How do CART models work

Table of contents

• What is CART?
• Information Gain and Entropy
• Growing a Tree
• Let Computer Do the Math
  • R
  • Python
• Numerical Explanatory Variables
• Other Splitting Metrics
  • Gini Impurity
  • Variance Reduction
• Summary
• References

# r import
library(tidyverse)
library(knitr)
library(kableExtra)
library(Rgraphviz)
library(rpart)
library(rattle)
library(reticulate)

# python import
import pandas as pd
from sklearn import tree
import matplotlib.pyplot as plt

What is CART?

Classification and Regression Trees (or CART for short) is a type of supervised machine learning algorithm [1] that is mostly used in classification problems [2], but can be applied for regression problems [3] as well. It can handle both categorical and numerical input variables.

In this tutorial, we are going to go step-by-step and build a tree for the classification problem. A way to solve the regression problem will be discussed at the end.

Imagine that we have medical data about patients and we have two drugs (Drug A and Drug B) that work differently for each patient based on their health metrics. We have such data for 15 patients. Now, the 16th patient comes in and we want to assign a drug to him based on this information. You can load the sample data set here.

patients_df <- read_csv("DecisionTree_SampleData.csv")
kable(patients_df, format = "markdown", caption = "<b>Patients Data</b>")

Our decision tree is going to look something like this:

In case you are wondering why is it called “a tree”, you have to look at it upside down:

Imagine that all our data is located at the root node of the tree. We want to split the data by most significant features (that will create new decision nodes) until we reach the leaf nodes with the most homogeneous target possible (ideally just one class is present in each leaf node).

The question is how do we split the data at each node?

Information Gain and Entropy

The main metrics to use for dealing with classification problems when building a tree are Information Gain (IG) and Gini impurity. We are going to use IG for this example, however, Gini impurity will be also discussed at the end.

IG is based on the concept of entropy [4] and information content from information theory [5]. First of all, let’s build an intuition behind the entropy. Simply saying, entropy is the measure of chaos or uncertainty in the data.

Imagine that I asked you to play the following simple game: I have two coins, one coin is unfair \(\left( P(\text{Heads}) = \frac{4}{5} \right)\), the other one is fair \(\left( P(\text{Heads}) = \frac{1}{2} \right)\). Your task is to pick a coin, make a prediction of an outcome (either Heads or Tails) and flip it. If your prediction is correct, I will give you $1. Which coin would you choose?

Well, I hope that you would choose the unfair coin. In this case, you are more certain about the outcome of the coin flip (on average it will come up Heads in \(80\%\) of the time). For the fair coin, you have the highest level of uncertainty, meaning that Heads and Tails can occur equally likely. Probability mass functions for both coins look as:

\[\begin{equation} p_{\text{unfair}} = \begin{cases} \frac{4}{5} & \text{Heads}\\ \frac{1}{5} & \text{Tails}\\ \end{cases} \end{equation}\]

\[\begin{equation} p_{\text{fair}} = \begin{cases} \frac{1}{2} & \text{Heads}\\ \frac{1}{2} & \text{Tails}\\ \end{cases} \end{equation}\]

We can calculate the entropy using the formula:

\[H(X) = - \sum_{i=1}^{n} p(x_i) \cdot log_2 \big( p(x_i) \big)\]

• \(n\) is the number of all possible outcomes for random variable \(X\);
• \(log_2\) is the logarithm with base 2 (\(log_2 2 = 1\), \(log_2 4 = 2\), etc).

Entropy for the unfair coin:

\[H(\text{unfair}) = - \left( \frac{4}{5} \cdot log_2 \frac{4}{5} + \frac{1}{5} \cdot log_2 \frac{1}{5} \right) \\ \approx 0.72\] Entropy for the fair coin:

\[H(\text{fair}) = - \left( \frac{1}{2} \cdot log_2 \frac{1}{2} + \frac{1}{2} \cdot log_2 \frac{1}{2} \right) \\ = 1\]

entropy <- function(p) {
  return(-sum(p*log(p, base = 2)))
}

p_unfair <- c(4/5, 1/5)
p_fair <- c(1/2, 1/2)

entropy(p_unfair)
## [1] 0.7219281
entropy(p_fair)
## [1] 1

As you can see, entropy for the fair coin is higher, meaning that the level of uncertainty is also higher. Note that entropy cannot be negative and the minimum value is \(0\) which means the highest level of certainty. For example, imagine the unfair coin with the probability \(P(\text{Heads}) = 1\) and \(P(\text{Tails}) = 0\). In such a way you are sure that the coin will come up Heads and entropy is equal to zero.

Coming back to our decision tree problem we want to split out data into nodes that have reduced the level of uncertainty. Basic algorithm look as follows:

1. For each node:

• Choose a feature from the data set.
• Calculate the significance (e.g., IG) of that feature in the splitting of data.
• Repeat for each feature.

2. Split the data by the feature that is the most significant splitter (e.g., highest IG).
3. Repeat until there are no features left.

Growing a Tree

We are going to split the data on training (first 15 patients) and test (16th patient) sets:

train_df <- patients_df[-16, ]
test_df <- patients_df[16, ]

temp_df <- train_df %>% 
  group_by(Drug) %>% 
  summarise(n = n(),
            p = round(n() / dim(train_df)[1], 2))

kable(temp_df, format = "markdown")

entropy(temp_df$p)
## [1] 0.9974016

train_df %>% 
  filter(BP == "normal") %>%
  select(-BP) %>% 
  kable(format = "markdown")

train_df %>% 
  filter(BP == "not normal") %>% 
  select(-BP) %>% 
  kable(format = "markdown")

train_df %>% 
  filter(BP == "not normal" & Gender == "M") %>% 
  select(-c(BP, Gender)) %>% 
  kable(format = "markdown")

train_df %>% 
  filter(BP == "not normal" & Gender == "F") %>% 
  select(-c(BP, Gender)) %>% 
  kable(format = "markdown") 

node0 <- "Blood Pressure"
node1 <- "Drug B (P=1.)"
node2 <- "Gender"
node3 <- "BMI"
node4 <- "BMI "
node5 <- "Drug A (P=0.66)"
node6 <- "Drug A (P=1.)"
node7 <- "Drug A (P=1.) "
node8 <- "Drug B (P=1.) "
nodeNames <- c(node0, node1, node2, node3, 
               node4, node5, node6, node7, node8)
        
# create new graph object
rEG <- new("graphNEL", nodes=nodeNames, edgemode="directed")
rEG <- addEdge(node0, node1, rEG)
rEG <- addEdge(node0, node2, rEG)
rEG <- addEdge(node2, node3, rEG)
rEG <- addEdge(node2, node4, rEG)
rEG <- addEdge(node3, node5, rEG)
rEG <- addEdge(node3, node6, rEG)
rEG <- addEdge(node4, node7, rEG)
rEG <- addEdge(node4, node8, rEG)

plot(rEG, attrs = at)
text(150, 300, "normal", col="black")
text(260, 300, "not normal", col="black")
text(200, 190, "male", col="black")
text(310, 190, "female", col="black")
text(100, 80, "normal", col="black")
text(230, 80, "not normal", col="black")
text(290, 80, "normal", col="black")
text(420, 80, "not normal", col="black")

test_df %>% 
  kable(format = "markdown") 

Let Computer Do the Math

model_dt <- rpart(
  formula = Drug ~ Gender + BMI + BP,
  data = train_df,
  method = "class",
  minsplit = 1,
  minbucket = 1,
  parms = list(
    split = "information" # for Gini impurity change to split = "gini"
  )
)

#plot decision tree model
fancyRpartPlot(model_dt, caption = NULL, type=4)

# return the class
predict(object = model_dt, newdata = test_df, type = "class")
##      1 
## Drug A 
## Levels: Drug A Drug B

# return the probability of each class
predict(object = model_dt, newdata = test_df, type = "prob")
##      Drug A    Drug B
## 1 0.8571429 0.1428571

Python

Unlike R, Python doesn’t allow categorical input variables, so we need to encode them:

patients_df = pd.read_csv("DecisionTree_SampleData.csv")
# drop id column
patients_df.drop('Patient ID', axis=1, inplace=True)

# convert to dummy variables
patients_df_labeled = pd.get_dummies(patients_df)
columns_to_drop = ['Gender_F', 'BMI_not normal', 'BP_not normal', 'Drug_???', 'Drug_Drug B']
patients_df_labeled.drop(
    columns_to_drop, 
    axis=1, 
    inplace=True)
unknown_drug = patients_df_labeled.iloc[15, :]
patients_df_labeled.drop(15, axis=0, inplace=True) # unknow drug row
patients_df_labeled.head().to_html(classes='table table-striped')

	Gender_M	BMI_normal	BP_normal	Drug_Drug A
0	1	0	0	1
1	1	0	1	0
2	1	0	0	1
3	1	1	0	1
4	1	0	0	1

For example Gender_M:1 means that subject is Male and Gender_M:0 means that subject is female.

To build a decision tree model we can use a DecisionTreeClassifier function from sklearn.tree module.

# split the data 
X = patients_df_labeled.drop('Drug_Drug A', axis=1)
y = patients_df_labeled['Drug_Drug A']

# fit the decision tree model
model = tree.DecisionTreeClassifier(
  criterion='entropy' # for Gini impurity change to criterion="gini"
)
model.fit(X, y)
## DecisionTreeClassifier(criterion='entropy')

# plot the tree
plt.figure(figsize=(10,10))
tree.plot_tree(
  model, filled=True, rounded=True,
  feature_names=X.columns.tolist(), proportion=False,
  class_names=['Drug B', 'Drug A']
)
plt.show()

Get a prediction for the 16th patient:

unknown_drug = pd.DataFrame(unknown_drug).T
unknown_drug.drop(['Drug_Drug A'], axis=1, inplace=True)

# return the class
model.predict(unknown_drug)
## array([1], dtype=uint8)

# return the probability of each class
model.predict_proba(unknown_drug)
## array([[0., 1.]])

Numerical Explanatory Variables

In our sample data set, we had only categorical variables (with 2 labels each). But what would the model do, if we had numerical value as an input?

Let’s look again on the Python decision tree and its decision node conditions, for example, Gender_M <= 0.5. Where does 0.5 come from? The answer overlaps with the answer on how the model handles numerical input values.

Recall the Gender_M column we created for the Python train set. It consisted of two numerical values (0 and 1). What the model does is filter the data by the average value among these two numbers. Average value was: \(\frac{0+1}{2}=0.5\). Now model has two sets of data: one set that matches the condition Gender_M <= 0.5 (Gender is female) and the other set that doesn’t (Gender is male).

Imagine that we have added a third category Non-binary to the Gender feature. First, we would encode the values and create a new column Gender_encoded, which would have three numerical values (0, 1 and 2). Say, Non-binary = 0, Male = 1, Female = 2. Now model would have two conditions for filtering the data:

• Gender_encoded <= 0.5 (Gender is non-binary)
• Gender_encoded <= 1.5 (Gender is non-binary OR male)

In a general way, we can say that if we have a numerical feature, the algorithm will convert it to “categorical” by choosing the average value as a threshold between actual input values:

• filter 1: feature <= threshold
• filter 2: feature > threshold

For example:

Threshold values are: 110 \(\left( \frac{100+120}{2} \right)\), 135 \(\left( \frac{120+150}{2} \right)\), 175 \(\left( \frac{150+200}{2} \right)\).

Other Splitting Metrics

Summary

We have looked at the example of the CART model for the binary classification problem. It is a relatively simple algorithm that doesn’t require much feature engineering. With the small data set and lots of free time, one can even build a model with “pen and paper”. Compared to other ML algorithms, CART can be considered as a “weak learner” due to the overfitting issue. However, they are often used as a base estimator in more complex bagging and boosting models (for example, Random Forest algorithm). Additionally, CART can help with the hidden data patterns, since they can be visualized.

References

• [1]: https://en.wikipedia.org/wiki/Supervised_learning
• [2]: https://en.wikipedia.org/wiki/Statistical_classification
• [3]: https://en.wikipedia.org/wiki/Regression_analysis
• [4]: https://en.wikipedia.org/wiki/Entropy_(information_theory)
• [5]: https://en.wikipedia.org/wiki/Information_theory