asymptotic statistical analysis on special manifolds...
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Asymptotic Statistical Analysis on Special Manifolds
(Stiefel and Grassmann manifolds)
Yasuko ChikuseFaculty of Engineering
Kagawa UniversityJapan
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[ I ] Stiefel manifold
Dimension of .
mkV , )( mk ≤
)1(21
, +−= kkkmV mk
≡def { －frames in ; －frame = a set
of orthonormal vectors in }kkk mR
mR
( ) .}';{ kIXXkmX =×∴=
3
Ex: (i) : orthogonal group.(ii) : unit hypersphere in .
( circle, sphere)
Applications in Earth Sciences, Medicine, Astronomy, Meteorology, Biology.
A large literature exists for analysis ofdirectional statistics.
vector cardiogramorbits of comets.
⎩⎨⎧ =
m
mm
VVmO
,1
,)(mR
2=m 3=m
⎪⎩
⎪⎨⎧
≤≤≤
=
:32
:1
mk
k
⎩⎨⎧
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[ II ] Grassmann manifold kmkG −, )( mk ≤
≡def { －planes in , i.e., －dimensional hyperplanes containing the origin in }
kk mR
mR
To each “ －plane” , corresponds a unique “ orthogonal projection matrix
idempotent of rank ” .
We carry out statistical analysis on .
k kmkG −∈ ,
mm×P kmkPk −∈ ,
kmkkmk PG −− ≡∴ ,,
kmkP −,
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Ex: ={axes or undirected lines through }
Applications: in the signal processing of radar with elements observing targets. A rather new sample space.
We confine our discussion mainly on in the following.
1,1 −mG→
0
m k
mkV ,
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[ III ] Population distributions on mkV ,
1. Uniform distribution (normalized invariant measure)
invariant under
).(),(,'
21
21
kOHmOHXHHX
∈∈→
][dX
7
where the differential form∫=mkV
dXdXdX,
),(/)(][
,'')( 111
→→
<
→
+
→
=
−
=∧∧∧= ij
ji
kijk
k
i
km
jxdxxdxdX for )(
1 kxxX ･･･=
choosing mk xx→
+
→
･･･1 s.t. ),()( 1k1 mOxxxx mk ∈→
+
→→→
･･････
∫ Γ=mkV k
kmk mdx,
),2/(/2)( 2/π with
).2/)1(()(||)()(1
4/)1(2/)1(
0−−ΓΠ=−=Γ
=
−+−
>∫ iadSSSetrak
i
kkka
Sk π
(multivariate Gamma function)
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2. Matrix Langevin distributionpdf
for matrix, w.r.t. with
hypergeometric function with matrix argument. Exponential type distribution. Parametrization of (svd), where
and (with first row positive) .0
),(),(,~
1
1,
≥≥≥
=Λ∈Θ∈Γ
k
kmk diagkOVλλλλ
･･･ ･･･
),,;(~( , Σmkm IMNX∵ d
);,( FkmL)4/';2/()/'()( 10 FFmFXFetrXfx =
F km× ].[dX).'|1
kIXXMF =Σ= −
:10 F
'ΓΛΘ=F
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⎩⎨⎧
Λ=ΛΘΛΓ=∈ΓΘ=
.)'()(
)'()'()()('
0
,0
etrXFetrXetrXFetrVX mk
∵∵Mode:
Concentrations:
Assume rank , unless otherwise stated.
(or ): uniform distribution.
Historical background: : directional disrtibution
von Mises distribution (1918)
Fisher distribution (1953)
Langevin (1905)
generalization by Watson & Williams (1956).
kF =
0=F 0=Λ
1=k
⎪⎩
⎪⎨
⎧
≥==
232
mmm
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Matrix-variatenormal distribution),;( 21,
if has the pdf
i.e., all elements of are i.i.d.
Let [i.e., ]
for
then
ΣΣMN km
)]2/'(exp[)2()(
Z),;0(~)(2/),(
,
ZZtrZ
IINkmZkmkm
kmkm
−=
×−πϕ
).1,0(NZ
( ),0)(,0)(),( 21
2/12
2/11
2/12
2/11
>×Σ>×Σ×
Σ−Σ=+ΣΣ= −−
kkmmkmM
MYZMZY
d
( ) ( ) .,;~ 21, ΣΣ× MNkmY kmd
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Hypergeometric functions with symmetric matrix argument.
Starting with for symmetric,
(Laplace transforms)
),()(00 SetrSF =
).;;(
))(;;(||)()(
1
111
0 112/)1(
YbbaaaF
dSYSbbaaFSSetra
qpqp
S qpqpma
m
･･････
･･････
+
>
+−
=
−Γ ∫
( )!)()(
)()(
);;(
1
1
0
11
lSC
bbaa
SbbaaF
q
p
ll
qpqp
λ
λλ
λλ
λ ･･････
･･････
トΣΣ=
∞
=,
mmS ×
12
where ordered partition of),...,( 1 mll=λ ,l
)(
,1)(),1()1()(
,2
1)(
),0(
0
1
11
SC
alaaaa
iaa
llll
l
l
m
i
i
m
im
i
λ
λ
=−++=
⎟⎠⎞
⎜⎝⎛ −
−Π=
=Σ≥≥≥
=
=
･･･
･･･
with
: zonal polynomial with
symmetric matrix
mm×.S
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Multi-mode with
and any
Concentrations:
3. Matrix Bingham distribution
Exponential type distribution. Parametrization of (sd), where
and
( ) ( ) ( )BmkFBXXetrXf x ;2/;2//' 11=
);,( BkmB
for symmetric matrix.mmB ×( ) ).'|
21,,;0~( 1
, kkkm IXXBINX =Σ−=Σ − ∵ withd
'ΓΛΓ=B( ),~ mO∈Γ ( ) ., 11 mmdiag λλλλ ≥≥=Λ
( ) ( )( )
⎪⎩
⎪⎨
⎧
Λ∈
×ΓΓΓ=ΓΓ
.
,
2
12121
kOH
kmH for
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4. A general form of distributions
pdf ( ));2/,;2/,(
)';;(
1111
11
BmbbkaaFBXXbbaaF
Xfqpqp
qpqpx ･･････
･･････
++
=
for symmetric matrix.mmB ×:
Ex: ( ) ( )( ) .|'|';
,''
01
00b
k BXXIBXXbF
BXXetrBXXF−−=
=
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5. Distributions with pdfof the form
where a subspace of of dimension
orthogonal projection matrix onto
( ),XPf ν
⎩⎨⎧
::
ν
νP
mR q
.ν
Ex: ( )( ) ( )
( )( ) ( ) ( )[ ]⎪
⎪⎩
⎪⎪⎨
⎧
=ΓΓΓΛΓΓΓ=ΓΛΓ=
=ΓΓΘΛΓ=ΓΛΘ=
,''''',';,
,''',';,
XPBXPetrXXetrBkmB
XPFetrXetrFkmL
νν
ν
with pdf ( )XFetr '∝
with pdf ( )BXXetr '∝
with
⎩⎨⎧ ΓΓ=
:'
ννP
subspace spanned by the the columns of .Γ
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6. A method to generate a new family of distributions
For an random matrix
What is the distribution of ?
using
where
km× ,Z( ) ( ) 2/12/12/1
2/1
'' ZZ
TH
THZZZZZZZZ
･･ == −
(polar decomposition of ).ZmkZ VH ,∈ : orientation of .Z
ZH
( ) ( ) [ ] ( ),2/
2/
ZZk
km
dTdHm
dZ Γ
=π
( )[ ]:
:)(,
Z
Z
dH
dTdZ Lebesgue measures,
normalized invariant measure on .,mkV
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Ex: (i) When
(ii) More generally, when
for
then
( )( ),,;0~ , kkm ImmNZ ×Σ then
( ) :|||| 2/1'2/ mZZ
kZH HHHf
Z
−−− ΣΣ=
Matrix angular central Gaussian [MACG( )] distribution.mI
Σ(MACG( ) = uniform distribution)
( ) ( ) ( ) ,''||'|| 12/12/ ZHZHgZZgZf kkZ
−−−− ΣΣ=ΣΣ=
( ),kOH ∈
( ).~ ΣMACGHZ d
d
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[ IV ] Inference for of Langevin distribution
Let a random sample from
then the log-likelihood is
where (svd), with
M.l.e.’s and
where
'ΓΛΘ=F
nXX ,,1 ( ).;, FkmL
,/1
nXX i
n
i=Σ=
( ) ( ) ( )[ ]( ) ( )[ ] ,4/;2/log'
4/;2/log',,;2
101'2
2101
Λ−ΘΛΓ=
Λ−=
mFXHHtrn
mFXFtrnXXFl
d
n
21 'HXHX d= ,~,1 mkVH ∈
( ) ( ) .0,, 112 >>>=∈ kkd xxxxdiagXkOH
( )kdiagHH λλ ˆˆˆ,ˆ,ˆ121 =Λ=Θ=Γ
( ) )(.,,1,ˆ4/ˆ;2/log 2
10 ＊ kixmFi
i
==∂
Λ∂λ
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[ V ] Asymptotic theorems in Inference and disrtibution theory
for
⎩⎨⎧ large sample n
large or small concentrations(near uniformity)
Λ
high dimension m
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(or ),
Testing
⎩⎨⎧
==Λ
−
02/1
1
0
:0:
FnFHH (uniformity) against
02/1 Λ=Λ −n
[ V.1 ] Large sample asymptotics
( )( )⎪⎩
⎪⎨⎧
=−
102/1
,
0,
,;
,;0~
HIIFmN
HIINXnmZ
kmkm
kmkm
Under , for large
under
⎪⎩
⎪⎨⎧
∴Λ ;,
~'1/;
20
2
20
nHxHx
ZtrZmtrkm
km
under
under for large
asymptotically Rayleigh-style, likelihood ratio, Rao score, locally best invariant tests.
:d .n
:d
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Solve (＊) by using the asymptotic expansion
for the with small or large:
.ˆ,,,1),ˆ(ˆ 3 Λ=Λ+= kiOmxiiλ for small
[ V.2 ] Small or large asymptoticsΛ
( )4/ˆ;2/ 210 ΛmF Λ̂
( ) ,,,1,ˆˆˆ
121
ˆ21 2
,1kixOkm
iji
k
ijji
==Λ++
Σ−−
− −
≠= λλλfor large .Λ̂
(when has rank 1, then )Λ ( ) ( )11
ˆ12
1ˆ −Λ+− 1
−= O
xmλ
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[ V.3 ] High dimensional asymptotics
(i) BackgroundApplications in compositional data and certain
permutation distributions.
For Langevin distributions with it is known that practically, concentrations for are larger than those for
3=m,1=k
.2=m
~);,( FkmL : uniform distribution ( ),0=Fas (see later) . ∞→m ( )?βmOF =∴
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(ii) Asymptotic expansions for distributions
For a random sample from , putfor fixed fixed.
nXX ,,1 … ( )FkmL ;,
( ),' kqXnmZ ×Ψ= ( ) qVqm mq ,,∈×Ψ
( ) ( )( ) ( )
( )( )
)},()]''
)'((21[41
'1{
2/3
2
2
,
,
−+ΨΨ−
Ψ++
Ψ+=
∑mOFFtr
ZFtrnZHanm
ZFtrmnZZf
kq
kqZ
トλ
λλ
ϕ
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where ( ) ( ) ( )( )( )⎪⎩
⎪⎨⎧
⋅
−= −
:,2/'2
,
2/kq,
kq
qk
HZZetrZ
λ
πϕ
Hermite polys. associated with the normal defined by( ),,;0, kqkq IIN
( )( ) ( )( ) ( ) ( )( )ZZZCZZH kqkqkq ,,, ' ϕϕ λλ ∂∂=
with ( ) ., ijij
ZZZ
Z =⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
=∂ for
not dependent on ,F( )( )~;,
,;0~ ,
FkmL
IINZ kqkq
∴
∴ :
: uniform distribution as
d
.∞→m
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(iii) Generalized Stam’s Limit Theorems
(iii, 1) Histrical backgroundde Finetti’s theorem (1929): an infinite, exchangeable
sequence of random variables is mixed i.i.d.
Poincare’s theorem (1912), or Stam’s (first) theorem (1982):asymptotic normality of the first coordinates of a random point (uniform) on as
If is orthogonally invariant, has a -mixture distribution of normal as [Diaconis, Eaton, Freedman, Lauritzen (1980’s)]
k.,,1 ∞→mV m
nXX ,,1 …nXX ,,1 …( ) ,,0 2σNσ .∞→n
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(iii, 2) Stam’s first theorem
Theorem 0. [Stam (1982)] Asymptotic normality of a finite number of coordinates of the uniform variate on as
Theorem 1. [Watson (1983)] If uniform on a finite number of elements of arei.i.d. as
,,1 mV .∞→m
~X d
,,mkV Xm( ) .,1,0 ∞→mN
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Extensions to more general distributions with pdf ( )XPf ν
where the orthogonal projection matrix ontoa subspace of dimension
Ex:
:νPν .q
( )( )( )( )⎩
⎨⎧
ΓΛΓ=ΓΛΘ=
,';,,';,
sdBkmBsvdFkmL
with pdf ( )XPFetr ν'∝with pdf ( ) ( )[ ]
'.'
ΓΓ=∝
ν
νν
PXPBXPetr
with
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( uniform case)
Theorem 2. [Chikuse (1991)] If
( ),,' , mfixedqVP mq ≤∈ΓΓΓ=ν
( ),~ XPfX ν
( ) ( ) ,,;0~' , kqkq IINkqXm ×Γ
with
then as .∞→m
↔= mq
d
d:
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Theorem 3. [Chikuse] Suppose that(1) When
(2) When the limiting distribution depends on
( ) .~ XPmfX νβ
( ) .,,;0~',21
, ∞→Γ< mIINXm kqkq β as↔⎟
⎠⎞
⎜⎝⎛ <=
210β(ex: Theorem 2.)
:
,21
=β ( ) .⋅f
Ex: (i) When then( ) ,',;,~ ΓΛΘ=FFmkmLX d
( ) ,,;'~' , kqkq IIFNXm ΓΓ as .∞→m(ii) When then ( ) ,',;,~ ΓΛΓ=BmBkmBX d
( )( ) ,,2;0~' 1, kqkq IINXm −Λ−Γ as .∞→m
d
d
:d
:d
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(iii, 3) Stam’s second theorem(limit orthogonality of )nXX ,,1 …
Theorem 0. [Stam (1982)] On
Theorem 1. [Watson (1983)] If is a random sample from the uniform distribution on then
are mutually
independent normal
nXX ,,1 …
.,1 mV
,,mkV
( ) ,1,' njikkXXm ji ≤<≤×
( ),,;0, kkkk IIN as .∞→m
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Extensions to non-uniform distributions on mkV ,
Theorem 2. [Chikuse (1993)] If is a randomsample from orthen are mutually independent normal
nXX ,,1 …
( )FkmL ;, ( ),;, BkmB,1,' njiXXm ji ≤<≤( ),,;0, kkkk IIN as .∞→m
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Theorem 3. [Chikuse](1) If is a r. sample from
or then are mutually independent normal
(2) If is a r. sample from then are mutually independent normalwhere
nXX ,,1 … ( ) ,21,;, ≤ββ FmkmL
( ) ,21,;, 2 <ββ BmkmB ,1,' njiXXm ji ≤<≤
( ),,;0, kkkk IIN as .∞→mnXX ,,1 … ( ) ,
21,;, 2 =ββ BmkmB
,1,' 2/1 njiXXm ji ≤<≤Σ−
( ),,;0, kkkk IIN as ,∞→m( ) .02 1 >−=Σ −BIm
(ex: Theorem 2.)↔⎟⎠⎞
⎜⎝⎛<=
210β
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Applications
Let be a random sample fromor
Then independent as
nXX ,,1 …( ) ,
21,;, ≤ββ FmkmL ( ) .
21,;, 2 <ββ BmkmB
.∞→m( )kkkkji IINXXm ,;0~ ,
' :d
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Ex: distance between 2 random points ( )2, =nYX
( ) ( )
,21,2~
'
1
2
⎟⎠⎞
⎜⎝⎛
−−=
mkNd
YXYXtrd
.∞→mas:d
,2||||, '1
'21 ji
ji
nnnnnn XtrXnkStrSSXXS ∑
<
+==++=
( ) .,0~ 1' kNXtrXmu jiij =
( )
,2
1,~||||
1||||
1
1
⎟⎠⎞
⎜⎝⎛ −
∴
++=∴ ∑<
−
mnnkNS
mOunkm
nkS
n
jiijn
.∞→mas:d
:d
(i)
where
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( )( ) ( )
,2~
.2
1,~
22
2
22''
qn
qijji
ji
nnktrMm
knnqXXXXtrm
χ
χ
⎟⎠⎞
⎜⎝⎛ −∴
−=∑
<
(ii)
( )( ) ,2
1
''2
2
1
'
∑
∑
<
=
+=
=
jiijjin
n
iiin
XXXXtrnn
ktrM
XXn
M
.∞→mas:
:
d
d
where
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References[1] C. Bingham, An antipodally symmetric distribution
on the sphere, Ann. Statist. 2 (1974) 1201-1225.
[2] Y. Chikuse, Statistics on Special Manifolds, Lecture Notes in Statistics, Vol. 174, Springer, New York, 2003.
[3] P. Diaconis, D. Freedman, A dozen de Finetti-style results in search of a theory, Ann. Insti Henri Poincare,
Probabilities et Statistique 23 (1987) 397-423.
37
[4] T. D. Downs, Orientation statistics, Biometrika 59 (1972) 665-676.
[5] A. T. James, Normal multivariate analysis and the orthogonalgroup, Ann. Math. Statist. 25 (1954) 40-75.
[6] P. E. Jupp, K. V. Mardia, Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions, Ann. Statist. 7 (1979) 599-606.
[7] C. G. Khatri, K. V. Mardia, The von Mises-Fisher matrix distribution in orientation statistics, J. Roy. Statist. Soc. B 39 (1977) 95-106.
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[8] R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982.
[9] M. J. Prentice, Antipodally symmetric distributions for orientation statistics, J. Statist. Plann. Inference 6 (1982) 205-214.
[10] A. J. Stam, Limit theorems for uniform distributions on spheres in high dimensional Euclidean spaces, J. Appl. Prob. 19 (1982) 221-228.
[11] G. S. Watson, Limit theorems on high dimensional spheres and Stiefel manifolds, in: S. Karlin, T. Amemiya, and L. A. Goodman (Eds.), Studies in Econometrics, Time Series, and Multivariate Statistics, Academic Press, New York, 1983, pp. 559-570.