Multivariate Statistical Analysis 93751009 93751503 Transformations To Near

Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯

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

Why do we need to transform the data? ? For regression or analysis of variance Objective A convenient statistical model Constant variance Suitable for the graph

l Power transformations How (univariate) (by. Tukey(1957), Box and Cox(1964))

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

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

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

Example Original Q-Q plot Transformed Q-Q plot

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

Example Original Q-Q plot Transformed Q-Q plot

l Power transformations How (multivariate)

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

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

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

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

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

Example Original chi-square plot Transformed chi-square plot

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

Example 罐頭 chi-square plot 課本 chi-square plot

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