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Principal Component Analysis (PCA) – Concept, Methodology, and Applications
1. Introduction
📌 Principal Component Analysis (PCA) is a dimensionality reduction technique used in statistics and machine learning.
📌 It transforms a large set of correlated variables into a smaller set of uncorrelated variables called Principal Components (PCs).
📌 The main goal of PCA is to retain the most important information while reducing complexity.
✔ Example: In a dataset with 10 economic indicators, PCA can reduce them to 2 or 3 principal components, capturing most of the variation in the data.
2. Concept of PCA
✔ PCA converts original correlated variables into a new set of uncorrelated components ranked by variance.
✔ The first principal component (PC1) captures the maximum variance in the data.
✔ The second principal component (PC2) captures the next highest variance, and so on.
✔ Principal components are orthogonal (uncorrelated) to each other.
3. Mathematical Formulation of PCA
✔ Suppose we have a dataset with p variables X1,X2,…,XpX_1, X_2, …, X_p.
✔ We transform them into principal components PC1,PC2,…,PCpPC_1, PC_2, …, PC_p: PC1=a11X1+a12X2+…+a1pXpPC_1 = a_{11}X_1 + a_{12}X_2 + … + a_{1p}X_p PC2=a21X1+a22X2+…+a2pXpPC_2 = a_{21}X_1 + a_{22}X_2 + … + a_{2p}X_p
where:
- aija_{ij} are the eigenvectors (loadings) of the covariance matrix.
- Eigenvalues represent the variance explained by each principal component.
- The principal components are obtained by diagonalizing the covariance matrix of the original variables.
4. Steps in PCA Computation
🔹 Step 1: Standardization of Data
✔ Ensure all variables are on the same scale by converting them to z-scores: Z=X−μσZ = \frac{X – \mu}{\sigma}
where:
- XX = original data,
- μ\mu = mean,
- σ\sigma = standard deviation.
🔹 Step 2: Compute the Covariance Matrix
✔ The covariance matrix shows the relationship between different variables.
✔ It helps identify correlated features that PCA will reduce.
🔹 Step 3: Compute Eigenvalues and Eigenvectors
✔ Eigenvalues indicate the variance explained by each principal component.
✔ Eigenvectors define the direction of the new feature space.
✔ The principal components are ranked by decreasing eigenvalues.
✔ Example Eigenvalues Output:
| Principal Component | Eigenvalue | % Variance Explained |
|---|---|---|
| PC1 | 5.2 | 52% |
| PC2 | 2.3 | 23% |
| PC3 | 1.1 | 11% |
| PC4 | 0.9 | 9% |
| PC5 | 0.5 | 5% |
✔ Interpretation: PC1 alone explains 52% of the variance, so we might keep only PC1 and PC2 to simplify our data.
🔹 Step 4: Choose the Number of Principal Components
✔ Use the “Scree Plot”, which plots eigenvalues vs. components, to identify the optimal number of PCs.
✔ The elbow point (where eigenvalues drop sharply) is the best cut-off.
✔ Example: If the scree plot shows a sharp decline after PC2, we keep PC1 and PC2 and ignore the rest.
🔹 Step 5: Transform the Original Data
✔ Convert the dataset into a new reduced feature space using the selected principal components.
5. Advantages of PCA
✅ Reduces dimensionality → Helps simplify models and avoid overfitting.
✅ Removes multicollinearity → Useful in regression and econometric analysis.
✅ Improves computational efficiency → Faster processing for machine learning.
✅ Enhances data visualization → Reduces high-dimensional data to 2D or 3D plots.
6. Limitations of PCA
❌ Loss of interpretability → Original variables are transformed into abstract components.
❌ Assumes linear relationships → May not work well for non-linear data.
❌ Sensitive to scaling → Requires proper standardization of data.
7. Applications of PCA
✔ Economics → Reducing macroeconomic indicators into key growth factors.
✔ Finance → Identifying major risk factors in stock market returns.
✔ Marketing → Segmenting customers based on buying behavior.
✔ Healthcare → Detecting key symptoms in disease diagnosis.
✔ Machine Learning → Feature extraction for classification problems.
8. Conclusion
✔ PCA is a powerful tool for reducing complexity in high-dimensional data.
✔ It helps identify key patterns while preserving most of the variance.
✔ Widely used in economics, finance, marketing, healthcare, and AI.