text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
hybrid model, there must be
a simulator S� so that
�
IdealF
S,Z ≈ ExecF
H,Z
Hence for every adversary A there exists a simulator S such that for every environment:
�
IdealF
S,
Z ≈ ExecQP
,A,Z
Thus, QP securely realizes F�.
2
Another implication of the theorem is that we can immediately deduce security in a
mu... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
Chapter
3:
Collecting
Data
population
is
a
collection
of
objects,
items,
humans/animals
(“units”)
about
A
which
information
is
sought.
A
sample
is
a
part
of
the
population
that
is
observed.
parameter
is
a
numerical ... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
sample
–
make
sure
you
know
what
you
can
calculate
and
what
you
can’t
calculate!
You
can’t
calculate
anything
from
the
population
if
you
only
have
a
sample.
sampling
frame
is
a
list
of
all
units
in
a
fin... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
sample
collector
(who
may
use
Judgment
sampling
Bias
is
possible
with
these
sampling
methods.
To
avoid
bias,
sample
.
sampling)
randomly
from
the
population.
quota
A
wi
bei
simple
random
sample
(SRS)
of
size
n
from
a
... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
SRS
from
each
one.
to
d
e.g.
o
pulati
o
statistics
customers
stratified
by
race
(some
races
are
rarer
than
others)
involves
dividing
the
population
into
homogeneous
eful
when
you
want
This
ons
as
well
as
on
the
wh
... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
3)
Draw
of
people
from
each
county
1-‐in-‐k
systematic
sampling
consists
of
selecting
every
kth
unit.
Useful
for
sampling
items
coming
off
assembly
lines.
MIT OpenCour... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
Lecture 7
Quantum Mechanical Measurements.
Symmetries, conserved quantities, and the labeling of states
Today’s Program:
1. Expectation values
2. Finding the momentum eigenfunctions and the dispersion relations for free particle.
3. Commutator and observables that commute
4. Symmetries and conserved quantities –... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
function space. That
u x
means that any wave function can be represented as a linear combination of
n :
u x
x
C u x
n n
n
Where the coefficients of the expansion are just as they are in the geometrical analogy
projections of the function
un
direction given by the inner products between the
o... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
1
Finding the eigenvectors and eigenvalues of operators, discuss the geometrical interpretation of
eigenvectors and eigenvalues – scaling.
Fourth Postulate (discrete non-degenerate): When the physical quantity a is measured on a
system in the normalized state
the probability P a n of obtaining the non-degenera... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
�� x,t0 un
* x x,t0 dx.
ˆ
n
Cnun x
, where C
x, t0
Substituting this into the equation above we get:
Aˆ x,t x,t * ˆ x
x A dx
ˆ
A
n
* *
n n
A C x dx
C u x ˆ u
n n
n
n
C u x C Au x dx
n
C u x C a u x dx
n n n
* *
n... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
Let’s consider an example: Particle flying in free space.
We have previously approximated particles in free space with plane waves but in reality they are
more like a linear superposition of multiple plane waves.
Let’s consider a particle described by a wavefunction:
1 ik0xi0t
1 2ik0x4i0t
x, t e
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
k0e
3ik0x3i0t
1
2 2
k0e
2ik0x
1
2k0e
2 2
3ik0x3i0t
dx
1
4
So on average the momentum is equal to: k0
What is the probability 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
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
How precisely can we measure a physical quantity? – Before we can answer this question
let’s consider a helpful mathematical concept:
The Commutator operator and commutation relations
The Commutator operator is defined as: Aˆ, Bˆ
AˆBˆ BˆAˆ
Two observables Aˆ and Bˆ are said to commute if: AˆBˆ BˆAˆ A... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
x
3 3 x
2m x
3
i
3 3 x
2m x
3
0
Do Hˆ
2
2
2m x 2
and pˆ i
x
share eigenfunctions?
The eigenfunctions for momentum are plane waves e
ikx : i
e
x
ikx k eikx
And from the previous lectures we remember that the plane waves are also eigenfunctions for the
free space Hamilt... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
x
xˆ, pˆ
xi
x
Then: xˆ, pˆ i
i x
i
x
5
It turns out that this relationship is very significant as it Mathematics it means that x and p cannot
be measured with absolute precision at the same time. In fact their uncertainties (or measurement... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
symmetries manifest themselves in equations?�Let us suppose that your system is
symmetric with respect to translations in x that would imply that any physical property could not
have an x dependence. In particular the energy would not have an explicit dependence on x thus:
H x, p
x
0 p const
dp
dt
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
� 0
States are labeled by specific values of their properties, which do not change with time –
these properties are called constants of motion. We learned that in QM physical properties are
represented by operators and that the values of properties obtained in measurements are
eigenvalues of the corresponding opera... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
��
t i
r
This type of differential equation is separable, i.e. we can look for a solution in the following
form: r
2
2m
1
r
i t
t
t
r
2
2m
V r
V r
t
r
1
t
Note that the left side... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
2
2m
r E uE ,
2 V
x
r
r
E
Then the solutions to time-dependent Schrodinger’s equation will have a form:
7
E
t
i
uE r E t uE r e
E
r, t
In general, since the Hamiltonian may have many eigenvalues and corresponding eige... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
�� x E x
2
2
2m x2
E uE
x
e
x
x
e
i
i
2 mE
2 x
2mE
2
x
Using the energy as a “label” doesn’t completely and uniquely specify a state.
What about momentum? – If momentum is a constant of motion then we can use it as an
additional label to uniquely specify... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
uniquely and completely specifies the
state.
8
Conservved Quantitiies Examplee II: Parity operator annd symmetrric potentialls
Definitioon of a parityy operator: ˆˆ x
x
What aree the eigenfuunctions and eigenvaluess of the parityy operator:
ˆ
u x u x
ˆ ˆ
ˆ
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
22
2 m2 x
2
2
2
2m x
1
2
2
2mm
m2
2 ˆ
x H x
Let’s cheeck the commmutator:
ˆ
ˆ
, Hˆ
x Hˆ x Hˆ ˆ
2
2 2
x ˆ
2
2mm x
2
m x
1
2
2
x
2 2
2
2m x
2
2m
2
x
mm2 x
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
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, llast
lecture wwe have showwn that SHOO eigenfunctiions are alwaays even or oodd.
9
MIT OpenCourseWare
http:... | 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 |
can be written in the form:
y [k] = CAk x[0] + C
k−1
�
�
Ak−i−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 underst... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
ejωt + e−jωt ) = 2u0e σt cos(ωt),
�
and the input u is a “half” of a sinusoid with exponentially-changing
amplitude.
E. Frazzoli (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)... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
will converge to zero.
The steady state response can be written as:
→
0, and the transient response
yss = G (s)e st ,
G (s) ∈ Cny ×nu ,
where G (s) = C (sI − A)−1B + D is a complex matrix.
The function G : s
describes how the system transforms an input e
→
st
G (s) is also known as the transfer function: it
into... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
− i − 1:
i=0
y [k] = CAk x[0] + C
k−1
�
Al Bu0z k−l−1 + Du0z k
l=0
= CAk x[0] + Cz k−1
�
k−1
�
�
(Az −1)i Bu0 + Du0z k .
i=0
E. Frazzoli (MIT)
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... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
(zI − A)−1Bu0 + C (zI − A)−1B + D u0z k .
�
�
��
Transient response
��
Steady−state response
� �
�
�
�
If the system is asymptotically stable, the transient response will converge to zero.
The steady state response can be written as:
yss[k] = G (z)z k ,
G (z) ∈ C,
where G (z) = C (zI − A)−1B + D is a comple... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
13
Models of continuous-time systems
CT
CT System
CT
x˙ (t) = Ax(t) + Bu(t)
y (t) = Cx(t) + Du(t)
⎡
1
. . .
0
0
. . .
A = ⎢
⎢
⎣
. . .
−a0 −a1
⎤
0
0
0
. . .
. . .
⎥
⎥
1
⎦
. . . −an−1
⎤⎡
0
. . .
⎥
B = ⎢
⎥
⎢
⎦
⎣
0
1
�
C = b0 b1
�
. . . bn−1
D = d
G (s) = C (sI − A)−1B + D
G (s) =
bn−1s n−1 ... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/3bf509c94d0977a81bf16f1d0e15c47e_MIT6_241JS11_lec09.pdf |
(z) = C (zI − A)−1B + D
G (z) =
bn−1z n−1 + . . . + b0
z n + an−1z n−1 + . . . + a0
+ d
E. Frazzoli (MIT)
Lecture 9: Transfer Functions
Mar 2, 2011
13 / 13
MIT OpenCourseWare
http://ocw.mit.edu
6.241J / 16.338J Dynamic Systems and Control
Spring 2011
For information about citing these materials or our Term... | 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 |
y stresses (forces per unit area in the
deformed geometry).
II. 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... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/3bfd1ab31d0d622c3d6f151fd0776ab0_MIT2_092F09_lec04.pdf |
−
Z L
0
dδu
dx
EA
du
dx
dx +
Z L
0
f Bδudx = 0
(A)
The equation above becomes:
Internal virtual work
}|
EA
z
Z L
{
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 k... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/3bfd1ab31d0d622c3d6f151fd0776ab0_MIT2_092F09_lec04.pdf |
εxx
εyy
εzz
γxy
γyz
γzx
εxx
εyy
εzz
γxy
γyz
γzx
;
εxx =
∂u
∂x
;
εzz =
∂u
∂z
We see that (B) is the generalized form of (A’). The principle of virtual work states 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 Princi... | 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 |
* +
pe 2τh
*
mh
p is analogous to n for holes, and so are τ and m*
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... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/3bfec0ef84475e74634685665d9eaee6_lecture_7.pdf |
1
E
−
2
T dE
e k
b
3
*
⎛ 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, ... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/3bfec0ef84475e74634685665d9eaee6_lecture_7.pdf |
in the ln term, 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
... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/3bfec0ef84475e74634685665d9eaee6_lecture_7.pdf |
∝ e 2kbT
− 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... | 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 |
{
}
j
i
82
≤
r
A = U DV ⊤ =
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... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
a n
×
n PSD matrix A, we have
λmax (A) = max x⊤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... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
called the Nuclear norm (or trace norm) of A.
kop is called the operator norm (or
A
k
q =
k
spectral norm) of A.
∞
∞
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” i... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
the regression function in this case
X = x] is a function from IRd to IRT . Clearly, f can be estimated
f (x) = IE[Y
independently for each coordinate, using the tools that we have developed in
the previous chapter. However, we will see that in several interesting scenar-
ios, some structure is shared across coordinate... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
for each of the n days are already known to headquarters and are stored in
d. In this case, it may be reasonable to assume that the
a matrix X
same subset of variables has an impact of the sales for each of the franchise,
though the magnitude of this impact may differ from franchise to franchise. As
a result, one may as... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
than in [LPTVDG11].
Indeed, rather than exploiting sparsity, observe that such a matrix Θ∗ has rank
k. This is the kind of structure that we will be predominantly using in this
chapter.
Rather than assuming that the columns of Θ∗ share the same sparsity
pattern, it may be more appropriate to assume that the matrix Θ∗ i... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
:
2 = u⊤
2
Akin to the sub-Gaussian sequence model, we have a direct observation
model where we observe the parameter of interest with additive noise. This
|2 = 1.
w
u
2
|
|
|
X X
⊤
u =n
4.2. Multivariate regression
86
|0 is small.
enables us to use thresholding methods for estimating Θ∗ when
However, this also follow... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
1
kop ≤
δ.
A
−
4σ
log(12)(d
∨
T ) + 2σ
2 log(1/δ)
p
p
d
Proof. This proof follows the same steps as Problem 1.4. Let
1 and
net for
that we 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⊤A... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
∈ N2.Together with the above display, it yields
t
8σ2
kop > t
A
12d+T exp
IP
≤
−
×
2
∼
subG(σ2) for any x
∈
δ
≤
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 o... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
A
1
−
≥
implies that S
Note that it follows from Weyl’s inequality that
|
> τ
λj| ≤
}
⊂ {
¯
Next define the oracle Θ =
ˆ
λj −
.
3τ
}
|
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 ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
|
∈
Plugging the above two displays in (4.4), we get
ˆΘsvt
k
−
Θ∗
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 exte... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
σ rank(Θ∗)
n
d
∨
(cid:16)
T + log(1/δ) .
(cid: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
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
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), this... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
(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
XΘ
2
F =
Y Y¯ 2
k − kF +
Y¯
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
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 penalty can also
be used. The estimator by nuclear norm penalization Θ is defined t... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
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-Gaussian random vectors such
that IE[XX ⊤] = Σ and X
Σ
subGd(
k
∼
kop). Then
d + log(1/δ)
n
∨
ˆΣ
k
Σ
kop .
k
Σ
kop
−
r
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
−
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
i=1
X
(cid:8)
Using polarization, we also have
(Xi⊤x)(Xi⊤y... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
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[Z 2
+]
∼
−
∼
subE(32) ,
and
Z 2
− −
IE[Z 2 ]
− ∼
subE(32)
Therefo... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
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
consistent estimator of Σ (in any norm as they are al... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
12)
Var(X ⊤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 ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
7 points in two dimension. This
image has become quite popular as it shows that gene expression levels can
recover the structure induced by geographic clustering. How is it possible to
“compress” half a million dimensions into only two? The answer is that the
data is intrinsically low dimensional. In this case, a plaus... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
matrix Σ
covariance model if it is of the form
∈
IRd
×
d is said to satisfy the spiked
Σ = θvv⊤ + Id ,
where θ > 0 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 poi... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
(Davis-Kahan sin(θ) theorem). Let Σ satisfy the spiked covari-
ance model and let Σ be any PSD estimator of Σ. Let v˜ denote the largest
˜
eigenvector of Σ. Then we have
˜
εv˜
|
−
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 t... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
is the largest eigenvector
˜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
− ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
+ 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. In the high-dimensi... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
� 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 = k
∼
Σ
v
(cid:3)
(cid:2)
|
min... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
, 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
|
−
v
2
2 ≤
|
2 sin2
∠(vˆ, v)
8
θ2
≤
S :
sup
S
|
|
=2k
ˆ
Σ(S)
k
Σ(S... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
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
,
for large enough C ensures that the desired bound holds with probability at
least 1
δ.
−
4.5. Problem set
99
4.5 PROBLEM SET
Problem 4.1... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
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 4.3.
be the any solution to the minimization problem
ˆ
Consider the multivari... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/3c56f629bc0fb207ecfb0d73a8b4eda8_MIT18_S997S15_Chapter4.pdf |
cid:76)(cid:82)(cid:81)(cid:68)(cid:79)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70)(cid:86)
Spring 2015
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms(cid:17) | 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 |
� = 1, �
If ei is a basis of H 2( ˇ
X, Z), ei in the K¨
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 ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
(since 2πiqˇi ∂qˇi
ei ∈ H 1,1 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 pro... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
x0 : . . . : x4) �→ (ξax0 : x1 : . . . : x4)
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 th... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
x3
We want to extend it to z =�
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 − ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
Ωψ be the 3-form on Xψ characterized uniquely by
(8)
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 sa... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
5ψdx3
fψ
�
dx0dx1dx2
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 + ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/3ca5f3b9037ae9486ab71549f1f679f1_MIT18_969s09_lec08.pdf |
+ x1 + x2 + x
(n!)5 .
5
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 : ... | 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 |
⎣
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
Six equa... | https://ocw.mit.edu/courses/8-02t-electricity-and-magnetism-spring-2005/3cbf58c60016d128c4a9d9fe0b555b1e_prs_w01d2.pdf |
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
An electric di... | 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 |
000, p. 495, figure 14.2.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The total energy of the system
• Kinetic energy K
• Potential energy V
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Polar Representation
Diagram of the Argand plane remove... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/3cc1140e37ba65a81f94ae6cd82e0c59_lec01b_bis_note.pdf |
1926…)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
When is a particle like a wave ?
Wavelength • momentum = Planck
Image of the double-slit experiment removed for copyright reasons.
See the simulation at http://www.kfunigraz.ac.at/imawww/vqm/movies.html:
"Samples from Visual Q... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/3cc1140e37ba65a81f94ae6cd82e0c59_lec01b_bis_note.pdf |
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Stationary Schrödinger’s Equation (I)
−
2
h
2
m
2
Ψ∇
r
r
*
),(
),(
trVtr
+
Ψ
r
),(
tr
=
i
h
r
),(
tr
Ψ∂
t
∂
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Stationary Schrödinger’s Equation (II)
−
⎡
⎢... | 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 |
• Linear Regression
(a function of past data)
• Prediction error
Lecture 6
6.435, System Identification
9
Prof. Munther A. Dahleh
Examples …. ARMAX
– ARMAX (Autoregressive moving average with exogenous input)
• Description
• Standard model
• More general, includes ARX model structure.
Lecture 6
6.435, System Identifi... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/3cd2a84f8e64c60fae99b9c09f342b86_lec6_6_435.pdf |
Describes the model structure
Define
It follows
Predictor:
Lecture 6
6.435, System Identification
18
Prof. Munther A. Dahleh
For a given
• Notice: can be stable even though is not!
•
Lecture 6
6.435, System Identification
19
Prof. Munther A. Dahleh
Predictor Models
Def: A predictor model is a linear time-invariant ... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/3cd2a84f8e64c60fae99b9c09f342b86_lec6_6_435.pdf |
.
is the map
is one particular model
Lecture 6
6.435, System Identification
24
Prof. Munther A. Dahleh
Example: ARX model structure
stable
Lecture 6
6.435, System Identification
25
Prof. Munther A. Dahleh
General Structure
Lecture 6
6.435, System Identification
26
Prof. Munther A. Dahleh
You need
to be differenti... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/3cd2a84f8e64c60fae99b9c09f342b86_lec6_6_435.pdf |
(local or global) if
it is identifiable (local or global) for all
Lecture 6
6.435, System Identification
31
Prof. Munther A. Dahleh
Central question II: Is the identified parameter equal to
the “true parameters” ?
Parametrized structure:
true system
Case I:
Case II:
Define:
Let for some . If is identifiable at ,
... | 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 |
a
graft lesion
vessel treated
ostial
experience
unscheduled case
lab device
antagonists
dissection post
rotablator
atherectomy
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
3/99-12/9... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
.40
0.30
0.20
0.10
0.00
LR
Score
aNN
ROC Area
0.840
LR:
Score: 0.855
aNN: 0.835
ROC = 0.50
0.00
0.20
0.40
0.60
0.80
1.00
1 - Specificity
Risk Score of Death: BWH Experience
Unadjusted Overall Mortality Rate = 2.1%
3000
2500
s
e
s
a
C
f
o
r
e
b
m
u
N
2000
1500
1000
500
0
Number
of Cases ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
e.g., 0.34)
“True”
Estimate
0.3
0.2
0.5
0.1
0
0
1
0
•
In practice, classification into category 0 or 1 is based on
Thresholded Results (e.g., if output or probability > 0.5
then consider “positive”)
– Threshold is arbitrary
threshold
normal
Disease
True
Negative (TN)
True
Positive (TP)
FN
FP
0
e... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
50
5
40
50
50
50
nl
TN
disease
TP
0.0
FN
FP
0.6
1.0
Sensitivity = 30/50 = .6
Specificity = 1
threshold
nl
D
“nl”
“D”
50
20
70
30
0
50
30
50
nl
TN
disease
TP
FN
0.0
0.7
1.0
4
.
0
d
l
o
h
s
e
r
h
T
6
.
0
d
l
o
h
s
e
r
h
T
7
.
0
d
l
o
h
s
e
r
h
T
“nl”
“D”
40
60
nl
D
40
0
10
50 ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
1
y
t
i
v
i
t
i
s
n
e
S
Area under ROC:
0.5
0
1 - Specificity
1
Perfect
discrimination
1
y
t
i
v
i
t
i
s
n
e
S
0
1 - Specificity
1
Perfect
discrimination
1
y
t
i
v
i
t
i
s
n
e
S
Area under ROC:
1
0
1 - Specificity
1
1
y
t
i
v
i
t
i
s
n
e
S
ROC
curve
Area = 0.86
0
1 - Specificity
1
What ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
8
0.2
0.5
0.7
0.9
concordant
discordant
concordant
concordant
concordant
All possible pairs 0-1
Systems’ estimates for
• Healthy
0.3
0.2
0.5
0.1
0.7
• Sick
0.8
0.2
0.5
0.7
0.9
concordant
tie
concordant
concordant
concordant
C - index
• Concordant
18
• Discordant
4
• Ties
3
C -index = Concordant... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
• “If the system is good in discrimination,
calibration can be fixed”
Calibration
• System can reliably estimate probability of
– a diagnosis
– a prognosis
• Probability is close to the “real” probability
What is the “real” probability?
• Binary events are YES/NO (0/1) i.e.,
probabilities are 0 or 1 for a given ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
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
0.1
0.2
0.2
0.3
0.5
0.5
0.7
0.7
0.8
0.9
sum of group = 0.5
sum of group = 1.3
sum of group = 3.1
χ2 = Σ [(observed - estimated)2/estim... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/3cff5fb64fbeca8fbffb9ca89c5574b0_hst951_5.pdf |
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