Regression Trees- activity 13

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)

# 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)

The histogram displays the distribution of wine quality ratings, showing how frequently each rating appears in the dataset. The x-axis represents the wine quality scores, which range from 3 to 9, while the y-axis indicates the frequency of each rating. The data appears to be concentrated around a few key ratings, with a peak at 5, followed by 6 and 7, suggesting that most wines in the dataset have average quality scores. The bars are shaded in gray, and the histogram provides a clear visual representation of how the ratings are spread. While the visualization is effective in conveying the distribution, a more distinct color scheme and refined axis labels could enhance readability.

# 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  

This output provides summary statistics for the wine dataset, showing key descriptive measures for each variable. The Min. and Max. values indicate the range of each feature, while the 1st Qu. (first quartile), Median, and 3rd Qu. (third quartile) give insight into the distribution of values. Notably, residual.sugar and total.sulfur.dioxide have wide ranges, suggesting potential outliers or significant variation. The quality variable, which is likely the target, has a median of 6, indicating that most wines are of moderate quality. The alcohol content also varies widely, which could impact the perception of quality.

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

Step 3: Training a model on the data

# regression tree using rpart
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 1.0004758 0.02446671
2 0.05098911      1 0.8449895 0.8480358 0.02335464
3 0.02796998      2 0.7940004 0.8062525 0.02281469
4 0.01970128      3 0.7660304 0.7882686 0.02202005
5 0.01265926      4 0.7463291 0.7616521 0.02086416
6 0.01007193      5 0.7336698 0.7621638 0.02097496
7 0.01000000      6 0.7235979 0.7469321 0.02043579

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 

Wines with alcohol < 10.85% tend to have lower quality. Wines with higher volatile acidity are more likely to be rated lower. Wines with low free sulfur dioxide (< 10.5 mg/L) tend to have lower quality.

#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)

The image shows a regression tree visualization used to predict wine quality based on various chemical properties. The root node at the top represents the overall mean wine quality score of 5.87, with the dataset split based on alcohol content at a threshold of 10.9. If the alcohol content is below this value, the data follows the left branch, while higher values go to the right. Further splits occur based on volatile acidity and free sulfur dioxide levels, refining the prediction of wine quality scores.

# a few adjustments to the diagram
rpart.plot(m.rpart, digits = 4, fallen.leaves = TRUE, type = 3, extra = 101)

The image presents a regression tree used to predict wine quality based on chemical properties. The root node splits the data based on alcohol content at a threshold of 10.85, creating two branches that further divide based on volatile acidity and free sulfur dioxide levels. Each node displays a predicted wine quality score along with the number of observations in that group. The leftmost branches generally predict lower quality scores, while the rightmost branches, associated with higher alcohol content and lower volatile acidity, predict higher scores. Adding more visual distinctions, such as color gradients or bolded decision splits, could enhance readability.

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
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
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]  Thu Feb 27 22:39:21 2025
---------------------------------

    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: 0.5 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

The Cubist model performed moderately well in predicting wine quality, with a correlation of 0.62 and a mean absolute error of 0.53. While the predictions are fairly close to the true values, there is still room for improvement.