Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.
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Transcript of Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.
![Page 1: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/1.jpg)
Multivariate Statistical Analysis
93751009 呂冠宏93751503 林其緯
![Page 2: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/2.jpg)
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
![Page 3: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/3.jpg)
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
![Page 4: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/4.jpg)
How (univariate)
Power transformations (byTukey(1957), Box and Cox(1964))
x
xx
ln
1)(
0
0
![Page 5: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/5.jpg)
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
![Page 6: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/6.jpg)
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
![Page 7: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/7.jpg)
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
![Page 8: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/8.jpg)
Example
Original Q-Q plot Transformed Q-Q plot
![Page 9: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/9.jpg)
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
![Page 10: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/10.jpg)
Example
Original Q-Q plot Transformed Q-Q plot
![Page 11: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/11.jpg)
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
![Page 12: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/12.jpg)
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
![Page 13: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/13.jpg)
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
![Page 14: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/14.jpg)
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
![Page 15: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/15.jpg)
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
![Page 16: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/16.jpg)
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
![Page 17: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/17.jpg)
Example
Original chi-square plot Transformed chi-square plot
![Page 18: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/18.jpg)
Example
chi-square plot (assumption 1)
chi-square plot (assumption 2)
![Page 19: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/19.jpg)
Example
罐頭 chi-square plot 課本 chi-square plot
![Page 20: Multivariate Statistical Analysis 93751009 呂冠宏 93751503 林其緯.](https://reader037.fdocument.pub/reader037/viewer/2022110402/56649e435503460f94b363d9/html5/thumbnails/20.jpg)
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.