text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
and F :
such that F is isomorphic to the functor Hom(X, ?). More precisely, if we
exists an object X
are given such an object X, together with an isomorphism � : F = Hom(X, ?), we say that the
functor F is represented by X (using �).
C ⊃
� C
∪
C
In a similar way, one can talk about representable functors from
ca... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. Its uniqueness is verified in a straightforward
manner.
.
Remark. In a similar way, if a category
is enriched over another category
of abelian groups or vector spaces), one can define the notion of a representable functor from
C
D
(say, the category
to
C
D
Example 6.10. Let A be an algebra. Then the forgetfu... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
× D
� D
⊃
⊃
D
C
Not every functor has a left or right adjoint, but if it does, it is unique and can be constructed
canonically (i.e., if we somehow found two such functors, then there is a canonical isomorphism
between them). This follows easily from the Yoneda lemma, as if F, G are a pair of adjoint functors
then F... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
�
Nondegeneracy of the inner product (on both sides)
If dim V =
, not
Left and right adjoint operators
Adjoint operators may not exist if dim V =
If they do, they are unique
W
= Hom(V, k)
given by f (v) = (u, v), u
V ⊕, f (v) = (u, v)
C ⊃ D
C ⊃
⊃
V ⊕
V ⊕
�
�
≤
≤
f
�
⊕
V
Example 6.13. 1. Let V be a fin... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
⊃
5. The left adjoint to the forgetful functor Assock
Vectk is the functor of tensor algebra:
T V . Also, if we denote by Commk the category of commutative algebras, then the left adjoint
⊃
V
to the forgetful functor Commk
�⊃
Vectk is the functor of the symmetric algebra: V
SV .
�⊃
⊃
One can give many more e... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
Example 6.15. The category of modules over an algebra A and the category of finite dimensional
modules over A are abelian categories.
Remark 6.16. The good thing about Definition 6.14 is that it allows us to visualize objects,
morphisms, kernels, and cokernels in terms of classical algebra. But the definition also has ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
cohomology of this complex is H i = Ker (di)/Im(di
Xi+1 (thus
−
n). The complex is said to be exact in the i-th term if
the cohomology is defined for 1
H i = 0, and is said to be an exact sequence if it is exact in all terms. A short exact sequence
is an exact sequence of the form
1), where di : Xi
⊃
⊃
i
∗
∗
... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
⊃
⊃
Y
⊃
Z,
F (X)
F (Y )
⊃
⊃
F (Z)
⊃
is exact. F is right exact if for any exact sequence
the sequence
X
Y
⊃
⊃
Z
⊃
0,
⊃
is exact. F is exact if it is both left and right exact.
⊃
F (X)
F (Y )
F (Z)
0
⊃
104
Definition 6.21. An abelian category
C
splits, i.e., is isomorphic to a sequence
is ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
Show that if (F, G) is a pair of adjoint additive functors between abelian categories,
then F is right exact and G is left exact.
Exercise. (a) Let Q be a quiver and i
Q a source. Let V be a representation of Q, and W a
representation of Qi (the quiver obtained from Q by reversing arrows at the vertex i). Prove th... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
is a free A-module, i.e., a direct sum of
�
(iv) The functor HomA(P, ?) on the category of A-modules is exact.
Proof. To prove that (i) implies (ii), take N = P . To prove that (ii) implies (iii), take M to be free
(this can always be done since any module is a quotient of a free module). To prove that (iii) implies... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
��rst establish the statement in the case when I 2 = 0. Note that in this case I is a
I, and e0a = ae0.
left and right module over A/I. Let e
We look for e in the form e = e
⊕
I. The equation for b is e0b + be0
be any lift of e0 to A. Then e
�
b = a.
⊕
+ b, b
= a
−
e
2
⊕
⊕
�
−
Set b = (2e0
1)a. Then
−
... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
k = m
1, and let us prove it for k = m. So we have an idempotent
em
A/I m, and we have to lift it to A/I m+1 . But (I m)2 = 0 in A/I m+1, so we are done.
−
1
−
�
Definition 7.4. A complete system of orthogonal idempotents in a unital algebra B is a collection
of elements e1, ..., en
B such that eiej = ζijei, an... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
orem 7.6. (i) For each i = 1, ..., n there exists a unique indecomposable finitely generated
projective module Pi such that dim Hom(Pi, Mj ) = ζij .
(ii) A =
n
=1(dim Mi)Pi.
�i
(iii) any indecomposable finitely generated projective module over A is isomorphic to P i for
some i.
0
�
n
=1 End(Mi), and Rad(A) is a ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
= Q1
either for l = 1 or for l = 2, so either Q1 = 0 or Q2 = 0.
�
Q2, then Hom(Ql, Mj ) = 0 for all j
Also, there can be no other indecomposable finitely generated projective modules, since any
such module has to occur in the decomposition of A. The theorem is proved.
References
[BGP] J. Bernstein, I. Gelfand, V.... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
7
Continuous-Time
Fourier Series
In representing and analyzing linear, time-invariant systems, our basic ap-
proach has been to decompose the system inputs into a linear combination of
basic signals and exploit the fact that for a linear system the response is the
same linear combination of the responses to the basic i... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/24eca8c9c36bf5931c022e17fd44dbfc_MITRES_6_007S11_lec07.pdf |
and complex exponentials leads to some ex-
tremely powerful concepts and results.
Before capitalizing on this property of complex exponentials in relation
to LTI systems, we must first address the question of how a signal can be rep-
resented as a linear combination of these basic signals. For periodic signals,
the rep... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/24eca8c9c36bf5931c022e17fd44dbfc_MITRES_6_007S11_lec07.pdf |
an increasingly better approximation to the square wave
except at the discontinuities; that is, as the number of terms becomes infinite,
the Fourier series converges to the square wave at every value of T except at
the discontinuities. However, for this example and more generally for period-
ic signals that are square-... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/24eca8c9c36bf5931c022e17fd44dbfc_MITRES_6_007S11_lec07.pdf |
The principle of
superposition for
linear systems.
x ( ]
x [n]
y (t)
y [n]
If:
x(t) = a, 0, (t) + a2 0 2 (t) +
ek(t)
k(t)
and system is linear
Then: y(t) = a, 0 1 (t) + a2 0 2 (t) +
Identical for discrete-time
If:
x= a, 1 +a2 02 +.
Then: y = a, 01 + a2+'...
Choose $k (t) or $k [n] so that:
TRANSPARENCY
7.2
Criteria f... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/24eca8c9c36bf5931c022e17fd44dbfc_MITRES_6_007S11_lec07.pdf |
9kn
skcomplex
=> Laplace transforms
zk complex
=> z-transforms
Continuous-Time Fourier Series
MARKERBOARD
7.1
C*T ;earier Seeres
Wct -'At+T.)-
TOU
e ?
Proos 0
"
C' cV
k
.t
pe i4-
'I
Per lea,
a.+
' Ak cs(k w.4+ ev
kms
oL'
F'13 GuO0MVe4 *
r
iv>'
.
~Pr, O;5,
*yj
-p
e -r- 1 .o or ',
The minus sign at the beginni... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/24eca8c9c36bf5931c022e17fd44dbfc_MITRES_6_007S11_lec07.pdf |
AVE
J7rak
-2/5
-2/3
0 1
2
3
k
ao =0;
ak = 1
1
(-1) k}
k#0
e odd harmonic
oak imaginary
eak = -a- k (antisymmetric)
sine series
x(t) = ao +
oo
, 2j aksinkwoOt
k=1
Continuous-Time Fourier Series
TRANSPARENCY
7.7
Fourier series
coefficients for a
symmetric periodic
square wave.
x(t) = E
ak ejkot
=27r
vo-To
k= - o... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/24eca8c9c36bf5931c022e17fd44dbfc_MITRES_6_007S11_lec07.pdf |
TRANSPARENCY
7.9
Partial sum
incorporating
(2N + 1) terms in the
Fourier series.
[The analysis equation
should read ak -
1/Tfx(t)e -jk"ot dt
TRANSPARENCY
7.10
Conditions for
convergence of the
Fourier series.
+00
x(t) =
ak ejkwot
synthesis
k =-00
a
=
kT To
XN (t)
N
k=-N
x(t) eikcoot analysis
ak e jkwot
eN (t) =x(t)... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/24eca8c9c36bf5931c022e17fd44dbfc_MITRES_6_007S11_lec07.pdf |
7.12
Representation of an
aperiodic signal as the
limiting form of a
periodic signal with
the period increasing
to infinity.
X(t) = x(t)
Iti< 2
As To -- oo
i(t) x(t)
- use Fourier series to represent x(t)
- let T0-o-ooto represent x(t)
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Signals and Systems
Professor A... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/24eca8c9c36bf5931c022e17fd44dbfc_MITRES_6_007S11_lec07.pdf |
§ 17. Channels with input constraints. Gaussian channels.
17.1 Channel coding with input constraints
Motivations: Let us look at
∼
∞
since supP IX
y, limitation of
infinite
realit
constraints on input distribution.
operations
(X; X + Z
second moment.
=
)
In
⇒
the additive Gaussian noise. Then the Shannon capacity is infi... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
• A code is an (n, M, (cid:15), P )-code if it is an (n, M, (cid:15))-code satisfying input constraint Fn ≜ {xn
∶
1
n ∑ (xk
c
)
≤ P
}
• Finite-
n
fundamen
tal limits:
M ∗
∗
{
n, (cid:15), P ) = max
(
M
{M
(
) =
Mmax n, (cid:15), P
max
• (cid:15)-capacity and Shannon capacity
∶ ∃(
∶ ∃(
)
}
-code
n, M, (cid:15), P
max-co... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
1 c Xk
}
Definition 17.4 (Admissible constraint). P is an admissible constraint if
c, and can
P
(
P0,
P . The set of admissible P ’s is denoted by
[ (
PX E c X
[
∞)
P0
,
)] ≤
, where
⇔ ∃
∞)
)
(
c x .
inf x
P0
or
D
≜
∶
∈A
0
∈ A
∃
x
be either
(
s.t. c x0
in the
) ≤
form
,
Clearly
∉ D
input constraint. So in the remaining
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
+
)
(
(X0; Y0)+
λf P1 .
λf P0
λI X1; Y1 . Hence f λP0 λP1
(⋅)
.
To extend these results to Ci P observe that for
vity of f
every
n
)] ≤
)+
∼
¯
(
P ↦
1
n
sup
PXn ∶E[c(X n)]≤P
I(X n; Y n)
is concave. Then taking lim inf n→∞ the same holds for Ci(P ).
An immediate consequence is that memoryless input is optimal for memory... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
) ≤
ˆ
sup
P nX ∶E[c(X n)]≤P
I(X n; Y n
) ≤
nf
(P )
Theorem
X ∀ > ∀
, γ
0, M , there exists an M, (cid:15) max-code with:
)
(
17.2 (Extended Feinstein’s
Lemma). Fix a random transformation PY X . PX , F
∀ ⊂
∀
∣
• Encoder satisfies the input constraint: f M F
] → ⊂ X
∶ [
;
• Probability of error bound:
(cid:15)PX (F ) ≤ P... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
+ [
X
X
]
j
)
= 1 − (cid:15)PX (F
)
From here, we can complete the proof by following the same steps as in the pro
lemma (Theorem 15.3).
of
of Feinstein’s
Theorem 17.3 (Achievability). For any information stable channel with input constraints and
P > P0 we have
C(P ) ≥ Ci(P )
(17.3)
Proof. Let us consider a special cas... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
„„„„„„„„„„„„„„„„„„„„„„„„
„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶
„„„„„„„„„„„„„„„„„„„„„„„„„
stationary
memoryless
WLLN and
y
b
n( (X; Y
I
assumption
+ exp
·„„„„„„„„„„„„„„„„„
(−nδ
)
‚„„„„„„„„„„„„„„„„„„¶
0
→
Also, since E[c(X)] < P , by WLLN, we have
P
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
15) P
)
≥
lim
0
→
δP
∶
sup
PX E[c(X)]<
sup
PX E c X P
∶ [ (
)]<
(I
(
X
; Y ) − 2δ
)
I
(
X
; Y
)
=
Ci P
( −) =
Ci P
( )
where the last equalit
that
P i X n; Y n) ≤ n(Ci − δ)) →
( (
for general information
is
y
from
the
con
i on (P0, ∞) by Proposition 17.1. Notice
stable channel, we just need to use the definition to show... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
P .
≤
)
n .
separable
2
) ⊥⊥
cost constraint:
A = B =
∼ N (
0, I
∣
Note: Here “white” = uncorrelated
Note: Complex AWGN channel is similarly defined: A = B = C, (x) = ∣x∣2
Theorem 17.5. For stationary (C)-AWGN channel, the channel capacity is equal to
capacity, and is given by:
, and Z
enden
indep
t.
∼
c
n
Gaussian
=
)
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
-ball of radius r in R is given by cnr
of an (cid:96)
for some
n
))
constan cn. Then
. Take the log
t
cn(nσ2)n/2
and divide by n, we get 1
σ2 )n/2
)
and
cn(n(P +σ2
= (1 + P
σ2 ).
2 log (1 + P
√
√
n
n
/
2
Theorem 17.5 applies to Gaussian noise. What if the noise is non-Gaussian and how sensitive is
the capacity formula ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
SNR in the high-SNR regime. On the other hand, if Z is discrete, then
2
PZ∥N EZ, σ2)) = ∞ and indeed in this case one can show that the capacity is infinite because
D(
(
the noise is “too weak”.
with
]
17.3.2 Parallel AWGN channel
Definition
A = B =
R
independent for each branch.
17.6 (Parallel AWGN). A parallel AWGN cha... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
and for the constraint Pk
0 by µk. We want to
First-order condition on Pk gives that
≥
the question b
ian multipliers
1
solve max
2
∑
oils down to the last maximization
∑ ≤
Pk P by λ
for
+ λ(P − ∑ Pk).
(
log 1
constrain
t
P
µkPk
k
2
σk
the
+
) −
1
2
1
σ2
k + Pk
= λ
−
µk, µkPk
=
0
therefore the optimal
solution
is
Pk = ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
AW
x2, PYj ∣Xj Yj Xj Zj, where Zj
0, σ2
=
+
∼ N (
A
)
j .
∶
WGN channel is defined as follows:
180
Theorem 17.8. Assume that for every T the following limits exist:
˜Ci(T ) =
1
lim
n→∞ n
˜P (T ) = lim
n→∞
n
∑
j=1
n
1
∑
n 1
j
=
1
2
log+ T
σ2
j
T − σ ∣+
∣
2
j
the
capacity of the non-stationary AWGN channel is given by th... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
(
X ; Yj)) =
j
e
log2
2
Pj
Pj + σ2
j
≤
log2 e
2
and thus
n
∑
j=1
1
n2
Var
(i(Xj; Yj)) < ∞ .
From here information stability follows via Theorem 16.9.
Note: Non-stationary AWGN is primarily interesting due to its relationship to the stationary
Additive Colored Gaussian noise channel in the following discussion.
17.5* St... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
U Cov(Zn
)U ∗ = Σ̃
Therefore
w have the
e
equiv
alent channel as follows:
Ỹ n = X̃n + Z̃n, Z̃n
j ∼ N (0, σ2
j ) indep across j
By Theorem 17.8, we have that
̃ =
C lim
1
n→∞ n
n
∑
j=1
n
1
∑
n =1
j
lim
n→∞
log+ T
σ2
j
=
1
2π
∫
0
2π
1
2
log+
T
fZ(ω)
∣ − σ2
T
j ∣+ = P (T )
dω. ( by Szeg¨o, Theorem 5.6)
Finally since U is u... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
, 2π
(
with ISI is given by
]. Then the capacity of the AWGN channel
C(T ) =
P (T ) =
2π
2π
1
2π
1
2π
∫
0
∫
0
log+(T ∣H(ω)∣2)dω
1
2
∣T −
1
H(ω)∣2
∣
+
∣ dω
Proof. (Sketch) At the decoder apply the inverse filter
equivalent channel then becomes a stationary colored-noise Gaussian channel:
with
frequency response ω ↦ 1
.
T... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
.
+
(
For details, see [Smi71] and [PW14, Section III].
183
17.8* Gaussian channels with fading
Fading channels are often used to model the urban signal propagation with multipath or shadowing.
The received signal Yi is modeled to be affected by multiplicative fading coefficient Hi and additive
noise Zi:
+
Yi HiXi Zi, Zi... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/24fb44b6a101e847c32ffe38bbeeea6e_MIT6_441S16_chapter_17.pdf |
Lecture #6: Background
There are two pictures of the acoustic field that have very clear physical interpretation: rays and normal
modes. The ray is perhaps the simplest to interpret, as the dominant paths the energy takes. The mode
picture is also simple for scientists and engineers to interpret, as an expansion of ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/25058a02940443d8cae6dde3271fdbb6_MIT2_682S12_bglec06.pdf |
rically symmetric ocean waveguide (see Fig. 5.1). (Remember that, by superposition,
we can create extended sources from the point solutions.) By cylindrical symmetry, the azimuthal part
of the solution factors out trivially and we quickly proceed to Eq. (5.5) which just has p (r, z) to look for.
At this point (thoug... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/25058a02940443d8cae6dde3271fdbb6_MIT2_682S12_bglec06.pdf |
pressure, p(r,z). Eq. 5.22
shows a simple asymptotic form of this. We are now “ready to roll!”
In section 5.2.1 of Frisk, we come to the “easy to understand, analytic toy solution” to the waveguide – a
very useful thing, and actually not even that horrible a first approximation to reality! Since this system is
a on... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/25058a02940443d8cae6dde3271fdbb6_MIT2_682S12_bglec06.pdf |
next section, “Modal intensity, interference wavelength, and cycle distance” should be a separate
chapter heading in my mind – it is simple, but very important. One of the most usual measurements in
ocean acoustics is p(r,z), from which one quickly processes the intensity, | ( )| . Frisk shows this in
modal form in ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/25058a02940443d8cae6dde3271fdbb6_MIT2_682S12_bglec06.pdf |
.7). This is
the so-called “Pekeris model,” and it is of enormous practical use! In Eqs. 5.68-5.71, and Fig. 5.8, Frisk
follows an old Clay and Medwin derivation to get the modal eigenvalue equation for this system in four
simple steps! Eq. (5.71) is another “commit to memory” gem, and contains a wealth of physics i... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/25058a02940443d8cae6dde3271fdbb6_MIT2_682S12_bglec06.pdf |
velocity, Cwater < C < Cbottom. The group
velocity is even more interesting, going from Cbott near modal cutoff frequency (the “ground wave”) to
a minimum with V< Cwater (the “Airy phase”) to Cwater at high frequencies (the “water wave”). Frisk
gives long, but very useful, analytical forms for and , which can be use... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/25058a02940443d8cae6dde3271fdbb6_MIT2_682S12_bglec06.pdf |
3 Spectral Clustering and Cheeger’s Inequality
3.1 Clustering
Clustering is one of the central tasks in machine learning. Given a set of data points, the purpose of
clustering is to partition the data into a set of clusters where data points assigned to the same cluster
correspond to similar data points (depending on t... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
Indeed,
Lloyd’s algorithm oftentimes gets stuck in local optima of (25). A few lectures from now we’ll discuss
convex relaxations for clustering, which can be used as an alternative algorithmic approach to Lloyd’s
algorithm, but since optimizing (25) is N P -hard there is not polynomial time algorithm that works
in the... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
5), such as
(cid:0)
K(cid:15)(u) = exp 1 u2(cid:1), by associating each point to a vertex and, for which pair of nodes, set the edge
weight as
2(cid:15)
wij = K(cid:15) ((cid:107)xi − xj(cid:107)) .
43
Figure 16: Examples of points separated in clusters.
Recall the construction of a matrix M = D−1W as the transition m... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
λt
2ϕ2(i)
.
..
λt
kϕk(i)
and clustering the points φt
tering.
(k 1)
− (1), φt
(k 1)
− (2), . . . , φt
(k 1)
− (n) ∈ Rk−1 using, for example, k-means clus-
44
Figure 17: Because the solutions of k-means are always convex clusters, it is not able to handle some
cluster structures.
3.3 Two clusters
We will mostly f... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
� (since cut(∅) = 0) which is a rather
meaningless choice of partition.
Remark 3.3 One way to circumvent this issue is to ask that |S| = |Sc| (let’s say that the number of
vertices n = |V | is even), corresponding to a balanced partition. We can then identify a partition with
{±1}n where yi = 1 is i ∈ S, and yi = −1 ot... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
Sc)
.
Note that Ncut(S) and h(S) are tightly related, in fact it is easy to see that:
h(S) ≤ Ncut(S) ≤ 2h(S).
13W is the matrix of weights and D the degree matrix, a diagonal matrix with diagonal entries Dii = deg(i).
46
Both h(S) and Ncut(S) favor nearly balanced partitions, Proposition 3.5 below will give an inter-
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
G = (V, E, W ) and a partition (S, Sc) of V , Ncut(S) corresponds
to the probability, in the random walk associated with G, that a random walker in the stationary
distribution goes to Sc conditioned on being in S plus the probability of going to S condition on being
in Sc, more explicitly:
Ncut(S) = Prob {X(t + 1) ∈ Sc... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
analogously to the balanced partition.
Recall that balanced partition can be written as
1
min
4 y∈{−1,1}n
1T y=0
yT LGy.
An intuitive way to relax the balanced condition is to allow the labels y to take values in two
different real values a and b (say yi = a if i ∈ S and yj = b if i ∈/ S) but not necessarily ±1. We can
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
)
= Ncut(S).
This means that finding the minimum Ncut corresponds to solving
min yT LGy
s. t. y ∈ {a, b}n for some a and b
yT
Dy = 1
yT D1 = 0.
(cid:50)
(26)
Since solving (26) is, in general, NP-hard, we consider a similar problem where the constraint that
y can only take two values is removed:
min yT LGy
s. t. y ∈ Rn
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
2 (LG)
and the minimizer is given by the second smallest eigenvector of LG = I − D− 2 W D− 2 , which is the
second largest eigenvector of D− 2 W D− 2 which we know is v2. This means that the optimal y in (27)
is given by ϕ2 = D− 1
2 v2. This confirms that this approach is equivalent to Algorithm 3.2.
its
1
1
1
1
1
Becau... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
[Chu10] for
four different proofs!). The proof that follows is an adaptation of the proof in this blog post [Tre11]
for the case of weighted graphs.
Proof. [of Lemma 3.8]
We will show that given y ∈ Rn satisfying
R
(y) :=
yT L
Gy
yT Dy
≤ δ,
50
and yT D1 = 0. there is a “rounding of it”, meaning a threshold τ and a corr... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
random with the distribution
Prob {τ ∈ [a, b]} =
(cid:90)
b
a
2
|τ |dτ,
where x1 ≤ a ≤ b ≤ xn.
It is not difficult to check that
Prob {τ ∈ [a, b]} =
(cid:26) (cid:12)
(cid:12)b2 − a2(cid:12)
(cid:12)
a2 + b2
if a and b have the same sign
if a and b have different signs
Let us start by estimating E cut(S).
E cut(S) = E
1
2... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
2
xj) = 2xT LGx
≤
2δxT Dx.
Also,
ij
wij(
|xi| + |x |)2 ≤
j
(cid:88)
ij
(cid:88)
ij
w 2x2 + 2x2. = 2
ij
j
i
deg(i)x2
(cid:33)
i + 2
(cid:32)
(cid:88)
i
(cid:88)
j
deg(j)xj = 4x Dx.
T
2
This means that
E cut(S) ≤
√
2δxT Dx
√
1
2
4xT Dx =
√
2δ xT Dx.
On the other hand,
E min{vol S, vol Sc} =
n
(cid:88)
deg(i) Prob... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
, since both random variables are positive,
E cut(S) ≤ E min{vol S, vol Sc
√
} 2δ,
or equivalently
(cid:104)
E
cut(S) − min{vol S, vol Sc
√
(cid:105)
} 2δ ≤ 0,
which guarantees, by the probabilistic method, the existence of S
such that
cut(S) ≤ min{vol S, vol Sc
√
} 2δ,
which is equivalent to
h(S) =
cut(S)
min{vol S, v... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
(cid:15),
cut(S
)
2.
cut(S)
vol( )
S
≥ 1
− (cid:15), for every S
⊂ V satisfying
3.5 Computing Eigenvectors
v
ol(
S) ≤ δ vol(V ).
Spectral clustering requires us to compute the second smallest eigenvalue of LG. One of the most
efficient ways of computing eigenvectors is through the power method. For simplicity we’ll consi... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
the case, what is important is that it never produces erroneous certificates
(and that it has a bounded-away-from-zero probably of succeeding, provided that M (cid:23) 0).
14Note that, in spectral clustering, an error on the calculation of ϕ2 propagates gracefully to the guarantee given by
Cheeger’s inequality.
53
The ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
k a positive integer, is the following true?
ρG(k) ≤ polylog(k)
λk.
(cid:112)
(30)
We note that (30) is known not to hold if we ask that the subsets form a partition (meaning that
every vertex belongs to at least one of the sets) [LRTV12]. Note also that no dependency on k would
contradict the Small-Set Expansion Hypot... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/251ccb47d3003505f06d27c8f91f831d_MIT18_S096F15_Ses8_11.pdf |
18.465, April 12, 2005
Some notes on location and scatter functionals
�
Recall that a sequence Qk of laws (probability measures), here on Rd, is
�
said to converge weakly to a law Q if f dQk → f dQ for every bounded
continuous function f . There exists a metric ρ on the set of all laws on Rd
which metrizes weak ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
majority, of laws P treated as
normal to an acceptable approximation in practice, such as laws P on R with
P ([0, ∞)) = 1, and laws discretized by rounding to finitely many decimal
places. The latter laws also cannot be obtained by replacement of up to half
a normal or other continuous law, but can be quite close to... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
asymptotics of the breakdown point (largest k/n for replacement, or
∗
1
k/(n + k) for adjunction, such that Tn doesn’t escape from all compact sets)
as n → ∞ are considered. Another type of definition is for functionals T
defined on laws P , which yield estimators when applied to empirical measures
Pn. In a function... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
(P )}0≤ε<∞ is a collection of subsets of P, then
{Nε(P ) : 0 ≤ ε < ∞, P ∈ P} will be called a suitable set of neighborhoods
iff for all P ∈ P, (a) N0(P ) = {P }, (b) For 0 ≤ ε < ε(cid:5) , Nε(P ) ⊂ Nε� (P ), and
(c) For ε > 0, ε(cid:5) > 0, Q ∈ Nε(P ), and ρ ∈ Nε� (Q), we have ρ ∈ Nε+ε� (P ).
These conditions clearl... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
of
dimension d − 1. I.i.d. samples from such laws are almost surely in general
position. On finite samples, breakdown (in the replacement or contamination
2
∗
sense) is usually considered for samples in general position. At other laws or
samples, the breakdown points may be lower.
Another issue is: for 0 < ε < ε ,... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
0 ≤ ε < ∞, P ∈ P
:= P(X, B)} a suitable set of neighbor-
hoods. Let T be a functional defined uniquely on a domain D ⊂ P with
values in Θ. Then for each P ∈ D, the explosion breakdown point of T at P
is
∗
ε (T, P ) := ε (T, P, {Nε}0≤ε<∞, Θ) := inf{ε ∈ [0, 1] :
for each compact K ⊂ Θ, T (Q) /∈ K for some Q ∈ D ∩ Nε(... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
3
∗
Definition. Let r (T, P ), the radius of continuity of T at P , be defined as
∗δ (T, P ) with the additional requirement that T (·) is weakly continuous at
Q for all Q ∈ Nε(P ). Define r ∗ and rd again analogously.
∗
C
∗
If neighborhoods Nε are defined by the total variation (replacement) dis-
tance d1 then the c... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
+ v, and any law Q ∈ D, the image measure P := Q ◦ f −1 ∈ D
also, with µ(P ) = Aµ(Q) + v or, respectively, Σ(P ) = AΣ(Q)A(cid:5). For d = 1,
σ(·) with 0 ≤ σ < ∞ will be called an affinely equivariant scale functional
iff σ2 satisfies the definition of affinely equivariant scatter functional. If we
have affinely equivariant ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
β = ( 1 − p)/(1 − p), with β = 1/2 only for continuous laws P . Such a
dependence on Θ naturally also occurs for other scale functionals, e.g. the
interquartile range.
∗
C
∗
2
Let T be an affinely equivariant location or scatter functional. Then εC ,
∗
δC , and r are all affinely invariant and 1/2 as a target maximal... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
rd > 0 are affinely
invariant. Thus, one may seek T for which these sets are as large as possible,
rather than making the values of ε as large as possible.
d
∗
d
∗
d
∗
∗
∗
∗
∗
∗
∗
∗
Location functionals which in some respects improve on the median and
still have δC = 1/2 at all laws have been proposed, especial... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
Rd. Let P ∈ D be such that P ◦ A−1 = P for each A ∈ A.
Then
(a) µ(P ) ∈ SA := {x ∈ Rd : Ax = x for all A ∈ A}.
(b) If SA is a singleton {xA}, then µ(P ) = xA.
(c) If for some v ∈ Rd , A consists of the one map x (cid:7)→ 2v −x, then µ(P ) = v.
(d) Let 2 ≤ n ≤ d + 1. Let V be a set of n points of Rd in general posi... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
equivariant location
functional. Then part (b) follows from part (a). For part (c), note that
x (cid:7)→ 2v − x has a unique fixed point v, so (c) follows from (b); here P is
symmetric around v.
For part (d), let x1, ..., xn be the points of V . Since they are in general
position, the vectors vj = xj − x1 for j = 2... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
than u and
v. For V the set of vertices of a regular simplex, we can take A as reflection
in the (d − 1)-dimensional hyperplane perpendicular to u − v and through
the midpoint of the line segment from u to v, so the affine transformations
A
�
n
i=2 si(xi −x1) : si ∈ R, i = 2, ..., n}, an (n−1)-dimensional
�
�
n
n
... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
A
is uniquely determined on H as stated.
�
n
6
There is an affine transformation AH of Rd such that AH x = x for all
x in H and AH y (cid:11)= y for all y not in H. Then AH induces the identity
permutatation of V , and we can assume it is the affine transformation chosen
to do so since P (H) = 1 and so P ◦ A−1 = P ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
envector of Σ(P ) ∈ Nd. Eigenvectors with distinct
= w in V , v − w and v − u are not
eigenvalues are orthogonal, but for v (cid:11)
= u (cid:11)
orthogonal, so they must have the same eigenvalue. Iterating, we find that
all such eigenvectors have the same eigenvalue c ≥ 0. Since they span Rd ,
Σ(P ) = cI follows. 2... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
ent reasons, they also found for another
class of estimators, as Maronna [26] did earlier for equivariant M-estimators
of location and scatter.
Here is a related consequence of Theorem 1, not directly about breakdown
points:
Proposition 2. For d = 1, 2, ..., there is a sequence {Qm}m≥3 of laws on
dR having densit... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
A as in Theorem 1(e) and (d),
ρ
r
since each A ∈ A is an orthogonal transformation preserving Ur , we have
r ◦ A−1 = ρr . Let µ(·) be an affinely equivariant location functional defined
ρ
at ρr . Then µ(ρr ) = e1/(d+1). Let Ma((x1, . . . , xd)(cid:5)) := (ax1, x2, . . . , xd)(cid:5) for
any a > 0 and τr := ρr ◦M −1. ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
) ≤ cd}, where cd is chosen so that if P is a normal distribution, Σ is
its covariance matrix. Then µ and Σ are affinely equivariant location and
scatter functionals of P respectively, because any affine transformation A
with Ax ≡ Bx + v takes ellipsoids to ellipsoids and multiplies all volumes by
the same (Jacobian) f... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
some restricted limit (the limit apparently cannot have a density). In the
given proof, the limit is concentrated in a union of d line segments parallel to
the x1 axis and so gives mass k/(d + 1) to some k-dimensional hyperplanes
for k = 1, . . . , d − 1.
1
∗
C
Proposition 6 will show that any affinely equivariant ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
any matrix,
possibly singular. It’s easily seen that if a functional is defined on all laws,
affinely equivariant, and weakly continuous, then it is singularly affine equiv-
ariant. For empirical measures Pn = n−1(δX1 +· · ·+δXn ), the classical sample
mean and covariance are evidently singularly affine equivariant. It tur... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
n, then µ(Pn) ≡
�
xdPn, the sample mean.
(b) If in addition µ(·) is defined for all n and all Pn on Rd, then as n varies,
µ(·) is not weakly continuous. Thus, there is no affinely equivariant, weakly
continuous location functional defined on all laws on Rd for d ≥ 2.
Proof. Part (a) follows from a result and proof of... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
= cn(X −X1(cid:5)
such that for any X, Σ(X −X1(cid:5)
applied to centered data matrices, Σ is proportional to the sample covariance
matrix.
(b) If Σ(·) is an affinely equivariant scatter functional defined for all n and
n on Rd for d ≥ 2, weakly continuous as a function of Pn, then Σ ≡ 0.
P
n)(X −X1(cid:5)
Proof. (... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
≡ ab(U, V ) follows for B11 = a, B22 = b,
and Bij = 0 otherwise. It remains to prove biadditivity (U, V + W ) ≡
(U, V )+(U, W ). For d ≥ 3 this is easy, letting X 3 = W , B11 = B22 = B23 = 1,
and Bij = 0 otherwise. For d = 2, we first get (U + V, V ) = (U, V ) + (V, V )
from B = (1
1
1). Symmetrically, (U, U + V ) =... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
+(cid:15)W +Z(cid:15)2−(cid:15)W −Z(cid:15)2 ,
letting first U = W + Y , V = Z, then U = W − Z, V = Y , then U = W ,
V = Z, and lastly U = W , V = Y . Applying (3) and dividing by 4 gives
(W, Y + Z) ≡ (W, Y ) + (W, Z), the desired biadditivity. So (·, ·) is indeed a
semi-inner product, in other words there is a C(n)... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
π(i) for a function π from {1, 2, . . . , d} into
itself with π(1) = j and π(2) = k. Then (BX)1 = X j and (BX)2 = X k .
Thus (X j , X k) = Σ12(BX) = Σjk(X), recalling (2) for j = k.
Let X ∈ Rd have ith component X
i and Y j := (X j )(cid:5). Then
Σjk(X − X1(cid:5)
n) = (Y j − X
j 1n, Y k − X 1n) = cn(Y j − X
k
j ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
, but Σ(Pn) don’t converge to 0 unless c1 = 0 and
so cn = 0 for all n, proving (b). 2
√
:=
So, the three properties of T : (a) affine equivariance, (b) weak continu-
ity on its domain D, and (c) being everywhere defined, cannot all hold for
location or scatter functionals on Rd for d ≥ 2 although they can for d = 1. ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
ers the possibility that continuity can be improved
to Fr´echet differentiability of some order or all orders with respect to some
norm metrizing weak convergence.
For some location and scatter functionals or estimators T on Rd for d ≥ 2,
there are ηk with 0 ≤ η0 ≤ η1 ≤ · · · ≤ ηd−1 < 1 and ηd−1 > 0 such that T (P )... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
scale-adjusted M-estimates
of location for d = 1 (Huber [20, §§6.5,6.6], Rousseeuw and Croux [32]) or for
d > 1 in the Stahel-Donoho functional, where univariate functionals µ, σ with
more continuity can be used, specifically, tν -functionals (Tyler [38], Maronna
and Yohai [27]).
Rousseeuw’s minimum-volume-ellipsoi... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
to Q weakly with
det Σ(Qk ) → 0. If there is no such y set κ(Σ) := 1. For d = 1, the collapse
and breakdown points of a scale functional σ(·) are defined as those of the
scatter functional σ2(·).
Remarks. For an affinely equivariant scatter functional on a non-empty do-
main D, the “no such y” case cannot occur. Hampe... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
and Tyler [21],
and thus have low breakdown point at laws with somewhat smaller values of
p. The following shows that even allowing det Σ = 0, there is still a tradeoff
between (definition-explosion) breakdown and collapse points.
Theorem 5. Let Σ be any affinely equivariant scatter functional with values
in Θ = Nd de... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
= 1, 2, . . ., let ρk := (1 − λ)P + λ(G ◦ M −1), so ρk ∈ D, and
d
∗
k
ζk := ρk ◦ M −1
1/k = (1 − λ)(P ◦ M −1
1/k ) + λG.
Then by affine equivariance, ζk ∈ D and det Σ(ζk ) = det Σ(ρk )/k2 → 0. Also,
:= (1 − λ)τ + λG where τ is a law concentrated
k converge weakly to ζ
ζ
in the hyperplane H := {x1 = 0}. Since G is a... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
the domain of µ(·) cannot be extended to contain any law 2(δa + δb)
with a (cid:11)= b and be weakly continuous at such a law.
∗
1
Proof. We can take a = 0 and b = 1. By part of the proof of Theorem 5 with
− 1 , there is a sequence ζm,k of
1
G := δ1, for any m = 1, 2, . . . and λm := 2
laws converging weakly as k ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
γ < 1. The second conclusion
bounds the usual contamination breakdown point at a general law F0, e.g. a
normal law, assuming the functional T is defined and continuous at a related
(1 − γ)-degenerate law. Such an assumption seems not to hold for many
location and scatter functionals given in the literature for γ ≤ 1... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
5). Let F0 be any law on Rd and F˜0
its projection into the linear subspace H := {x : x1 = 0} via M0. Suppose
15
that for some γ ∈ (0, 1) and law ρ on Rd, the law P := (1−γ)F˜0+γρ ∈ D and
if T = Σ, Σ(P ) is non-singular, or if T = µ, µ(P ) /∈ H. Then εR(T, P ) ≤ γ.
If in addition, T (·) is weakly continuous at P o... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
M −1) → +∞.
a
a
For T = µ, |µ(Qa ◦ M −1)| → +∞, and the second conclusion follows. 2
a
For γ = 1/(d + 1), the hypothesis µ(P ) /∈ H of Theorem 7 holds by
Theorem 1(d) with n = d + 1 and P = Pd+1 an empirical measure if P ∈ D.
3 Univariate trimming and the shorth
Let J be a probability density function on [0, 1] su... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
)J(y)dy
and the J-trimmed variance of Q, i.e. the variance of QJ ,
σ2
J (Q) :=
� ∞
−∞
2
x dQJ (x) − µJ (Q)2
16
2 (Q) if desired by a constant
c >
exist and are finite. One may multiply σ
1
J
so that if Q is the standard normal distribution then cσ2 (Q) = 1.
It is
straightforward to verify that with or without... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
0 and β + γ = 2α. Here β and γ can be chosen to minimize the
variance of the resulting distribution [3], the distance of points in its support
from the median [23], or otherwise. Location and scale functionals based on
such trimming will still give collapse point 1 − 2α and can increase δ∗ to 2α,
for α < 1/4. Asymme... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
Mα(P ) the set of midpoints (a + b)/2.
Then Kα(P ) and Mα(P ) are compact, nonempty sets. If Iα(P ) consists of
:= (a + b)/2,
just one interval [a, b], let µSh,α(P )
which for α = 1/2 Davies [7, p. 1856] calls the “middle of the shortest half”
functional; Rousseeuw and Leroy [33, p. 169] call it the “least median o... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
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