Question

After a data set was partitioned, the first partition contains 43 cases that belong to Class 1 and 12 cases that belong to Class 0, and the second partition contains 24 cases that belong to Class 1 and 121 cases that belong to Class 0.

a. Compute and list the Gini impurity index for the root node. (Round your answer to 4 decimal places.)
b. Compute and list the Gini impurity index for partition 1. (Round your answer to 4 decimal places.)
c. Compute and list the Gini impurity index for partition 2. (Round your answer to 4 decimal places.)
d. Compute and list the Gini impurity index for the split. (Round your answer to 4 decimal places.)

Answer 1

To compute the Gini impurity index for a dataset partition, calculate the probability of each class and use a formula to find the impurity index.

Step-by-step explanation:

To compute the Gini impurity index for the root node, we need to calculate the probability of each class in the entire dataset.

1. Class 0 probability = total number of Class 0 cases / total number of cases
2. Class 1 probability = total number of Class 1 cases / total number of cases
3. Gini impurity index = 1 - (Class 0 probability)^2 - (Class 1 probability)^2

For partition 1 and partition 2, we calculate the Gini impurity index using the same formula, but considering only the cases in each partition.

To calculate the Gini impurity index for the split, we consider the weighted average of the Gini impurity index of each partition, where the weight is the proportion of cases in each partition.

Answer 2

The Gini impurity index is a metric for evaluating the purity of a dataset. For the presented case, the Gini impurity for the root node is 0.4445, for partition 1 it's 0.3412, for partition 2 it's 0.2746, and the Gini impurity for the split is 0.2894, all rounded to four decimal places.

Step-by-step explanation:

The Gini impurity index is a measure of how often a randomly chosen element from the set would be incorrectly labeled if it was randomly labeled according to the distribution of labels in the subset. To calculate the Gini impurity for a binary classification:

1. Compute the probabilities of each class in the partition.
2. Use the formula Gini impurity = 1 - (p² + q²), where p and q are the probabilities of the two classes.

For the root node, we consider all cases. There are 43 Class 1 + 12 Class 0 + 24 Class 1 + 121 Class 0 = 200 total cases. Class 1 = 43 + 24 = 67 cases, and Class 0 = 12 + 121 = 133 cases. The probabilities are:

• p(Class 1) = 67 / 200 = 0.335
• p(Class 0) = 133 / 200 = 0.665

Using the formula, we get the Gini impurity for the root node: 1 - (0.335² + 0.665²) = 0.4445, rounded to four decimal places.

For partition 1, we have 43 + 12 = 55 cases, with:

• p(Class 1) = 43 / 55 ≈ 0.7818
• p(Class 0) = 12 / 55 ≈ 0.2182

The Gini impurity for partition 1: 1 - (0.7818² + 0.2182²) = 0.3412, rounded to four decimal places.

For partition 2, we have 24 + 121 = 145 cases, with:

• p(Class 1) = 24 / 145 ≈ 0.1655
• p(Class 0) = 121 / 145 ≈ 0.8345

The Gini impurity for partition 2: 1 - (0.1655² + 0.8345²) = 0.2746, rounded to four decimal places.

To compute the Gini impurity for the split, we take a weighted sum of the Gini impurities of each partition:

Gini split = (55/200) * 0.3412 + (145/200) * 0.2746 = 0.2894, rounded to four decimal places.