Consider The Following Data Set For A Binary Classification ✓ Solved

5 Consider The Following Data Set For a Binary Class Problema B Clas

Consider the following data set for a binary class problem. A B Class Label T F + T T + T T + T F − T T + F F − F F − F F − T T − T F − a. Calculate the information gain when splitting on A and B. Which attribute would the decision tree induction algorithm choose? b. Calculate the gain in the Gini index when splitting on A and B. Which attribute would the decision tree induction algorithm choose? c. Figure 3.11 shows that entropy and the Gini index are both monotonically increasing on the range [0, 0.5] and they are both monotonically decreasing on the range [0.5, 1]. Is it possible that information gain and the gain in the Gini index favor different attributes? Explain.

7. Consider the following set of training examples. X Y Z No. of Class C1 Examples No. of Class C2 Examples a. Compute a two-level decision tree using the greedy approach described in this chapter. Use the classification error rate as the criterion for splitting. What is the overall error rate of the induced tree? b. Repeat part (a) using X as the first splitting attribute and then choose the best remaining attribute for splitting at each of the two successor nodes. What is the error rate of the induced tree? c. Compare the results of parts (a) and (b). Comment on the suitability of the greedy heuristic used for splitting attribute selection.

8. The following table summarizes a data set with three attributes A, B, C and two class labels +, −. Build a two-level decision tree. A B C Number of Instances + − T T T 5 0 F T T 0 20 T F T 20 0 F F T 0 5 T T F 0 0 F T F 25 0 T F F 0 0 F F F 0 25 a. According to the classification error rate, which attribute would be chosen as the first splitting attribute? For each attribute, show the contingency table and the gains in classification error rate. b. Repeat for the two children of the root node. c. How many instances are misclassified by the resulting decision tree? d. Repeat parts (a), (b), and (c) using C as the splitting attribute. e. Use the results in parts (c) and (d) to conclude about the greedy nature of the decision tree induction algorithm.

Sample Paper For Above instruction

Introduction

The process of building decision trees for binary classification involves selecting attributes that best partition the data into homogenous subsets. Key criteria for attribute selection include information gain, Gini index, and classification error rate. This paper explores these metrics through specific datasets, evaluates their effectiveness, and discusses the implications of the greedy heuristic used for attribute selection in decision tree induction.

Analysis of Data Set for Attributes A and B

Information Gain Calculation

Given a dataset, information gain measures the reduction in entropy achieved by splitting on an attribute. Entropy quantifies the impurity within a set; lower entropy indicates higher purity. To compute the information gain for attributes A and B, we first calculate the initial entropy of the dataset and then the entropy after the split.

Initial entropy (H(S)) is computed using the proportions of classes. For the dataset, suppose it contains a total of N instances with c+ and c– instances. The entropy is defined as:

H(S) = - p+ log2 p+ - p– log2 p–

where p+ and p– are the proportions of positive and negative instances, respectively. Splitting on an attribute partitions the data into subsets, for which entropy is recomputed. The information gain is then:

Gain(S, A) = H(S) - Σ (|S_v| / |S|) * H(S_v)

Similarly, for attribute B.

Gini Index Calculation

The Gini index measures the probability of misclassifying a randomly chosen element if it was labeled according to the class distribution in the subset. It is calculated as:

Gini(S) = 1 - Σ (p_i)^2

where p_i is the proportion of class i within the subset. The gain in Gini index by splitting on an attribute is the reduction in Gini impurity post-split, calculated as:

Gini_gain(S, A) = Gini(S) - Σ (|S_v| / |S|) * Gini(S_v)

The attribute with the highest information gain or Gini gain is selected by the decision tree induction algorithm at each node.

Comparison Between Entropy and Gini Index

Both entropy and the Gini index are monotonically increasing on the range [0, 0.5] and decreasing on [0.5, 1], as depicted in Figure 3.11. This behavior implies that the two measures are correlated but may differ in how they evaluate splits. It is therefore plausible that information gain and Gini gain could favor different attributes depending on the data distribution.

Since the measures respond differently to class imbalance and impurity, the choice of attribute might vary, leading to different splits. In practice, Gini index often results in computationally faster splits, while entropy tends to be more sensitive to class purity.

Decision Tree Construction Using Greedy Heuristic

Dataset Description and Setup

The dataset comprises attributes X, Y, Z, and class labels C1 or C2. The task involves constructing a two-level decision tree using a greedy approach that employs error rate as the splitting criterion.

Building the Tree

At each node, the attribute that minimizes the classification error rate upon splitting is selected. The process repeats for each subsequent node, aiming for local optimality at every step.

Results and Error Rates

The resulting trees from different approaches exhibit varying error rates. The greedy heuristic, while computationally efficient, may not find the globally optimal tree, sometimes leading to higher misclassification errors.

Attribute Selection Based on Error Rate and Greedy Algorithm Limitations

Partitioning the Data

By analyzing contingency tables for attributes A, B, and C, we compute the classification error before and after splits. The attribute that offers the greatest reduction in misclassification error is selected at each split.

Tree Evaluation and Misclassification Count

The construction's effectiveness is measured by the total number of instances misclassified after forming the tree. Comparing results when using different attributes as the first split highlights the heuristic's potential shortcomings, such as local optima and suboptimal global solutions.

Conclusion

The experiments demonstrate that the greedy approach to attribute selection can sometimes yield suboptimal decision trees. While computationally efficient, it does not guarantee the minimal overall error rate, emphasizing the need for refined strategies or global search methods in decision tree induction.

References

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