1 multivariate normal distribution shyh-kang jeng department of electrical engineering/ graduate...
TRANSCRIPT
![Page 1: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/1.jpg)
11
Multivariate Normal DistributionMultivariate Normal Distribution
Shyh-Kang JengShyh-Kang JengDepartment of Electrical Engineering/Department of Electrical Engineering/
Graduate Institute of Communication/Graduate Institute of Communication/
Graduate Institute of Networking and Graduate Institute of Networking and MultimediaMultimedia
![Page 2: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/2.jpg)
22
Multivariate Normal DistributionMultivariate Normal DistributionGeneralized from univariate normal Generalized from univariate normal densitydensityBase of many multivariate analysis Base of many multivariate analysis techniquestechniquesUseful approximation to Useful approximation to ““truetrue”” population distributionpopulation distributionCentral limit distribution of many Central limit distribution of many multivariate statistics multivariate statistics Mathematical tractableMathematical tractable
![Page 3: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/3.jpg)
33
Univariate Normal DistributionUnivariate Normal Distribution
xexf
N
x 2/]/)[(
2
2
2
2
1)(
),(
![Page 4: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/4.jpg)
44
Table 1, AppendixTable 1, Appendix
![Page 5: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/5.jpg)
55
Square of DistanceSquare of Distance(Mahalanobis distance) (Mahalanobis distance)
)('
)())((
1
122
μxμx
xxx
![Page 6: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/6.jpg)
66
pp-dimensional Normal Density-dimensional Normal Density
],,,['
vector random from sample a is
,,2,1,
2
1)(
),(
21
2/)()'(2/12/
1
p
i
p
p
XXX
pix
ef
N
X
x
Σx
Σμ
μxΣμx
![Page 7: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/7.jpg)
77
Example 4.1 Bivariate NormalExample 4.1 Bivariate Normal
)1(
1,
),Corr(/
)Var(),Var(
)(),(
2122211
2122211
1112
1222
2122211
1
2221
1211
2122111212
222111
2211
ΣΣ
XX
XX
XEXE
![Page 8: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/8.jpg)
88
Example 4.1 Squared DistanceExample 4.1 Squared Distance
22
22
11
1112
2
22
22
2
11
11212
22
11
11221112
22111222
2122211
2211
1
21
1
)1(
1],[
)('
xxxx
x
x
xx
μxΣμx
![Page 9: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/9.jpg)
99
Example 4.1 Density FunctionExample 4.1 Density Function
]}2
[)1(2
1exp{
)1(2
1),(
22
22
11
1112
2
22
22
2
11
11212
2122211
21
xx
xx
xxf
![Page 10: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/10.jpg)
1010
Example 4.1 Bivariate DistributionExample 4.1 Bivariate Distribution
11 = 22, 12 = 0
![Page 11: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/11.jpg)
1111
Example 4.1 Bivariate DistributionExample 4.1 Bivariate Distribution
11 = 22, 12 = 0.75
![Page 12: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/12.jpg)
1212
ContoursContours
pi
c
c
contourdensityyprobabilitConstant
iii
i
,,2,1,
:axes
at centered ellipsoidan of surface
)-()'(such that all
i
21
eΣe
e
μ
μxΣμxx
![Page 13: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/13.jpg)
1313
Result 4.1Result 4.1
definite positive
for ),(1/for ),(
1
definite positive :
1
1-
1-
Σ
ΣeΣe
eeΣeΣe
Σ
![Page 14: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/14.jpg)
1414
Example 4.2 Bivariate ContourExample 4.2 Bivariate Contour
]2
1,
2
1[',
]2
1,
2
1[',
rseigenvecto and seigenvalue
normal, Bivariate
212112
112111
2211
e
e
![Page 15: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/15.jpg)
1515
Example 4.2 Positive CorrelationExample 4.2 Positive Correlation
![Page 16: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/16.jpg)
1616
Probability Related to Probability Related to Squared DistanceSquared Distance
1y probabilit has
)()()'(
satisfying values of ellipsoid Solid21pμxΣμx
x
![Page 17: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/17.jpg)
1717
Probability Related to Probability Related to Squared DistanceSquared Distance
![Page 18: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/18.jpg)
1818
Result 4.2Result 4.2
),( bemust
every for )','(:'
)','(
:'
),(:
2211
ΣμX
aΣaaμaXa
Σaaμa
Xa
ΣμX
p
pp
p
N
N
N
XaXaXa
N
![Page 19: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/19.jpg)
1919
Example 4.3 Marginal DistributionExample 4.3 Marginal Distribution
),(
:in ofon distributi Marginal
),()','(:'
','
'],0,,0,1['
),(:]',,,[
111
111
1
21
iii
i
pp
N
X
NN
X
NXXX
X
ΣaaμaXa
Σaaμa
Xaa
ΣμX
![Page 20: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/20.jpg)
2020
Result 4.3Result 4.3
),(:
)',(:
),(:
11
2121
1111
ΣdμdX
AΣAAμAX
ΣμX
p
q
pqpq
pp
pp
p
N
N
XaXa
XaXa
XaXa
N
![Page 21: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/21.jpg)
2121
Proof of Result 4.3: Part 1Proof of Result 4.3: Part 1
)',(:every for valid
))'('),('(:)('
))'()'(,)'((:)'(
'
,')(' ncombinatiolinear Any
AAΣAμAXb
bAAΣbAμbAXb
bAΣAbμAbXAb
bAa
XaAXb
qN
N
N
![Page 22: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/22.jpg)
2222
Proof of Result 4.3: Part 2Proof of Result 4.3: Part 2
),(:
arbitrary is
)',''(:''
)','(:'
'')('
ΣdμdX
a
ΣaadaμadaXa
ΣaaμaXa
daXadXa
pN
N
N
![Page 23: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/23.jpg)
2323
Example 4.4 Linear CombinationsExample 4.4 Linear Combinations
322211
2
33232213222312
13222312221211
32
21
3
2
1
32
21
3
, with verifiedbe can
)',(:
2
2'
110
011
),(:
XXYXXY
N
X
X
X
XX
XX
N
AAΣAμAX
AAΣ
Aμ
AX
ΣμX
![Page 24: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/24.jpg)
2424
Result 4.4Result 4.4
),(:
|
|
,,
),(:
1111
2221
1211
2
1
)1)((2
)1(1
ΣμX
ΣΣ
ΣΣ
Σ
μ
μ
μ
X
X
X
ΣμX
q
qp
q
p
N
N
4.3Result in |Set :Proof))(()()(
qpqqqpq0IA
![Page 25: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/25.jpg)
2525
Example 4.5 Subset DistributionExample 4.5 Subset Distribution
4424
2422
4
221
4424
242211
4
21
4
21
5
,:
,,
),(:
N
X
X
N
X
ΣμX
ΣμX
![Page 26: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/26.jpg)
2626
Result 4.5Result 4.5
22
11
2
1
2
1
2222211111
1221
2221
1211
2
1
2
1
)(21
)1(2
)1(1
|'
|
,:
tindependen ),(:),,(: (c)
0 ifonly and ift independen:,
|
|
,:--- (b)
),Cov( t,independen :, (a)
21
21
2121
Σ0
0Σ
μ
μ
X
X
ΣμXΣμX
ΣXX
ΣΣ
ΣΣ
μ
μ
X
X
0XXXX
qqqq
N
NN
N
![Page 27: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/27.jpg)
2727
Example 4.6 IndependenceExample 4.6 Independence
) also and oft independen is (
tindependen are and
tindependennot :,
200
031
014
),(:
213
32
11
21
3
XXX
XX
X
XX
N
X
Σ
ΣμX
![Page 28: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/28.jpg)
2828
Result 4.6Result 4.6
211
221211
221
2211
221
22
2221
1211
2
1
2
1
covariance
and )(mean with normal
is given ofon distributi lconditiona
0,
|
|
,),,(:
ΣΣΣΣ
μxΣΣμ
xXX
Σ
ΣΣ
ΣΣ
Σ
μ
μ
μΣμ
X
X
X
pN
![Page 29: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/29.jpg)
2929
Proof of Result 4.6Proof of Result 4.6
22
211
221211
22
221
221211
12212
|
'|
'
covariance with normaljoint
:
)(
)(
,
|0
|
Σ0
0ΣΣΣΣ
AAΣ
μX
μXΣΣμX
μXA
I
ΣΣI
A
![Page 30: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/30.jpg)
3030
Proof of Result 4.6Proof of Result 4.6
)),((
:given
),0(:)(
))()((
)
|)()((
)()(/),()|(t independen ,
tindependen are and )(
211
221211221
22121
221
211
221211221
221211
221
221211221
221211
22
221
221211221
221211
22221
221211
ΣΣΣΣμxΣΣμ
xXX
ΣΣΣΣμXΣΣμX
μxΣΣμxμXΣΣμX
xX
μxΣΣμxμXΣΣμX
μXμXΣΣμX
q
q
N
N
f
f
APBPBAPBAPBA
![Page 31: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/31.jpg)
3131
Example 4.7 Conditional BivariateExample 4.7 Conditional Bivariate
)),(()|(
thatshow
),(:
22
212
112222
12121
2212
1211
2
12
2
1
xNxxf
Nx
x
![Page 32: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/32.jpg)
3232
Example 4.1 Density FunctionExample 4.1 Density Function
]}2
[)1(2
1exp{
)1(2
1),(
22
22
11
1112
2
22
22
2
11
11212
2122211
21
xx
xx
xxf
![Page 33: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/33.jpg)
3333
Example 4.7Example 4.7
)1(2/))(/(
21211
22121
2221211
2122211
22
222
2
2222
12112
1211
22
22
11
1112
2
22
22
2
11
11212
21211
222221211
)1(2
1
)(/),()|(
2)1(2)1(2
)(
2
1)(
)1(2
1
]2[)1(2
1
xxe
xfxxfxxf
xxx
xxxx
![Page 34: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/34.jpg)
3434
Result 4.7Result 4.7
ondistributi theof percentile
)th (100upper thedenotes )( where
,1 is )}()()'(:{
ellipsoid solid theinsidey probabilit The (b)
: )()'( (a)
0),,(:
2p
2p
2p
1
2p
1
μXΣμXx
μXΣμX
ΣΣμX pN
![Page 35: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/35.jpg)
3535
22 Distribution Distribution
)2,12/ withondistributi(Gamma
0,0
0,)2/(2
1)(
(d.f.) freedom of degrees :,
)1,0(:);,(:
,),,(:),,(:
2
22/12/22/2
2
1
2
2
2222
2111
2
ef
x
NX
ZNX
NXNX
i i
ii
i
iii
![Page 36: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/36.jpg)
3636
22 Distribution Curves Distribution Curves
![Page 37: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/37.jpg)
3737
Table 3, AppendixTable 3, Appendix
![Page 38: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/38.jpg)
3838
Proof of Result 4.7 (a)Proof of Result 4.7 (a)
2
1
21
2
2
1
1
1
'
'
2'2
1'1
1
2
2
1
'
'
1
1
:)()'(),1,0(:
/
/
/
'
)',0(:)(,)(1
)()'(1
)()'(
p
p
iii
p
pp
iiii
pp
p
p
ii
p
ii
i
ii
p
i i
ZNZ
NZ
μXΣμX
Ieee
ee
e
e
e
AAΣ
AAΣμXAZμXe
μXeeμXμXΣμX
![Page 39: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/39.jpg)
3939
Proof of Result 4.7 (b)Proof of Result 4.7 (b)
1)()()'(
by ddistribute
variablerandom new )()'(
),(:by ellipsoid the toassigned
yprobabilit theis)()'(
21
2
1
21
p
p
p
P
N
cP
μXΣμX
μXΣμX
ΣμX
μXΣμX
![Page 40: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/40.jpg)
4040
Result 4.8Result 4.8
n
jj
n
jj
nn
n
j
n
jjjjpnn
jp
n
b
c
bbb
ccNccc
N
1
2
1
2
122112
1 1
222111
j
21
)()'(
)'()(
matrix covariancewith
normaljoint are and
)(,:
),(:
tindependenmutually :,,,
ΣΣcb
ΣcbΣ
VXXXV
ΣμXXXV
ΣμX
XXX
![Page 41: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/41.jpg)
4141
Proof of Result 4.8Proof of Result 4.8
ΣAAΣ
ΣΣAAΣ
AAΣAμV
VAX
III
IIIA
Σ00
0Σ0
00Σ
Σ
μ
μ
μ
μ
ΣμXXXX
X
X
X
X
X
)(:' rmsofdiagonalteoff
)(,)(:' of termsdiagonalblock
)',(:,
,
),(:],,,['
1
1
2
1
2
22
1
21
21
2
1
''2
'1
n
jjj
n
jj
n
jj
pn
n
n
npn
bc
bc
Nbbb
ccc
N
![Page 42: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/42.jpg)
4242
Example 4.8 Linear CombinationsExample 4.8 Linear Combinations
312123
22
21
321
1
34321
2223'
3'
)','(:'
201
011
113
,
1
1
3
),( identicalt independen:,,,
aaaaaaa
aaa
N
N
Σaa
μa
ΣaaμaXa
Σμ
ΣμXXXX
![Page 43: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/43.jpg)
4343
Example 4.8 Linear CombinationsExample 4.8 Linear Combinations
0ΣVVXXXXV
ΣΣΣ
μμμ
ΣμXXXXV
V
V
VV
4
12143212
4
1
2
4
1
343211
)(),Cov(,3
201
011
113
)(
2
2
6
2
),(:2
1
2
1
2
1
2
1
1
1
11
jjj
jj
jjj
bc
c
c
N
![Page 44: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/44.jpg)
4444
Multivariate Normal LikelihoodMultivariate Normal Likelihood
estimates likelihood Maximum
estimation likelihood Maximum
likelihood
,,, fixedfor and offunction a as
2
1
,,,
ofdensity Joint
),( from sample random:,,,
21
2/)()'(
2/2/21
21
1
1
n
nnpn
pn
n
jjj
e
N
xxxΣμ
ΣXXX
ΣμXXX
μxΣμx
![Page 45: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/45.jpg)
4545
Maximum-likelihood EstimationMaximum-likelihood Estimation
![Page 46: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/46.jpg)
4646
Trace of a MatrixTrace of a Matrix
k
i
k
jij
k
iiiij
kk
a
cc
caa
1 1
2
1
1)(
)'tr()e(
)tr()tr()d(
)tr()tr()c(
)tr()tr()(tr)b(
)tr()tr()a(
scalar a is ;)tr(
AA
AABB
BAAB
BABA
AA
AA
![Page 47: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/47.jpg)
4747
Result 4.9Result 4.9
k
ii
k
kk
1
)tr( b)(
)'tr()'tr(' (a)
vector1 :
matrix symetric :
A
AxxAxxAxx
x
A
![Page 48: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/48.jpg)
4848
Proof of Result 4.9 (a)Proof of Result 4.9 (a)
)'tr()')tr(()'tr(
)tr()tr(
)tr(
)tr()tr(
matrix :,matrix :
1 11 1
1 1
AxxxAxAxx
BCCB
BC
CBBC
CB
m
i
k
jjiij
k
j
m
iijji
m
i
k
jjiij
cbbc
cb
mkkm
![Page 49: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/49.jpg)
4949
Proof of Result 4.9 (b)Proof of Result 4.9 (b)
k
ii
kdiag
1
21
)tr()'tr(
)'tr()tr(
,,,
','
ΛΛPP
ΛPPA
Λ
IPPΛPPA
![Page 50: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/50.jpg)
5050
Likelihood FunctionLikelihood Function
2/''tr
2/2/
1
1
1
1
1
1
1
1
1
2
1),(
''
'
'
'tr'
n
jjj n
nnp
n
jjj
n
jjj
n
jjj
n
jjj
n
jjj
eL
n
μxμxxxxxΣ
ΣΣμ
μxμxxxxx
μxxxμxxx
μxμx
μxμxΣμxΣμx
![Page 51: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/51.jpg)
5151
Result 4.10Result 4.10
BΣ
Σ
BΣ
B
BΣ
b
ebe
b
pp
pp
bppb
bb
2/1for only holding
equality with , definite positive allfor
211
scalar positive :
matrix definite positive symmetric:
)(
2/)tr( 1
![Page 52: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/52.jpg)
5252
Proof of Result 4.10Proof of Result 4.10
IBΣBBΣ
BΣ
BBΣ
ΣBBΣBΣ
BΣB
BΣBBBΣBΣ
BΣ
BΣ
bb
ebebebe
eee
bppb
bbbbb
p
i
bipb
bp
ii
b
p
ii
p
i
i
i
p
ii
2such that )2/1( when attained is boundupper
211
2at 2 maximum a has
11
/,tr
positive all , of seigenvalue:
trtrtr
2/112/1
2/)tr(2/
1
2/2/
12/)tr(
11
1i
1
2/112/1
2/112/12/12/111
1
11
![Page 53: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/53.jpg)
5353
Result 4.11 Maximum Likelihood Result 4.11 Maximum Likelihood Estimators of Estimators of and and
SXXXXΣ
Xμ
ΣμXXX
n
n
n
N
j
n
jj
pn
1'
1ˆ
ˆ
),( from sample random:,,,
1
21
![Page 54: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/54.jpg)
5454
Proof of Result 4.11Proof of Result 4.11
SxxxxΣ
ΣΣμ
xμ
μxΣμxxxxxΣ
Σμ
xxxxΣ
n
n
n
eL
n
L
n
jjj
nnp
n
jjj
n
jjj
1'
1ˆ
2
1),ˆ(
ˆ
'2
1'tr
2
1
:),( ofExponent
1
'tr
2/2/
1
1
1
1
1
![Page 55: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/55.jpg)
5555
Invariance PropertyInvariance Property
)Var( of MLE1
ˆ
ˆ of MLE
ˆˆ'ˆ' of MLE
:Examples
)( ofestimator likelihood maximum:)ˆ(
ofestimator likelihood maximum:ˆ
1
2
11
i
n
jijiii
iiii
XXXn
hh
μΣμμΣμ
![Page 56: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/56.jpg)
5656
Sufficient StatisticsSufficient Statistics
population normal
temultivaria a of statistics sufficient are and
and through ,,,
nsobservatio ofset wholeon the depends
2
1
,,,
ofdensity Joint
21
2/''tr
2/2/
21
1
1
Sx
Sxxxx
Σ
XXX
μxμxxxxxΣ
n
n
nnp
n
n
jjj
e
![Page 57: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/57.jpg)
5757
Distribution of Sample MeanDistribution of Sample Mean
4.8Result cf.
)/,(:
:case teMultivaria
)/,(:
1 :case Univariate
),( from sample random:,,,
2
21
nN
nNX
p
N
p
pn
ΣμX
ΣμXXX
![Page 58: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/58.jpg)
5858
Sampling Distribution of Sampling Distribution of SS
)|)1((on distributiWishart :)1(
),(:
:case teMultivaria
),0(:,)1(
:)1(
1 :case Univariate
),( from sample random:,,,
11
'
2
1
222
21
2
1
22
21
ΣSZZS
Σ0XXZ
ΣμXXX
n-Wn-
N
NZZsn
XXsn
p
N
n
n
jjj
pjj
j
n
jj
n
n
jj
pn
![Page 59: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/59.jpg)
5959
Wishart DistributionWishart Distribution
)'|'(:')|(:
)|(:
)|(:),|(:
:Properties
definite positive:
21
2)|(
2121
2211
1
2/)1(4/)1(2/)1(
2/tr2/)2(
1
21
21
1
CCΣCACCACΣAA
ΣAAAA
ΣAAΣAA
A
Σ
AΣA
AΣ
mm
mm
mm
p
i
nppnp
pn
n
WW
W
WW
in
ew
![Page 60: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/60.jpg)
6060
Univariate Central Limit TheoremUnivariate Central Limit Theorem
size sample largefor normalnearly also is
on distributi normalnearly a has
ty variabilisame the
ely approximat having variablesrandom :
,,, causes
tindependen ofnumber large aby determined:
21
21
X
X
VVVX
V
VVV
X
n
i
n
![Page 61: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/61.jpg)
6161
Result 4.12 Law of Large NumbersResult 4.12 Law of Large Numbers
nε-μY-εP
n
YYYY
YE
YYY
n
i
n
as 1][
0, prescribedany for is,That
to
yprobabilitin converges
)( with normal) benot (may population
a from nsobservatiot independen:,.,
21
21
![Page 62: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/62.jpg)
6262
Result 4.12 Multivariate CasesResult 4.12 Multivariate Cases
ΣS
μX
μX
XXX
y toprobabilitin converges
y toprobabilitin converges
)(mean with
normal) temultivaria benot (may population
from nsobservatiot independen ,,,
i
21
E
n
![Page 63: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/63.jpg)
6363
Result 4.13 Central Limit TheoremResult 4.13 Central Limit Theorem
normal)nearly is populationparent the
when moderatefor ion approximat good (quite
size sample largefor
),(ely approximat is )(
covariance
finite and mean with population
a fromn observatiot independen:,,, 21
n
pn
Nn p
n
Σ0μX
Σ
μ
XXX
![Page 64: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/64.jpg)
6464
Limit Distribution of Limit Distribution of Statistical DistanceStatistical Distance
n-p
n
n
n-p
n
pnn
N
p
p
p
largefor
ely approximat:)()'(
large is
y whenprobabilithigh with toclose
largefor
ely approximat :)()'(
size sample largefor )1
,(nearly :
21
21
μXSμX
ΣS
μXΣμX
ΣμX
![Page 65: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/65.jpg)
6565
Evaluating Normality of Univariate Evaluating Normality of Univariate Marginal DistributionsMarginal Distributions
sticcharacterith for theon distributi
normal assumedan from departure indicate
628.0)046.0)(954.0(3954.0ˆor
396.1)317.0)(683.0(3683.0ˆeither
)22(in lying data ofportion :ˆ
)(in lying data ofportion :ˆ
data, ofsymmetry checkingAfter
2
1
2
1
i
nnp
nnp
sx,sxp
sx,sxp
i
i
iiiiiii
iiiiiii
![Page 66: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/66.jpg)
6666
Evaluating Normality of Univariate Evaluating Normality of Univariate Marginal DistributionsMarginal Distributions
0026.0]3ˆ[
),( is ˆ ofon distributi The
),( large, is When
on distributi binomial
:intervalan within samples ofNumber
n
pqppP
n
pqpN
n
yp
npqnpNqpy
nn
qpy
n
yny
yny
![Page 67: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/67.jpg)
6767
Q-QQ-Q Plot Plot
ondistributi normal a
from are data theif since linear,
ely approximat are they if see to,Plot
2/1
2
1][
/)2
1(/: ofPortion
20 e.g., large, tomoderate anddistict be Let
on nsobservatio:
)()(
)()(
2/)(
)(
)(
)()2()1(
2)(
jj
jj
zq
j
j
j
in
qx
xq
n
jdzeqZP
njnjxx
nnx
Xxxx
j
![Page 68: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/68.jpg)
6868
Example 4.9Example 4.9
![Page 69: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/69.jpg)
6969
Example 4.9Example 4.9
![Page 70: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/70.jpg)
Histogram of MidTerm Scores Histogram of MidTerm Scores of Students of This Course in of Students of This Course in
2006 2006
7070
![Page 71: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/71.jpg)
Q-Q Plot of MidTerm Scores of Q-Q Plot of MidTerm Scores of Students of This Course in 2006Students of This Course in 2006
7171n = 33, rQ = 0.946652
![Page 72: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/72.jpg)
7272
Example 4.10 Radiation Data of Example 4.10 Radiation Data of Closed-Door Microwave OvenClosed-Door Microwave Oven
![Page 73: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/73.jpg)
7373
Measurement of StraightnessMeasurement of Straightness
4.2 Tablein value
eappropriat thebelow falls if cesignifican
of levelat hypothesisnormality Reject the
)()(
))((
1
2)(
1
2)(
1)()(
Q
n
jj
n
jj
n
jjj
Q
r
qqxx
qqxx
r
![Page 74: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/74.jpg)
7474
Table 4.2 Table 4.2 Q-QQ-Q Plot Correlation Plot Correlation Coefficient TestCoefficient Test
![Page 75: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/75.jpg)
7575
Example 4.11Example 4.11
hypothesisnormality reject not Do9351.0
10.0,10
994.0,795.8
472.8,584.8
0,770.0 4.9, Example from dataFor
10
1
2)(
10
1
2)()(
10
1)(
Q
Qj
j
jjj
jj
r
n
rq
xxqxx
qx
![Page 76: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/76.jpg)
7676
Evaluating Bivariate NormalityEvaluating Bivariate Normality
)5.0()()'(such that all
bygiven ellipse in the lie
nsobservatio sample of 50%roughly ifCheck
22
1 xxSxxx
![Page 77: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/77.jpg)
7777
Example 4.12Example 4.12
![Page 78: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/78.jpg)
7878
Example 4.12Example 4.12
conclusion reach the tosmall toois )10( size sample However,
normality bivariatereject 50%an Greater th
39.1 with are nsobservatio 10 ofout Seven
39.134.4]4224,974.126[',
102927
309.62
128831.0003293.0
003293.0000184.0'
2927
309.62
39.1)5.0(
1030.1476.255
76.25520.005,10,
2927
309.62
2
221
5
2
1
2
12
22
5
n
d
dxx
x
x
x
xd
Sx
![Page 79: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/79.jpg)
7979
Chi-Square PlotChi-Square Plot
)/)2
1(1()/)
2
1(( that Note
1 slope havingorigin the
throughlinestraight a resemble shouldplot The
)),/)2
1((( allGraph
freedom of degrees on with distributi square-chi
theof quantile /)2
1(100:)/)
2
1((
distance squared Order the
distance squared :)()(
2,
2)(,
,
2)(
2)2(
2)1(
12
njnjq
dnjq
p
njnjq
ddd
xxSxxd
ppc
jpc
pc
n
![Page 80: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/80.jpg)
8080
Example 4.13 Chi-Square Plot Example 4.13 Chi-Square Plot for Example 4.12for Example 4.12
![Page 81: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/81.jpg)
8181
Example 4.13 Chi-Square Plot Example 4.13 Chi-Square Plot for Example 4.12for Example 4.12
![Page 82: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/82.jpg)
8282
Chi-Square Plot for Computer Chi-Square Plot for Computer Generated 4-variate Normal DataGenerated 4-variate Normal Data
![Page 83: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/83.jpg)
8383
Steps for Detecting OutliersSteps for Detecting OutliersMake a dot plot for each variableMake a dot plot for each variableMake a scatter plot for each pair of Make a scatter plot for each pair of variablesvariablesCalculate the standardized values. Calculate the standardized values. Examine them for large or small Examine them for large or small valuesvaluesCalculated the squared statistical Calculated the squared statistical distance. Examine for unusually distance. Examine for unusually large values. In chi-square plot, large values. In chi-square plot, these would be points farthest from these would be points farthest from the origin.the origin.
![Page 84: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/84.jpg)
8484
Helpful Transformation to Helpful Transformation to Near NormalityNear Normality
Original ScaleOriginal Scale Transformed ScaleTransformed Scale
Counts, Counts, yy
Proportions, Proportions,
Correlations, Correlations, rr
p
pp
ˆ1
ˆlog
2
1)ˆlogit(
y
r
rrz
1
1log
2
1)( sFisher'
p̂
![Page 85: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/85.jpg)
8585
Box and Cox’s Box and Cox’s Univariate TransformationsUnivariate Transformations
n
jj
n
jj
n
jj
xn
x
xxxn
n
λ
x
xx
1
)(_____
)(
11
2_____)()(
)(
1
ln11
ln2
)(
maximize to Choose
0,ln
0,1
![Page 86: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/86.jpg)
8686
Example 4.16 (Example 4.16 () vs. ) vs.
![Page 87: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/87.jpg)
8787
Example 4.16 Example 4.16 Q-QQ-Q Plot Plot
![Page 88: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/88.jpg)
8888
Transforming Multivariate Transforming Multivariate ObservationsObservations
p
jpjjj
n
jjkk
n
jjk
n
jkjk
k
p
pxxx
xn
x
xxxn
n
p
ˆ1
ˆ1
ˆ1
'
1
ln11
ln2
maximize toSelect
sticscharacteri the
for ssformationpower tran:,,,
)ˆ(
2
)ˆ(2
1
)ˆ(1)ˆ(
1
)(_____
)(
11
2_____)()(
k
21
21
λx
![Page 89: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/89.jpg)
8989
More Elaborate ApproachMore Elaborate Approach
p
jpjjj
p
k
n
jjkkp
p
p
pxxx
xn
p
111'
from computed is )(
ln1)(ln2
,,,
maximize to',,,Select
sticscharacteri the
for ssformationpower tran:,,,
)(
2
)(2
1
)(1)(
1 121
21
21
21
λx
λS
λS
λ
![Page 90: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/90.jpg)
9090
Example 4.17 Original Example 4.17 Original Q-QQ-Q Plot for Plot for Open-Door DataOpen-Door Data
![Page 91: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/91.jpg)
9191
Example 4.17 Example 4.17 Q-QQ-Q Plot of Plot of Transformed Open-Door DataTransformed Open-Door Data
![Page 92: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/92.jpg)
9292
Example 4.17 Contour Plot ofExample 4.17 Contour Plot of for Both Radiation Datafor Both Radiation Data),( 21
![Page 93: 1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking](https://reader036.vdocuments.mx/reader036/viewer/2022062421/56649d985503460f94a831f0/html5/thumbnails/93.jpg)
9393
Transform for Data Including Transform for Data Including Large Negative ValuesLarge Negative Values
2,01log
2,0)2/(11
0,0)1log(
0,0/11
2)(
xx
xx
xx
xx
x