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Factor Analysis and Inference for Structured Covariance Matrices

Factor Analysis and Inference for Structured Covariance Matrices. Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia. History. Early 20 th -century attempt to define and measure intelligence

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Factor Analysis and Inference for Structured Covariance Matrices

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  1. Factor Analysis and Inference for Structured Covariance Matrices Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia

  2. History • Early 20th-century attempt to define and measure intelligence • Developed primarily by scientists interested in psychometrics • Advent of computers generated a renewed interest • Each application must be examined on its own merits

  3. Essence of Factor Analysis • Describe the covariance among many variables in terms of a few underlying, but unobservable, random factors. • A group of variables highly correlated among themselves, but having relatively small correlations with variables in different groups represent a single underlying factor

  4. Example 9.8Examination Scores

  5. Orthogonal Factor Model

  6. Orthogonal Factor Model

  7. Orthogonal Factor Model

  8. Orthogonal Factor Model

  9. Orthogonal Factor Model

  10. Example 9.1: Verification

  11. Example 9.2: No Solution

  12. Ambiguities of L When m>1

  13. Principal Component Solution

  14. Principal Component Solution

  15. Residual Matrix

  16. Determination of Number of Common Factors

  17. Example 9.3Consumer Preference Data

  18. Example 9.3Determination of m

  19. Example 9.3Principal Component Solution

  20. Example 9.3Factorization

  21. Example 9.4Stock Price Data • Weekly rates of return for five stocks • X1: Allied Chemical • X2: du Pont • X3: Union Carbide • X4: Exxon • X5: Texaco

  22. Example 9.4Stock Price Data

  23. Example 9.4Principal Component Solution

  24. Example 9.4Residual Matrix for m=2

  25. Maximum Likelihood Method

  26. Result 9.1

  27. Factorization of R

  28. Example 9.5: Factorization ofStock Price Data

  29. Example 9.5ML Residual Matrix

  30. Example 9.6Olympic Decathlon Data

  31. Example 9.6Factorization

  32. Example 9.6PC Residual Matrix

  33. Example 9.6ML Residual Matrix

  34. A Large Sample Test for Number of Common Factors

  35. A Large Sample Test for Number of Common Factors

  36. Example 9.7Stock Price Model Testing

  37. Example 9.8Examination Scores

  38. Example 9.8Maximum Likelihood Solution

  39. Example 9.8Factor Rotation

  40. Example 9.8Rotated Factor Loading

  41. Varimax Criterion

  42. Example 9.9: Consumer-Preference Factor Analysis

  43. Example 9.9Factor Rotation

  44. Example 9.10 Stock Price Factor Analysis

  45. Example 9.11Olympic Decathlon Factor Analysis

  46. Example 9.11Rotated ML Loadings

  47. Factor Scores

  48. Weighted Least Squares Method

  49. Factor Scores of Principal Component Method

  50. Orthogonal Factor Model

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