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 )ExampleHow do we transform the data??

(The multivariate case )Example

Why do we need to transform the data??

Objective

A convenient statistical model

Constant variance Suitable for the graph

For regression or analysis of variance

How (univariate)

Power transformations (byTukey(1957), Box and Cox(1964))

x

xx

ln

1)(

0

0

How (univariate)

nxxx ,,, 21

)(ix

Given the observations

Then the log-likelihood function of the is :

Assumption:

There exist a for which is for some and),( 2N 2

nxxx 21,

n

i

Ji

xxL xnn

n

1

2)(2

2)|( log)(2

1log

2log

2log

21

1

1

2 )( and ),,( where

n

iixJ

How (univariate)

n

ii

n

ii x

nx

n 1

2)(2

1

)( )ˆ(1

ˆ and 1

ˆ

Jnl loglog

2)(

2ˆ

Then we have :

Thus for fixed ,the maximized log-likelihood is,

(expect for a constant)

n

i

J

xniy

1i

ˆy re whe log

2

2

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 )(l

Example

Original Q-Q plot Transformed Q-Q plot

ExampleIn Example 4.10 (open door)

We perform a power transformations of the data

Then we must find the value of maximizing the function )(l

Example

Original Q-Q plot Transformed Q-Q plot

How (multivariate)

Power transformations

p

ip

i

i

ip

i

i

i

p

p

x

x

x

x

x

x

x

1

1

1

2

2

1

1

)(

)(2

)(1

)(

2

1

2

1

),,,( 21 ipiii xxxx

),,,( 21 p

How (multivariate)

Given the observations

nxxx

,,, 21

Assumption 1:

There exist a for which is for some and )(ix

)I,( n

2N 2

Then the log-likelihood function of the is :nxxx

,,, 21

n

i

Jii

xxL xxnnp

n

1

)(')(2

2)|( log)()(2

1loglog

2log

21

n

i

p

jij

n

i i

i jxx

xJ

1

1

11

)(2 )(

and ),,( where

How (multivariate)

Then we have :

n

iii xxxxx

1

)()()()(2)( )()'(2n

1ˆ and ˆ

Thus for fixed , the maximized log-likelihood is,

(expect for a constant)

Jnl loglog)(2ˆ

n

i

p

jij

n

i i

i jxx

xJ

1

1

11

)(

)(

where

n

iij

jjp

jp xn

xxxxx1

)()()()(2

)(1

)( 1 and ),,,( where 21

How (multivariate)

Assumption 2:

There exist a for which is for some and )(ix

),( N

Then the log-likelihood function of the is :nxxx

,,, 21

n

i

Jii

xxL xxnnp

n

1

)(1')(2)|( log)()(2

1log

2log

2log 1

n

i

p

jij

n

i i

i jxx

xJ

1

1

11

)(2 )(

and ),,( where

How (multivariate)

Then we have :

Thus for fixed , the maximized log-likelihood is,

(expect for a constant)

Jnl loglog

2)(

ˆ

n

i

p

jij

n

i i

i jxx

xJ

1

1

11

)(

)(

where

'))((n

1ˆ and ˆ1

)()()()()(

n

iii xxxxx

n

iij

jjp

jp xn

xxxxx1

)()()()(2

)(1

)( 1 and ),,,( where 21

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

),( 21 )(l

12)(l

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. 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.