Regression Trees

Regression Trees and Model Trees

Understanding regression trees and model trees

Example: Calculating SDR

# set up the data
tee <- c(1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7, 7, 7, 7)
at1 <- c(1, 1, 1, 2, 2, 3, 4, 5, 5)
at2 <- c(6, 6, 7, 7, 7, 7)
bt1 <- c(1, 1, 1, 2, 2, 3, 4)
bt2 <- c(5, 5, 6, 6, 7, 7, 7, 7)

Standard Deviation Reduction # The goal is to find the split that reduces the standard deviation the most. # Higher SDR = A more “pure” (consistent) grouping of wine ratings.

# compute the SDR
sdr_a <- sd(tee) - (length(at1) / length(tee) * sd(at1) + length(at2) / length(tee) * sd(at2))
sdr_b <- sd(tee) - (length(bt1) / length(tee) * sd(bt1) + length(bt2) / length(tee) * sd(bt2))

# compare the SDR for each split
sdr_a

[1] 1.202815

sdr_b

[1] 1.392751

Exercise No 3: Estimating Wine Quality

Step 2: Exploring and preparing the data

wine <- read.csv("whitewines.csv")

# examine the wine data
str(wine)

'data.frame':   4898 obs. of  12 variables:
 $ fixed.acidity       : num  6.7 5.7 5.9 5.3 6.4 7 7.9 6.6 7 6.5 ...
 $ volatile.acidity    : num  0.62 0.22 0.19 0.47 0.29 0.14 0.12 0.38 0.16 0.37 ...
 $ citric.acid         : num  0.24 0.2 0.26 0.1 0.21 0.41 0.49 0.28 0.3 0.33 ...
 $ residual.sugar      : num  1.1 16 7.4 1.3 9.65 0.9 5.2 2.8 2.6 3.9 ...
 $ chlorides           : num  0.039 0.044 0.034 0.036 0.041 0.037 0.049 0.043 0.043 0.027 ...
 $ free.sulfur.dioxide : num  6 41 33 11 36 22 33 17 34 40 ...
 $ total.sulfur.dioxide: num  62 113 123 74 119 95 152 67 90 130 ...
 $ density             : num  0.993 0.999 0.995 0.991 0.993 ...
 $ pH                  : num  3.41 3.22 3.49 3.48 2.99 3.25 3.18 3.21 2.88 3.28 ...
 $ sulphates           : num  0.32 0.46 0.42 0.54 0.34 0.43 0.47 0.47 0.47 0.39 ...
 $ alcohol             : num  10.4 8.9 10.1 11.2 10.9 ...
 $ quality             : int  5 6 6 4 6 6 6 6 6 7 ...

# the distribution of quality ratings
hist(wine$quality)

# summary statistics of the wine data
summary(wine)

 fixed.acidity    volatile.acidity  citric.acid     residual.sugar     chlorides      
 Min.   : 3.800   Min.   :0.0800   Min.   :0.0000   Min.   : 0.600   Min.   :0.00900  
 1st Qu.: 6.300   1st Qu.:0.2100   1st Qu.:0.2700   1st Qu.: 1.700   1st Qu.:0.03600  
 Median : 6.800   Median :0.2600   Median :0.3200   Median : 5.200   Median :0.04300  
 Mean   : 6.855   Mean   :0.2782   Mean   :0.3342   Mean   : 6.391   Mean   :0.04577  
 3rd Qu.: 7.300   3rd Qu.:0.3200   3rd Qu.:0.3900   3rd Qu.: 9.900   3rd Qu.:0.05000  
 Max.   :14.200   Max.   :1.1000   Max.   :1.6600   Max.   :65.800   Max.   :0.34600  
 free.sulfur.dioxide total.sulfur.dioxide    density             pH          sulphates     
 Min.   :  2.00      Min.   :  9.0        Min.   :0.9871   Min.   :2.720   Min.   :0.2200  
 1st Qu.: 23.00      1st Qu.:108.0        1st Qu.:0.9917   1st Qu.:3.090   1st Qu.:0.4100  
 Median : 34.00      Median :134.0        Median :0.9937   Median :3.180   Median :0.4700  
 Mean   : 35.31      Mean   :138.4        Mean   :0.9940   Mean   :3.188   Mean   :0.4898  
 3rd Qu.: 46.00      3rd Qu.:167.0        3rd Qu.:0.9961   3rd Qu.:3.280   3rd Qu.:0.5500  
 Max.   :289.00      Max.   :440.0        Max.   :1.0390   Max.   :3.820   Max.   :1.0800  
    alcohol         quality     
 Min.   : 8.00   Min.   :3.000  
 1st Qu.: 9.50   1st Qu.:5.000  
 Median :10.40   Median :6.000  
 Mean   :10.51   Mean   :5.878  
 3rd Qu.:11.40   3rd Qu.:6.000  
 Max.   :14.20   Max.   :9.000  

Data Spliting # We are using the first 75% for training and the last 25% for testing. # IMPORTANT: This assumes the CSV isn’t sorted by quality. # If it were, our model would only “see” one type of wine during training.

wine_train <- wine[1:3750, ]
wine_test <- wine[3751:4898, ]

Step 3: Training a model on the data

# regression tree using rpart
# 'quality ~ .' means we predict quality using ALL other variables.
library(rpart)
m.rpart <- rpart(quality ~ ., data = wine_train)

# get basic information about the tree
m.rpart

n= 3750 

node), split, n, deviance, yval
      * denotes terminal node

 1) root 3750 2945.53200 5.870933  
   2) alcohol< 10.85 2372 1418.86100 5.604975  
     4) volatile.acidity>=0.2275 1611  821.30730 5.432030  
       8) volatile.acidity>=0.3025 688  278.97670 5.255814 *
       9) volatile.acidity< 0.3025 923  505.04230 5.563380 *
     5) volatile.acidity< 0.2275 761  447.36400 5.971091 *
   3) alcohol>=10.85 1378 1070.08200 6.328737  
     6) free.sulfur.dioxide< 10.5 84   95.55952 5.369048 *
     7) free.sulfur.dioxide>=10.5 1294  892.13600 6.391036  
      14) alcohol< 11.76667 629  430.11130 6.173291  
        28) volatile.acidity>=0.465 11   10.72727 4.545455 *
        29) volatile.acidity< 0.465 618  389.71680 6.202265 *
      15) alcohol>=11.76667 665  403.99400 6.596992 *

# get more detailed information about the tree
summary(m.rpart)

Call:
rpart(formula = quality ~ ., data = wine_train)
  n= 3750 

          CP nsplit rel error    xerror       xstd
1 0.15501053      0 1.0000000 1.0005747 0.02447080
2 0.05098911      1 0.8449895 0.8477424 0.02339683
3 0.02796998      2 0.7940004 0.8053836 0.02282606
4 0.01970128      3 0.7660304 0.7880536 0.02220038
5 0.01265926      4 0.7463291 0.7660650 0.02135318
6 0.01007193      5 0.7336698 0.7559186 0.02102829
7 0.01000000      6 0.7235979 0.7554624 0.02099190

Variable importance
             alcohol              density     volatile.acidity            chlorides 
                  34                   21                   15                   11 
total.sulfur.dioxide  free.sulfur.dioxide       residual.sugar            sulphates 
                   7                    6                    3                    1 
         citric.acid 
                   1 

Node number 1: 3750 observations,    complexity param=0.1550105
  mean=5.870933, MSE=0.7854751 
  left son=2 (2372 obs) right son=3 (1378 obs)
  Primary splits:
      alcohol              < 10.85    to the left,  improve=0.15501050, (0 missing)
      density              < 0.992035 to the right, improve=0.10915940, (0 missing)
      chlorides            < 0.0395   to the right, improve=0.07682258, (0 missing)
      total.sulfur.dioxide < 158.5    to the right, improve=0.04089663, (0 missing)
      citric.acid          < 0.235    to the left,  improve=0.03636458, (0 missing)
  Surrogate splits:
      density              < 0.991995 to the right, agree=0.869, adj=0.644, (0 split)
      chlorides            < 0.0375   to the right, agree=0.757, adj=0.339, (0 split)
      total.sulfur.dioxide < 103.5    to the right, agree=0.690, adj=0.155, (0 split)
      residual.sugar       < 5.375    to the right, agree=0.667, adj=0.094, (0 split)
      sulphates            < 0.345    to the right, agree=0.647, adj=0.038, (0 split)

Node number 2: 2372 observations,    complexity param=0.05098911
  mean=5.604975, MSE=0.5981709 
  left son=4 (1611 obs) right son=5 (761 obs)
  Primary splits:
      volatile.acidity    < 0.2275   to the right, improve=0.10585250, (0 missing)
      free.sulfur.dioxide < 13.5     to the left,  improve=0.03390500, (0 missing)
      citric.acid         < 0.235    to the left,  improve=0.03204075, (0 missing)
      alcohol             < 10.11667 to the left,  improve=0.03136524, (0 missing)
      chlorides           < 0.0585   to the right, improve=0.01633599, (0 missing)
  Surrogate splits:
      pH                   < 3.485    to the left,  agree=0.694, adj=0.047, (0 split)
      sulphates            < 0.755    to the left,  agree=0.685, adj=0.020, (0 split)
      total.sulfur.dioxide < 105.5    to the right, agree=0.683, adj=0.011, (0 split)
      residual.sugar       < 0.75     to the right, agree=0.681, adj=0.007, (0 split)
      chlorides            < 0.0285   to the right, agree=0.680, adj=0.003, (0 split)

Node number 3: 1378 observations,    complexity param=0.02796998
  mean=6.328737, MSE=0.7765472 
  left son=6 (84 obs) right son=7 (1294 obs)
  Primary splits:
      free.sulfur.dioxide  < 10.5     to the left,  improve=0.07699080, (0 missing)
      alcohol              < 11.76667 to the left,  improve=0.06210660, (0 missing)
      total.sulfur.dioxide < 67.5     to the left,  improve=0.04438619, (0 missing)
      residual.sugar       < 1.375    to the left,  improve=0.02905351, (0 missing)
      fixed.acidity        < 7.35     to the right, improve=0.02613259, (0 missing)
  Surrogate splits:
      total.sulfur.dioxide < 53.5     to the left,  agree=0.952, adj=0.214, (0 split)
      volatile.acidity     < 0.875    to the right, agree=0.940, adj=0.024, (0 split)

Node number 4: 1611 observations,    complexity param=0.01265926
  mean=5.43203, MSE=0.5098121 
  left son=8 (688 obs) right son=9 (923 obs)
  Primary splits:
      volatile.acidity    < 0.3025   to the right, improve=0.04540111, (0 missing)
      alcohol             < 10.05    to the left,  improve=0.03874403, (0 missing)
      free.sulfur.dioxide < 13.5     to the left,  improve=0.03338886, (0 missing)
      chlorides           < 0.0495   to the right, improve=0.02574623, (0 missing)
      citric.acid         < 0.195    to the left,  improve=0.02327981, (0 missing)
  Surrogate splits:
      citric.acid          < 0.215    to the left,  agree=0.633, adj=0.141, (0 split)
      free.sulfur.dioxide  < 20.5     to the left,  agree=0.600, adj=0.063, (0 split)
      chlorides            < 0.0595   to the right, agree=0.593, adj=0.047, (0 split)
      residual.sugar       < 1.15     to the left,  agree=0.583, adj=0.023, (0 split)
      total.sulfur.dioxide < 219.25   to the right, agree=0.582, adj=0.022, (0 split)

Node number 5: 761 observations
  mean=5.971091, MSE=0.5878633 

Node number 6: 84 observations
  mean=5.369048, MSE=1.137613 

Node number 7: 1294 observations,    complexity param=0.01970128
  mean=6.391036, MSE=0.6894405 
  left son=14 (629 obs) right son=15 (665 obs)
  Primary splits:
      alcohol              < 11.76667 to the left,  improve=0.06504696, (0 missing)
      chlorides            < 0.0395   to the right, improve=0.02758705, (0 missing)
      fixed.acidity        < 7.35     to the right, improve=0.02750932, (0 missing)
      pH                   < 3.055    to the left,  improve=0.02307356, (0 missing)
      total.sulfur.dioxide < 191.5    to the right, improve=0.02186818, (0 missing)
  Surrogate splits:
      density              < 0.990885 to the right, agree=0.720, adj=0.424, (0 split)
      volatile.acidity     < 0.2675   to the left,  agree=0.637, adj=0.253, (0 split)
      chlorides            < 0.0365   to the right, agree=0.630, adj=0.238, (0 split)
      residual.sugar       < 1.475    to the left,  agree=0.575, adj=0.126, (0 split)
      total.sulfur.dioxide < 128.5    to the right, agree=0.574, adj=0.124, (0 split)

Node number 8: 688 observations
  mean=5.255814, MSE=0.4054895 

Node number 9: 923 observations
  mean=5.56338, MSE=0.5471747 

Node number 14: 629 observations,    complexity param=0.01007193
  mean=6.173291, MSE=0.6838017 
  left son=28 (11 obs) right son=29 (618 obs)
  Primary splits:
      volatile.acidity     < 0.465    to the right, improve=0.06897561, (0 missing)
      total.sulfur.dioxide < 200      to the right, improve=0.04223066, (0 missing)
      residual.sugar       < 0.975    to the left,  improve=0.03061714, (0 missing)
      fixed.acidity        < 7.35     to the right, improve=0.02978501, (0 missing)
      sulphates            < 0.575    to the left,  improve=0.02165970, (0 missing)
  Surrogate splits:
      citric.acid          < 0.045    to the left,  agree=0.986, adj=0.182, (0 split)
      total.sulfur.dioxide < 279.25   to the right, agree=0.986, adj=0.182, (0 split)

Node number 15: 665 observations
  mean=6.596992, MSE=0.6075098 

Node number 28: 11 observations
  mean=4.545455, MSE=0.9752066 

Node number 29: 618 observations
  mean=6.202265, MSE=0.6306098 

#install.packages("rpart.plot")

# use the rpart.plot package to create a visualization
library(rpart.plot)

# a basic decision tree diagram
rpart.plot(m.rpart, digits = 3)

# a few adjustments to the diagram
# 'digits = 4' controls the precision of the predicted ratings in the leaves.
# 'fallen.leaves = TRUE' forces the final predictions to the bottom for readability
rpart.plot(m.rpart, digits = 4, fallen.leaves = TRUE, type = 3, extra = 101)

Step 4: Evaluate model performanc

# generate predictions for the testing dataset
p.rpart <- predict(m.rpart, wine_test)

# compare the distribution of predicted values vs. actual values
summary(p.rpart)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  4.545   5.563   5.971   5.893   6.202   6.597 

summary(wine_test$quality)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  3.000   5.000   6.000   5.901   6.000   9.000 

# compare the correlation
cor(p.rpart, wine_test$quality)

[1] 0.5369525

# function to calculate the mean absolute error
MAE <- function(actual, predicted) {
  mean(abs(actual - predicted))  
}

# mean absolute error between predicted and actual values
MAE(p.rpart, wine_test$quality)

[1] 0.5872652

# mean absolute error between actual values and mean value
# This tells us how many 'points' our rating prediction is off on average.
# If MAE = 0.59, our model is usually within 0.6 units of the real quality score.

# Compare this to the 'Null Model' (just guessing the average 5.87)
# If our model's MAE isn't significantly lower than 0.67, the tree isn't helpful!
mean(wine_train$quality) # result = 5.87

[1] 5.870933

MAE(5.87, wine_test$quality)

[1] 0.6722474

Step 5: Improving model performance

#install.packages("plyr")
#install.packages("Cubist")

# train a Cubist Model Tree
# Unlike rpart (which gives one number), Cubist creates a linear equation 
# for every leaf. This allows for much more granular, "sloped" predictions.

# x = wine_train[-12] removes the 'quality' column so the model doesn't 
# use the answer to predict the answer!
library(Cubist)
m.cubist <- cubist(x = wine_train[-12], y = wine_train$quality)

# display basic information about the model tree
m.cubist

Call:
cubist.default(x = wine_train[-12], y = wine_train$quality)

Number of samples: 3750 
Number of predictors: 11 

Number of committees: 1 
Number of rules: 25 

# display the tree itself
summary(m.cubist)

Call:
cubist.default(x = wine_train[-12], y = wine_train$quality)


Cubist [Release 2.07 GPL Edition]  Tue Feb 24 00:18:42 2026
---------------------------------

    Target attribute `outcome'

Read 3750 cases (12 attributes) from undefined.data

Model:

  Rule 1: [21 cases, mean 5.0, range 4 to 6, est err 0.5]

    if
    free.sulfur.dioxide > 30
    total.sulfur.dioxide > 195
    total.sulfur.dioxide <= 235
    sulphates > 0.64
    alcohol > 9.1
    then
    outcome = 573.6 + 0.0478 total.sulfur.dioxide - 573 density
              - 0.788 alcohol + 0.186 residual.sugar - 4.73 volatile.acidity

  Rule 2: [28 cases, mean 5.0, range 4 to 8, est err 0.7]

    if
    volatile.acidity > 0.31
    citric.acid <= 0.36
    residual.sugar <= 1.45
    total.sulfur.dioxide <= 97
    alcohol > 9.1
    then
    outcome = 168.2 + 4.75 citric.acid + 0.0123 total.sulfur.dioxide
              - 170 density + 0.057 residual.sugar - 6.4 chlorides + 0.84 pH
              + 0.14 fixed.acidity

  Rule 3: [171 cases, mean 5.1, range 3 to 6, est err 0.3]

    if
    volatile.acidity > 0.205
    chlorides <= 0.054
    density <= 0.99839
    alcohol <= 9.1
    then
    outcome = 147.4 - 144 density + 0.08 residual.sugar + 0.117 alcohol
              - 0.87 volatile.acidity - 0.09 pH - 0.01 fixed.acidity

  Rule 4: [37 cases, mean 5.3, range 3 to 6, est err 0.5]

    if
    free.sulfur.dioxide > 30
    total.sulfur.dioxide > 235
    alcohol > 9.1
    then
    outcome = 19.5 - 0.013 total.sulfur.dioxide - 2.7 volatile.acidity
              - 10 density + 0.005 residual.sugar + 0.008 alcohol

  Rule 5: [64 cases, mean 5.3, range 5 to 6, est err 0.3]

    if
    volatile.acidity > 0.205
    residual.sugar > 17.85
    then
    outcome = -23.6 + 0.233 alcohol - 5.2 chlorides - 0.75 citric.acid
              + 28 density - 0.81 volatile.acidity - 0.19 pH
              - 0.002 residual.sugar

  Rule 6: [56 cases, mean 5.3, range 4 to 7, est err 0.6]

    if
    fixed.acidity <= 7.1
    volatile.acidity > 0.205
    chlorides > 0.054
    density <= 0.99839
    alcohol <= 9.1
    then
    outcome = 40.6 + 0.374 alcohol - 1.62 volatile.acidity
              + 0.026 residual.sugar - 38 density - 0.21 pH
              - 0.01 fixed.acidity

  Rule 7: [337 cases, mean 5.3, range 3 to 7, est err 0.4]

    if
    fixed.acidity <= 7.8
    volatile.acidity > 0.305
    chlorides <= 0.09
    free.sulfur.dioxide <= 82.5
    total.sulfur.dioxide > 130
    total.sulfur.dioxide <= 235
    sulphates <= 0.64
    alcohol <= 10.4
    then
    outcome = -32.1 + 0.233 alcohol - 9.7 chlorides
              + 0.0038 total.sulfur.dioxide - 0.0081 free.sulfur.dioxide
              + 35 density + 0.81 volatile.acidity

  Rule 8: [30 cases, mean 5.5, range 3 to 7, est err 0.5]

    if
    fixed.acidity > 7.1
    volatile.acidity > 0.205
    chlorides > 0.054
    density <= 0.99839
    alcohol <= 9.1
    then
    outcome = 244 - 1.56 fixed.acidity - 228 density
              + 0.0252 free.sulfur.dioxide - 7.3 chlorides
              - 0.19 volatile.acidity + 0.003 residual.sugar

  Rule 9: [98 cases, mean 5.5, range 4 to 8, est err 0.5]

    if
    volatile.acidity > 0.155
    chlorides > 0.09
    total.sulfur.dioxide <= 235
    sulphates <= 0.64
    then
    outcome = 55.9 - 3.85 volatile.acidity - 52 density
              + 0.023 residual.sugar + 0.092 alcohol + 0.35 pH
              + 0.05 fixed.acidity + 0.3 sulphates
              + 0.001 free.sulfur.dioxide

  Rule 10: [446 cases, mean 5.6, range 4 to 8, est err 0.5]

    if
    fixed.acidity <= 7.8
    volatile.acidity > 0.155
    volatile.acidity <= 0.305
    chlorides <= 0.09
    free.sulfur.dioxide <= 82.5
    total.sulfur.dioxide > 130
    total.sulfur.dioxide <= 235
    sulphates <= 0.64
    alcohol > 9.1
    alcohol <= 10.4
    then
    outcome = 15.1 + 0.35 alcohol - 3.09 volatile.acidity - 14.7 chlorides
              + 1.16 sulphates - 0.0022 total.sulfur.dioxide
              + 0.11 fixed.acidity + 0.45 pH + 0.5 citric.acid - 14 density
              + 0.006 residual.sugar

  Rule 11: [31 cases, mean 5.6, range 3 to 8, est err 0.8]

    if
    volatile.acidity > 0.31
    citric.acid > 0.36
    free.sulfur.dioxide <= 30
    total.sulfur.dioxide <= 97
    then
    outcome = 3.2 + 0.0584 total.sulfur.dioxide + 7.77 volatile.acidity
              + 0.328 alcohol - 9 density + 0.003 residual.sugar

  Rule 12: [20 cases, mean 5.7, range 3 to 8, est err 0.9]

    if
    free.sulfur.dioxide > 82.5
    total.sulfur.dioxide <= 235
    sulphates <= 0.64
    alcohol > 9.1
    then
    outcome = -8.9 + 109.3 chlorides + 0.948 alcohol

  Rule 13: [331 cases, mean 5.8, range 4 to 8, est err 0.5]

    if
    volatile.acidity > 0.31
    free.sulfur.dioxide <= 30
    total.sulfur.dioxide > 97
    alcohol > 9.1
    then
    outcome = 89.8 + 0.0234 free.sulfur.dioxide + 0.324 alcohol
              + 0.07 residual.sugar - 90 density - 1.47 volatile.acidity
              + 0.48 pH

  Rule 14: [116 cases, mean 5.8, range 3 to 8, est err 0.6]

    if
    fixed.acidity > 7.8
    volatile.acidity > 0.155
    free.sulfur.dioxide > 30
    total.sulfur.dioxide > 130
    total.sulfur.dioxide <= 235
    sulphates <= 0.64
    alcohol > 9.1
    then
    outcome = 6 + 0.346 alcohol - 0.41 fixed.acidity - 1.69 volatile.acidity
              - 2.9 chlorides + 0.19 sulphates + 0.07 pH

  Rule 15: [115 cases, mean 5.8, range 4 to 7, est err 0.5]

    if
    volatile.acidity > 0.205
    residual.sugar <= 17.85
    density > 0.99839
    alcohol <= 9.1
    then
    outcome = -110.2 + 120 density - 3.46 volatile.acidity - 0.97 pH
              - 0.022 residual.sugar + 0.088 alcohol - 0.6 citric.acid
              - 0.01 fixed.acidity

  Rule 16: [986 cases, mean 5.9, range 3 to 9, est err 0.6]

    if
    volatile.acidity <= 0.31
    free.sulfur.dioxide <= 30
    alcohol > 9.1
    then
    outcome = 280.4 - 282 density + 0.128 residual.sugar
              + 0.0264 free.sulfur.dioxide - 3 volatile.acidity + 1.2 pH
              + 0.65 citric.acid + 0.09 fixed.acidity + 0.56 sulphates
              + 0.015 alcohol

  Rule 17: [49 cases, mean 6.0, range 5 to 8, est err 0.5]

    if
    volatile.acidity > 0.155
    residual.sugar > 8.8
    free.sulfur.dioxide > 30
    total.sulfur.dioxide <= 130
    pH <= 3.26
    alcohol > 9.1
    then
    outcome = 173.5 - 169 density + 0.055 alcohol + 0.38 sulphates
              + 0.002 residual.sugar

  Rule 18: [114 cases, mean 6.1, range 3 to 9, est err 0.6]

    if
    volatile.acidity > 0.31
    citric.acid <= 0.36
    residual.sugar > 1.45
    total.sulfur.dioxide <= 97
    alcohol > 9.1
    then
    outcome = 302.3 - 305 density + 0.0128 total.sulfur.dioxide
              + 0.096 residual.sugar + 1.94 citric.acid + 1.05 pH
              + 0.17 fixed.acidity - 6.7 chlorides
              + 0.0022 free.sulfur.dioxide - 0.21 volatile.acidity
              + 0.013 alcohol + 0.09 sulphates

  Rule 19: [145 cases, mean 6.1, range 5 to 8, est err 0.6]

    if
    volatile.acidity > 0.155
    free.sulfur.dioxide > 30
    total.sulfur.dioxide <= 195
    sulphates > 0.64
    then
    outcome = 206 - 209 density + 0.069 residual.sugar + 0.38 fixed.acidity
              + 2.79 sulphates + 0.0155 free.sulfur.dioxide
              - 0.0051 total.sulfur.dioxide - 1.71 citric.acid + 1.04 pH

  Rule 20: [555 cases, mean 6.1, range 3 to 9, est err 0.6]

    if
    total.sulfur.dioxide > 130
    total.sulfur.dioxide <= 235
    sulphates <= 0.64
    alcohol > 10.4
    then
    outcome = 108 + 0.276 alcohol - 109 density + 0.05 residual.sugar
              + 0.77 pH - 1.02 volatile.acidity - 4.2 chlorides
              + 0.78 sulphates + 0.08 fixed.acidity
              + 0.0016 free.sulfur.dioxide - 0.0003 total.sulfur.dioxide

  Rule 21: [73 cases, mean 6.2, range 4 to 8, est err 0.4]

    if
    volatile.acidity > 0.155
    citric.acid <= 0.28
    residual.sugar <= 8.8
    free.sulfur.dioxide > 30
    total.sulfur.dioxide <= 130
    pH <= 3.26
    sulphates <= 0.64
    alcohol > 9.1
    then
    outcome = 4.2 + 0.147 residual.sugar + 0.47 alcohol + 3.75 sulphates
              - 2.5 volatile.acidity - 5 density

  Rule 22: [244 cases, mean 6.3, range 4 to 8, est err 0.6]

    if
    citric.acid > 0.28
    residual.sugar <= 8.8
    free.sulfur.dioxide > 30
    total.sulfur.dioxide <= 130
    pH <= 3.26
    then
    outcome = 40.1 + 0.278 alcohol + 1.3 sulphates - 39 density
              + 0.017 residual.sugar + 0.001 total.sulfur.dioxide + 0.17 pH
              + 0.03 fixed.acidity

  Rule 23: [106 cases, mean 6.3, range 4 to 8, est err 0.6]

    if
    volatile.acidity <= 0.155
    free.sulfur.dioxide > 30
    then
    outcome = 139.1 - 138 density + 0.058 residual.sugar + 0.71 pH
              + 0.92 sulphates + 0.11 fixed.acidity - 0.73 volatile.acidity
              + 0.055 alcohol - 0.0012 total.sulfur.dioxide
              + 0.0007 free.sulfur.dioxide

  Rule 24: [137 cases, mean 6.5, range 4 to 9, est err 0.6]

    if
    volatile.acidity > 0.155
    free.sulfur.dioxide > 30
    total.sulfur.dioxide <= 130
    pH > 3.26
    sulphates <= 0.64
    alcohol > 9.1
    then
    outcome = 114.2 + 0.0142 total.sulfur.dioxide - 107 density
              - 11.8 chlorides - 1.57 pH + 0.124 alcohol + 1.21 sulphates
              + 1.16 volatile.acidity + 0.021 residual.sugar
              + 0.04 fixed.acidity

  Rule 25: [92 cases, mean 6.5, range 4 to 8, est err 0.6]

    if
    volatile.acidity <= 0.205
    alcohol <= 9.1
    then
    outcome = -200.7 + 210 density + 5.88 volatile.acidity + 23.9 chlorides
              - 2.83 citric.acid - 1.17 pH


Evaluation on training data (3750 cases):

    Average  |error|                0.5
    Relative |error|               0.67
    Correlation coefficient        0.66


    Attribute usage:
      Conds  Model

       84%    93%    alcohol
       80%    89%    volatile.acidity
       70%    61%    free.sulfur.dioxide
       63%    50%    total.sulfur.dioxide
       44%    70%    sulphates
       26%    44%    chlorides
       22%    76%    fixed.acidity
       16%    87%    residual.sugar
       11%    86%    pH
       11%    45%    citric.acid
        8%    97%    density


Time: 0.2 secs

# generate predictions for the model
p.cubist <- predict(m.cubist, wine_test)

# summary statistics about the predictions
summary(p.cubist)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  3.677   5.416   5.906   5.848   6.238   7.393 

# correlation between the predicted and true values
cor(p.cubist, wine_test$quality)

[1] 0.6201015

# mean absolute error of predicted and true values
# (uses a custom function defined above)
MAE(wine_test$quality, p.cubist) 

[1] 0.5339725