han liu medicres world congress 2015
TRANSCRIPT
From High Dimensional Data to Big Data
Han Liu
Acknowledgement
2
Ethan Fang Princeton University
Fang Han JHU Biostatistics
Cun-hui Zhang Rutgers Statistics
ŚƩƉǁǁǁƉƌŝŶĐĞƚŽŶĞĚƵŚĂŶůŝƵ
Big Data Movement
3
^ƚĂƟƐƟĐĂů ĐŚĂůůĞŶŐĞƐ ŽĨ ŝŐ ĂƚĂ
Big Data Movement
3
DĂƐƐŝǀĞ н ,ŝŐŚ ŝŵĞŶƐŝŽŶĂů
^ƚĂƟƐƟĐĂů ĐŚĂůůĞŶŐĞƐ ŽĨ ŝŐ ĂƚĂ
Big Data Movement
3
DĂƐƐŝǀĞ н ,ŝŐŚ ŝŵĞŶƐŝŽŶĂů н ŽŵƉůĞdž н EŽŝƐLJ
^ƚĂƟƐƟĐĂů ĐŚĂůůĞŶŐĞƐ ŽĨ ŝŐ ĂƚĂ
Big Data Movement
3
DĂƐƐŝǀĞ н ,ŝŐŚ ŝŵĞŶƐŝŽŶĂů н ŽŵƉůĞdž н EŽŝƐLJ
ƵƌƌĞŶƚ ůŝƚĞƌĂƚƵƌĞ
^ƚĂƟƐƟĐĂů ĐŚĂůůĞŶŐĞƐ ŽĨ ŝŐ ĂƚĂ
Big Data Movement
3
DĂƐƐŝǀĞ н ,ŝŐŚ ŝŵĞŶƐŝŽŶĂů н ŽŵƉůĞdž н EŽŝƐLJ
dŚŝƐ ƚĂůŬƵƌƌĞŶƚ ůŝƚĞƌĂƚƵƌĞ
^ƚĂƟƐƟĐĂů ĐŚĂůůĞŶŐĞƐ ŽĨ ŝŐ ĂƚĂ
Big Data Movement
3
DĂƐƐŝǀĞ н ,ŝŐŚ ŝŵĞŶƐŝŽŶĂů н ŽŵƉůĞdž н EŽŝƐLJ
dŚŝƐ ƚĂůŬƵƌƌĞŶƚ ůŝƚĞƌĂƚƵƌĞ
^ƚĂƟƐƟĐĂů ĐŚĂůůĞŶŐĞƐ ŽĨ ŝŐ ĂƚĂ
ůŝƩůĞ ƐĞŵŝƉĂƌĂŵĞƚƌŝĐŝƚLJ ŐŽĞƐ Ă ůŽŶŐ ǁĂLJ
Outline
4
• ,ŝŐŚ ĚŝŵĞŶƐŝŽŶĂů ŵƵůƟǀĂƌŝĂƚĞ ƐƚĂƟƐƟĐƐ
• ƐƟŵĂƟŽŶ ŵĞƚŚŽĚ ĂŶĚ ƚŚĞŽƌLJ
• ^ĞŵŝƉĂƌĂŵĞƚƌŝĐ dƌĂŶƐĞůůŝƉƟĐĂů ŵŽĚĞůŝŶŐ
High Dimensional Multivariate Analysis
5
DĞƚŚŽĚ ƐƟŵĂƚĞ ƐŽŵĞ ĐŽǀĂƌŝĂŶĐĞ ĨƵŶĐƟŽŶĂů
High Dimensional Multivariate Analysis
5
DĞƚŚŽĚ ƐƟŵĂƚĞ ƐŽŵĞ ĐŽǀĂƌŝĂŶĐĞ ĨƵŶĐƟŽŶĂů
W^s > Y'ĂƵƐƐŝĂŶ ŐƌĂƉŚŝĐĂů ŵŽĚĞů'ĂƵƐƐŝĂŶ ƌĂŶĚŽŵ ĚĞƐŝŐŶ ƌĞŐƌĞƐƐŝŽŶWƌŝŶĐŝƉĂů ĐŽŵƉŽŶĞŶƚ ƌĞŐƌĞƐƐŝŽŶ · · ·
High Dimensional Multivariate Analysis
5
DĞƚŚŽĚ ƐƟŵĂƚĞ ƐŽŵĞ ĐŽǀĂƌŝĂŶĐĞ ĨƵŶĐƟŽŶĂů
dŚĞŽƌLJ :ƵƐƟĮĞĚ ƵŶĚĞƌ 'ĂƵƐƐŝĂŶ ŵŽĚĞů
W^s > Y'ĂƵƐƐŝĂŶ ŐƌĂƉŚŝĐĂů ŵŽĚĞů'ĂƵƐƐŝĂŶ ƌĂŶĚŽŵ ĚĞƐŝŐŶ ƌĞŐƌĞƐƐŝŽŶWƌŝŶĐŝƉĂů ĐŽŵƉŽŶĞŶƚ ƌĞŐƌĞƐƐŝŽŶ · · ·
High Dimensional Multivariate Analysis
5
ůŐŽƌŝƚŚŵµ,
^ĂŵƉůĞ ŵĞĂŶ ĐŽǀĂƌŝĂŶĐĞ
DĞƚŚŽĚ ƐƟŵĂƚĞ ƐŽŵĞ ĐŽǀĂƌŝĂŶĐĞ ĨƵŶĐƟŽŶĂů
dŚĞŽƌLJ :ƵƐƟĮĞĚ ƵŶĚĞƌ 'ĂƵƐƐŝĂŶ ŵŽĚĞů
W^s > Y'ĂƵƐƐŝĂŶ ŐƌĂƉŚŝĐĂů ŵŽĚĞů'ĂƵƐƐŝĂŶ ƌĂŶĚŽŵ ĚĞƐŝŐŶ ƌĞŐƌĞƐƐŝŽŶWƌŝŶĐŝƉĂů ĐŽŵƉŽŶĞŶƚ ƌĞŐƌĞƐƐŝŽŶ · · ·
High Dimensional Multivariate Analysis
5
^ƉĂƌƐŝƚLJ ƌĞŐƵůĂƌŝnjĂƟŽŶ
ůŐŽƌŝƚŚŵµ,
^ĂŵƉůĞ ŵĞĂŶ ĐŽǀĂƌŝĂŶĐĞ
DĞƚŚŽĚ ƐƟŵĂƚĞ ƐŽŵĞ ĐŽǀĂƌŝĂŶĐĞ ĨƵŶĐƟŽŶĂů
dŚĞŽƌLJ :ƵƐƟĮĞĚ ƵŶĚĞƌ 'ĂƵƐƐŝĂŶ ŵŽĚĞů
W^s > Y'ĂƵƐƐŝĂŶ ŐƌĂƉŚŝĐĂů ŵŽĚĞů'ĂƵƐƐŝĂŶ ƌĂŶĚŽŵ ĚĞƐŝŐŶ ƌĞŐƌĞƐƐŝŽŶWƌŝŶĐŝƉĂů ĐŽŵƉŽŶĞŶƚ ƌĞŐƌĞƐƐŝŽŶ · · ·
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d
' = (s, )
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d
' = (s, )
yũ yŬ
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d yũ yŬ | ƌĞƐƚ
' = (s, )
yũ yŬ
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d (ϭ)ũŬ = Ϭyũ yŬ | ƌĞƐƚ
' = (s, )
yũ yŬ
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d (ϭ)ũŬ = Ϭyũ yŬ | ƌĞƐƚ
' = (s, )
min ƚƌ
log || +
ũ,Ŭ |ũŬ|
'ůĂƐƐŽ
yũ yŬ
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d (ϭ)ũŬ = Ϭyũ yŬ | ƌĞƐƚ
' = (s, )
^ĂŵƉůĞ ĐŽǀĂƌŝĂŶĐĞ
min ƚƌ
log || +
ũ,Ŭ |ũŬ|
'ůĂƐƐŽ
yũ yŬ
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d (ϭ)ũŬ = Ϭyũ yŬ | ƌĞƐƚ
' = (s, )
EĞŐĂƟǀĞ 'ĂƵƐƐŝĂŶ ůŽŐͲůŝŬĞůŝŚŽŽĚ
^ĂŵƉůĞ ĐŽǀĂƌŝĂŶĐĞ
min ƚƌ
log || +
ũ,Ŭ |ũŬ|
'ůĂƐƐŽ
yũ yŬ
Gaussian Graphical Model
6
y E(µ,)
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d (ϭ)ũŬ = Ϭyũ yŬ | ƌĞƐƚ
' = (s, )
EĞŐĂƟǀĞ 'ĂƵƐƐŝĂŶ ůŽŐͲůŝŬĞůŝŚŽŽĚ
ϭͲƌĞŐƵůĂƌŝnjĂƟŽŶ
^ĂŵƉůĞ ĐŽǀĂƌŝĂŶĐĞ
min ƚƌ
log || +
ũ,Ŭ |ũŬ|
'ůĂƐƐŽ
yũ yŬ
Sparse Principal Component Analysis
7
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d
Sparse Principal Component Analysis
7
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d
Ͳ >ĞĂĚŝŶŐ ĞŝŐĞŶǀĞĐƚŽƌ ŽĨ
Sparse Principal Component Analysis
7
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d
Ͳ >ĞĂĚŝŶŐ ĞŝŐĞŶǀĞĐƚŽƌ ŽĨ
maxxϮ=ϭ xd x
ƐƵďũĞĐƚ ƚŽ ĂƌĚ(x) k
^ƉĂƌƐĞ W
Sparse Principal Component Analysis
7
Ŷ ƉŽŝŶƚƐ ŽĨ y = (yϭ, . . . , yĚ)d
Ͳ >ĞĂĚŝŶŐ ĞŝŐĞŶǀĞĐƚŽƌ ŽĨ
^W ƌĞůĂdžĂƟŽŶ Žƌ ƚƌƵŶĐĂƚĞĚ ƉŽǁĞƌ ŵĞƚŚŽĚ
maxxϮ=ϭ xd x
ƐƵďũĞĐƚ ƚŽ ĂƌĚ(x) k
^ƉĂƌƐĞ W
High Dimensional Theory
8
dŚĞŽƌLJ ŝƐ ǁĞůů ƵŶĚĞƌƐƚŽŽĚ ƵŶĚĞƌ 'ĂƵƐƐŝĂŶ ŵŽĚĞů
High Dimensional Theory
8
dŚĞŽƌLJ ŝƐ ǁĞůů ƵŶĚĞƌƐƚŽŽĚ ƵŶĚĞƌ 'ĂƵƐƐŝĂŶ ŵŽĚĞů
= ŽW(ϭ)WĂƌĂŵĞƚĞƌ ĞƐƟŵĂƟŽŶ ĐŽŶƐŝƐƚĞŶĐLJ
High Dimensional Theory
8
ƐƟŵĂƚŽƌ dƌƵƚŚ
dŚĞŽƌLJ ŝƐ ǁĞůů ƵŶĚĞƌƐƚŽŽĚ ƵŶĚĞƌ 'ĂƵƐƐŝĂŶ ŵŽĚĞů
= ŽW(ϭ)WĂƌĂŵĞƚĞƌ ĞƐƟŵĂƟŽŶ ĐŽŶƐŝƐƚĞŶĐLJ
High Dimensional Theory
8
ƐƟŵĂƚŽƌ dƌƵƚŚ
dŚĞŽƌLJ ŝƐ ǁĞůů ƵŶĚĞƌƐƚŽŽĚ ƵŶĚĞƌ 'ĂƵƐƐŝĂŶ ŵŽĚĞů
= ŽW(ϭ)WĂƌĂŵĞƚĞƌ ĞƐƟŵĂƟŽŶ ĐŽŶƐŝƐƚĞŶĐLJ
PƐƚƌƵĐƚƵƌĞ() = ƐƚƌƵĐƚƵƌĞ()
ϭ Ž(ϭ)
^ƚƌƵĐƚƵƌĞ ƌĞĐŽǀĞƌLJ ĐŽŶƐŝƐƚĞŶĐLJ
High Dimensional Theory
8
ƐƟŵĂƚŽƌ dƌƵƚŚ
dŚĞŽƌLJ ŝƐ ǁĞůů ƵŶĚĞƌƐƚŽŽĚ ƵŶĚĞƌ 'ĂƵƐƐŝĂŶ ŵŽĚĞů
= ŽW(ϭ)WĂƌĂŵĞƚĞƌ ĞƐƟŵĂƟŽŶ ĐŽŶƐŝƐƚĞŶĐLJ
PƐƚƌƵĐƚƵƌĞ() = ƐƚƌƵĐƚƵƌĞ()
ϭ Ž(ϭ)
^ƚƌƵĐƚƵƌĞ ƌĞĐŽǀĞƌLJ ĐŽŶƐŝƐƚĞŶĐLJ
EƵŵĞƌŝĐĂů ƌĂƚĞƐ ŽĨ ĐŽŶǀĞƌŐĞŶĐĞ ĨŽƌ ĂƩĂŝŶŝŶŐ
ŽŵƉƵƚĂƟŽŶĂů ĐŽŵƉůĞdžŝƚLJ
Theoretical Foundations
9
R4
R3
R2
R5
R1 µ µmax = KW
log ĚŶ
max = KW
log ĚŶ
Ϯ = KW
ĚŶ
supvϬƐ
|vd( )v|vϮϮ
= KW
Ɛ log ĚŶ
infvϬƐ
|vd v|vϮϮ
> Ϭ ǁŝƚŚ ŚŝŐŚ ƉƌŽď
Real Data are non-Gaussian
10
>Ăďϭ >ĂďϮ >Ăďϯ >Ăď ŵ
ŝŽůŽŐŝĐĂů ĐŽŶƚĞdžƚƐ ! "## $##ƌĞ
ĂƐƚĐĂŶ
ĐĞƌ
ǁŝŶŐΖƐ
ƐĂƌĐŽŵ
Ă
^ƚĞŵ ĐĞůů
(> ϮK)
DzZϭ
'ĞŶĞƐ(> ϭϮK)
'ĞŶĞ ĞdžƉƌĞƐƐŝŽŶ ĚĂƚĂ,/WͲĐŚŝƉ ĚĂƚĂ
ZĞŐƵůĂƚ
ŽƌLJĨĂĐƚ
Žƌ
;> ϭKͿ
Real Data are non-Gaussian
10
>Ăďϭ >ĂďϮ >Ăďϯ >Ăď ŵ
ŝŽůŽŐŝĐĂů ĐŽŶƚĞdžƚƐ ! "## $##ƌĞ
ĂƐƚĐĂŶ
ĐĞƌ
ǁŝŶŐΖƐ
ƐĂƌĐŽŵ
Ă
^ƚĞŵ ĐĞůů
(> ϮK)
DzZϭ
'ĞŶĞƐ(> ϭϮK)
'ĞŶĞ ĞdžƉƌĞƐƐŝŽŶ ĚĂƚĂ,/WͲĐŚŝƉ ĚĂƚĂ
ZĞŐƵůĂƚ
ŽƌLJĨĂĐƚ
Žƌ
;> ϭKͿ
Real Data are non-Gaussian
10
ZĞůĂdž 'ĂƵƐƐŝĂŶ ĂƐƐƵŵƉƟŽŶ ǁŝƚŚŽƵƚ ůŽƐŝŶŐ ƐƚĂƟƐƟĐĂů ĂŶĚĐŽŵƉƵƚĂƟŽŶĂů ĞĸĐŝĞŶĐŝĞƐ
>Ăďϭ >ĂďϮ >Ăďϯ >Ăď ŵ
ŝŽůŽŐŝĐĂů ĐŽŶƚĞdžƚƐ ! "## $##ƌĞ
ĂƐƚĐĂŶ
ĐĞƌ
ǁŝŶŐΖƐ
ƐĂƌĐŽŵ
Ă
^ƚĞŵ ĐĞůů
(> ϮK)
DzZϭ
'ĞŶĞƐ(> ϭϮK)
'ĞŶĞ ĞdžƉƌĞƐƐŝŽŶ ĚĂƚĂ,/WͲĐŚŝƉ ĚĂƚĂ
ZĞŐƵůĂƚ
ŽƌLJĨĂĐƚ
Žƌ
;> ϭKͿ
Transelliptical Distribution
11
'ĂƵƐƐŝĂŶ ůůŝƉƟĐĂů ĐŽƉƵůĂ
Transelliptical Distribution
11
'ĂƵƐƐŝĂŶ ůůŝƉƟĐĂů ĐŽƉƵůĂ
ƌĂŶĚŽŵ ǀĞĐƚŽƌ y = (yϭ, . . . , yĚ)d ŝƐ ƚƌĂŶƐĞůůŝƉƟĐĂůĞĮŶŝƟŽŶ ;,ĂŶ ĂŶĚ >ŝƵ ϮϬϭϮͿ
y dĚ(µ,, ĨũĚũ=ϭ, Ő)
ŝĨ ƚŚĞƌĞ ĞdžŝƐƚ ƐƚƌŝĐƚůLJ ŝŶĐƌĞĂƐŝŶŐ ĨƵŶĐƐ Ĩϭ, . . . , ĨĚ ƐƵĐŚ ƚŚĂƚz =
Ĩϭ(yϭ), . . . , ĨĚ(yĚ)
ŚĂƐ ĂŶ ĞůůŝƉƟĐĂůůLJ ĚĞŶƐŝƚLJ
ϭ||ϭ/Ϯ Ő
ϭ
Ϯ (y µ)dϭ(y µ)
Transelliptical Distribution
11
'ĂƵƐƐŝĂŶ ůůŝƉƟĐĂů ĐŽƉƵůĂ
ƌďŝƚƌĂƌLJ ƵŶŝǀĂƌŝĂƚĞ ĨƵŶĐƟŽŶ
ƌĂŶĚŽŵ ǀĞĐƚŽƌ y = (yϭ, . . . , yĚ)d ŝƐ ƚƌĂŶƐĞůůŝƉƟĐĂůĞĮŶŝƟŽŶ ;,ĂŶ ĂŶĚ >ŝƵ ϮϬϭϮͿ
y dĚ(µ,, ĨũĚũ=ϭ, Ő)
ŝĨ ƚŚĞƌĞ ĞdžŝƐƚ ƐƚƌŝĐƚůLJ ŝŶĐƌĞĂƐŝŶŐ ĨƵŶĐƐ Ĩϭ, . . . , ĨĚ ƐƵĐŚ ƚŚĂƚz =
Ĩϭ(yϭ), . . . , ĨĚ(yĚ)
ŚĂƐ ĂŶ ĞůůŝƉƟĐĂůůLJ ĚĞŶƐŝƚLJ
ϭ||ϭ/Ϯ Ő
ϭ
Ϯ (y µ)dϭ(y µ)
Visualization
12
Special Cases
13
dƌĂŶƐĞůůŝƉƟĐĂůdĚ(µ,, ĨũĚ
ũ=ϭ, Ő)
Special Cases
13
dƌĂŶƐĞůůŝƉƟĐĂůdĚ(µ,, ĨũĚ
ũ=ϭ, Ő)
Ĩũ(ƚ) = ƚ, ũ
Special Cases
13
dƌĂŶƐĞůůŝƉƟĐĂůdĚ(µ,, ĨũĚ
ũ=ϭ, Ő)
ůůŝƉƟĐĂůĚ(µ,, Ő)
Ĩũ(ƚ) = ƚ, ũ
Special Cases
13
dƌĂŶƐĞůůŝƉƟĐĂůdĚ(µ,, ĨũĚ
ũ=ϭ, Ő)
ůůŝƉƟĐĂůĚ(µ,, Ő)
Ĩũ(ƚ) = ƚ, ũ
Ő(ƚ) = Ğƚ
Special Cases
13
dƌĂŶƐĞůůŝƉƟĐĂůdĚ(µ,, ĨũĚ
ũ=ϭ, Ő)
ůůŝƉƟĐĂůĚ(µ,, Ő)
EŽŶƉĂƌĂŶŽƌŵĂůEWEĚ(µ,, ĨũĚ
ũ=ϭ)
Ĩũ(ƚ) = ƚ, ũ
Ő(ƚ) = Ğƚ
Special Cases
13
'ĂƵƐƐŝĂŶ
dƌĂŶƐĞůůŝƉƟĐĂůdĚ(µ,, ĨũĚ
ũ=ϭ, Ő)
ůůŝƉƟĐĂůĚ(µ,, Ő)
EŽŶƉĂƌĂŶŽƌŵĂůEWEĚ(µ,, ĨũĚ
ũ=ϭ)
ƐƚƌŝĐƚ ĞdžƚĞŶƐŝŽŶ ŽĨ 'ĂƵƐƐŝĂŶdŚĞŽƌĞŵ ϭ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ
Ĩũ(ƚ) = ƚ, ũ
Ő(ƚ) = Ğƚ
14
Identifiability Conditions
Ĩϭ(yϭ), . . . , ĨĚ(yĚ)
Ě(µ,, Ő)
14
Identifiability Conditions
Ĩϭ(yϭ), . . . , ĨĚ(yĚ)
Ě(µ,, Ő)
EŽƚ ŝĚĞŶƟĮĂďůĞ
14
Identifiability Conditions
Ĩϭ(yϭ), . . . , ĨĚ(yĚ)
Ě(µ,, Ő)
EŽƚ ŝĚĞŶƟĮĂďůĞ
KƉƟŽŶ ϭ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿµ = Ϭ ĚŝĂŐ() = IĚ
14
Identifiability Conditions
Ĩϭ(yϭ), . . . , ĨĚ(yĚ)
Ě(µ,, Ő)
EŽƚ ŝĚĞŶƟĮĂďůĞ
KƉƟŽŶ ϭ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿµ = Ϭ ĚŝĂŐ() = IĚ
KƉƟŽŶ Ϯ ;>ŝƵ >ĂīĞƌƚLJ tĂƐƐĞƌŵĂŶ ϮϬϬϵͿ
Eyũ = EĨũ(yũ) = µũ Var(yũ) = Var[Ĩũ(yũ)] = ũũ
14
Identifiability Conditions
Ĩϭ(yϭ), . . . , ĨĚ(yĚ)
Ě(µ,, Ő)
EŽƚ ŝĚĞŶƟĮĂďůĞ
KƉƟŽŶ ϭ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿµ = Ϭ ĚŝĂŐ() = IĚ
ĚŽƉƚĞĚ ŝŶ ƚŚŝƐ ƚĂůŬ
KƉƟŽŶ Ϯ ;>ŝƵ >ĂīĞƌƚLJ tĂƐƐĞƌŵĂŶ ϮϬϬϵͿ
Eyũ = EĨũ(yũ) = µũ Var(yũ) = Var[Ĩũ(yũ)] = ũũ
15
Hierarchical Representation
dƌĂŶƐĞůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ ŚĂƐ Ă ůĂƚĞŶƚ 'ĂƵƐƐŝĂŶ ƌĞƉƌĞƐĞŶƚĂƟŽŶ
15
Hierarchical Representation
dƌĂŶƐĞůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ ŚĂƐ Ă ůĂƚĞŶƚ 'ĂƵƐƐŝĂŶ ƌĞƉƌĞƐĞŶƚĂƟŽŶ
(yϭ, . . . , yĚ) dĚ(µ,, ĨũĚũ=ϭ, Ő)
dŚĞŽƌĞŵ Ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ
15
Hierarchical Representation
dƌĂŶƐĞůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ ŚĂƐ Ă ůĂƚĞŶƚ 'ĂƵƐƐŝĂŶ ƌĞƉƌĞƐĞŶƚĂƟŽŶ
(ϭ, . . . , Ě) Ě(µ,, Ő)
yũ = Ĩϭũ (ũ)
(yϭ, . . . , yĚ) dĚ(µ,, ĨũĚũ=ϭ, Ő)
dŚĞŽƌĞŵ Ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ
15
Hierarchical Representation
dƌĂŶƐĞůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ ŚĂƐ Ă ůĂƚĞŶƚ 'ĂƵƐƐŝĂŶ ƌĞƉƌĞƐĞŶƚĂƟŽŶ
(ϭ, . . . , Ě) Ě(µ,, Ő)
yũ = Ĩϭũ (ũ)
(zϭ, . . . , zĚ) EĚ(µ,)
ũĚ= µũ + Ő(zũ µũ)
(yϭ, . . . , yĚ) dĚ(µ,, ĨũĚũ=ϭ, Ő)
dŚĞŽƌĞŵ Ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ
15
Hierarchical Representation
dƌĂŶƐĞůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ ŚĂƐ Ă ůĂƚĞŶƚ 'ĂƵƐƐŝĂŶ ƌĞƉƌĞƐĞŶƚĂƟŽŶ
(ϭ, . . . , Ě) Ě(µ,, Ő)
yũ = Ĩϭũ (ũ)
(zϭ, . . . , zĚ) EĚ(µ,)
ũĚ= µũ + Ő(zũ µũ)
ZĂŶĚŽŵ ǀĂƌŝĂďůĞ ĚĞƚĞƌŵŝŶĞĚ ďLJ Ő(·)
(yϭ, . . . , yĚ) dĚ(µ,, ĨũĚũ=ϭ, Ő)
dŚĞŽƌĞŵ Ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ
15
Hierarchical Representation
dƌĂŶƐĞůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ ŚĂƐ Ă ůĂƚĞŶƚ 'ĂƵƐƐŝĂŶ ƌĞƉƌĞƐĞŶƚĂƟŽŶ
(ϭ, . . . , Ě) Ě(µ,, Ő)
yũ = Ĩϭũ (ũ)
(zϭ, . . . , zĚ) EĚ(µ,)
ũĚ= µũ + Ő(zũ µũ)
ZĂŶĚŽŵ ǀĂƌŝĂďůĞ ĚĞƚĞƌŵŝŶĞĚ ďLJ Ő(·)
,ĞĂǀLJ ƚĂŝů н ƚĂŝů ĚĞƉĞŶĚĞŶĐĞ
(yϭ, . . . , yĚ) dĚ(µ,, ĨũĚũ=ϭ, Ő)
dŚĞŽƌĞŵ Ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ
15
Hierarchical Representation
dƌĂŶƐĞůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ ŚĂƐ Ă ůĂƚĞŶƚ 'ĂƵƐƐŝĂŶ ƌĞƉƌĞƐĞŶƚĂƟŽŶ
(ϭ, . . . , Ě) Ě(µ,, Ő)
yũ = Ĩϭũ (ũ)
(zϭ, . . . , zĚ) EĚ(µ,)
ũĚ= µũ + Ő(zũ µũ)
ZĂŶĚŽŵ ǀĂƌŝĂďůĞ ĚĞƚĞƌŵŝŶĞĚ ďLJ Ő(·)
,ĞĂǀLJ ƚĂŝů н ƚĂŝů ĚĞƉĞŶĚĞŶĐĞ
ƐLJŵŵĞƚƌŝĐŝƚLJ(yϭ, . . . , yĚ) dĚ(µ,, ĨũĚũ=ϭ, Ő)
dŚĞŽƌĞŵ Ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ
15
Hierarchical Representation
dƌĂŶƐĞůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ ŚĂƐ Ă ůĂƚĞŶƚ 'ĂƵƐƐŝĂŶ ƌĞƉƌĞƐĞŶƚĂƟŽŶ
(ϭ, . . . , Ě) Ě(µ,, Ő)
yũ = Ĩϭũ (ũ)
(zϭ, . . . , zĚ) EĚ(µ,)
ũĚ= µũ + Ő(zũ µũ)
ZĂŶĚŽŵ ǀĂƌŝĂďůĞ ĚĞƚĞƌŵŝŶĞĚ ďLJ Ő(·)
ZĞƵƐĞ ŝŶƚĞƌƉƌĞƚĂƟŽŶƐ ŽĨ 'ĂƵƐƐŝĂŶ ĐŽǀĂƌŝĂŶĐĞ ĨƵŶĐƟŽŶĂůƐ
,ĞĂǀLJ ƚĂŝů н ƚĂŝů ĚĞƉĞŶĚĞŶĐĞ
ƐLJŵŵĞƚƌŝĐŝƚLJ(yϭ, . . . , yĚ) dĚ(µ,, ĨũĚũ=ϭ, Ő)
dŚĞŽƌĞŵ Ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ
Transelliptical Graphical Model
16
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
'
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
(ϭ)ϯϰ = Ϭ'
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
(ϭ)ϯϰ = Ϭ'
zϭ
zϮ
zϯ
zϰ
zĚ
ůůŝƉƟĐĂů
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
(ϭ)ϯϰ = Ϭ'
ϭ
Ϯ
ϯ
ϰ
Ě
'ĂƵƐƐŝĂŶ
zϭ
zϮ
zϯ
zϰ
zĚ
ůůŝƉƟĐĂů
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
(ϭ)ϯϰ = Ϭ'
ϭ
Ϯ
ϯ
ϰ
Ě
'ĂƵƐƐŝĂŶ
zϭ
zϮ
zϯ
zϰ
zĚ
ůůŝƉƟĐĂů
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
(ϭ)ϯϰ = Ϭ'
ϭ
Ϯ
ϯ
ϰ
Ě
'ĂƵƐƐŝĂŶ
zϭ
zϮ
zϯ
zϰ
zĚ
ůůŝƉƟĐĂů
ϯ ĂŶĚ ϰ ĂƌĞ ĐŽŶĚŝŶĚĞƉĞŶĚĞŶƚ
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
(ϭ)ϯϰ = Ϭ'
ϭ
Ϯ
ϯ
ϰ
Ě
'ĂƵƐƐŝĂŶ
zϭ
zϮ
zϯ
zϰ
zĚ
ůůŝƉƟĐĂů
ϯ ĂŶĚ ϰ ĂƌĞ ĐŽŶĚŝŶĚĞƉĞŶĚĞŶƚ
zϯ ĂŶĚ zϰ ĂƌĞ ĐŽŶĚƵŶĐŽƌƌĞůĂƚĞĚ
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Transelliptical Graphical Model
16
yϭ
yϮ
yϯ
yϰ
yĚ
(ϭ)ϯϰ = Ϭ'
ϭ
Ϯ
ϯ
ϰ
Ě
'ĂƵƐƐŝĂŶ
zϭ
zϮ
zϯ
zϰ
zĚ
ůůŝƉƟĐĂů
ϯ ĂŶĚ ϰ ĂƌĞ ĐŽŶĚŝŶĚĞƉĞŶĚĞŶƚ
zϯ ĂŶĚ zϰ ĂƌĞ ĐŽŶĚƵŶĐŽƌƌĞůĂƚĞĚ
yϯ ĂŶĚ yϰ ĂƌĞ ĐŽŶĚƵŶĐŽƌƌĞůĂƚĞĚ ǁƌ ƚƌĂŶŬ ĐŽƌƌĞůĂƟŽŶ
dŚĞŽƌĞŵ ϯ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ y dĚ(µ,, ĨũĚũ=ϭ, Ő)
Semiparametric Inference
17
>Ğƚ xϭ, . . . ,xŶ ďĞ Ŷ ĚĂƚĂ ĨƌŽŵ dĚ(µ,, ĨũĚũ=ϭ, Ő)
Semiparametric Inference
17
>Ğƚ xϭ, . . . ,xŶ ďĞ Ŷ ĚĂƚĂ ĨƌŽŵ dĚ(µ,, ĨũĚũ=ϭ, Ő)
&ŝŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů /ŶĮŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů
Semiparametric Inference
17
>Ğƚ xϭ, . . . ,xŶ ďĞ Ŷ ĚĂƚĂ ĨƌŽŵ dĚ(µ,, ĨũĚũ=ϭ, Ő)
;DĂũŽƌ ŝŶƚĞƌĞƐƚͿ&ŝŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů /ŶĮŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů
Semiparametric Inference
17
>Ğƚ xϭ, . . . ,xŶ ďĞ Ŷ ĚĂƚĂ ĨƌŽŵ dĚ(µ,, ĨũĚũ=ϭ, Ő)
;EƵŝƐĂŶĐĞͿ;DĂũŽƌ ŝŶƚĞƌĞƐƚͿ&ŝŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů /ŶĮŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů
Semiparametric Inference
17
>Ğƚ xϭ, . . . ,xŶ ďĞ Ŷ ĚĂƚĂ ĨƌŽŵ dĚ(µ,, ĨũĚũ=ϭ, Ő)
;EƵŝƐĂŶĐĞͿ;DĂũŽƌ ŝŶƚĞƌĞƐƚͿ
ZĞƋƵŝƌĞŵĞŶƚ &ŝŶĚ µ, ƚŚĂƚ ĂƌĞ ŝŶǀĂƌŝĂŶƚ ƚŽ ĨũĚũ=ϭ ĂŶĚ Ő
&ŝŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů /ŶĮŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů
Semiparametric Inference
17
>Ğƚ xϭ, . . . ,xŶ ďĞ Ŷ ĚĂƚĂ ĨƌŽŵ dĚ(µ,, ĨũĚũ=ϭ, Ő)
;EƵŝƐĂŶĐĞͿ;DĂũŽƌ ŝŶƚĞƌĞƐƚͿ
ZĞƋƵŝƌĞŵĞŶƚ &ŝŶĚ µ, ƚŚĂƚ ĂƌĞ ŝŶǀĂƌŝĂŶƚ ƚŽ ĨũĚũ=ϭ ĂŶĚ Ő
&ŝŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů /ŶĮŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů
ůŐŽƌŝƚŚŵµ,
džŝƐƟŶŐ ŚŝŐŚ ĚŝŵĞŶƐŝŽŶĂů ŵĞƚŚŽĚƐ
Semiparametric Inference
17
>Ğƚ xϭ, . . . ,xŶ ďĞ Ŷ ĚĂƚĂ ĨƌŽŵ dĚ(µ,, ĨũĚũ=ϭ, Ő)
;EƵŝƐĂŶĐĞͿ;DĂũŽƌ ŝŶƚĞƌĞƐƚͿ
ZĞƋƵŝƌĞŵĞŶƚ &ŝŶĚ µ, ƚŚĂƚ ĂƌĞ ŝŶǀĂƌŝĂŶƚ ƚŽ ĨũĚũ=ϭ ĂŶĚ Ő
^ĂŵƉůĞ ŵĞĂŶ ĐŽǀĂƌŝĂŶĐĞ
&ŝŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů /ŶĮŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů
ůŐŽƌŝƚŚŵµ,
džŝƐƟŶŐ ŚŝŐŚ ĚŝŵĞŶƐŝŽŶĂů ŵĞƚŚŽĚƐ
Semiparametric Inference
17
>Ğƚ xϭ, . . . ,xŶ ďĞ Ŷ ĚĂƚĂ ĨƌŽŵ dĚ(µ,, ĨũĚũ=ϭ, Ő)
;EƵŝƐĂŶĐĞͿ;DĂũŽƌ ŝŶƚĞƌĞƐƚͿ
ZĞƋƵŝƌĞŵĞŶƚ &ŝŶĚ µ, ƚŚĂƚ ĂƌĞ ŝŶǀĂƌŝĂŶƚ ƚŽ ĨũĚũ=ϭ ĂŶĚ Ő
^ĂŵƉůĞ ŵĞĂŶ ĐŽǀĂƌŝĂŶĐĞ
&ŝŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů /ŶĮŶŝƚĞ ĚŝŵĞŶƐŝŽŶĂů
ůŐŽƌŝƚŚŵµ,
džŝƐƟŶŐ ŚŝŐŚ ĚŝŵĞŶƐŝŽŶĂů ŵĞƚŚŽĚƐ
Technical Requirements
18
R4
R3
R2
R5
R1 µ µmax = KW
log ĚŶ
max = KW
log ĚŶ
Ϯ = KW
ĚŶ
ZĞƋƵŝƌĞŵĞŶƚ µ, ĐŽŶǀĞƌŐĞƐ ƚŽ µ, ǁŝƚŚ Ă ĨĂƐƚ ƌĂƚĞ
supvϬƐ
|vd( )v|vϮϮ
= KW
Ɛ log ĚŶ
infvϬƐ
|vd v|vϮϮ
> Ϭ ǁŝƚŚ ŚŝŐŚ ƉƌŽď
Estimating Mean
19
ĂŶƚŽŶŝΖƐ DͲƐƟŵĂƚŽƌ ;ĂŶƚŽŶŝ ϮϬϭϭͿ
Estimating Mean
19
ĂŶƚŽŶŝΖƐ DͲƐƟŵĂƚŽƌ ;ĂŶƚŽŶŝ ϮϬϭϭͿ
µũ ŝƐ ƚŚĞ ƐŽůƵƟŽŶ ƚŽŶ
ŝ=ϭ · (xŝũ µũ)
= Ϭ
Estimating Mean
19
ĂŶƚŽŶŝΖƐ DͲƐƟŵĂƚŽƌ ;ĂŶƚŽŶŝ ϮϬϭϭͿ
dƵŶŝŶŐ ƉĂƌĂŵĞƚĞƌ
µũ ŝƐ ƚŚĞ ƐŽůƵƟŽŶ ƚŽŶ
ŝ=ϭ · (xŝũ µũ)
= Ϭ
Estimating Mean
19
ĂŶƚŽŶŝΖƐ DͲƐƟŵĂƚŽƌ ;ĂŶƚŽŶŝ ϮϬϭϭͿ
dƵŶŝŶŐ ƉĂƌĂŵĞƚĞƌ
µũ ŝƐ ƚŚĞ ƐŽůƵƟŽŶ ƚŽŶ
ŝ=ϭ · (xŝũ µũ)
= Ϭ
Ϭ ϱϱ
Estimating Mean
19
ĂŶƚŽŶŝΖƐ DͲƐƟŵĂƚŽƌ ;ĂŶƚŽŶŝ ϮϬϭϭͿ
dƵŶŝŶŐ ƉĂƌĂŵĞƚĞƌ
µũ ŝƐ ƚŚĞ ƐŽůƵƟŽŶ ƚŽŶ
ŝ=ϭ · (xŝũ µũ)
= Ϭ
Ϭ ϱϱ
(ƚ) = logϭ + ƚ + ƚϮ/Ϯ
ƚ
Estimating Mean
19
ĂŶƚŽŶŝΖƐ DͲƐƟŵĂƚŽƌ ;ĂŶƚŽŶŝ ϮϬϭϭͿ
dƵŶŝŶŐ ƉĂƌĂŵĞƚĞƌ
µũ ŝƐ ƚŚĞ ƐŽůƵƟŽŶ ƚŽŶ
ŝ=ϭ · (xŝũ µũ)
= Ϭ
Ϭ ϱϱ
(ƚ) = logϭ + ƚ + ƚϮ/Ϯ
ƚ
µ µmax = KW
log ĚŶ
dŚĞŽƌĞŵ ;ĂŶƚŽŶŝ ϮϬϭϭͿ /Ĩ EyϮ
ũ < D
R1
Estimating Covariance
20
Estimating Covariance
20
ũŬ = ũŬ · ũ · Ŭ
^ƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶWĂŝƌǁŝƐĞ ĐŽƌƌĞůĂƟŽŶ
Estimating Covariance
20
ũŬ = ũŬ · ũ · Ŭ
^ƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶWĂŝƌǁŝƐĞ ĐŽƌƌĞůĂƟŽŶ
ƉƉůLJ ĂŶƚŽŶŝΖƐ ĞƐƟŵĂƚŽƌ ƚŽ ĞƐƟŵĂƚĞ ϮŶĚͲŽƌĚĞƌ ŵŽŵĞŶƚƐƟŵĂƟŶŐ ŵĂƌŐŝŶĂů ƐƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶ ũ
Estimating Covariance
20
ũŬ = ũŬ · ũ · Ŭ
^ƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶWĂŝƌǁŝƐĞ ĐŽƌƌĞůĂƟŽŶ
ƉƉůLJ ĂŶƚŽŶŝΖƐ ĞƐƟŵĂƚŽƌ ƚŽ ĞƐƟŵĂƚĞ ϮŶĚͲŽƌĚĞƌ ŵŽŵĞŶƚƐƟŵĂƟŶŐ ŵĂƌŐŝŶĂů ƐƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶ ũ
ũŬ = sin
Ϯ ũŬ
;Ϳ
Estimating Covariance
20
ũŬ = ϮŶ(Ŷϭ)
ŝ<ŝƐŐŶ(xŝũ xŝũ) · ƐŐŶ(xŝŬ xŝŬ)
ũŬ = ũŬ · ũ · Ŭ
^ƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶWĂŝƌǁŝƐĞ ĐŽƌƌĞůĂƟŽŶ
ƉƉůLJ ĂŶƚŽŶŝΖƐ ĞƐƟŵĂƚŽƌ ƚŽ ĞƐƟŵĂƚĞ ϮŶĚͲŽƌĚĞƌ ŵŽŵĞŶƚƐƟŵĂƟŶŐ ŵĂƌŐŝŶĂů ƐƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶ ũ
ũŬ = sin
Ϯ ũŬ
;Ϳ
Estimating Covariance
20
ũŬ = ϮŶ(Ŷϭ)
ŝ<ŝƐŐŶ(xŝũ xŝũ) · ƐŐŶ(xŝŬ xŝŬ)
ũŬ = ũŬ · ũ · Ŭ
^ƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶWĂŝƌǁŝƐĞ ĐŽƌƌĞůĂƟŽŶ
ƉƉůLJ ĂŶƚŽŶŝΖƐ ĞƐƟŵĂƚŽƌ ƚŽ ĞƐƟŵĂƚĞ ϮŶĚͲŽƌĚĞƌ ŵŽŵĞŶƚƐƟŵĂƟŶŐ ŵĂƌŐŝŶĂů ƐƚĂŶĚĂƌĚ ĚĞǀŝĂƟŽŶ ũ
dŚĞŽƌĞŵ ;>ŝƵ ,ĂŶ ŚĂŶŐ ϮϬϭϮͿ /Ĩ Eyϰũ < D
R2 max = KW
log ĚŶ
ũŬ = sin
Ϯ ũŬ
;Ϳ
21
ŶĂůLJnjŝŶŐ max
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƐŚŽǁ maxũŬũŬ ũŬ
= KW
log ĚŶ
21
ŶĂůLJnjŝŶŐ max
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƐŚŽǁ maxũŬũŬ ũŬ
= KW
log ĚŶ
ũŬ = sin
ϮE ũŬ
&Žƌ ĂŶLJ ůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ <ƌƵƐŬĂů;ϭϵϱϴͿ ƐŚŽǁƐ
21
WĞĂƌƐŽŶΖƐĐŽƌƌĞůĂƟŽŶ ĐŽĞĨ
WŽƉƵůĂƟŽŶ<ĞŶĚĂůůΖƐ ƚĂƵ
ŶĂůLJnjŝŶŐ max
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƐŚŽǁ maxũŬũŬ ũŬ
= KW
log ĚŶ
ũŬ = sin
ϮE ũŬ
&Žƌ ĂŶLJ ůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ <ƌƵƐŬĂů;ϭϵϱϴͿ ƐŚŽǁƐ
21
WĞĂƌƐŽŶΖƐĐŽƌƌĞůĂƟŽŶ ĐŽĞĨ
DŽŶŽƚŽŶĞ ƚƌĂŶƐ ŝŶǀĂƌŝĂŶƚ
WŽƉƵůĂƟŽŶ<ĞŶĚĂůůΖƐ ƚĂƵ
ŶĂůLJnjŝŶŐ max
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƐŚŽǁ maxũŬũŬ ũŬ
= KW
log ĚŶ
ũŬ = sin
ϮE ũŬ
&Žƌ ĂŶLJ ůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ <ƌƵƐŬĂů;ϭϵϱϴͿ ƐŚŽǁƐ
21
WĞĂƌƐŽŶΖƐĐŽƌƌĞůĂƟŽŶ ĐŽĞĨ
DŽŶŽƚŽŶĞ ƚƌĂŶƐ ŝŶǀĂƌŝĂŶƚ
WŽƉƵůĂƟŽŶ<ĞŶĚĂůůΖƐ ƚĂƵ
ŶĂůLJnjŝŶŐ max
ůƐŽ ŚŽůĚƐ ĨŽƌ ƚŚĞ ƚƌĂŶƐĞůůŝƉƟĐĂů ĨĂŵŝůLJ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƐŚŽǁ maxũŬũŬ ũŬ
= KW
log ĚŶ
ũŬ = sin
ϮE ũŬ
&Žƌ ĂŶLJ ůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ <ƌƵƐŬĂů;ϭϵϱϴͿ ƐŚŽǁƐ
21
WĞĂƌƐŽŶΖƐĐŽƌƌĞůĂƟŽŶ ĐŽĞĨ
DŽŶŽƚŽŶĞ ƚƌĂŶƐ ŝŶǀĂƌŝĂŶƚ
WŽƉƵůĂƟŽŶ<ĞŶĚĂůůΖƐ ƚĂƵ
ŶĂůLJnjŝŶŐ max
ůƐŽ ŚŽůĚƐ ĨŽƌ ƚŚĞ ƚƌĂŶƐĞůůŝƉƟĐĂů ĨĂŵŝůLJ
maxũŬũŬ ũŬ
maxũŬ ũŬ E ũŬ
= KW
log ĚŶ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƐŚŽǁ maxũŬũŬ ũŬ
= KW
log ĚŶ
ũŬ = sin
ϮE ũŬ
&Žƌ ĂŶLJ ůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ <ƌƵƐŬĂů;ϭϵϱϴͿ ƐŚŽǁƐ
21
WĞĂƌƐŽŶΖƐĐŽƌƌĞůĂƟŽŶ ĐŽĞĨ
DŽŶŽƚŽŶĞ ƚƌĂŶƐ ŝŶǀĂƌŝĂŶƚ
hͲƐƚĂƟƐƟĐ ƚŚĞŽƌLJ
WŽƉƵůĂƟŽŶ<ĞŶĚĂůůΖƐ ƚĂƵ
ŶĂůLJnjŝŶŐ max
ůƐŽ ŚŽůĚƐ ĨŽƌ ƚŚĞ ƚƌĂŶƐĞůůŝƉƟĐĂů ĨĂŵŝůLJ
maxũŬũŬ ũŬ
maxũŬ ũŬ E ũŬ
= KW
log ĚŶ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƐŚŽǁ maxũŬũŬ ũŬ
= KW
log ĚŶ
ũŬ = sin
ϮE ũŬ
&Žƌ ĂŶLJ ůůŝƉƟĐĂů ĚŝƐƚƌŝďƵƟŽŶ <ƌƵƐŬĂů;ϭϵϱϴͿ ƐŚŽǁƐ
Transelliptical Theory
22
Transelliptical Theory
22
dŚĞŽƌĞŵ ;,ĂŶ >ŝƵ ϮϬϭϯͿ ƐƐƵŵŝŶŐ Ě ŝŶĐƌĞĂƐĞƐ ǁŝƚŚ Ŷ ĐŽŶƐŝĚĞƌ
• ƐƉĂƌƐĞ ƉƌŝŶĐŝƉůĞ ĐŽŵƉŽŶĞŶƚ ĂŶĂůLJƐŝƐ• ŐƌĂƉŚŝĐĂů ŵŽĚĞů ĞƐƟŵĂƟŽŶ• ƐƉĂƌƐĞ ĐŽƌƌĞůĂƟŽŶ ŵĂƚƌŝdž ĞƐƟŵĂƟŽŶ
;hŶƐƵƉĞƌǀŝƐĞĚͿ
Transelliptical Theory
22
ZĂƚĞ ŽĨ dƌĂŶƐĞůůŝƉƟĐĂů ŵĞƚŚŽĚ = ZĂƚĞ ŽĨ 'ĂƵƐƐŝĂŶ ŵĞƚŚŽĚ
dŚĞŽƌĞŵ ;,ĂŶ >ŝƵ ϮϬϭϯͿ ƐƐƵŵŝŶŐ Ě ŝŶĐƌĞĂƐĞƐ ǁŝƚŚ Ŷ ĐŽŶƐŝĚĞƌ
• ƐƉĂƌƐĞ ƉƌŝŶĐŝƉůĞ ĐŽŵƉŽŶĞŶƚ ĂŶĂůLJƐŝƐ• ŐƌĂƉŚŝĐĂů ŵŽĚĞů ĞƐƟŵĂƟŽŶ• ƐƉĂƌƐĞ ĐŽƌƌĞůĂƟŽŶ ŵĂƚƌŝdž ĞƐƟŵĂƟŽŶ
;hŶƐƵƉĞƌǀŝƐĞĚͿ
Transelliptical Theory
22
ZĂƚĞ ŽĨ dƌĂŶƐĞůůŝƉƟĐĂů ŵĞƚŚŽĚ = ZĂƚĞ ŽĨ 'ĂƵƐƐŝĂŶ ŵĞƚŚŽĚ
• ƐƉĂƌƐĞ > ĂŶĚ ƋƵĂĚƌĂƟĐ ĂŶĂůLJƐŝƐ• ƐƉĂƌƐĞ ƌĂŶĚŽŵ ĚĞƐŝŐŶ ƌĞŐƌĞƐƐŝŽŶ• ƉƌŝŶĐŝƉĂů ĐŽŵƉŽŶĞŶƚ ƌĞŐƌĞƐƐŝŽŶ
;^ƵƉĞƌǀŝƐĞĚͿ
dŚĞŽƌĞŵ ;,ĂŶ >ŝƵ ϮϬϭϯͿ ƐƐƵŵŝŶŐ Ě ŝŶĐƌĞĂƐĞƐ ǁŝƚŚ Ŷ ĐŽŶƐŝĚĞƌ
• ƐƉĂƌƐĞ ƉƌŝŶĐŝƉůĞ ĐŽŵƉŽŶĞŶƚ ĂŶĂůLJƐŝƐ• ŐƌĂƉŚŝĐĂů ŵŽĚĞů ĞƐƟŵĂƟŽŶ• ƐƉĂƌƐĞ ĐŽƌƌĞůĂƟŽŶ ŵĂƚƌŝdž ĞƐƟŵĂƟŽŶ
;hŶƐƵƉĞƌǀŝƐĞĚͿ
Transelliptical Theory
22
ZĂƚĞ ŽĨ dƌĂŶƐĞůůŝƉƟĐĂů ŵĞƚŚŽĚ = ZĂƚĞ ŽĨ 'ĂƵƐƐŝĂŶ ŵĞƚŚŽĚ
ZĂƚĞ ŽĨ dƌĂŶƐĞůůŝƉƟĐĂů ŵĞƚŚŽĚ =
sZĂƚĞ ŽĨ 'ĂƵƐƐŝĂŶ ŵĞƚŚŽĚ
• ƐƉĂƌƐĞ > ĂŶĚ ƋƵĂĚƌĂƟĐ ĂŶĂůLJƐŝƐ• ƐƉĂƌƐĞ ƌĂŶĚŽŵ ĚĞƐŝŐŶ ƌĞŐƌĞƐƐŝŽŶ• ƉƌŝŶĐŝƉĂů ĐŽŵƉŽŶĞŶƚ ƌĞŐƌĞƐƐŝŽŶ
;^ƵƉĞƌǀŝƐĞĚͿ
dŚĞŽƌĞŵ ;,ĂŶ >ŝƵ ϮϬϭϯͿ ƐƐƵŵŝŶŐ Ě ŝŶĐƌĞĂƐĞƐ ǁŝƚŚ Ŷ ĐŽŶƐŝĚĞƌ
• ƐƉĂƌƐĞ ƉƌŝŶĐŝƉůĞ ĐŽŵƉŽŶĞŶƚ ĂŶĂůLJƐŝƐ• ŐƌĂƉŚŝĐĂů ŵŽĚĞů ĞƐƟŵĂƟŽŶ• ƐƉĂƌƐĞ ĐŽƌƌĞůĂƟŽŶ ŵĂƚƌŝdž ĞƐƟŵĂƟŽŶ
;hŶƐƵƉĞƌǀŝƐĞĚͿ
Transelliptical Theory
22^ƉĂƌƐŝƚLJ ůĞǀĞů
ZĂƚĞ ŽĨ dƌĂŶƐĞůůŝƉƟĐĂů ŵĞƚŚŽĚ = ZĂƚĞ ŽĨ 'ĂƵƐƐŝĂŶ ŵĞƚŚŽĚ
ZĂƚĞ ŽĨ dƌĂŶƐĞůůŝƉƟĐĂů ŵĞƚŚŽĚ =
sZĂƚĞ ŽĨ 'ĂƵƐƐŝĂŶ ŵĞƚŚŽĚ
• ƐƉĂƌƐĞ > ĂŶĚ ƋƵĂĚƌĂƟĐ ĂŶĂůLJƐŝƐ• ƐƉĂƌƐĞ ƌĂŶĚŽŵ ĚĞƐŝŐŶ ƌĞŐƌĞƐƐŝŽŶ• ƉƌŝŶĐŝƉĂů ĐŽŵƉŽŶĞŶƚ ƌĞŐƌĞƐƐŝŽŶ
;^ƵƉĞƌǀŝƐĞĚͿ
dŚĞŽƌĞŵ ;,ĂŶ >ŝƵ ϮϬϭϯͿ ƐƐƵŵŝŶŐ Ě ŝŶĐƌĞĂƐĞƐ ǁŝƚŚ Ŷ ĐŽŶƐŝĚĞƌ
• ƐƉĂƌƐĞ ƉƌŝŶĐŝƉůĞ ĐŽŵƉŽŶĞŶƚ ĂŶĂůLJƐŝƐ• ŐƌĂƉŚŝĐĂů ŵŽĚĞů ĞƐƟŵĂƟŽŶ• ƐƉĂƌƐĞ ĐŽƌƌĞůĂƟŽŶ ŵĂƚƌŝdž ĞƐƟŵĂƟŽŶ
;hŶƐƵƉĞƌǀŝƐĞĚͿ
Proof of the Theory
23
Proof of the Theory
23
R4
R3
R2
R5
R1 µ µmax = KW
log ĚŶ
max = KW
log ĚŶ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƉƌŽǀĞ Zϭ Ͳ Zϱ
supvϬƐ
|vd( )v|vϮϮ
= KW
Ɛ log ĚŶ
infvϬƐ
|vd v|vϮϮ
> Ϭ ǁŝƚŚ ŚŝŐŚ ƉƌŽď
Ϯ = KW
Ělog ĚŶ
Proof of the Theory
23
ĂŶƚŽŶŝ ;ϮϬϭϭͿ
R4
R3
R2
R5
R1 µ µmax = KW
log ĚŶ
max = KW
log ĚŶ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƉƌŽǀĞ Zϭ Ͳ Zϱ
supvϬƐ
|vd( )v|vϮϮ
= KW
Ɛ log ĚŶ
infvϬƐ
|vd v|vϮϮ
> Ϭ ǁŝƚŚ ŚŝŐŚ ƉƌŽď
Ϯ = KW
Ělog ĚŶ
Proof of the Theory
23
ĂŶƚŽŶŝ ;ϮϬϭϭͿ
ŽƌŽůůĂƌLJ ŽĨ Zϰ
R4
R3
R2
R5
R1 µ µmax = KW
log ĚŶ
max = KW
log ĚŶ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƉƌŽǀĞ Zϭ Ͳ Zϱ
supvϬƐ
|vd( )v|vϮϮ
= KW
Ɛ log ĚŶ
infvϬƐ
|vd v|vϮϮ
> Ϭ ǁŝƚŚ ŚŝŐŚ ƉƌŽď
Ϯ = KW
Ělog ĚŶ
Proof of the Theory
23
ĂŶƚŽŶŝ ;ϮϬϭϭͿ
ŽƌŽůůĂƌLJ ŽĨ Zϰ
dŚĞŽƌLJ ŽĨ hͲƐƚĂƟƐƟĐ
R4
R3
R2
R5
R1 µ µmax = KW
log ĚŶ
max = KW
log ĚŶ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƉƌŽǀĞ Zϭ Ͳ Zϱ
supvϬƐ
|vd( )v|vϮϮ
= KW
Ɛ log ĚŶ
infvϬƐ
|vd v|vϮϮ
> Ϭ ǁŝƚŚ ŚŝŐŚ ƉƌŽď
Ϯ = KW
Ělog ĚŶ
Proof of the Theory
23
ĂŶƚŽŶŝ ;ϮϬϭϭͿ
ŽƌŽůůĂƌLJ ŽĨ Zϰ
dŚĞŽƌLJ ŽĨ hͲƐƚĂƟƐƟĐ
R4
R3
R2
R5
R1 µ µmax = KW
log ĚŶ
max = KW
log ĚŶ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƉƌŽǀĞ Zϭ Ͳ Zϱ
supvϬƐ
|vd( )v|vϮϮ
= KW
Ɛ log ĚŶ
infvϬƐ
|vd v|vϮϮ
> Ϭ ǁŝƚŚ ŚŝŐŚ ƉƌŽď
Ϯ = KW
Ělog ĚŶ
DĂƚƌŝdž ĐŽŶĐĞŶƚƌĂƟŽŶ
Proof of the Theory
23
ĂŶƚŽŶŝ ;ϮϬϭϭͿ
ŽƌŽůůĂƌLJ ŽĨ Zϰ
dŚĞŽƌLJ ŽĨ hͲƐƚĂƟƐƟĐ
dĞĐŚŶŝĐĂůR4
R3
R2
R5
R1 µ µmax = KW
log ĚŶ
max = KW
log ĚŶ
WƌŽŽĨ dŚĞ ŬĞLJ ŝƐ ƚŽ ƉƌŽǀĞ Zϭ Ͳ Zϱ
supvϬƐ
|vd( )v|vϮϮ
= KW
Ɛ log ĚŶ
infvϬƐ
|vd v|vϮϮ
> Ϭ ǁŝƚŚ ŚŝŐŚ ƉƌŽď
Ϯ = KW
Ělog ĚŶ
DĂƚƌŝdž ĐŽŶĐĞŶƚƌĂƟŽŶ
Empirical Results
24
&Žƌ ŶŽŶ'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
Empirical Results
24
^ĂŵƉůĞ y EWEĚ(, Ĩ) ǁŝƚŚ Ŷ = ϮϬϬ, Ě = ϰϬ ĂŶĚ Ĩũ
&Žƌ ŶŽŶ'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
Empirical Results
24
true graph transelliptical glasso
^ĂŵƉůĞ y EWEĚ(, Ĩ) ǁŝƚŚ Ŷ = ϮϬϬ, Ě = ϰϬ ĂŶĚ Ĩũ
&Žƌ ŶŽŶ'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
Empirical Results
24
true graph transelliptical glasso
FP
FN
^ĂŵƉůĞ y EWEĚ(, Ĩ) ǁŝƚŚ Ŷ = ϮϬϬ, Ě = ϰϬ ĂŶĚ Ĩũ
&Žƌ ŶŽŶ'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
Empirical Results
24
true graph transelliptical glasso
FP
FN
^ĂŵƉůĞ y EWEĚ(, Ĩ) ǁŝƚŚ Ŷ = ϮϬϬ, Ě = ϰϬ ĂŶĚ Ĩũ
KƌĂĐůĞ ŐƌĂƉŚ ƉŝĐŬ ƚŚĞ ďĞƐƚ ƚƵŶŝŶŐ ƉĂƌĂŵĞƚĞƌ ďLJ ƚŚĞ ƚƌƵƚŚ
&Žƌ ŶŽŶ'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
Efficiency Loss
25
&Žƌ 'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
Efficiency Loss
25
ŽŵƉƵƚĂƟŽŶĂůůLJ ŶŽ ĞdžƚƌĂ ĐŽƐƚ
^ƚĂƟƐƟĐĂůůLJ ƐĂŵƉůĞ y EĚ(Ϭ,) ǁŝƚŚ Ŷ = ϴϬ, Ě = ϭϬϬ
&Žƌ 'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
Efficiency Loss
25
ROC curve for graph recovery
1-FP
1-FN
ŽŵƉƵƚĂƟŽŶĂůůLJ ŶŽ ĞdžƚƌĂ ĐŽƐƚ
^ƚĂƟƐƟĐĂůůLJ ƐĂŵƉůĞ y EĚ(Ϭ,) ǁŝƚŚ Ŷ = ϴϬ, Ě = ϭϬϬ
&Žƌ 'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
Efficiency Loss
25
ROC curve for graph recovery
1-FP
1-FN
ŽŵƉƵƚĂƟŽŶĂůůLJ ŶŽ ĞdžƚƌĂ ĐŽƐƚ
^ƚĂƟƐƟĐĂůůLJ ƐĂŵƉůĞ y EĚ(Ϭ,) ǁŝƚŚ Ŷ = ϴϬ, Ě = ϭϬϬ
&Žƌ 'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
dƌĂŶƐĞůůŝƉƟĐĂů'ůĂƐƐŽ
Efficiency Loss
25
ROC curve for graph recovery
1-FP
1-FN
ŽŵƉƵƚĂƟŽŶĂůůLJ ŶŽ ĞdžƚƌĂ ĐŽƐƚ
^ƚĂƟƐƟĐĂůůLJ ƐĂŵƉůĞ y EĚ(Ϭ,) ǁŝƚŚ Ŷ = ϴϬ, Ě = ϭϬϬ
&Žƌ 'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
dƌĂŶƐĞůůŝƉƟĐĂů'ůĂƐƐŽ
Efficiency Loss
25
ROC curve for graph recovery
1-FP
1-FN
ŽŵƉƵƚĂƟŽŶĂůůLJ ŶŽ ĞdžƚƌĂ ĐŽƐƚ
^ƚĂƟƐƟĐĂůůLJ ƐĂŵƉůĞ y EĚ(Ϭ,) ǁŝƚŚ Ŷ = ϴϬ, Ě = ϭϬϬ
ůŵŽƐƚ ŶŽ ĞĸĐŝĞŶĐLJ ůŽƐƐ
&Žƌ 'ĂƵƐƐŝĂŶ ĚĂƚĂ ƚƌĂŶƐĞůůŝƉƟĐĂů 'ĂƵƐƐŝĂŶ
dƌĂŶƐĞůůŝƉƟĐĂů'ůĂƐƐŽ
26
Summary
^ĐĂůĂďůĞ ƐĞŵŝƉĂƌĂŵĞƚƌŝĐ ŵĞƚŚŽĚ н ,ŝŐŚ ĚŝŵĞŶƐŝŽŶĂů ƚŚĞŽƌLJ
26
Summary
^ĐĂůĂďůĞ ƐĞŵŝƉĂƌĂŵĞƚƌŝĐ ŵĞƚŚŽĚ н ,ŝŐŚ ĚŝŵĞŶƐŝŽŶĂů ƚŚĞŽƌLJ
• ^ĞŵŝƉĂƌĂŵĞƚƌŝĐ ŵŽĚĞůŝŶŐ ǁŝƚŚ ƉĂƌĂŵĞƚƌŝĐ ƌĂƚĞ
• Ɛ ƐĐĂůĂďůĞ ĂƐ ƚŚĞ ƉĂƌĂŵĞƚƌŝĐ ŵĞƚŚŽĚ
• >ĂƌŐĞůLJ ƵŶĞdžƉůŽƌĞĚ ĂƌĞĂ ǁŝƚŚ ŵĂŶLJ ĨƵƚƵƌĞ ĞdžƚĞŶƐŝŽŶƐ
26
Summary
^ĐĂůĂďůĞ ƐĞŵŝƉĂƌĂŵĞƚƌŝĐ ŵĞƚŚŽĚ н ,ŝŐŚ ĚŝŵĞŶƐŝŽŶĂů ƚŚĞŽƌLJ
^ŽŌǁĂƌĞ ŚƵŐĞ ĂŶĚ ƐŵĂƌƚ ĂƌĞ ĂǀĂŝůĂďůĞ ŽŶ ZE
• ^ĞŵŝƉĂƌĂŵĞƚƌŝĐ ŵŽĚĞůŝŶŐ ǁŝƚŚ ƉĂƌĂŵĞƚƌŝĐ ƌĂƚĞ
• Ɛ ƐĐĂůĂďůĞ ĂƐ ƚŚĞ ƉĂƌĂŵĞƚƌŝĐ ŵĞƚŚŽĚ
• >ĂƌŐĞůLJ ƵŶĞdžƉůŽƌĞĚ ĂƌĞĂ ǁŝƚŚ ŵĂŶLJ ĨƵƚƵƌĞ ĞdžƚĞŶƐŝŽŶƐ