Numerical Problems- Classification Measures

Example:

Consider a Bayes classifier, which miss classifies 8 out of 30 test cases .Then find the accuracy and error rate.

🧠 Given

• Total test cases = 30
• Misclassified cases = 8

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🔹 Step 1: Correct Predictions

$$\text{Correct Predictions} = 30 - 8 = 22$$

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🔹 Step 2: Compute Accuracy

$\text{Accuracy}=\frac{22}{30}=0.733$

👉 Accuracy = 0.733 (or 73.3%)

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🔹 Step 3: Compute Error Rate

$\text{Error Rate}=\frac{8}{30}=0.267$

👉 Error Rate = 0.267 (or 26.7%)

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🎯 Final Answer

Measure	Value
Accuracy	0.733 (73.3%)
Error Rate	0.267 (26.7%)

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📌 Key Insight

$$\text{Error Rate} = 1 - \text{Accuracy}$$

Example:

Imagine an application that is developed to recognize cats and dogs. This identifies eight dogs in a picture containing 12 dogs and some cats. Of the eight dogs identified, five actually are dogs while the rest are cats. Compute the True Positive, False Positive and False Negative values.

📊 Confusion Matrix

🧠 Recall from the problem:

• TP = 5 (dogs correctly identified)
• FP = 3 (cats identified as dogs)
• FN = 7 (dogs missed)

We are not given total number of cats, so TN cannot be determined.

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📋 Matrix Representation

	Predicted Dog	Predicted Cat
Actual Dog	TP = 5	FN = 7
Actual Cat	FP = 3	TN = ?

Example:

For a classifier, the confusion matrix is given.What is the precision, recall and accuracy of that classifier?

📊 Performance Metrics from Confusion Matrix

Given:

• TP = 9, FP = 1, FN = 9, TN = 5
• Total samples $n = 24$
  

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🔹 1. Precision (Positive Class)

$\text{Precision}_P = \frac{TP}{TP + FP} = \frac{9}{9 + 1} = \frac{9}{10} = 0.9$

👉 Precision (Positive) = 0.9

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🔹 2. Precision (Negative Class)

$\text{Precision}_N = \frac{TN}{TN + FN} = \frac{5}{5 + 9} = \frac{5}{14} = 0.357$

👉 Precision (Negative) ≈ 0.357

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🔹 3. Recall (Positive Class / Sensitivity / TPR)

$\text{Recall}_P = \frac{TP}{TP + FN} = \frac{9}{9 + 9} = \frac{9}{18} = 0.5$

👉 Recall (Positive) = 0.5

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🔹 4. Recall (Negative Class / Specificity / TNR)

${\text{Recall}}_N=\frac{TN}{TN+FP}=\frac{5}{5+1}=\frac{5}{6}=0.83$

👉 Recall (Negative) ≈ 0.83

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🔹 5. Accuracy

$\text{Accuracy} = \frac{TP + TN}{n} = \frac{9 + 5}{24} = \frac{14}{24} = 0.583$

👉 Accuracy ≈ 0.583 (58.3%)

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📊 Final Summary

Metric	Value
Precision (Positive)	0.90
Precision (Negative)	0.357
Recall (Positive)	0.50
Recall (Negative)	0.83
Accuracy	0.583

Example

📊 Confusion Matrix Interpretation

	Actual Spam	Actual Not Spam
Predicted Spam	TP = 70	FP = 30
Predicted Not Spam	FN = 70	TN = 330

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🧠 1. Precision (for Spam)

$\text{Precision} = \frac{TP}{TP + FP} = \frac{70}{70 + 30} = \frac{70}{100} = 0.7$

👉 Precision = 0.7 (70%)

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🧠 2. Recall (for Spam)

$\text{Recall} = \frac{TP}{TP + FN} = \frac{70}{70 + 70} = \frac{70}{140} = 0.5$

👉 Recall = 0.5 (50%)