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)

# 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))

sdr_b

[1] 1.392751

Exercise No 3: Estimating Wine Quality

Step 2: Exploring and preparing the data

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

Error in file(file, "rt") : cannot open the connection

# 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 ...

# 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  

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 1.0008330 0.02447515
2 0.05098911      1 0.8449895 0.8481054 0.02336586
3 0.02796998      2 0.7940004 0.8029105 0.02268910
4 0.01970128      3 0.7660304 0.7776164 0.02147106
5 0.01265926      4 0.7463291 0.7577502 0.02069972
6 0.01007193      5 0.7336698 0.7540073 0.02060305
7 0.01000000      6 0.7235979 0.7485112 0.02048703

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")

trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/rpart.plot_3.1.4.tar.gz'
Content type 'application/x-gzip' length 993302 bytes (970 KB)
==================================================
downloaded 970 KB


The downloaded source packages are in
    ‘/tmp/RtmpSBhFhd/downloaded_packages’

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

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. 
  5.563   5.563   5.563   5.563   5.563   5.563 

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

[1] NA

# 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] NA

MAE(5.87, wine_test$quality)

[1] NA

Step 5: Improving model performance

install.packages("plyr")

trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/plyr_1.8.9.tar.gz'
Content type 'application/x-gzip' length 823755 bytes (804 KB)
==================================================
downloaded 804 KB


The downloaded source packages are in
    ‘/tmp/RtmpGGgkl7/downloaded_packages’

# 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: 7 

# 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  3 01:25:38 2026
---------------------------------

    Target attribute `outcome'

*** Ignoring cases with unknown or N/A target value

Read 1599 cases (12 attributes) from undefined.data

Model:

  Rule 1: [630 cases, mean 5.3, range 3 to 8, est err 0.4]

    if
    alcohol <= 9.8
    then
    outcome = 5 - 0.79 volatile.acidity - 0.099 alcohol
              + 0.052 fixed.acidity - 0.31 citric.acid + 0.33 sulphates
              + 0.29 pH - 0.0031 free.sulfur.dioxide
              - 0.0007 total.sulfur.dioxide - 0.4 chlorides

  Rule 2: [589 cases, mean 5.3, range 3 to 8, est err 0.4]

    if
    sulphates <= 0.92
    alcohol <= 9.8
    then
    outcome = 5.5 + 1.28 sulphates - 0.9 volatile.acidity - 0.33 citric.acid
              + 0.029 fixed.acidity - 0.033 alcohol
              - 0.0008 total.sulfur.dioxide - 0.0023 free.sulfur.dioxide
              - 0.4 chlorides - 0.1 pH

  Rule 3: [80 cases, mean 5.3, range 3 to 7, est err 0.7]

    if
    volatile.acidity > 0.31
    total.sulfur.dioxide <= 14
    sulphates <= 0.63
    alcohol > 9.8
    then
    outcome = 0.5 + 0.549 alcohol - 1.61 volatile.acidity + 0.36 sulphates
              - 0.18 pH - 0.0005 total.sulfur.dioxide - 0.07 citric.acid
              + 0.001 free.sulfur.dioxide

  Rule 4: [340 cases, mean 5.6, range 4 to 7, est err 0.5]

    if
    volatile.acidity > 0.31
    total.sulfur.dioxide > 14
    sulphates <= 0.63
    alcohol > 9.8
    then
    outcome = 5.1 + 2.85 sulphates + 0.19 alcohol - 0.74 citric.acid
              - 0.69 volatile.acidity - 0.74 pH
              - 0.0027 total.sulfur.dioxide + 0.0013 free.sulfur.dioxide

  Rule 5: [407 cases, mean 6.1, range 3 to 8, est err 0.6]

    if
    volatile.acidity > 0.31
    sulphates > 0.63
    alcohol > 9.8
    then
    outcome = 7.6 + 0.309 alcohol - 0.0073 total.sulfur.dioxide - 1.12 pH
              - 0.81 volatile.acidity - 0.079 fixed.acidity + 0.22 sulphates
              + 0.002 free.sulfur.dioxide

  Rule 6: [71 cases, mean 6.2, range 5 to 8, est err 0.5]

    if
    volatile.acidity <= 0.31
    sulphates <= 0.73
    alcohol > 9.8
    then
    outcome = 131.4 + 4.85 volatile.acidity - 124 density - 1.35 pH
              + 0.056 fixed.acidity + 0.54 sulphates + 0.036 alcohol
              + 0.021 residual.sugar

  Rule 7: [85 cases, mean 6.5, range 5 to 8, est err 0.4]

    if
    volatile.acidity <= 0.31
    sulphates > 0.73
    then
    outcome = 17 + 0.39 alcohol + 0.113 fixed.acidity
              + 0.25 volatile.acidity - 16 density + 0.14 sulphates


Evaluation on training data (1599 cases):

    Average  |error|                0.4
    Relative |error|               0.62
    Correlation coefficient        0.62


    Attribute usage:
      Conds  Model

       96%   100%    alcohol
       71%   100%    sulphates
       45%   100%    volatile.acidity
       19%    93%    total.sulfur.dioxide
              96%    pH
              93%    free.sulfur.dioxide
              81%    fixed.acidity
              74%    citric.acid
              55%    chlorides
               7%    density
               3%    residual.sugar


Time: 0.0 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. 
  5.865   5.865   5.865   5.865   5.865   5.865 

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

[1] NA

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

[1] NA