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Discriminant Analysis – Concept, Methodology, and Applications
1. Introduction
📌 Discriminant Analysis (DA) is a classification technique used in statistics and machine learning.
📌 It helps differentiate between two or more groups based on predictor variables.
📌 Used when groups are known in advance (supervised learning).
📌 Common in economics, finance, marketing, medicine, and social sciences.
✔ Example: A bank wants to classify loan applicants into “high-risk” and “low-risk” categories based on income, credit score, and debt. Discriminant Analysis finds a mathematical rule to classify new applicants.
2. Concept of Discriminant Analysis
✔ Given a dataset with p predictor variables and known group labels, Discriminant Analysis finds a discriminant function that best separates the groups.
✔ It assumes each group follows a multivariate normal distribution with different means but equal covariances.
3. Types of Discriminant Analysis
🔹 (1) Linear Discriminant Analysis (LDA)
✔ Used when groups have similar variance-covariance structure.
✔ Finds a linear combination of variables that maximizes group separation.
✔ Example: Classifying students as “pass” or “fail” based on exam scores and study hours.
✔ Mathematical Representation: D=b1X1+b2X2+…+bpXp+CD = b_1 X_1 + b_2 X_2 + … + b_p X_p + C
where:
- DD = Discriminant score
- X1,X2,…,XpX_1, X_2, …, X_p = Predictor variables
- b1,b2,…,bpb_1, b_2, …, b_p = Coefficients (weights)
- CC = Constant
🔹 (2) Quadratic Discriminant Analysis (QDA)
✔ Used when covariance matrices differ across groups.
✔ More flexible than LDA but requires more data.
✔ Example: Medical diagnosis where different diseases have varying variability in symptoms.
🔹 (3) Fisher’s Discriminant Analysis
✔ Maximizes the difference between group means while minimizing variance within groups.
✔ Projects data onto a new axis where separation is maximum.
✔ Used in pattern recognition and image processing.
4. Steps in Discriminant Analysis
🔹 Step 1: Define the Groups
✔ Identify the dependent variable (category labels).
✔ Example: Classifying firms as “profitable” or “non-profitable”.
🔹 Step 2: Select Predictor Variables
✔ Choose independent variables that may influence classification.
✔ Example: Revenue, debt ratio, and market share for company profitability classification.
🔹 Step 3: Compute Discriminant Function
✔ Find coefficients b1,b2,…,bpb_1, b_2, …, b_p that maximize group separation.
✔ Use Fisher’s criterion: maxBetween-group varianceWithin-group variance\max \frac{\text{Between-group variance}}{\text{Within-group variance}}
✔ Solve using eigenvalues and eigenvectors of the covariance matrix.
🔹 Step 4: Classification Rule
✔ Calculate discriminant scores for new observations.
✔ Assign observations to the group with the highest score.
✔ Example Rule for Two Groups:
If D>0D > 0, assign to Group 1,
If D<0D < 0, assign to Group 2.
🔹 Step 5: Validate the Model
✔ Test accuracy using cross-validation.
✔ Use classification matrices to measure errors.
✔ Example Confusion Matrix Output:
| Actual \ Predicted | Group 1 | Group 2 |
|---|---|---|
| Group 1 | 90% | 10% |
| Group 2 | 12% | 88% |
✔ Accuracy = (90% + 88%) / Total = 89%.
5. Advantages of Discriminant Analysis
✅ Interpretable Model → Easy to understand decision-making process.
✅ Effective for Small Data → Works well with limited samples.
✅ Handles Multiple Groups → Can classify more than two categories.
✅ Robust to Non-Linearity → Works even if assumptions are slightly violated.
6. Limitations of Discriminant Analysis
❌ Assumes Normality → Requires data to be normally distributed.
❌ Sensitive to Outliers → Extreme values affect accuracy.
❌ Assumes Equal Covariance (LDA) → May fail if groups have different variances.
7. Applications of Discriminant Analysis
✔ Economics → Predicting business failures and economic downturns.
✔ Finance → Classifying credit risk (good vs. bad borrowers).
✔ Marketing → Customer segmentation based on purchasing behavior.
✔ Healthcare → Diagnosing diseases based on symptoms.
✔ Fraud Detection → Identifying fraudulent transactions.
8. Conclusion
✔ Discriminant Analysis is a powerful classification technique for supervised learning.
✔ LDA is used when groups have equal variance, while QDA works when variances differ.
✔ Widely applied in finance, economics, marketing, and medicine.