1 multivariate normal distribution shyh-kang jeng department of electrical engineering/ graduate...
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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
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
33
Univariate Normal DistributionUnivariate Normal Distribution
xexf
N
x 2/]/)[(
2
2
2
2
1)(
),(
44
Table 1, AppendixTable 1, Appendix
55
Square of DistanceSquare of Distance(Mahalanobis distance) (Mahalanobis distance)
)('
)())((
1
122
μxμx
xxx
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
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
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
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
1010
Example 4.1 Bivariate DistributionExample 4.1 Bivariate Distribution
11 = 22, 12 = 0
1111
Example 4.1 Bivariate DistributionExample 4.1 Bivariate Distribution
11 = 22, 12 = 0.75
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
1313
Result 4.1Result 4.1
definite positive
for ),(1/for ),(
1
definite positive :
1
1-
1-
Σ
ΣeΣe
eeΣeΣe
Σ
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
1515
Example 4.2 Positive CorrelationExample 4.2 Positive Correlation
1616
Probability Related to Probability Related to Squared DistanceSquared Distance
1y probabilit has
)()()'(
satisfying values of ellipsoid Solid21pμxΣμx
x
1717
Probability Related to Probability Related to Squared DistanceSquared Distance
1818
Result 4.2Result 4.2
),( bemust
every for )','(:'
)','(
:'
),(:
2211
ΣμX
aΣaaμaXa
Σaaμa
Xa
ΣμX
p
pp
p
N
N
N
XaXaXa
N
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
2020
Result 4.3Result 4.3
),(:
)',(:
),(:
11
2121
1111
ΣdμdX
AΣAAμAX
ΣμX
p
q
pqpq
pp
pp
p
N
N
XaXa
XaXa
XaXa
N
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
2222
Proof of Result 4.3: Part 2Proof of Result 4.3: Part 2
),(:
arbitrary is
)',''(:''
)','(:'
'')('
ΣdμdX
a
ΣaadaμadaXa
ΣaaμaXa
daXadXa
pN
N
N
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
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
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
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
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
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
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
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
3131
Example 4.7 Conditional BivariateExample 4.7 Conditional Bivariate
)),(()|(
thatshow
),(:
22
212
112222
12121
2212
1211
2
12
2
1
xNxxf
Nx
x
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
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
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
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
3636
22 Distribution Curves Distribution Curves
3737
Table 3, AppendixTable 3, Appendix
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
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
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
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
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
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
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
4545
Maximum-likelihood EstimationMaximum-likelihood Estimation
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
4747
Result 4.9Result 4.9
k
ii
k
kk
1
)tr( b)(
)'tr()'tr(' (a)
vector1 :
matrix symetric :
A
AxxAxxAxx
x
A
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
6868
Example 4.9Example 4.9
6969
Example 4.9Example 4.9
Histogram of MidTerm Scores Histogram of MidTerm Scores of Students of This Course in of Students of This Course in
2006 2006
7070
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
7272
Example 4.10 Radiation Data of Example 4.10 Radiation Data of Closed-Door Microwave OvenClosed-Door Microwave Oven
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
7474
Table 4.2 Table 4.2 Q-QQ-Q Plot Correlation Plot Correlation Coefficient TestCoefficient Test
7575
Example 4.11Example 4.11
hypothesisnormality reject not Do9351.0
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