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Understanding Principal Components: Data Reduction and Interpretation

Learn how Principal Component Analysis explains variance in sets of variables through linear combinations, with applications in data reduction and interpretation. This comprehensive guide covers the mathematical proofs, examples, and geometric interpretations of principal components. Discover how to determine the number of principal components using scree plots and interpret the results for practical datasets such as stock returns and biological measurements. Ensure data quality by checking for linear dependencies and outliers.

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Understanding Principal Components: Data Reduction and Interpretation

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  1. Principal Components Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia

  2. Concept of Principal Components x2 x1

  3. Principal Component Analysis • Explain the variance-covariance structure of a set of variables through a few linear combinations of these variables • Objectives • Data reduction • Interpretation • Does not need normality assumption in general

  4. Principal Components

  5. Result 8.1

  6. Proof of Result 8.1

  7. Result 8.2

  8. Proof of Result 8.2

  9. Proportion of Total Variance due to the kth Principal Component

  10. Result 8.3

  11. Proof of Result 8.3

  12. Example 8.1

  13. Example 8.1

  14. Example 8.1

  15. Geometrical Interpretation

  16. Geometric Interpretation

  17. Standardized Variables

  18. Result 8.4

  19. Proportion of Total Variance due to the kth Principal Component

  20. Example 8.2

  21. Example 8.2

  22. Principal Components for Diagonal Covariance Matrix

  23. Principal Components for a Special Covariance Matrix

  24. Principal Components for a Special Covariance Matrix

  25. Sample Principal Components

  26. Sample Principal Components

  27. Example 8.3

  28. Example 8.3

  29. Scree Plot to Determine Number of Principal Components

  30. Example 8.4: Pained Turtles

  31. Example 8.4

  32. Example 8.4: Scree Plot

  33. Example 8.4: Principal Component • One dominant principal component • Explains 96% of the total variance • Interpretation

  34. Geometric Interpretation

  35. Standardized Variables

  36. Principal Components

  37. Proportion of Total Variance due to the kth Principal Component

  38. Example 8.5: Stocks Data • Weekly rates of return for five stocks • X1: Allied Chemical • X2: du Pont • X3: Union Carbide • X4: Exxon • X5: Texaco

  39. Example 8.5

  40. Example 8.5

  41. Example 8.6 • Body weight (in grams) for n=150 female mice were obtained after the birth of their first 4 litters

  42. Example 8.6

  43. Comment • An unusually small value for the last eigenvalue from either the sample covariance or correlation matrix can indicate an unnoticed linear dependency of the data set • One or more of the variables is redundant and should be deleted • Example: x4 = x1 + x2 + x3

  44. Check Normality and Suspect Observations • Construct scatter diagram for pairs of the first few principal components • Make Q-Q plots from the sample values generated by each principal component • Construct scatter diagram and Q-Q plots for the last few principal components

  45. Example 8.7: Turtle Data

  46. Example 8.7

  47. Large Sample Distribution for Eigenvalues and Eigenvectors

  48. Confidence Interval for li

  49. Approximate Distribution of Estimated Eigenvectors

  50. Example 8.8

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