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Multivariate Statistical Analysis

Multivariate Statistical Analysis. 93751009 呂冠宏 93751503 林其緯. Transformations To Near Normality. Why do we need to transform the data?? How do we transform the data?? (The univariate case ) Example How do we transform the data?? (The multivariate case ) Example.

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Multivariate Statistical Analysis

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  1. Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯

  2. Transformations To Near Normality • Why do we need to transform the data?? • How do we transform the data?? (The univariate case ) • Example • How do we transform the data?? (The multivariate case ) • Example

  3. Why do we need to transform the data?? For regression or analysis of variance

  4. How (univariate) • Power transformations (byTukey(1957), Box and Cox(1964))

  5. How (univariate) Given the observations Assumption: There exist a for which is for some and Then the log-likelihood function of the is :

  6. How (univariate) Then we have : Thus for fixed ,the maximized log-likelihood is, (expect for a constant)

  7. Example In Example 4.10 (closed door) We perform a power transformations of the data Then we must find the value of maximizing the function

  8. Example Original Q-Q plot Transformed Q-Q plot

  9. Example In Example 4.10 (open door) We perform a power transformations of the data Then we must find the value of maximizing the function

  10. Example Original Q-Q plot Transformed Q-Q plot

  11. How (multivariate) • Power transformations

  12. How (multivariate) Given the observations Assumption 1: There exist a for which is for some and Then the log-likelihood function of the is :

  13. How (multivariate) Then we have : Thus for fixed , the maximized log-likelihood is, (expect for a constant)

  14. How (multivariate) Assumption 2: There exist a for which is for some and Then the log-likelihood function of the is :

  15. How (multivariate) Then we have : Thus for fixed , the maximized log-likelihood is, (expect for a constant)

  16. Example In Example 4.10 (closed door and open door) We perform a power transformations of the data (by assumption 2) Then we must find the value of maximizing the function

  17. Example Original chi-square plot Transformed chi-square plot

  18. Example chi-square plot (assumption 1) chi-square plot (assumption 2)

  19. Example 罐頭chi-square plot 課本chi-square plot

  20. References • Box, G. E. P., and Cox, D. R. (1964) “An analysis of transformations.” Journal of the Royal Statistical Society, 26, 825-840. • Hernandez, F., and Johnson, R. A. (1980) “The large-sample behavior of transformations to normality.” Journal of the American Statistical Association, 75, 855-861. • Sanford, W. (2001) “Yeo-Johnson Power Transformations.” Supported by National Science Foundation Grant DUE 97-52887.

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