text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
(p, q)
(by Cauchy-Schwarz)
≤
=
Z
2
−
Z
min(p, q)
Z
Z
1 + TV(IP, Q) (cid:3)
1
−
TV(IP, Q)2
(cid:1)(cid:0)
= (cid:2)
= 1
(cid:0)
−
min(p, q)
TV(IP, Q)
(cid:1)
The two displays yield
KL(IP, Q)
2
2
−
where we used the fact that 0
x
[0, 1].
≥
∈
1
TV(IP,
−
TV(IP, Q)
p
≤
Q
)2
TV(IP,
≥
1 and √
1
≤
)2
Q ,
x
−
≤
1
−
x/2 for
Pins... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
1
2
h
1
= 1
2
TV(IP0, IP1)
i
KL(IP1, IP0)
i
θ1 −
2σ2
θ 2
0
2
|
r
2α
h
1
p
−
−
≥
n
−
i
|
h
i
(cid:17)
(Prop.-def. 5.4)
(Lemma 5.8)
(Example 5.7)
Clearly the result of Theorem 5.9 matches the upper bound for Θ = IRd
only for d = 1. How about larger d? A quick inspection of our proof shows
that our technique, in its prese... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
Fano’s inequality. We use it in a
form that is tailored to our purposes and that is due to Lucien Birg´e [Bir83]
and builds upon an original result in [IH81].
Theorem 5.10 (Fano’s inequality). Let P1, . . . , PM , M
Pk,
tributions such that Pj ≪
j, k. Then
∀
2 be probability dis-
≥
inf max Pj ψ(X) = j
ψ 1
M
j
≤
≤
(cid:... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
�(X)
|
X)
X)) + IP(Z = ψ(X)
|
j=ψ(X)
X
X)]
X)
IP(Z = ψ(X)
|
log
−
IP(Z = j
IP(Z = ψ(X)
|
X)]
X)
|
h
qj log(qj) ,
ψ(X)
j=X
IP(Z = j X)
|
IP(Z = ψ(X)
|
X)
i
where
and
M
1
j=
X
h(x) = x log(x) + (1
x) log(1
x)
−
−
qj =
X)
IP(Z = j
IP(Z = ψ(X)
|
|
X)
j=ψ(X) qj = 1. It implies by Jensen’s inequality that
is such that qj ≥
0... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
x)]
dPZ (x)
o
dPZ (x)
IP(Z = j
|
X = x) log[IP(Z = j
|
1 dPj (x) log
M dPZ
1 dPj (x)
M dPZ
(cid:16)
dPj(x)
M
k=1 dPk(x)
(cid:17)o
dPj(x)
(cid:17)
log
(cid:16)
P
log
dPj(x)
dPk(x)
(cid:16)
log M (by Jensen)
dPj (x)
(cid:17)
−
Z n X
j=1
M
=
=
≤
=
M
j=1
X Z n
1
M
X Z
j=1
M1
M 2
j,k=1
X Z
M
1
M 2
KL(Pj, Pk)
log M ,
−
j,k=1... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
that
ˆ
inf sup IPθ θ
| −
ˆθ θ
Θ
(cid:1)
(cid:0)
uction to hypothesis testing that
from the red
θ 2
2
|
2α .
≥
φ
∈
1
≥ 2 −
inf max IPθ
ψ
j
≤
M
≤
1
1
≥ 2 −
2α
ψ = j
j
(cid:2)
(cid:3)
6
6
6
6
6
5.4. Lower bounds based on many hypotheses
113
If follows from (ii) and Example 5.7 that
KL(IPj, IP
θk
n θ
k) = | − |
j
2σ2
2
2
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
size
d with respect to the Hamming
of a packing of the discrete hypercube
distance defined by
0, 1
p
p
}
{
d
ρ(ω, ω′) =
1I(ωi = ωj′ ) ,
i=1
X
ω, ω′
∀
0, 1
d
}
∈ {
Lemma 5.12 (Varshamov-Gilbert). For any γ
0, 1
vectors ω1, . . . ωM ∈ {
1
2 −
γ d for all j = k ,
(i) ρ(ωj, ωk)
d such that
≥
}
(0, 1/2), there exist binary
∈... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
6
5.5. Application to the Gaussian sequence model
114
as soon as
M (M
1) < 2 exp 2γ2d
−
(cid:16)
A sufficient condition for the above inequality
2γ d
e 2 . For this value of M , we have
(cid:17)
ld is to take M =
to ho
2
eγ d
⌊
⌋ ≥
IP
j = k , ρ(ωj, ωk)
γ d > 0
∀
(cid:0)
(cid:1)
(cid:1)
(cid:0)
ist
here ex
and by virtue ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
4
β2σ2d
≥ 16n
β2σ2d
n
≤
≤
32β2σ2
n
log(M ) =
2ασ2
n
log(M ) ,
for β = α . Applying now Theorem 5.11 yields
√
4
α σ2d
ˆ
inf sup IPθ θ
| − | ≥ 256 n
ˆθ θ IRd
∈
(cid:0)
2
θ 2
It implies the following corollary.
1
≥ 2 −
2α .
(cid:1)
Corollary 5.13. The minimax rate of estimation of over IRd in the Gaussian
sequence model i... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
k and d such that
1
d/8. There exist binary vectors ω1, . . . ωM ∈ {
d such that
0, 1
≤
≤
k
}
k
2
(i) ρ(ωi, ωj)
for all i = j ,
(ii) log(M )
log(1 + ) .
≥
k
8
≥
ωj|0 = k for all j .
d
2k
(iii)
|
Proof. Take ω1, . . . , ωM independently and uniformly at random from the set
C0(k) =
ω
{
0, 1
d :
}
∈ {
ω
|0 = k
}
,
|
of k-... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
(cid:0)
ωj = x : ρ(ωj, x) <
k
2
IP
d
∃
(cid:0)
M
0,1
X
∈{
}
|0=k
x
|
IP
ωj = x : ρ(ωj, x) <
(cid:0)
d
j=1
0,1
X X
∈{
}
|0=k
x
|
(cid:1)
k
2
(cid:1)
ω = x0 : ρ(ω, x0) <
k
2
(cid:1)
6
6
6
6
6
5.5. Application to the Gaussian sequence model
116
where ω has the same distribution as ω1 and x0 is any k-sparse vector in
d. T... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
k
2
≤
k
IP
Zi >
k
= IE exp s
Zi
sk
e− 2
k
2
i=1
(cid:0) X
The above MGF can be controlled by induction on k as follows:
i=1
(cid:0) X
h
(cid:1)
(cid:1)
(cid:0)
(cid:1)i
k
IE exp s
Zi
= IE
exp
s
h
i
=1
(cid:0) X
(cid:1)i
h
(cid:0)
= IE
exp
s
k
1
−
i=1
X
1
k
−
i=1
X
1
k
−
Zi
IE exp
sZk
Z1, . . . , Zk=1
(cid:1)
(cid:0)
Zi... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
+ )
d
2k
d
2k
(cid:17)
e
xp log M
log(1 + )
e
xp
≤
≤
< 1 .
d
2k
(cid:17)
(cid:17)
(for d
8k)
≥
(cid:16)
(cid:16)
(cid:16)
k
4
k
4
−
d
2k
log M < log(1 + )
If we take M such that
Apply the sparse Varshamov-Gilbert lemma to obtain ω1, . . . , ωM with
k/2 for all j = k. Let
k log(1 + d ) and such that ρ(ωj, ωk)
log(M )
8
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
the following corollary.
k log(1 + )
1
d
2k ≥ 2 −
2α .
(cid:1)
Corollary 5.15. Recall that
of IRd. The minimax rate of estimation over
model is φ(
least squares estimator θls
IR denotes the set of all k-sparse vectors
B0(k) in the Gaussian sequence
B0(k)) = σ k log(ed/k). Moreover, it is attained by the constrained
0(k... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
can check the conditions of
(i)
θj −
|
θ 2
k|2 =
R2
k2
ρ(ωj, ωk)
R2
≥ 2k ≥
4R min
R
8
, β2σ
log(ed/√n)
8n
(ii)
θj −
|
2
θk|2 ≤
2R2
k
≤
4Rβσ
r
log(ed/√n)
n
(cid:0)
≤
2ασ2
n
log(M ) ,
.
(cid:1)
for β small enough if d
Applying now Theorem 5.11 yields
≥
Ck for some constant C > 0 chosen large enough.
inf sup IP
ˆθ θ
IRd
∈... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
≤
0
|
−
θ∗
|
2
2 =
θ∗
|
2
2 ≤ |
|
θ∗
|
1 = R2 .
2
5.5. Application to the Gaussian sequence model
119
2
Remark 5.17. Note that the inequality
1 appears to be quite loose.
Nevertheless, it is tight up to a multiplicative constant for the vectors of the
σ log d ,
form θj = ω R
j
n
k
2/β
we have k
that are employed in th... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
2ε
|
θi −
θj| ≥
θ1, . . . , θN }
(b) Show that for any x
θi|2 ≤
2ε.
−
x
|
∈ B2(0, 1), there exists i = 1, . . . , N such that
(c) Use (b) to conclude that there exists a constant C′ > 0 such that N
C′/εd .
≥
Problem 5.4. Show that the rate φ = σ2d/n is the minimax rate of estimation
over:
(a) The Euclidean Ball of IRd ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
Optimization.
John Wiley & Sons, Inc., Hoboken, NJ, third edition, 2008. With
an appendix on the life and work of Paul Erdo˝s.
Dennis S. Bernstein. Matrix mathematics. Princeton University
Press, Princeton, NJ, second edition, 2009. Theory, facts, and
formulas.
Patrick Billingsley. Probability and measure. Wiley Series... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
ography
122
[CT07]
[CZ12]
[CZZ10]
[DDGS97]
[EHJT04]
[FHT10]
[FR13]
[Gru03]
[GVL96]
[HTF01]
[IH81]
Emmanuel Candes and Terence Tao. The Dantzig selector: sta-
tistical estimation when p is much larger than n. Ann. Statist.,
35(6):2313–2351, 2007.
T. Tony Cai and Harrison H. Zhou. Minimax estimation of large
covariance m... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
. Convex polytopes, volume 221 of Graduate
Texts in Mathematics. Springer-Verlag, New York, second edi-
tion, 2003. Prepared and with a preface by Volker Kaibel, Victor
Klee and Gu¨nter M. Ziegler.
Gene H. Golub and Charles F. Van Loan. Matrix computa-
tions. Johns Hopkins Studies in the Mathematical Sciences.
Johns Ho... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
03]
[Tsy09]
St´ephane Mallat. A wavelet tour of signal processing. Else-
vier/Academic Press, Amsterdam, third edition, 2009. The
sparse way, With contributions from Gabriel Peyr´e.
Harry Markowitz. Portfolio selection. The journal of finance,
7(1):77–91, 1952.
Arkadi Nemirovski. Topics in non-parametric statistics. In ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
er, 2003.
Alexandre B. Tsybakov. Introduction to nonparametric estima-
tion. Springer Series in Statistics. Springer, New York, 2009.
Revised and extended from the 2004 French original, Translated
by Vladimir Zaiats.
MIT OpenCourseWare
http://ocw.mit.edu
18.S997 High-dimensional Statistics
Spring 2015
For information ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
Turbulent Flow and Transport
9 Dispersion in Pipe and Channel flow
9.1
Dispersion in laminar pipe flow. Purely diffusive dispersion, purely
convective dispersion, and Taylor (or Taylor−Aris) dispersion. Scaling laws that define
the conditions under which the various types of dispersion occur. Radial concentration ... | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/5091c145ca7a3fe1d6d759cae9b803e2_9_Taylor_dispersion.pdf |
channel flow." Ann. Rev. Fluid Mech., Vol. 5 (1973): 59−78.
Chatwin, P. C., and P. J. Sullivan. J. Fluid Mech., 120 (1982):347−358.
Smith, R. J. Fluid Mech., 215 (1990):195−207. | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/5091c145ca7a3fe1d6d759cae9b803e2_9_Taylor_dispersion.pdf |
6.088 Intro to C/C++
Day 4: Object-oriented programming in C++
Eunsuk Kang and Jean Yang
Today’s topics
Why objects?
Object-oriented programming (OOP) in C++
�classes
�fields & methods
�objects
�representation invariant
2
Why objects?
At the end of the day...
computers just manipulate 0’s and 1’s
Figure by MIT... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
body.
However, some of the cells are resistant to drugs and
may survive.
What are objects?
Characteristics?
Responsibilities?
11
Write a program that simulates the growth of virus
population in humans over time. Each virus cell
reproduces itself at some time interval. Patients may
undergo drug treatment to in... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
Virus(float newReproductionRate, float newResistance);
Virus* reproduce(float immunity);
bool survive(float immunity);
};
20
class name
field
class Virus {
float reproductionRate; // rate of reproduction, in %
float resistance;
static const float defaultReproductionRate = 0.1;
// resistance against drugs, in %
p... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
, float newResistance); public
Virus* reproduce(float immunity);
bool survive(float immunity);
};
private: can only be accessed inside the class
public: accessible by anyone
25
How do we decide private vs. public?
Client
depends
Interface
satisfies
Implementation
interface: parts of class that change infrequentl... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
.
Virus* Virus::reproduce(float immunity) {
float prob = (float) rand() / RAND_MAX; // generate number between 0 and 1
// If the patient's immunity is too strong, it cannot reproduce
if (immunity > prob)
return NULL;
// Does the virus reproduce this time?
if (prob > reproductionRate)
return NULL;
// No!
return... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
Virus::survive(float immunity) {
// If the patient's immunity is too strong,
// then this cell cannot survive
if (immunity > resistance)
return false;
return true;
}
35
Working with objects
Patient class declaration
#include “Virus.h”
#define MAX_VIRUS_POP 1000
class Patient {
Virus* virusPop[MAX_VIRUS_POP]... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
at the end of scope
41
Objects on heap
To allocate an object on heap:
�use “new” keyword (analogous to “malloc”)
To deallocate:
�use “delete” keyword (analogous to “free”)
Patient* p = new Patient(0.1, 5);
...
delete p;
42
Dynamic object creation: Example
Patient::Patient(float initImmunity, int initNumVirusCell... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
~Patient();
void takeDrug();
bool simulateStep();
};
What are the representation invariants for Patient?
48
Rep. invariant violation
void Patient::takeDrug(){
immunity = immunity + 0.1;
}
What’s wrong with this method?
49
Preserving rep. invariant
bool Patient::checkRep() {
return (immunity >= 0.0) && (immunity... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/5097a61eb3a3ccf8c6022521cfb1c560_MIT6_088IAP10_lec04.pdf |
MEASURE AND INTEGRATION: LECTURE 1
Preliminaries. We need to know how to measure the “size” or “vol
ume” of subsets of a space X before we can integrate functions f : X →
R or f : X C.→
We’re familiar with volume in Rn . What about more general spaces
X? We need a measure function µ : {subsets of X} → [0, ∞].
For t... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/509c71b048eab7f1971e8f1cea40392e_18125_lec1.pdf |
, τY ) be a topological spaces. Then f : X
Y is→
continuous if f −1(U ) ∈ τX for all U ∈ τY . “Inverse images of open sets
are open.”
i=1
i=1
Let (X, M) be a measure space (i.e., M is a σalgebra for the space
X). Then f : X → Y is measurable if f −1(U ) ∈ M for all U ∈ τY .
“Inverse images of open sets are measura... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/509c71b048eab7f1971e8f1cea40392e_18125_lec1.pdf |
by f (x) = u(x) × v(x). Then
Proof. Define f : X
h = Φf˙. We just need to show (NTS) that f is measurable. Let
R ⊂ R2 be a rectangle of the form I1 × I2 where each Ii ⊂ R(i = 1, 2) is
an open interval. Then f −1(R) = u−1(I1) ∩ v−1(I2). Let x ∈ f −1(R) so
that f (x) ∈ R. Then u(x) ∈ I1 and v(x) ∈ I2. Since u is meas... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/509c71b048eab7f1971e8f1cea40392e_18125_lec1.pdf |
, z �→ Im z, and
z �→ | |
z , respectively.
(c) If f, g are real measurable, then so are f + g and f g. (Also
holds for complex measurable functions.)
(d) If E ⊂ X is measurable (i.e., E ∈ M), then the characteristic
function of E,
χE (x) =
�
1
0
if x ∈ E;
otherwise.
Proposition 0.3. Let F be any collection ... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/509c71b048eab7f1971e8f1cea40392e_18125_lec1.pdf |
A ∈ M, and so ∪iAi
∪iAi ∈ M
∗.
∈ M
. It follows that
�
Borel Sets. By the previous proposition, if X is a topological space,
then there exists a smallest σalgebra B containing the open sets. Ele
ments of B are called Borel sets.
If f : (X, B) → (Y, τ ) and f −1(U ) ∈ B for all U ∈ τ , then f is
called Borel measur... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/509c71b048eab7f1971e8f1cea40392e_18125_lec1.pdf |
1
Instruction Set Evolution
in the Sixties:
GPR, Stack, and Load-Store
Architectures
Arvind
Computer Science and Artificial Intelligence Laboratory
M.I.T.
Based on the material prepared by
Arvind and Krste Asanovic
6.823 L3- 2
Arvind
The Sixties
• Hardware costs started dropping
- memories beyond 32K words see... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
organization because stacks are
convenient for:
1. expression evaluation;
2. subroutine calls, recursion, nested interrupts;
3. accessing variables in block-structured
languages.
• B6700, a later model, had many more innovative
features
– tagged data
– virtual memory
– multiple processors and memories
Septembe... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
part of the processor state
⇒ stack must be bounded and small
≈ number of Registers,
not the size of main memory
• Conceptually stack is unbounded
⇒
a part of the stack is included in the
processor state; the rest is kept in the
main memory
September 14, 2005
Stack Size and Memory References
6.823 L3- ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
and Expression
Evaluation
a b c * + a d c * + e - /
6.823 L3- 12
Arvind
program
push a
push b
push c
*
+
push a
push d
push c
*
+
push e
-
/
stack (size = 2)
R0
R0 R1
R0 R1 R2
R0
R1
R0
R0 R1
R0 R1 R2
R0 R1 R2 R3
R0 R1 R2
R1
R0
R0 R1 R2
R1
R0
R0
a and c are
“loaded” twice
⇒
not the best ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
–
–
lexical addressing
<
display registers to speed up accesses
to stack frames
ll , d
>
Proc P
Proc Q
Proc R
Q
R
Q
September 14, 2005
3
2
ll = 1
display
registers
stack
static
links
automatic loading of display registers?
6.823 L3- 14
Arvind
dynamic
links
R
Q
R
Q
P
Stack Machines: Esse... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
Died by 1980
6.823 L3- 17
Arvind
1. Stack programs are not smaller if short
(Register) addresses are permitted.
2. Modern compilers can manage fast register space
better than the stack discipline.
3. Lexical addressing is a useful abstract model for
compilers but hardware support for it (i.e.
display) is not ne... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
0 has some special properties
– 4 Floating Point 64-bit Registers
– A Program Status Word (PSW)
• PC, Condition codes, Control flags
• A 32-bit machine with 24-bit addresses
– No instruction contains a 24-bit address !
• Data Formats
– 8-bit bytes, 16-bit half-words, 32-bit words,
64-bit double-words
September ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
ISSCC 2001)
– 0.18µm CMOS, 7 layers copper wiring
– 770MHz systems shipped in 2000
• Single-issue 7-stage CISC pipeline
• Redundant datapaths
– every instruction performed in two parallel datapaths and
results compared
• 256KB L1 I-cache, 256KB L1 D-cache on-chip
• 20 CPUs + 32MB L2 cache per Multi-Chip Module ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
per instruction
complicates exception & interrupt handling
September 14, 2005
IBM 360: Branches & Condition Codes
• Arithmetic and logic instructions set condition
6.823 L3- 26
Arvind
codes
– equal to zero
– greater than zero
– overflow
– carry...
• I/O instructions also set condition codes
– channel busy
•... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
823 L3- 29
Arvind
• Separate instructions to manipulate three types of reg.
8 60-bit data registers (X)
8 18-bit address registers (A)
8 18-bit index registers (B)
• All arithmetic and logic instructions are reg-to-reg
3
6
opcode i
3
j
3
k
Ri ← (Rj) op (Rk)
• Only Load and Store instructions refer to mem... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
- 32
Arvind
Full Employment for Architects
• Good news: “Ideal” instruction set changes continually
– Technology allows larger CPUs over time
– Technology constraints change (e.g., now it is power)
– Compiler technology improves (e.g., register allocation)
– Programming styles change (assembly, HLL, object-orien... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
Simple Probabilistic
Reasoning
6.873/HST951
Harvard-MIT Division of Health Sciences and Technology
HST.951J: Medical Decision Support
Change over 30 years
• 1970’s: human knowledge, not much data
• 2000’s: vast amounts of data, traditional human
knowledge (somewhat) in doubt
• Could we “re-discover” all of me... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
Specificity
-
FN
+
FP
T
Receiver Operator Characteristic
(ROC) Curve
TPR (sensitivity)
1
T
0
0
FPR (1-specificity)
1
What makes a better test?
TPR (sensitivity)
1
superb
OK
worthless
0
0
FPR (1-specificity)
1
How certain are we after a test?
T+
TP=p(T+|D+)
D+
p(D+)
FN=p(T-|D+)
D?
T
T+
p(D-)=1-p... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
antibiotics
Perform pyelogram
Perform arteriography
Measure patient’s
temperature
Determine if there is
proteinuria
What happens when we act?
• Treatment: leads to few possible outcomes
– different outcomes have different probabilities
• probabilities depend on distribution of disease probabilities
– value of out... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
oli
Renal infarction (bilateral)
Renal vein thrombosis
Malignant hypertension & nephrosclerosis
Scleroderma
ARF’s Database: P(obs|D)
Conditional probabilities for
Proteinuria
Diseases
Probabilities
Trace
to 2+
3+ to
4+
0
ATN
FARF
OBSTR
AGN
CN
HS
PYE
AE
RI
RVT
VASC
SCL
CGAE
MH
0.1
0.8
0.7
0.01
0.01
0... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
sex
transfusion within one day
jaundice or ascites
ischemia of extremities or aortic
aneurism
• atrial fibrillation or recent MI
Invasive tests and treatments
• Tests
• Treatments
– biopsy
– retrograde
pyelography
– transfemoral
arteriography
–
–
–
–
–
–
–
–
steroids
conservative therapy
iv-fluids
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
30
0.25
0.05
0.35
0.05
0.20
0.60
0.35
0.20
0.20
0.90
0.90
0.25
0.85
0.60
0.60
0.90
0.25
0.90
Modeling test:
transfemoral arteriography
p(clot)
cost
0.01 500
0.01 800
0.01 500
0.01 500
0.01 500
0.01 800
0.01 500
0.03 800
0.85 500
0.50 500
0.01 500
0.01 500
0.01 500
0.01 500
atn
farf... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
How many questions do you need to ask
to distinguish among n items? log2(n)
• Entropy of a probability distribution is a
measure of how certainly the distribution
identifies a single answer; or how many
more questions are needed to identify it
Entropy of a distribution
Hi (P1,K, Pn ) = �
- Pj log2 Pj
n
j =1
P
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
,1)
– Can compute second order probability
P(p)distribution
“real” p
= average
p
Assumptions in ARF
• Exhaustive, mutually exclusive set of
diseases
• Conditional independence of all questions,
tests, and treatments
• Cumulative (additive) disutilities of tests
and treatments
• Questions have no modeled disu... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
3.052 Nanomechanics of Materials and Biomaterials Tuesday 02/27/07
I
Prof. C. Ortiz, MIT-DMSE
LECTURE 6: AFM IMAGING II :
ARTIFACTS AND APPLICATIONS
Outline :
LAST TIME : BASIC PRINCIPLES OF ATOMIC FORCE MICROSCOPY ................................................ 2
FACTORS AFFECTING RESOLUTION .................... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/50b9921af4b6e9f6f039b4489ab10c2a_lec6.pdf |
electron microscopes, samples do not need to be coated
or stained, minimal damage, 2) Unlike electron microscopes, samples can be
imaged in fluid environments (near-physiological conditions), 3) Unlike STM
samples do not need to be conductive, 4) Sub-nm resolutions have been
achieved on biological samples (detailed... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/50b9921af4b6e9f6f039b4489ab10c2a_lec6.pdf |
Fadhesion
0
Distance, D (nm)
CANTILEVER
THERMAL NOISE
Lindsay Scanning Tunneling Microscopy
and Spectroscopy 1993, 335.
kt
Shao, et al. Ultramicroscopy 1996,
66, 141.
=
cantilever
m
δt(max) m
δt(max)
m
PROBE TIP SHARPNESS
Sheng, et al. J. Microscopy 1999, 196, 1.
Image removed due to
copyright restrictions.
Image r... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/50b9921af4b6e9f6f039b4489ab10c2a_lec6.pdf |
θ
2D Height
Profile
-Deep feature
-Depth underestimated
-Need high aspect ratio probe tip.
4
3.052 Nanomechanics of Materials and Biomaterials Tuesday 02/27/07
AFM IMAGING OF BIOLOGICAL MACROMOLECULES: DNA
Prof. C. Ortiz, MIT-DMSE
Tapping Mode image of nucleosomal DNA.
Courtesy of Y... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/50b9921af4b6e9f6f039b4489ab10c2a_lec6.pdf |
vier, Inc., http://www.sciencedirect.com. Used with permission.
6
3.052 Nanomechanics of Materials and Biomaterials Tuesday 02/27/07
SUPPORTED LIPID BILAYERS
Prof. C. Ortiz, MIT-DMSE
http://en.wikipedia.org/wiki/Image:Cell_membrane_detailed_diagram.svg
http://faculty.virginia.edu/ta... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/50b9921af4b6e9f6f039b4489ab10c2a_lec6.pdf |
4. Cyclic groups
Lemma 4.1. Let G be a group and let Hi, i ∈ I be a collection of
subgroups of G.
Then the intersection
is a subgroup of G
H = Hi,
i∈I
Proof. First note that H is non-empty, as the identity belongs to every
Hi. We have to check that H is closed under products and inverses.
Suppose that g and h ... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
be the smallest subset of G, closed under taking products
and inverses.
As H is closed under taking products and inverses, it is clear that
H must contain K. On the other hand, as K is a subgroup of G, K
must contain H.
But then H = K.
D
Definition 4.4. Let G be a group. We say that a subset S of G gen
erates G,... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
not one. As the order of g divides the order of
G and this is prime, it follows that the order of g is equal to the order
of G.
But then G = (g) and G is cyclic.
D
It is interesting to go back to the problem of classifying groups of
finite order and see how these results change our picture of what is
going on.
N... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
.
∗
e
a
b
c
e a b c
e a b c
a e ?
e
b
c
e
Now ? must in fact be c, simply by a process of elimination. In fact
we must put c somewhere in the row that contains a and we cannot
put it in the last column, as this already contains c. Continuing in this
way, it turns out there is only one way to fill in th... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
First an easy lemma about the order of an element.
Lemma 4.9. Let G be a group and let g ∈ G be an element of G.
Then the order of g is the smallest positive number k, such that
ka = e.
Proof. Replacing G by the subgroup (g) generated by g, we might as
well assume that G is cyclic, generated by g.
Suppose that gl... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
= e is infinity. Thus we may assume that
the order of g is finite.
Suppose that |G| < k. Then there must be some repetitions in the
set
{ e, g, g , g , g , . . . , g k−1 }.
4
3
2
Thus ga = g
b for some a = b between 0 and k − 1. Suppose that a < b.
Then gb−a = e. But this contradicts the fact that k is the smalles... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
.
As G is generated by a, there are integers m and n such that g = am
and h = an . Then
gh = a m a n
m+n
= a
n+m
= a
= hg.
4
MIT OCW: 18.703 Modern AlgebraProf. James McKernan
Thus G is abelian. Hence (1).
(2) and (3) follow from (4.9).
D
Note that we can easily write down a cyclic group of order n. The ... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
there are two ways to calculate
'
[1] + [5].
One way is to add 1 and 5 and take the equivalence class. [1] + [5] =
[6]. On the other hand we could compute [1] + [5] = [4] + [−1] = [3].
Of course [6] = [3] = [0] so we are okay.
So now suppose that a' is equal to a modulo n and b' is equal to b
modulo n. This means... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
b] = [a · b].
Once again we need to check that this is well-defined. Exercise left
for the reader.
Do we get a group? Again associativity is easy, and [1] plays the
role of the identity. Unfortunately, inverses don’t exist. For example
[0] does not have an inverse. The obvious thing to do is throw away
zero. But e... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
rime to n. But then [ab] ∈ Un. Hence
multiplication is well-defined.
6
MIT OCW: 18.703 Modern AlgebraProf. James McKernan
This rule of multiplication is clearly associative. Indeed suppose that
[a], [b] and [c] ∈ Un. Then
([a] · [b]) · [c] = [ab] · c
= [(ab)c]
= [a(bc)]
= [a] · [bc]
= [a] · ([b] · [c]).
So mu... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
The Euler ϕ function is multiplicative.
That is, if m and n are coprime positive integers,
ϕ(mn) = ϕ(m)ϕ(n).
Proof. We will prove this later in the course.
D
7
MIT OCW: 18.703 Modern AlgebraProf. James McKernan
Given (4.15), and the fact that any number can be factored, it suffices
to compute ϕ(pk), where p is pr... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
a prime number. Then
ϕ(p k) = p k − p k−1 .
Example 4.18. What is the order of U5000?
5000 = 5 · 1000 = 5 · (10)3 = 54 · 23 .
Now
and
ϕ(23) = 23 − 22 = 4,
ϕ(54) = 54 − 53 = 53(4) = 125 · 4.
As the Euler-phi function is multiplicative, we get
ϕ(5000) = 4 · 4 · 125 = 24 · 53 = 2000.
It is also interesting to se... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
has order two. But
then U8 cannot be cyclic.
9
MIT OCW: 18.703 Modern AlgebraProf. James McKernan MIT OpenCourseWare
http://ocw.mit.edu
18.703 Modern Algebra
Spring 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
18.404/6.840 Lecture 22
Last time:
- Finished NL = coNL
- Time and Space Hierarchy Theorems
Today: (Sipser §9.2)
- A “natural” intractable problem
- Oracles and P versus NP
1
Review: Hierarchy Theorems
Theorems:
SPACE ! " #
⊆, SPACE " #
for space constructible ".
TIME ! " #... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
PTIME ⊆ EXPSPACE
≠
≠
Space Hierarchy Theorem
Defn: & is EXPTIME-complete if
1) & ∈ EXPTIME
2)
Same for EXPSPACE-complete
For all ( ∈ EXPTIME, ( ≤* &
Theorem: If B is EXPTIME-complete then & ∉ P
Theorem: If B is EXPSPACE-complete then & ∉ PSPACE (and & ∉ P)
intractable
Next will exhibit an EXPSPACE-complete pr... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
Showing ! ≤# $%&'(↑
Theorem: $%&'(↑ is EXPSPACE-complete
Proof continued: Let ! ∈ EXPSPACE decided by TM + in space 2 -.
Give a polynomial-time reduction / mapping ! to $%&'(↑.
.
/ 0 = 23, 25
0 ∈ ! iff 6 23 = 6 25
all strings except a rejecting computation history for + on 0.
Construct 23 so that 6 23
Construct ... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
history for 3 on 4.
*2 = *+,-./0,10 ∪ *+,-.789: ∪ *+,-.1:;:<0
Rejecting computation history for 3 on 4:
2 AL
= >424? ⋯ 4A ˽
… ˽ # ababa
H2 = Hstart
2 AL
⋯
H?
abababa #
⋯
#
2 AL
⋯
= reject ⋯
Hreject
*+,-./0,10 generates all strings that do not start with Hstart = =>424? ⋯ 4A ˽ … ˽
*+,-./0,10 = M> ∪ M2 ∪... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
! ≤# $%&'(↑ (*+,-./012 & *+,-.425267)
Construct *8 to generate all strings except a rejecting computation history for 9 on :.
*8 = *+,-.<7,47 ∪ *+,-./012 ∪ *+,-.425267
Rejecting computation history for 9 on ::
2 BM
>?:8:@ ⋯ :B
˽ … ˽
#
2 BM
ababa ⋯ abababa
#
⋯
#
I8 = Istart
I@
2 BM
⋯ >reject ⋯
Ireject
267 ... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
oracle for !}
Thus NP ⊆ P+#,
NP = P+#,? Probably No because coNP ⊆ P+#,
Defn: NP# = ( ( is decidable in nondeterministic polynomial time with an oracle for !}
Recall MIN-FORMULA = 7 7 is a minimal Boolean formula }
Example: MIN−FORMULA ∈ NP+#,
“On input 7
1. Guess shorter formula 9
2. Use &!' oracle to solve th... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
= coNP7"(
(c) MIN-FORMULA ∈ P()*+
NP()*+ = coNP()*+
(d)
9
Check-in 22.3
Quick review of t... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
Transition to the Systems Age
• Beginning ~ 1940 (according to Blanchard &
Fabrycky)
• Rescuing Prometheus
• Thomas P. Hughes, Prof. of History and
Sociology of Technology, U. of Penn.
• Tells the story of four major projects
– SAGE
– Atlas
– CA/T
– ARPANET
Figure removed for copyright reasons.
Schematic of SAGE ... | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/50d4d56330e3943da83a34c61c690a16_lec2.pdf |
project…”
– www.bigdig.com/
• >7 Miles of tunnels
• Projected to cost
$14.6B
• 87% Complete
Key Aspects of the CA/T
• Greater “messy complexity” than either SAGE or Atlas
(T. Hughes)
• Bechtel / Parsons Brinkerhoff coordinates
• ~1/3 of budget spent on remediation
• Highly publicized mistakes
– Voids in concret... | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/50d4d56330e3943da83a34c61c690a16_lec2.pdf |
res commercial SE and contrasts in with
government SE | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/50d4d56330e3943da83a34c61c690a16_lec2.pdf |
Support Vector Machines
Stephan Dreiseitl
University of Applied Sciences
Upper Austria at Hagenberg
Harvard-MIT Division of Health Sciences and Technology
HST.951J: Medical Decision Support
Overview
• Motivation
• Statistical learning theory
• VC dimension
• Optimal separating hyperplanes
• Kernel functions
•... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
+1,-1}
• Fundamental question: Is learning
consistent?
• Can we infer performance on test set
(generalization error) from performance
on training set?
Statistical learning theory
Average error on a data set D for model
with parameter α:
n
Remp(α ) = 2
1
n ∑| y(α, xi ) − ti |
i =1
Expected error of same mod... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
/ h) + 1) − log(η / 4)
n
• Fix data set and order classifiers
according to their VC dimension
• For each classifier, train and calculate
right-hand side
• Best classifier minimizes right-hand side
Structural risk minimization
R(α ) ≤ Remp(α ) +
h(log(2n / h) + 1) − log(η / 4)
n
Model
Remp
VC conf.
Upper bo... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
+1 resp. g(o) = -1
Margin =|g(x)|/||w|| + |g(o)|/||w||
= 2 /||w||
Largest (optimal) margin:
maximize 2 /||w|| equiv. to
minimize ||w||2
subject to ti (w • xi + w0) –1 ≥ 0
Optimal hyperplanes
• Optimal hyperplane has largest margin
(“large margin classifiers”)
• Parameter estimation problem turned into
constrained o... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
y2
2
x2
2 + 2x1x2 y1 y2 + x2
⋅
2
2 y2
2
y
1
1 y2
y
2
2
y2
2 y1
= x1
= ( x1 y1 + x2 y2)2
2
y1
x1
⋅
=
x2 y2
Nonlinear SVM
• Recall: Input data xi enters calculation
only via dot products xi ·xj or Φ(xi)·Φ(xj)
• Kernel trick: ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
,y) = (x ·y)2
SVM examples
Linearly separable
C=100
SVM examples
C=100
C=1
SVM examples
Linear function
Quad. polynomial
SVM examples
Quad. poly., C=10
Quad. poly., C=100
SVM examples
Cubic polynomial
Gaussian, σ = 1
SVM examples
Quad. polynomial
Cubic polynomial
SVM examples
Cubic polynomial
Degr... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
and Knowledge Discovery. 1998; 2(2):121-
167.
• Christianini N, Shawe-Taylor J. An
introduction to support vector machines.
Cambridge University Press 2000.
• Vapnik V. Statistical learning theory. Wiley
Interscience 1998. | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
Soft Lithography and
Materials Properties in MEMS
Carol Livermore
Massachusetts Institute of Technology
* With thanks to Steve Senturia and Joel Voldman, from
whose lecture notes some of these materials are
adapted.
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectro... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
romechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 3
Sample process: multilayer SU-8 microfluidics
> Spin coat, prebake, expose, and
postbake first layer
> Spin coat, prebake, ex... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
7J/2.372J Spring 2007, Lecture 5 - 5
Cracking in SU-8
> SU-8 shrinks in developer, causing cracks and loss of
adhesion
cracks
100 μm
Courtesy of Joel Voldman. Used with permission.
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
ography from a master (Si, SU-8, etc)
> Used as a conformable stamp for patterning onto other surfaces
> Good for sealing microfluidic devices; can be sealed to many
materials
> Can be spin-coated
> Possible to dry etch
> Low cost pattern replication
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Desi... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
and peel
Master
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 10
UV light
... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 12
Microcontact printing on non-planar surfaces
Figure 2 on p. 186 in Rogers, J. A., R. J. Jackman, and G. M. Whitesides. "Constructing Single- and
Multiple-helical Microcoils and Characte... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 14
Parylene
> A vapor-deposited polymer that provides very conformal
coatings
> Thickness range: submicron to about 75 microns
> Chemically resistant, relatively inpermeable
• Component encapsulation
> Low friction film can act as a dry lubricant
> Low-defect d... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
> Material Properties in MEMS
• Role of material properties in MEMS
• Some examples
• Determining material properties
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute o... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
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