text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
2k k k
k 2k k
2 2
1
4
ik0x3i0t
ik0x3i0t
2ik0x
k0e
k0e
1
2
1
4
1
4
1
2
1
1
2
1
1
ik0xi0t
2
2 2
e
0
0
0
0
0
0
0
1 ik0xi0t
1 2ik0x4i0t
e
2
x 2
e
1
e
2
ik0xi0t
dx
2k0e
2ik0x4i0t
1
2
ik0xi0t
k0e
dx
k0e
3ik0x3i0t ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
for this particle to be flying left (in the opposite direction of x axis)?
Here we need to find the probability that particle has negative momentum. Indeed from plane
wave expansion we find that our particle can have momentum of k0 since it has a plane wave
component eik0 xi0k . The coefficient in front of this... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
AˆBˆ BˆAˆ 0
Geometrical interpretation: Recall from your linear algebra recitation and the problem set 1.
When two matrices commute they share eigenvectors.
Example: Free space Hamiltonian and momentum: Hˆ
2 2
2m x
2 , i
x
x
2
2
2
2m x
i
x
x ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
i
e
x
ikx k eikx
And from the previous lectures we remember that the plane waves are also eigenfunctions for the
free space Hamiltonian:
eikx
eikx
2 2
2m x 2
2k 2
2m
Fundamental theorem of algebra: If two operators A and B commute one can construct a basis
of the state space with eigenfunct... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
measurement
error) must obey the following relationship:
x p
2
xˆ, pˆ
2
x p
This relationship is called Heisenberg’s Uncertainty Principle. In fact it also holds for the
uncertainties of energy and time:
E t
2
Symmetries, conserved quantities and constants of motion – how do we ident... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
fest Theorem:
d
dt
d
dt
Aˆ
1 Aˆ, Hˆ
i
Aˆ
t
Aˆ x x x
1
i
Aˆ, Hˆ
x
x
Aˆ
t
x
Consequently in order for a physical quantity to be a constant of motion the corresponding
observable has to obey the following relationships:
6
A... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
��
0
Reminder:
d
dt
Hˆ 0 E const uE x
x, t
i
uE
x e
t
E
E
Schrodinger’s equation:
2
2m
r i
V
r, t
t
r, t
t
. Let’s substitute it into the Schrodinger’s equation above:
, t r
t i... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
i t
t
t
r
2
2m
V r
V r
t
r
1
t
Note that the left side of the equation only depends on position r and the right side of the
equation only depends on time t. This can only be true when both sides of the equation are
constant E – for energy. Then the equation above split... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
corresponding eigenfunctions,
the solution for this system is a linear combination of all the possible solutions corresponding to
different energies:
r, t
CEuE
E
i
E
t
r e
, where CE are the coefficients that can be determined from the initial
and boundary conditions.
This is a very ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
�� 0
x
pˆ, Hˆ
pˆ
t
0
d
dt
pˆ 0
Therefore the momentum is a conserved quantity and its eigenvalues can be used to label the
states. Then the unique labels for the eigenfunctions above would be:
uE,k x e , k
ikx
uE,k eikx
x
2mE
2
In fact we represent all the e... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
1
The eigeenfunctions of the paritty operator all are eitheer odd or evven.
f (−x)= f (x) even
f
f (−x)−= f (x) odd�
f
Does Hammiltonian foor Simple Haarmonic Osc illator commmute with thee parity operrator?
ˆ
H x
2
2
2 m2 x 2 HˆH x
2m x
1
22
22
2
x
11
22 ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
��
x
2 2
2
2m x
2
2m
2
x
mm2 x
1
2
2
2
22
x
2mm
2
2
x
2
m2 x x 0
1
2
1
2
m2
2
x x
This meaans that one can always ffind a set of eigenfunctioons commonn to Hˆ and ˆ . In fact, ll... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
6.241 Dynamic Systems and Control
Lecture 9: Transfer Functions
Readings: DDV, Chapters 10, 11, 12
Emilio Frazzoli
Aeronautics and Astronautics
Massachusetts Institute of Technology
March 2, 2011
E. Frazzoli (MIT)
Lecture 9: Transfer Functions
Mar 2, 2011
1 / 13
Asymptotic Stability (Preview)
We have seen tha... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
1Bu[i] + Du[t]
�
i=0
or
y (t) = C exp(At)x(0) + C
� t
0
exp(A(t − τ ))Bu(τ ) dτ + Du(t).
However, the convolution integral (CT) and the sum in the DT equation are
hard to interpret, and do not offer much insight.
In order to gain a better understanding, we will study the response to
elementary inputs of a for... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
oli (MIT)
Lecture 9: Transfer Functions
Mar 2, 2011
4 / 13
Output response to elementary inputs (1/2)
Recall that,
y (t) = CeAt x(0) + C
� t
0
e A(t−τ )Bu(τ ) dτ + Du(t).
Plug in u(t) = u0est :
y (t) = CeAt x(0) + C
� t
0
e A(t−τ )Bu0e sτ dτ + Du0e st
�� t
�
= CeAt x(0) + C
e(sI −A)τ dτ e At Bu0 + Du0e st... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
transforms an input e
→
st
G (s) is also known as the transfer function: it
into the output G (s)e .
st
E. Frazzoli (MIT)
Lecture 9: Transfer Functions
Mar 2, 2011
6 / 13
Laplace Transform
The (one-sided) Laplace transform F : C
defined as
� +∞
F (s) =
f (t)e−st dt,
→
C of a sequence f : R
≥0 →
R is
for al... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
Lecture 9: Transfer Functions
Mar 2, 2011
8 / 13
Matrix geometric series
Recall the formula for the sum of a geometric series:
k−1
�
i
m =
i=0
1 − mk
1 − m
.
For a matrix:
k−1
�
M i = I + M + M 2 + . . . M k−1 .
i=0
k−1
�
M i (I − M) = (I + M + M 2 + . . . M k−1)(I − M) = I − M k .
i=0
i.e.,
k−1
�
M... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
a complex number.
The function G : z
describes how the system transforms an input z
→
k
into the output G (z)z .
k
G (z) is also known as the (pulse, or discrete) transfer function: it
E. Frazzoli (MIT)
Lecture 9: Transfer Functions
Mar 2, 2011
10 / 13
Z-Transform
The (one-sided) z-transform F : C → C of a seq... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
⎥
⎢
⎦
⎣
0
1
�
C = b0 b1
�
. . . bn−1
D = d
G (s) = C (sI − A)−1B + D
G (s) =
bn−1s n−1 + . . . + b0
s n + an−1s n−1 + . . . + a0
+ d
E. Frazzoli (MIT)
Lecture 9: Transfer Functions
Mar 2, 2011
12 / 13
Models of discrete-time systems
DT DT System
DT
x[k + 1] = Ax[k] + Bu[k]
y [k] = Cx[k] + Du[k]
⎡
... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
Control
Spring 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
2.092/2.093 — Finite Element Analysis of Solids & Fluids I
Fall ‘09
Lecture 4 - The Principle of Virtual Work
Prof. K. J. Bathe
MIT OpenCourseWare
Su = Surface on which displacements are prescribed
Sf = Surface on which loads are applied
Su ∪ Sf = S
; Sf ∩ Su = ∅
Given the system geometry (V, Su, Sf ), loads (... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/3bfd1ab31d0d622c3d6f151fd0776ab0_MIT2_092F09_lec04.pdf |
τij nj = fi
Sf on Sf
Compatibility: ui = u S
i
u on Su and all displacements must be continuous.
•
•
Stress-strain laws
This is known as the differential formulation.
Example
Reading assignment: Section 3.3.4
• Equilibrium
EA
EA
d2u
+ f B = 0
dx2
�
du
�
�
�
dx x=L
= R
2
(a)
(b)
Lecture 4
The Prin... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/3bfd1ab31d0d622c3d6f151fd0776ab0_MIT2_092F09_lec04.pdf |
dx =
dδu
dx
du
dx
0
External virtual work
}|
{
f Bδudx
z
Z L
+
0
Virtual work due to
boundary forces
z }| {
(cid:12)
(cid:12)
Rδu
(cid:12)L
dx are the virtual strains, du
where dδu
on Su, since we do not know the external forces on Su. To solve EA d2u
where d2u
dx2 exists ( du
where only u is continuous.
dx are the rea... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/3bfd1ab31d0d622c3d6f151fd0776ab0_MIT2_092F09_lec04.pdf |
that for any compatible
virtual displacement field imposed on the body in its state of equilibrium, the total internal virtual work is
4
Lecture 4
The Principle of Virtual Work
2.092/2.093, Fall ‘09
equal to the total external virtual work. Note that this variational formulation is equivalent to the differential
for... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/3bfd1ab31d0d622c3d6f151fd0776ab0_MIT2_092F09_lec04.pdf |
Metals and Insulators
• Covalent bonds, weak U seen by e-, with EF
being in mid-band area: free e-, metallic
• Covalent or slightly ionic bonds, weak U to
medium U, with EF near band edge
– EF in or near kT of band edge: semimetal
– EF in gap: semiconductor
• More ionic bonds, large U, EF in very large
gap, insulator... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/3bfec0ef84475e74634685665d9eaee6_lecture_7.pdf |
Note that in both photon stimulated promotion as well as thermal
promotion, an equal number of holes and electrons are produced, i.e. n=p
©1999 E.A. Fitzgerald
5
Thermal Promotion of Carriers
• We have already developed how electrons are promoted in energy with T:
Fermi-Dirac distribution
Just need to fold this i... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/3bfec0ef84475e74634685665d9eaee6_lecture_7.pdf |
2m
⎞ 2
1
e
⎟⎟
2 ⎜⎜
2
2π ⎝ h ⎠
3
2
E
k
b
F
T
E
−
kb
e
g
T
NC
EF −Eg
n = N C e kbT
7
©1999 E.A. Fitzgerald
Density of Thermally Promoted of Carriers
• A similar derivation can be done for holes, except the density of states
for holes is used
• Even though we know that n=p, we will derive a separate expre... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/3bfec0ef84475e74634685665d9eaee6_lecture_7.pdf |
the
Fermi level sits about in the center of the band gap
3
kbT
⎛
⎞
2 (me
p or n = ni = 2
⎟
⎜
2
⎝ 2πh ⎠
* )3
* mv
4
−E
g
e 2kbT
©1999 E.A. Fitzgerald
9
Law of Mass Action for Carrier Promotion
3
kbT
⎛
⎞
2 = np = 4
ni
⎟
⎜
2
⎝ 2πh ⎠
(me
* )3
* mh
2
E
−
kb
e
g
T
;
2 = NC NV e
ni
k
b
E
−
g
... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/3bfec0ef84475e74634685665d9eaee6_lecture_7.pdf |
− Eg
This can be a
measurement
for Eg
For Si, Eg=1.1eV, and let μe and μh be approximately equal at 1000cm2/V-sec (very good Si!)
σ~1010cm-3*1.602x10-19*1000cm2/V-sec=1.6x10-6 S/m, or a resistivity ρ of about 106 ohm-m max
•One important note: No matter how pure Si is, the material will always be a poor
insulato... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/3bfec0ef84475e74634685665d9eaee6_lecture_7.pdf |
r
e
t
p
a
h
4
C 4
Matrix estimation
Over the past decade or so, matrices have entered the picture of high-dimensional
statistics for several reasons. Perhaps the simplest explanation is that they are
the most natural extension of vectors. While this is true, and we will see exam-
ples where the extension from vectors t... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
� =
λj ujvj⊤ ,
j=1
X
,
λ1, . . . , λr}
r diagonal matrix with positive diagonal entries
IR that are orthonormal and V is
r} ∈
IRn that are also orthonormal. Moreover,
{
m
×
where D is a r
U is a matrix with columns
a matrix with columns
it holds that
u1, . . . , u
{
v1, . . . , vr} ∈
j uj ,
{
AA⊤uj = λ2
and
A⊤Avj = λ2
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
�Ax .
x
∈S
n−1
Norms and inner product
Let A =
aij}
{
in the following notation.
and B =
{
be two real matrices. Their size will be implicit
bij}
Vector norms
The simplest way to treat a matrix is to deal with it as if it were a vector. In
particular, we can extend ℓq norms to matrices:
|q =
A
|
a q
ij
|
|
1/q
(cid:17)... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
.
∞
∞
We are going to employ these norms to assess the proximity to our matrix
of interest. While the interpretation of vector norms is clear by extension from
the vector case, the meaning of “
kop is small” is not as transparent. The
B
following subsection provides some inequalities (without proofs) that allow a
bette... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
However, we will see that in several interesting scenar-
ios, some structure is shared across coordinates and this information can be
leveraged to yield better prediction bounds.
The model
Throughout this section, we consider the following multivariate linear regres-
sion model:
Y = XΘ∗ + E ,
(4.1)
∈
T is the matrix of... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
franchise. As
a result, one may assume that the matrix Θ∗ has each of its T columns that
is row sparse and that they share the same sparsity pattern, i.e., Θ∗ is of the
form:
IRn
∈
∈
×
0 0
0 0
• •
•
•
0 0
.
.
.
.
.
.
0 0
• •
•
•
0 0
.
.
.
.
.
.
0 0
,
Θ =
indicates a potentially nonzero entry.
∗ ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
matrix Θ∗ is low rank
or approximately so. As a result, while the matrix may not be sparse at all,
the fact that it is low rank still materializes the idea that some structure is
shared across different tasks. In this more general setup, it is assumed that the
columns of Θ∗ live in a lower dimensional space. Going back ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
|0 is small.
enables us to use thresholding methods for estimating Θ∗ when
However, this also follows from Problem 4.1. The reduction to the vector case
in the sGMM is just as straightforward. The interesting analysis begins when
Θ∗ is low-rank, which is equivalent to sparsity in its unknown eigenbasis.
Θ∗
|
Consider t... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
can always choose
− , it holds
u
N2 be a 1/4-net for
N1 be a 1/4-
It follows from Lemma 1.18
. Moreover, for any
T
−
S
12d and
1
|N | ≤
2
|N | ≤
S
1, v
T 1
2T
1.
1
−
−
d
∈ S
∈ S
u⊤Av
≤
≤
≤
It yields
1
max x⊤Av + max u⊤Av
4 u
x
∈N1
d−1
y
max max x⊤Ay + max max x⊤Av + max u⊤Av
x
1
∈N
∈S
∈S
1
4 x
1
max max x⊤Ay + max max ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
≤
k
(cid:0)
(cid:1)
(cid:0)
(cid:1)
or
f
4σ
log(12)(d
t
≥
∨
T ) + 2σ
2 log(1/δ) .
p
p
The following theorem holds.
Theorem 4.3. Consider the multivariate linear regression model (4.1) under
the assumption ORT or, equivalently, the sub-Gaussian matrix model (4.2).
Then, the singular value thresholding estimator Θsvt wit... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
τ
}
|
S λj ujvj⊤ and note that
and Sc
λj|
⊂ {
j :
j :
|
j
λj| ≤ k
F
kop ≤
τ . It
.
A
ˆΘsvt
k
Θ∗
2
F
k
−
∈
ˆ2
Θsvt
k
P
≤
¯
Θ
k
−
2
F + 2
¯
Θ
k
−
Θ∗
2
F
k
(4.4)
Using Cauchy-Schwarz, we control the first term as follows
ˆΘsvt
k
¯Θ
k
2
F ≤
−
ˆ
rank(Θsvt
¯Θ)
k
−
ˆΘsvt
¯Θ
2
op ≤
k
S
2
|
|k
−
ˆ
Θsvt
¯Θ
2
op
k
−
4.2. Multivar... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
�
2
F ≤
k
144
τ 2 +
2
λj |
Sc |
jX∈
S
jX
∈
2) and on Sc,
Since on S, τ 2 = min(τ 2,
|
ˆΘsvt
k
λj|
Θ∗
2
F ≤
k
−
432
3 min(τ 2,
2), it yields,
λj|
|
|
2
λj|
≤
min(τ 2,
j
X
rank(Θ∗)
2)
λj
|
|
432
τ 2
≤
j=1
X
= 432 rank(Θ∗)τ 2 .
In the next subsection, we extend our analysis to the case where X does not
necessarily satisfy... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
:17)
Proof. We begin as usual by noting that
Y
X ˆ
Θrk
2
F + 2nτ 2 rank(Θrk)
ˆ
k
−
which is equivalent to
k
Y
−
≤ k
XΘ∗
2
kF + 2nτ 2 rank(Θ∗) ,
XΘ∗
X ˆ r
Θ k
2
F ≤
Next, by Young’s inequality, we have
ˆ
E, Θ
h
X rk X
−
−
k
2
k
Θ∗
i −
2nτ rank(Θ ) + 2nτ rank(Θ∗) .
ˆ rk
2
2
2
E, X ˆΘrk
h
−
XΘ∗
= 2
E, U
h
2 +
i
i
1
2 k
ˆ
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
k
XΘ∗
k
−
Note that rank(N )
equality, we get
≤
E, U
h
2 =
i
Φ⊤E, N/
h
Φ⊤E
k
≤ k
≤
rank(N )
k
2
op
Φ⊤E
N
2
kF i
k
2
N
1
2
k
op k
2
N
F
k
k
Φ⊤E
rank(Θrk) + rank(Θ∗) .
2
op
ˆ
k
k
Next, note that Lemma 4.2 yields
≤ k
(cid:2)
Φ⊤E
2
op ≤
nτ 2 rank(Θrk) + rank(Θ∗) .
nτ 2 so that (cid:3)
k
ˆ
k
E, U
h
2
i
≤
Together with (4.5)... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
)
≤
min
IRd×T k −
k
Θ
∈
rank(Θ)
Y XΘ 2
F
k
≤
can be solved efficiently. To that end, let Y = X(X⊤X)†X⊤Y denote the orthog-
onal projection of Y onto the image space of X: this is a linear operator from
IRd
T . By the Pythagorean theorem, we get for any Θ
T into IRn
IRd
T ,
×
×
×
¯
∈
k
¯
Next consider the SVD of Y:
−
k
Y
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
k.
−
Once XΘrk has been found, one may obtain a corresponding Θrk by least
2
F over matrices of rank at most
k
¯
7→ k
XΘ
Y
ˆ
ˆ
squares but this is not necessary for our results.
Remark 4.5. While the rank penalized estimator can be computed efficiently,
it is worth pointing out that a convex relaxation for the rank penal... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
contains information about the moments of order 2 of the random
vector X. A natural candidate to estimate Σ is the empirical covariance matrix
ˆΣ defined by
≻
∈
(cid:2)
(cid:3)
n
ˆΣ =
1
n
XiX
i⊤ .
Using the tools of Chapter 1, we c
i=1
X
an prove the following result.
Theorem 4.6. Let X1, . . . , Xn be n i.i.d. sub-Gaus... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
k
Idkop ≤
0, by a union bound,
−
x,y
ˆ
2 max x⊤(Σ
∈N
Id)y
−
I
dkopk
−
Σ1/2
kop
be a 1/4-net for
Let
Lemma 4.2 that
N
d
−
1 such that
|N | ≤
12d. It follows from the proof of
So that for any t
≥
ˆ
IP Σ
k
(cid:0)
It holds,
Idkop > t
−
≤
(cid:1)
IP
x⊤ ˆ(Σ
x,y
X
∈N
(cid:0)
Id)y
>
t/2
.
(4.6)
−
(cid:1)
ˆ
x⊤(Σ
Id)y =
−
n1
n
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
− 2
ed Cauchy-Sc(cid:0)
(cid:0)
exp
(cid:0)
IE
i
(cid:1)(cid:1)
Z 2
− −
IE[Z 2 ]
−
1/2
,
(cid:16)
(cid:0)
where in the last inequality, we u(cid:1)s(cid:1)(cid:3)
subG
d(1), we have Z+, Z
(cid:17)
hwarz. Next(cid:1),(cid:1)(cid:3)s
ince X
subG(2), and it follows from Lemma 1.12 that
(cid:0)
(cid:2)
(cid:2)
∼
Z 2
+ −
IE... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
In particular, the right hand sid
−
−
≤
∧
(cid:17)
h
i
(
)
.
144d exp
n
2
2
t
32
t
32
t
32
≥
2d
n
2
n
(cid:16)
es our proof.
This conclud
log(144) + log(1/δ)
log(144) + log(1/δ)
2d
n
∨
(cid:17) (cid:16)
2
n
(4.7)
(0, 1) if
∈
1/2
(cid:17)
Theorem 4.6 indicates that for fixed d, the empirical covariance matrix is a
consis... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
⊤u)
d
=
(cid:12)
(cid:12)
δ.
.
Σ
k
kop
r
(cid:16)
d + log(1/δ)
n
d + log(1/δ)
n
∨
(cid:17)
with probability 1
−
The above fact is useful in the Markowitz theory of portfolio section for
IRd such that
example [Mar52], where a portfolio of assets is a vector u
|1 = 1 and the risk of a portfolio is given by the variance V... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
” half a million dimensions into only two? The answer is that the
data is intrinsically low dimensional. In this case, a plausible assumption is
that all the 1, 387 points live close to a two-dimensional linear subspace. To see
how this assumption (in one dimension instead of two for simplicity) translates
into the str... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
and v
∈ S
d
−
1. The vector v is called the spike.
4.4. Principal component analysis
95
Courtesy of Macmillan Publishers Ltd. Used with permission.
Figure 4.1. Projection onto two dimensions of 1, 387 points from gene expression data.
Source: Gene expression blog.
v
Figure 4.2. Points are close to a one dimensional sp... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
|
−
v
2
2 ≤
|
2 sin2
∠(v˜, v)
8
˜
Σ Σ 2 .
θ2 k − kop
≤
min
1
∈{±
ε
}
(cid:0)
(cid:1)
1, it holds under the spiked covariance model
Proof. Note that for any u
that
d
−
∈ S
u⊤Σu = 1 + θ(v⊤u)2 = 1 + θ cos2(∠(u, v)) .
Therefore,
v⊤Σv
−
v˜⊤Σv˜ = θ[1
−
cos2(∠(v˜, v))] = θ sin2(∠(v˜, v)) .
Next, observe that
v⊤Σv
v˜⊤
−
˜
≤
=
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
˜of Σ and in the last one, we used the fact that the matrix v˜v˜⊤
vv⊤ has rank
at most 2.
−
Next, we have that
v˜v˜⊤
k
−
vv⊤
k
2
F = 2(1
−
(v⊤v˜)2) = 2 sin2(∠(v˜, v)) .
Therefore, we have proved that
2 ∠
θ sin ( (v˜, v))
˜
2 Σ
≤ k
Σ
kop sin(∠(v˜, v)) ,
−
so that
sin(∠(v˜, v))
2 ˜
θ
≤ k
Σ Σ
− kop .
To conclude the proof... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
)
d + log(1/δ)
n
d + log(1/δ)
n
∨
(cid:17)
This result justifies the use of the empirical covariance matrix Σ as a re-
placement for the true covariance matrix Σ when performing PCA in low di-
n,
mensions, that is when d
the above result is uninformative. As before, we resort to sparsity to overcome
this limitation.
n. ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
�
Xn be
IRd such that IE XX ⊤ = Σ and X
Theorem 4.10. Let X1, . . . ,
n i.i.d. copies of a sub-Gaussian random
subGd(
vector X
kop). Assume
k
further that Σ = θvv⊤ + Id satisfies the spiked covariance model for v such
d/2. Then, the k-sparse largest eigenvector vˆ of the empirical
that
≤
covariance matrix satisfies,
∈
|0... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
that
i
d matrix M , we defined the matrix M (S) to be the
1, . . . , d
}
⊂ {
Σ(S), vˆ(S)vˆ(S)⊤
vv = Σ(S)
v(S)v(S)⊤
IRd, x(S)
| × |
IR|
×
−
−
−
−
∈
∈
S
ˆ
⊤
i
h
S
|
v⊤Σv
vˆ⊤Σvˆ
−
≤ k
ˆ
Σ(S)
Σ(S)
kopk
−
vˆ(S)vˆ(S)⊤
−
v(S)v(S)⊤
k1 .
Following the same steps as in the proof of Theorem 4.8, we get now that
min
1
∈{±
ε
}
εvˆ
|... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
i
−
2
(cid:17)
where we used (4.7) in the second inequality. Using Lemma 2.7, we get that
the right-hand side above is further bounded by
exp
n
2
(
−
t
32
t
32
∧
2
(cid:17)
) + 2k log(144) + k log
ed
2k
(cid:0)
(cid:1)i
(cid:16)
Choosing now t such that
h
C
t
≥
r
k log(ed/k) + log(1/δ)
n
∨
k log(ed/k) + log(1/δ)
n
,
fo... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
�M
XΘ
∗
2
kF .
−
σ2 rank(Θ∗)
n
T )
(d
∨
with probability .99.
2. Show that there exists a matrix n
some estimator Θ and
ˆ
×
n matrix P such that P M = XΘ for
ˆ
ˆ
1
n
k
X ˆΘ
XΘ∗
2
kF .
−
σ2 rank(Θ∗)
n
T )
(d
∨
with probability .99.
3. Comment on the above results in light of the results obtain in Section 4.2.
Problem 4.... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.969 Topics in Geometry: Mirror Symmetry
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
MIRROR SYMMETRY: LECTURE 8
DENIS AUROUX
Last time: 18.06 Linear Algebra.
Today: 18.02 Multivariable Calculus. / 18.04 C... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
¨
complexified K¨ahler moduli space: if [B + iω] =
�
e∗
i
B + iω.
ahler cone, we obtain coordinates on the
tˇiei, let ˇqi = exp(2πitˇi), tˇi =
�
Example. Returning to our example, ˇq = exp(2πi T 2 B + iω).
Conjecture 1 (Mirror Symmetry). Let f : X → (D∗)S be a family of Calabi-
Yau 3-folds with LCSL at 0. Then ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
etc.).
�p = �
�m(p)
∂ Xˇ
∂
i ∂q
,
Xˇ
,
Ω) and the RHS to a (1, 1)
∂
= ∂tˇi
=
1
�
2
DENIS AUROUX
Remark. A more grown-up version of mirror symmetry would give you an equiv-
T X) with its usual product structure and H ∗( ˇX, C)
alence between H ∗(X,
with the quantum twisted product structure as Frobenius alg... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
gives Xψ = Xξφ, so let z = (5ξ)−5 . Then z
0, i.e. ψ → ∞, gives a toric
degeneration of Xψ to {x0x1x2x3x4 = 0}. This is maximally unipotent, as the
monodromy on H 3 is given by
→
∼
(4)
⎜
⎜
⎝
⎛
1 1 0 0
0 1 1 0
0 0 1 1
0 0 0 1
⎞
⎟
⎟
⎠
so it is LCSL. We want to compute the periods of the holomorphic volume form
ψ. T... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
�
should be given by the root of fψ which tends to 0 as ψ → ∞. We need to show
that there is only one such value (giving us a simple degeneration rather than a
branched covering). Explicitly, set x3 = (ψx0x1x2)1/4y:
(6)
i.e.
(7)
fψ = 0 ⇔ x0
5 + x1
5 + x 5
2 + (ψx0x1x2)5/4 y 5 + 1 − 5(ψx0x1x2)5/4 y
y =
y5
5
+
... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
Ωψ ∧ dfψ = 5ψdx0 ∧ dx1 ∧ dx2 ∧ dx3
∂fψ
at each point of Xψ. At a point where ∂x3
and
= 0, (
x0, x1, x2) are local coordinates,
(9)
Ωψ =
5ψdx0 ∧ dx1 ∧ dx2
∂fψ
∂x3
=
5ψdx0 ∧ dx1 ∧ dx2
4 − 5ψx0x1x2
5x3
Defining it in terms of other coordinates, we get the same formula on restrictions.
We need to extend this to ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
dx2
where fψ has a unique pole at x3. The residue is precisely
5ψ
(∂f /∂x3)
, giving us
(13)
=
�
5ψ
T0 (∂f /∂x3)
dx0dx1dx2 =
�
T0
Ωψ
�
DENIS AUROUX
�
T0
Ωψ =
1
2πi
�
(5ψ)−1(x5
0 + x5
dx0dx1dx2dx3
x0x1x2x3
dx0dx1dx2dx3
2 + x5
1 + x5
�
1 − (5ψ)−1 x0
3 + 1) − x0x1x2x3
5 + x2
5 + x1
x0x1x2x3
5
5
dx0... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
5
3
1
2
4
So
(14)
(15)
�
Ωψ = −(2πi)3
∞
�
n)!
(5
5(5
ψ)5n
(n!)
T0
In terms of z = (5ψ)−5, the period is proportional to
n=0
(16)
Set an = (5n)! Then
(n!)5 .
φ0(z) =
∞ (5n)! n
�
z
(n!)5
n=0
(17)
(n + 1)4 an+1 =
�
d
(5n + 5)!
(n!)5(n + 1)
�
cnzn) =
Setting Θ = z dz : Θ(
(18)
= 5(5n + 4)(5n +... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
Electric field lines in the space
surrounding a charge distribution show:
PRS02
1. Directions of the forces that exist in
space at all times.
2. Only directions in which static charges
would accelerate when at points on
those lines
3. Only directions in which moving
charges would accelerate when at
points on ... | https://ocw.mit.edu/courses/8-02t-electricity-and-magnetism-spring-2005/3cbf58c60016d128c4a9d9fe0b555b1e_prs_w01d2.pdf |
⎥
4 ⎦
⎣
3/ 2
ˆi
ˆi
E-Field of Two Equal Charges
PRS02
G
1. E =
ˆj
2k qs
e
⎡ 2 d ⎤
⎢ s +
⎥
4 ⎦
⎣
2
3/ 2
There are a several ways to see this. For
example, consider d→0. Then,
G
E → ke
2q ˆj
2s
which is what we want (sitting above a
point charge with charge 2 q)
E-Field of Five Equal Charges
PRS02
S... | https://ocw.mit.edu/courses/8-02t-electricity-and-magnetism-spring-2005/3cbf58c60016d128c4a9d9fe0b555b1e_prs_w01d2.pdf |
More rapidly than 1/r2
3) More slowly than 1/r2
4) Who knows?
PRS02
E-Field of a Dipole
(2) It falls off more rapidly
We know this must be a case by
thinking about what a dipole looks like
from a large distance. To first order, it
isn’t there (net charge is 0), so the E-
Field must decrease faster.
PRS02
A... | https://ocw.mit.edu/courses/8-02t-electricity-and-magnetism-spring-2005/3cbf58c60016d128c4a9d9fe0b555b1e_prs_w01d2.pdf |
3.012 Fund of Mat Sci: Bonding – Lecture 1 bis
WAVE MECHANICS
Photo courtesy of Malene Thyssen, www.mtfoto.dk/malene/
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last Time
1. Players: particles (protons and neutrons in
the nuclei, electrons) and electromagnetic
fields (photons)... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/3cc1140e37ba65a81f94ae6cd82e0c59_lec01b_bis_note.pdf |
• Potential energy V
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Polar Representation
Diagram of the Argand plane removed for copyright reasons.
See Mortimer, R. G. Physical Chemistry. 2nd ed.
San Diego, CA: Elsevier, 2000, p. 1011, figure B.6.
3.012 Fundamentals of Materials Sc... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/3cc1140e37ba65a81f94ae6cd82e0c59_lec01b_bis_note.pdf |
h = 6.626 x 10-34 J s = 2π a.u.)
See animation at http://www.kfunigraz.ac.at/imawww/vqm/movies.html
Select “Samples from Visual Quantum Mechanics” > “Double-slit experiment”
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Time-dependent Schrödinger’s equation
(Newton’s 2nd law for qu... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/3cc1140e37ba65a81f94ae6cd82e0c59_lec01b_bis_note.pdf |
inger’s Equation (II)
−
⎡
⎢
⎣
2
h
2
m
2
+∇
r
)(
rV
⎤
r
)(
r
ϕ =
⎥
⎦
r
rE
)(
ϕ
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005) | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/3cc1140e37ba65a81f94ae6cd82e0c59_lec01b_bis_note.pdf |
System Identification
6.435
SET 6
– Parametrized model structures
– One-step predictor
– Identifiability
Munther A. Dahleh
Lecture 6
6.435, System Identification
1
Prof. Munther A. Dahleh
Models of LTI Systems
• A complete model
u = input
y = output
e = noise (with PDF).
Lecture 6
6.435, System Identification
2
Prof... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/3cd2a84f8e64c60fae99b9c09f342b86_lec6_6_435.pdf |
Examples …. ARMAX
– ARMAX (Autoregressive moving average with exogenous input)
• Description
• Standard model
• More general, includes ARX model structure.
Lecture 6
6.435, System Identification
10
Prof. Munther A. Dahleh
Examples …. ARMAX
• One step predictor
or
• Pseudo-linear Regression
past predictions
where
or s... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/3cd2a84f8e64c60fae99b9c09f342b86_lec6_6_435.pdf |
A predictor model is a linear time-invariant stable
filter that defines a predictor
Def: A complete probabilistic model of a linear time-invariant
system is a pair of a predictor model
and
the PDF associated with the prediction error (noise).
In most situations, is not complete known. We may
work with means & vari... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/3cd2a84f8e64c60fae99b9c09f342b86_lec6_6_435.pdf |
. Munther A. Dahleh
Model Structures
Proposition: The parametrization
(= set of parameters of A , B , C , D , F )
restricted to the set
is a model structure
has no zeros outside the unit disc
Lecture 6
6.435, System Identification
28
Prof. Munther A. Dahleh
Proof: Follows from the previous general derivative. Notice... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/3cd2a84f8e64c60fae99b9c09f342b86_lec6_6_435.pdf |
OE:
Suppose . Then
& iff (B, F ) are coprime.
(To do this cleanly, need to consider where &
are the delay powers in both B & F )
Lecture 6
6.435, System Identification
34
Prof. Munther A. Dahleh
Theorem:
Identifiability
is identifiable at iff
1)
There are no common factors of
2)
3)
”
”
Lecture 6
6.435, System Ident... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/3cd2a84f8e64c60fae99b9c09f342b86_lec6_6_435.pdf |
Harvard-MIT Division of Health Sciences and Technology
HST.951J: Medical Decision Support, Fall 2005
Instructors: Professor Lucila Ohno-Machado and Professor Staal Vinterbo
6.873/HST.951 Medical Decision Support
Spring 2004
Evaluation
Lucila Ohno-Machado
Outline
Calibration and
Discrimination
• AUCs
• H-L stat... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
ectomy
angiojet
max pre stenosis
max post stenosis
no reflow
Cases
Women
Age > 74yrs
Acute MI
Primary
Shock
Study Population
Development Set
1/97-2/99
Validation Set
3/99-12/99
2,804
1,460
909 (32.4%)
433 (29.7%)
p=.066
595 (21.2%)
308 (22.5%)
p=.340
250
156
(8.9%)
(5.6%)
144
95
(9.9%)
(6... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
C
f
o
r
e
b
m
u
N
2000
1500
1000
500
0
Number
of Cases
53.6%
Mortality
Risk
21.5%
12.4%
2.2%
0 to 2
3 to 4
5 to 6
7 to 8
9 to 10
>10
Risk Score Category
60%
50%
40%
30%
20%
10%
0%
Evaluation Indices
General indices
• Brier score (a.k.a. mean squared error)
2
Σ(e - o )
i
i
n
e = estim... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
“nl”
“D”
Sens = TP/TP+FN
40/50 = .8
Spec = TN/TN+FP
45/50 = .9
PPV = TP/TP+FP
40/45 = .89
NPV = TN/TN+FN
45/55 = .81
Accuracy = TN +TP
85/100 = .85
nl
D
“nl”
45
10
“D”
5
40
“nl”
“D”
Sensitivity = 50/50 = 1
Specificity = 40/50 = 0.8
threshold
“nl”
“D”
nl
D
40
0
10
50
50
50
40
60
nl
TN
diseas... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
50
70
30
“nl”
“D”
1
y
t
i
v
i
t
i
s
n
e
S
ROC
curve
0
1 - Specificity
1
1
y
t
i
v
i
t
i
s
n
e
S
ROC
curve
s
d
l
o
h
s
e
r
h
T
l
l
A
0
1 - Specificity
1
45 degree line:
no discrimination
1
y
t
i
v
i
t
i
s
n
e
S
0
1 - Specificity
1
45 degree line:
no discrimination
1
y
t
i
v
i
t
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
sick):
0.3
0.2
0.5
0.1
0.7
0.8
0.2
0.5
0.7
0.9
Estimates per class
• Healthy (real outcome is 0)
• Sick (real outcome is1)
0.3
0.2
0.5
0.1
0.7
0.8
0.2
0.5
0.7
0.9
All possible pairs 0-1
• Healthy
0.3
0.2
0.5
0.1
0.7
<
• Sick
0.8
0.2
0.5
0.7
0.9
concordant
discordant
concordant
concordant
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
0.835
ROC = 0.50
0.00
0.20
0.40
0.60
0.80
1.00
1 - Specificity
Calibration Indices
Discrimination and Calibration
• Discrimination measures how much the
system can discriminate between cases
with gold standard ‘1’ and gold standard ‘0’
• Calibration measures how close the
estimates are to a “real” prob... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
1.3
sum of group = 3.1
0
0
1
0
0
1
0
1
1
1
sum = 1
sum = 1
sum = 3
Regression line
Linear
Regression
and
450 line
1
p
u
o
r
g
r
e
p
s
e
u
l
a
v
d
e
v
r
e
s
b
o
f
o
g
v
A
0
Avg of estimates per group 1
Goodness-of-fit
Sort systems’ estimates, group, sum, chi-square
Estimated
Observed
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
3 sum = 0.8
0.5
0.5 sum = 1.0
0.7
0.7
0.8
0.9 sum = 3.1
Observed
0
0
1
0 sum = 1
0
1 sum = 1
0
1
1
1 sum = 3
Decomposition of Error
The “ideal” model generates data D.
A “learned” model is learned from D.
Once learned, model M is fixed.
After learning, I and M are conditionally
independent given D. ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
2 ⎤
) ⎥⎦
⎡ 1
+ ⎢⎣ 2 ∑ B P
(
2
1
) − ∑ B P
(
2
)2 ⎤
⎥⎦
=
Decomposition of Error
=
1
2
+
1
2
− ∑
)
B P A P
(
(
)
+ ∑
P
( AB
)
−∑
P
( AB
)
+
1 ∑
2
( A P
2)
−
1 ∑
2
( A P
(B P
2)
−
1 ∑
2
(B P
2)
=
2)
+
1 ∑
2
1
[∑ A P
(
2
2
) − ∑ AB
P
(
) + ∑ B P
(
2
) ]=
− = 1[∑
)
B P A P
(
(
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
3.23 Fall 2007 – Lecture 9
BAND STRUCTURE
3.23 Electronic, Optical and Magnetic Properties of Materia... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/3d028542d1fa3737380cb3c7ff4636bc_clean9.pdf |
⎟
⎟
⎟⎟
⎜⎜
⎝⎝ CCq G+ 2
⎠
⎠
⎟
⎟
⎟
2 ⎟
( q + 2G) ⎟
⎠
V 2
− G
V
V− G
2
h
2m
C
C
⎛
− 2 ⎞
q G
⎜
⎟
⎜
q G ⎟
−
⎟ = E ⎜ C
⎟
⎜
⎟
C
⎜
q G ⎟
+
⎟⎟
⎜⎜
⎝ CCq G+ 2
⎠
⎝
⎠
q
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
4
Free electron dispersions, 1-d
3.23 Elec... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/3d028542d1fa3737380cb3c7ff4636bc_clean9.pdf |
(
)q
2
h
2m
V
G
V
2G
V−3G
VV −2G
V−G
(
)
q G +
2
2
h
2m
V
G
−
VV −3G
V−4G
⎞
⎟
⎟
⎟
CC −22GG
⎞⎞
⎛ q
⎛
⎟⎟
⎜
⎟
⎟⎜ Cq G ⎟
⎟⎜ C
⎟
⎟⎜
⎟
⎟⎜ C
q G+ ⎟
⎟⎜
⎟
⎟⎝ Cq G 2 ⎠
⎟
2 ⎟
(q + 2G) ⎟
⎠
V−2G
V
−G
+
q
2
h
2m
−
CC −22G
⎛ q G ⎞⎞
⎛
⎜
⎟
⎜ Cq G ⎟
= E
⎜
⎟
Cq
⎜
⎟
⎜ C +
q G ⎟
⎜
⎟
⎝ C +
⎠
q G 2... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/3d028542d1fa3737380cb3c7ff4636bc_clean9.pdf |
of Materials - Nicola Marzari (MIT, Fall 2007)
Fermi energy
2
-0.939
Copper
A
1
Z
3
Q_
S
1
D
1
Z
2
Z
1
Z
3
Z
4
Z
1
1'
1
3
2'
5
2
3
1
A
1
2'
3
A
3
Q+
Q_
Q+
Q_
3
A
3
Q+
1
A
1
D
2
D
D
5
2'
D
1
S
3
S
S
1
-0.539
G
12
G
25'
G
1
S
4 S
2 K2
K4
K3
K1
K1
1
X
W
L
G K
Figure by MIT OpenCourseWare.
9
ρ r r
( ) = ∑r
n k,
fn k... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/3d028542d1fa3737380cb3c7ff4636bc_clean9.pdf |
Adjustable Voltage Power Supply
+15V
0.1uf
270Ω
1N758
10v
10K
V+
V-
.
Vo
1uf
6.091 IAP 2008 Lecture 4
Appendix p1
555 Block Diagram
ready
Threshold
Control Voltage
Trigger
VCC
8
6
5
2
5k
5k
5k
+
Comp
A
_
+
Comp
B
_
R
S
Flip
Flop
Q
Inhibit/
Reset
7
3
Discharge
Output
+15
1
Gnd
4
Reset
R
C
1k
Figure by MIT OpenCourseWar... | https://ocw.mit.edu/courses/6-091-hands-on-introduction-to-electrical-engineering-lab-skills-january-iap-2008/3d13008aaf5b26f9be91462c463c0414_lec4b.pdf |
Key Concepts for this section
1: Lorentz force law, Field, Maxwell’s equation
2: Ion Transport, Nernst-Planck equation
3: (Quasi)electrostatics, potential function,
4: Laplace’s equation, Uniqueness
5: Debye layer, electroneutrality
Goals of Part II:
(1) Understand when and why electromagnetic (E and B)
interaction ... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3d3260381b93d72c4b8ac060fa3c3fc7_fields_lec6.pdf |
( )
x
2
RT
zF
ln
+
1 e
−
κ
x
tanh
−
1 e
−
κ
x
tanh
⎡
⎢
⎢
⎢
⎢
⎣
⎤Φ⎛
zF
⎞
0
⎟
⎜
⎥
4
RT
⎠
⎝
⎥
Φ⎛
zF
⎞
⎥
0
⎟
⎜
⎥
4
RT
⎠
⎝
⎦
κ
,
⎛
= ⎜
⎝
2
2
2
z F c
0
ε
RT
1/ 2
⎞
⎟
⎠
Debye-Huckel approximation
Φ = Φ
( )
x
e κ−
x
0
When
zF
Φ <<
0
RT
8
6
4
2
( )
c x
c
0
Φ
zF
RT
0
=
2
c- (counterion)
c+ (co-ion)
1.15
1.1
1.05
1
0.95
( )
c x... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3d3260381b93d72c4b8ac060fa3c3fc7_fields_lec6.pdf |
⎝
c
2
c
1
⎞
⎟
⎠
RT
zF
ln
⎛
⎜
⎝
c
2
c
1
⎞
⎟
⎠
Nernst Equilibrium potential
Diffusion of charged particles -> generate electric field
-> stops diffusion of ions | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3d3260381b93d72c4b8ac060fa3c3fc7_fields_lec6.pdf |
6.02 Fall 2012
Lecture #14
• Spectral content via the DTFT
6.02 Fall 2012
Lecture 14 Slide #1
Demo: “Deconvolving” Output of
Channel with Echo
x[n]
Channel,
h1[.]
y[n]
z[n]
Receiver
filter, h2[.]
Suppose channel is LTI with
h 1[n]=δ[n]+0.8δ[n-1]
H1(Ω) = ?? = ∑h1[m]e
− jΩm
m
So:
= 1+ 0.8e–jΩ = 1 ... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/3d71e96b6e7942d74c15fb2f99244985_MIT6_02F12_lec14.pdf |
DTFT) for
Spectral Representation of General x[n]
If we can write
h[n] =
∫ H (Ω)e
1
2π<2π>
then we can write
x[n] =
∫ X(Ω)e
1
2π<2π>
jΩn
dΩ
where H (Ω) = ∑h[m]e
− jΩm
Any contiguous
interval of length
2
m
jΩn
dΩ
where
X(Ω) = ∑ x[m]e
− jΩm
m
This Fourier representation expresses x[n] as
a weighted comb... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/3d71e96b6e7942d74c15fb2f99244985_MIT6_02F12_lec14.pdf |
�)X(Ω)
Compare with y[n]=(h*x)[n]
Again, convolution in time
has mapped to
multiplication in frequency
6.02 Fall 2012
Lecture 14 Slide #7
Magnitude and Angle
Y (Ω) = H (Ω)X(Ω)
| Y (Ω) |= |H (Ω) |. | X(Ω) |
and
< Y (Ω) = < H (Ω)+ < X(Ω)
6.02 Fall 2012
Lecture 14 Slide #8
Core of the Story ... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/3d71e96b6e7942d74c15fb2f99244985_MIT6_02F12_lec14.pdf |
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