0.1250 1.0000000 1.1428571
[342] {affluence=1,
unemployment=1,
prop.white=1,
lon.class=West} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[343] {affluence=1,
unemployment=1,
urban.stress=1,
prop.white=1} => {lon.class=West} 0.1250 1.0000000 3.2000000
[344] {unemployment=1,
urban.stress=1,
prop.white=1,
lon.class=West} => {affluence=1} 0.1250 1.0000000 4.0000000
[345] {affluence=1,
unemployment=1,
urban.stress=1,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[346] {affluence=1,
unemployment=1,
urban.stress=1,
prop.white=1} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[347] {affluence=1,
urban.stress=1,
prop.white=1,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[348] {unemployment=1,
urban.stress=1,
prop.white=1,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[349] {affluence=1,
urban.stress=1,
lon.class=West,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[350] {affluence=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[351] {affluence=1,
urban.stress=1,
prop.white=1,
lat.class=Central} => {lon.class=West} 0.1250 1.0000000 3.2000000
[352] {urban.stress=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[353] {unemployment=1,
urban.stress=1,
lon.class=West,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[354] {unemployment=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[355] {unemployment=1,
urban.stress=1,
prop.white=1,
lon.class=West} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[356] {unemployment=1,
urban.stress=1,
prop.white=1,
lat.class=Central} => {lon.class=West} 0.1250 1.0000000 3.2000000
[357] {urban.stress=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[358] {unemployment=1,
urban.stress=2,
lon.class=Central,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[359] {unemployment=1,
urban.stress=2,
prop.white=1,
lat.class=Central} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[360] {unemployment=1,
urban.stress=2,
prop.white=1,
lon.class=Central} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[361] {unemployment=1,
prop.white=1,
lon.class=Central,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[362] {urban.stress=2,
prop.white=1,
lon.class=Central,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[363] {affluence=2,
urban.stress=1,
lon.class=Central,
lat.class=South} => {prop.white=1} 0.1250 1.0000000 1.1428571
[364] {affluence=2,
prop.white=1,
lon.class=Central,
lat.class=South} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[365] {affluence=2,
urban.stress=1,
prop.white=1,
lat.class=South} => {lon.class=Central} 0.1250 1.0000000 2.6666667
[366] {urban.stress=1,
prop.white=1,
lon.class=Central,
lat.class=South} => {affluence=2} 0.1250 1.0000000 2.6666667
[367] {affluence=3,
urban.stress=2,
lon.class=East,
lat.class=Central} => {unemployment=2} 0.1250 1.0000000 1.7777778
[368] {affluence=3,
unemployment=2,
lon.class=East,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[369] {unemployment=2,
urban.stress=2,
lon.class=East,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[370] {affluence=3,
unemployment=2,
urban.stress=2,
lat.class=Central} => {lon.class=East} 0.1250 1.0000000 3.2000000
[371] {affluence=3,
urban.stress=2,
lon.class=East,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[372] {affluence=3,
prop.white=1,
lon.class=East,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[373] {urban.stress=2,
prop.white=1,
lon.class=East,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[374] {affluence=3,
unemployment=2,
urban.stress=2,
lon.class=East} => {prop.white=1} 0.1875 1.0000000 1.1428571
[375] {affluence=3,
urban.stress=2,
prop.white=1,
lon.class=East} => {unemployment=2} 0.1875 1.0000000 1.7777778
[376] {affluence=3,
unemployment=2,
prop.white=1,
lon.class=East} => {urban.stress=2} 0.1875 1.0000000 2.6666667
[377] {unemployment=2,
urban.stress=2,
prop.white=1,
lon.class=East} => {affluence=3} 0.1875 1.0000000 2.6666667
[378] {affluence=3,
unemployment=2,
urban.stress=2,
prop.white=1} => {lon.class=East} 0.1875 1.0000000 3.2000000
[379] {affluence=3,
unemployment=2,
lon.class=East,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[380] {affluence=3,
prop.white=1,
lon.class=East,
lat.class=Central} => {unemployment=2} 0.1250 1.0000000 1.7777778
[381] {unemployment=2,
prop.white=1,
lon.class=East,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[382] {affluence=3,
unemployment=2,
prop.white=1,
lat.class=Central} => {lon.class=East} 0.1250 1.0000000 3.2000000
[383] {unemployment=2,
urban.stress=2,
lon.class=East,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[384] {urban.stress=2,
prop.white=1,
lon.class=East,
lat.class=Central} => {unemployment=2} 0.1250 1.0000000 1.7777778
[385] {unemployment=2,
prop.white=1,
lon.class=East,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[386] {unemployment=2,
urban.stress=2,
prop.white=1,
lat.class=Central} => {lon.class=East} 0.1250 1.0000000 3.2000000
[387] {affluence=2,
urban.stress=1,
lon.class=West,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[388] {affluence=2,
unemployment=2,
lon.class=West,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[389] {affluence=2,
unemployment=2,
urban.stress=1,
lon.class=West} => {lat.class=North} 0.1250 1.0000000 3.2000000
[390] {affluence=2,
urban.stress=1,
lon.class=West,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[391] {affluence=2,
prop.white=1,
lon.class=West,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[392] {affluence=2,
urban.stress=1,
prop.white=1,
lon.class=West} => {lat.class=North} 0.1250 1.0000000 3.2000000
[393] {affluence=2,
unemployment=2,
lon.class=West,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[394] {affluence=2,
prop.white=1,
lon.class=West,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[395] {affluence=2,
unemployment=2,
prop.white=1,
lon.class=West} => {lat.class=North} 0.1250 1.0000000 3.2000000
[396] {unemployment=2,
urban.stress=1,
lon.class=West,
lat.class=North} => {prop.white=1} 0.1875 1.0000000 1.1428571
[397] {urban.stress=1,
prop.white=1,
lon.class=West,
lat.class=North} => {unemployment=2} 0.1875 1.0000000 1.7777778
[398] {unemployment=2,
prop.white=1,
lon.class=West,
lat.class=North} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[399] {unemployment=2,
urban.stress=1,
prop.white=1,
lon.class=West} => {lat.class=North} 0.1875 1.0000000 3.2000000
[400] {affluence=2,
unemployment=2,
urban.stress=1,
lon.class=West} => {prop.white=1} 0.1250 1.0000000 1.1428571
[401] {affluence=2,
urban.stress=1,
prop.white=1,
lon.class=West} => {unemployment=2} 0.1250 1.0000000 1.7777778
[402] {affluence=2,
unemployment=2,
prop.white=1,
lon.class=West} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[403] {affluence=2,
urban.stress=1,
lon.class=Central,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[404] {affluence=2,
unemployment=2,
lon.class=Central,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[405] {unemployment=2,
urban.stress=1,
lon.class=Central,
lat.class=North} => {affluence=2} 0.1250 1.0000000 2.6666667
[406] {affluence=2,
urban.stress=1,
lon.class=Central,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[407] {affluence=2,
prop.white=1,
lon.class=Central,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[408] {urban.stress=1,
prop.white=1,
lon.class=Central,
lat.class=North} => {affluence=2} 0.1250 1.0000000 2.6666667
[409] {affluence=2,
unemployment=2,
lon.class=Central,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[410] {affluence=2,
prop.white=1,
lon.class=Central,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[411] {unemployment=2,
prop.white=1,
lon.class=Central,
lat.class=North} => {affluence=2} 0.1250 1.0000000 2.6666667
[412] {affluence=2,
unemployment=2,
urban.stress=1,
lat.class=North} => {prop.white=1} 0.2500 1.0000000 1.1428571
[413] {affluence=2,
urban.stress=1,
prop.white=1,
lat.class=North} => {unemployment=2} 0.2500 1.0000000 1.7777778
[414] {affluence=2,
unemployment=2,
prop.white=1,
lat.class=North} => {urban.stress=1} 0.2500 1.0000000 1.7777778
[415] {unemployment=2,
urban.stress=1,
prop.white=1,
lat.class=North} => {affluence=2} 0.2500 0.8000000 2.1333333
[416] {affluence=2,
unemployment=2,
urban.stress=1,
prop.white=1} => {lat.class=North} 0.2500 0.8000000 2.5600000
[417] {unemployment=2,
urban.stress=1,
lon.class=Central,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[418] {urban.stress=1,
prop.white=1,
lon.class=Central,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[419] {unemployment=2,
prop.white=1,
lon.class=Central,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[420] {affluence=2,
unemployment=2,
urban.stress=1,
lon.class=Central} => {prop.white=1} 0.1875 1.0000000 1.1428571
[421] {affluence=2,
unemployment=2,
prop.white=1,
lon.class=Central} => {urban.stress=1} 0.1875 1.0000000 1.7777778
[422] {unemployment=2,
urban.stress=1,
prop.white=1,
lon.class=Central} => {affluence=2} 0.1875 1.0000000 2.6666667
[423] {affluence=3,
unemployment=2,
urban.stress=2,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[424] {affluence=3,
unemployment=2,
prop.white=1,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[425] {unemployment=2,
urban.stress=2,
prop.white=1,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[426] {affluence=1,
unemployment=1,
urban.stress=1,
lon.class=West,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[427] {affluence=1,
unemployment=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[428] {affluence=1,
unemployment=1,
urban.stress=1,
prop.white=1,
lon.class=West} => {lat.class=Central} 0.1250 1.0000000 2.6666667
[429] {affluence=1,
unemployment=1,
urban.stress=1,
prop.white=1,
lat.class=Central} => {lon.class=West} 0.1250 1.0000000 3.2000000
[430] {affluence=1,
urban.stress=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {unemployment=1} 0.1250 1.0000000 4.0000000
[431] {unemployment=1,
urban.stress=1,
prop.white=1,
lon.class=West,
lat.class=Central} => {affluence=1} 0.1250 1.0000000 4.0000000
[432] {affluence=3,
unemployment=2,
urban.stress=2,
lon.class=East,
lat.class=Central} => {prop.white=1} 0.1250 1.0000000 1.1428571
[433] {affluence=3,
urban.stress=2,
prop.white=1,
lon.class=East,
lat.class=Central} => {unemployment=2} 0.1250 1.0000000 1.7777778
[434] {affluence=3,
unemployment=2,
prop.white=1,
lon.class=East,
lat.class=Central} => {urban.stress=2} 0.1250 1.0000000 2.6666667
[435] {unemployment=2,
urban.stress=2,
prop.white=1,
lon.class=East,
lat.class=Central} => {affluence=3} 0.1250 1.0000000 2.6666667
[436] {affluence=3,
unemployment=2,
urban.stress=2,
prop.white=1,
lat.class=Central} => {lon.class=East} 0.1250 1.0000000 3.2000000
[437] {affluence=2,
unemployment=2,
urban.stress=1,
lon.class=West,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[438] {affluence=2,
urban.stress=1,
prop.white=1,
lon.class=West,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[439] {affluence=2,
unemployment=2,
prop.white=1,
lon.class=West,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[440] {affluence=2,
unemployment=2,
urban.stress=1,
prop.white=1,
lon.class=West} => {lat.class=North} 0.1250 1.0000000 3.2000000
[441] {affluence=2,
unemployment=2,
urban.stress=1,
lon.class=Central,
lat.class=North} => {prop.white=1} 0.1250 1.0000000 1.1428571
[442] {affluence=2,
urban.stress=1,
prop.white=1,
lon.class=Central,
lat.class=North} => {unemployment=2} 0.1250 1.0000000 1.7777778
[443] {affluence=2,
unemployment=2,
prop.white=1,
lon.class=Central,
lat.class=North} => {urban.stress=1} 0.1250 1.0000000 1.7777778
[444] {unemployment=2,
urban.stress=1,
prop.white=1,
lon.class=Central,
lat.class=North} => {affluence=2} 0.1250 1.0000000 2.6666667
Now, the fun part comes with setting our own parameters to create the list of item-sets! The new object rules2.LA now consists of item-sets that have a minimum support of $0.5$, and minimum confidence of $0.5$.
In [4]:
rules2.LA <- apriori(life.in.LA[,3:8],parameter = list(supp=0.50, conf=0.5),control = list(verbose=F))
rules2.LA.sorted <- sort(rules2.LA, by=c("lift"), decreasing=TRUE) # sort the list by lift (in descending order)
inspect(rules2.LA.sorted) # print out the results
lhs rhs support confidence lift
[1] {prop.white=1} => {unemployment=2} 0.5625 0.6428571 1.142857
[2] {prop.white=1} => {urban.stress=1} 0.5625 0.6428571 1.142857
[3] {unemployment=2} => {prop.white=1} 0.5625 1.0000000 1.142857
[4] {urban.stress=1} => {prop.white=1} 0.5625 1.0000000 1.142857
[5] {} => {unemployment=2} 0.5625 0.5625000 1.000000
[6] {} => {urban.stress=1} 0.5625 0.5625000 1.000000
[7] {} => {prop.white=1} 0.8750 0.8750000 1.000000
We have a list of 7 rules in above, but are they all independent rules?<br><br>
The answer is $\textbf{NO!}$ For example, we have rules {prop.white=1} => {unemployment=2}, and {unemployment=2} => {prop.white=1}, which are simply reciprocals of each other. Hence the rule {unemployment=2} => {prop.white=1} is redundant.<br><br>
Which of the 2 rules should we remove? {prop.white=1} => {unemployment=2}, or {unemployment=2} => {prop.white=1}? <br>
This is not an easy question to answer. To make a point, consider two situations
Example 1: We asked people about their favourite sport and a team/player. We found that $100\%$ of people who said that the Ottawa Senators were their favourite team gave hockey as their favourite sport. We also found that $40\%$ of hockey lovers said they cheer for Sens.
Example 2: Last year, $40\%$ of successful candidates for a company were graduate students, and $100\%$ of graduate students who applied to this company got a job offer.
In these scenarios, which statements are more interesting?<br><br> In what follows, we attempt to remove all redundant rules. As a result, we remove $2$ rules.
In [7]:
subset.matrix <- as.matrix(is.subset(rules2.LA.sorted, rules2.LA.sorted))
subset.matrix[lower.tri(subset.matrix, diag=T)] <- NA
subset.matrix
class(subset.matrix)
(redundant.LA <- colSums(subset.matrix, na.rm=T) >= 1)
which(redundant.LA)
# remove redundant rules
rules2.LA.pruned <- rules2.LA.sorted[!redundant.LA]
inspect(rules2.LA.pruned)
| {unemployment=2,prop.white=1} | {urban.stress=1,prop.white=1} | {unemployment=2,prop.white=1} | {urban.stress=1,prop.white=1} | {unemployment=2} | {urban.stress=1} | {prop.white=1} | |
|---|---|---|---|---|---|---|---|
| {unemployment=2,prop.white=1} | NA | FALSE | TRUE | FALSE | FALSE | FALSE | FALSE |
| {urban.stress=1,prop.white=1} | NA | NA | FALSE | TRUE | FALSE | FALSE | FALSE |
| {unemployment=2,prop.white=1} | NA | NA | NA | FALSE | FALSE | FALSE | FALSE |
| {urban.stress=1,prop.white=1} | NA | NA | NA | NA | FALSE | FALSE | FALSE |
| {unemployment=2} | NA | NA | NA | NA | NA | FALSE | FALSE |
| {urban.stress=1} | NA | NA | NA | NA | NA | NA | FALSE |
| {prop.white=1} | NA | NA | NA | NA | NA | NA | NA |
'matrix'
- {unemployment=2,prop.white=1}
- FALSE
- {urban.stress=1,prop.white=1}
- FALSE
- {unemployment=2,prop.white=1}
- TRUE
- {urban.stress=1,prop.white=1}
- TRUE
- {unemployment=2}
- FALSE
- {urban.stress=1}
- FALSE
- {prop.white=1}
- FALSE
- {unemployment=2,prop.white=1}
- 3
- {urban.stress=1,prop.white=1}
- 4
lhs rhs support confidence lift
[1] {prop.white=1} => {unemployment=2} 0.5625 0.6428571 1.142857
[2] {prop.white=1} => {urban.stress=1} 0.5625 0.6428571 1.142857
[3] {} => {unemployment=2} 0.5625 0.5625000 1.000000
[4] {} => {urban.stress=1} 0.5625 0.5625000 1.000000
[5] {} => {prop.white=1} 0.8750 0.8750000 1.000000
3. VISUALIZING ASSOCIATION RULES <a name=visual></a>¶
So far, we sought interesting associations between attributes, and they were presented in tabular format. Now, let's look at the graphical representation of the association rule (by using package arulesViz). Will we able to extract information?<br>
Unfortunately, the graphical representations do not bring much insight with the parameters given above. The first plot is very, very messy as we have $444$ item-sets to graph. The second graph has much clearner looking; empty-sets direct to $3$ single item-sets (urban.stress=1, unemployment=2, and prop.white=1). We also see loops among urban.stress, unemployment, and prop.white. These loops correspond to what we considered to be redundant. The third plot is even simpler; the loops are removed from the plot.
In [8]:
library(arulesViz)
plot(rules.LA, method="graph", control=list(type="items")) # This one is super messy
plot(rules2.LA, method="graph", control=list(type="items")) # This one is simple
plot(rules2.LA.pruned, method="graph", control=list(type="items")) # This is even simpler
In [22]:
# load and explore data
class=c(rep("3rd",52),rep("1st",118),rep("2nd",154),rep("3rd",387),rep("Crew",670),rep("1st",4),rep("2nd",13.01),rep("3rd",89),rep("Crew",3),rep("1st",5),rep("2nd",11),rep("3rd",13),rep("1st",1),rep("2nd",13),rep("3rd",14),rep("1st",57),rep("2nd",14),rep("3rd",75),rep("Crew",192),rep("1st",140),rep("2nd",80),rep("3rd",76),rep("Crew",20))
sex=c(rep("Male",35),rep("Female",17),rep("Male",1329),rep("Female",109),rep("Male",29),rep("Female",28),rep("Male",338),rep("Female",316))
age=c(rep("Child",52),rep("Adult",1438),rep("Child",57),rep("Adult",654))
survived=c(rep("No",1490),rep("Yes",711))
titanic=data.frame(class,sex,age,survived)
str(titanic)
summary(titanic)
table(titanic$age,titanic$survived)
table(titanic$class,titanic$survived)
'data.frame': 2201 obs. of 4 variables: $ class : Factor w/ 4 levels "1st","2nd","3rd",..: 3 3 3 3 3 3 3 3 3 3 ... $ sex : Factor w/ 2 levels "Female","Male": 2 2 2 2 2 2 2 2 2 2 ... $ age : Factor w/ 2 levels "Adult","Child": 2 2 2 2 2 2 2 2 2 2 ... $ survived: Factor w/ 2 levels "No","Yes": 1 1 1 1 1 1 1 1 1 1 ...
class sex age survived 1st :325 Female: 470 Adult:2092 No :1490 2nd :285 Male :1731 Child: 109 Yes: 711 3rd :706 Crew:885
No Yes
Adult 1438 654
Child 52 57
No Yes
1st 122 203
2nd 167 118
3rd 528 178
Crew 673 212
- Look for rules where the consequent consists of either "Survived" levels (i.e., we're trying to figure out what lead to either survival or lack of survival). This can be arranged by adding the
appearanceparameters to the apriori command, as below:
rules2 <- apriori(titanic,parameter = list(minlen=2, supp=0.005, conf=0.8),appearance = list(rhs=c("survived=No", "survived=Yes"),default="lhs"),control = list(verbose=F))
- Set the parameters to reduce the number of rules: rules of size at least 2, with support at least greater than 0.005 and confidence greater than 0.8.
5. OTHER REFERENCES <a name=others></a>¶
- http://www.saedsayad.com/association_rules.htm
- http://www.rdatamining.com/examples/association-rules
- https://cran.r-project.org/web/packages/arulesViz/vignettes/arulesViz.pdf
- http://www.lynda.com/R-tutorials/Up-Running-R/120612-2.html?
- http://michael.hahsler.net/research/arules_RUG_2015/demo/ (very good example)