text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
Pj ψ(X) = j
ψ 1
M
j
≤
≤
(cid:2)
(cid:3)
1
≥ −
1
M
2
M
j,k=1
KL
(Pj, Pk) + log 2
P
log(M
1)
−
6
6
6
5.4. Lower bounds based on many hypotheses
111
where the infimum is taken over all tests with values in
1, . . . , M
.
}
{
1, . . . , M
be a random variable such that IP(Z = i) = 1/M
PZ . Note that PZ is a mixture distrib... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
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 and
qj log(qj ) =
j=ψ(X)
X
P
−
j=ψ(X)
X
qj log
1
qj
(cid:16) (cid:17)
log
≥ −
qj
qj
j=ψ(X)
(cid:16) X (cid:17)
log(M
=
−
1) .
−
By the same convexity argument, we get h(x)
log 2. It yields
≥ −... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
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
X
Together with (5.8), it yields
1
M 2
M
1
j,k=
X
Since
KL(Pj , Pk)
log M
log 2
−
≥ −
−
1
IP(Z = ψ(X)) log(M )
−
IP(Z = ψ(X)) =
M1
M
j=1
X
this implies the desired result.
Pj(ψ(X) = j)
max Pj (ψ(X) = j) ,
1 j
≤ ≤
M
≤
Fano’s ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
(IPj, IP
θk
n θ
k) = | − |
j
2σ2
2
2
≤
α log(M ) .
Moreover, since M
5,
≥
1
M
2
M
j,k=1
KL
(IPj, IPk) + log 2
P
log(M 1)
−
α log(M ) + log 2
≤ lo
g(M
1)
−
≤
2α + .
1
2
The proof then follows from Fano’s inequality.
2
≤
Theorem 5.11 indicates that we must take φ
ασ log(M ). Therefore, the
2n
larger the M , the larger th... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
2), there exist binary
∈
ii) M = e d
(
γ
2
⌊
2
γ d
e 2
(cid:1)
.
(cid:0)
⌋ ≥
Proof. Let ωj,i, 1
≤
with parameter 1/2 and observe that
d, 1
≤
≤
≤
j
i
M to be i.i.d Bernoulli random variables
ρ(ωj, ωk) = X
d
−
∼
Bin(d, 1/2) .
Therefore it follows from a union bound that
IP
j = k , ρ(ωj, ωk) <
∃
(cid:2)
1
2
−
γ
d
M (M 1)
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
�
−
1
2
ω , . . . ω
1
M
0, 1
}
∈ {
d
that
5.5 APPLICATION TO THE GAUSSIAN SEQUENCE MODEL
We are now in a position to apply Theorem 5.11 by choosing θ1, . . . , θM
based on ω1, . . . , ωM from the Varshamov-Gilbert Lemma.
Lower bounds for estimation
⌋ ≥
ed/16
Take γ = 1/4 and apply the Varshamov-Gilbert Lemma to obtain ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
imator θls = Y.
ˆ
Note that this rate is minimax over sets Θ that are strictly smaller than IRd
(see Problem 5.4). Indeed, it is minimax over any subset of IRd that contains
θ1, . . . , θM .
6
6
5.5. Application to the Gaussian sequence model
115
Lower bounds for sparse estimation
It appears from Table 5.1 that when e... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
∈
be k random variables such that U1 is drawn uniformly at (cid:0)ra(cid:1)ndom
1, . . . , d
}
{
and for any i = 2, . . . , k, conditionally on U1, . . . , Ui 1, the ran-
from
1, . . . , d
}
dom variable Ui is drawn uniformly at random from
.
1}
Then define
U1, . . . , Ui
1, . . . , d
}\{
{
{
−
−
k
1 if i
0 otherwise .
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
equally likely.
}
Note that
ρ(ω, x0)
k
−
≥
k
Zi ,
i=1
X
supp(x0)).
Indeed the left hand side is the number of
where Zi = 1I(Ui ∈
coordinates on which the vectors ω, x0 disagree and the right hand side is
the number of coordinates in supp(x0) on which the two vectors disagree. In
Ber(k/d) and for any i = 2, . . . , d, c... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
s
(cid:1)
Zi
h
i=1
X
(cid:0)
(cid:1)i(cid:0)
2k
d
(es
−
1) + 1
(cid:1)
≤
..
.
2k
d
≤
= 2k
(cid:0)
k
1) + 1
(es
−
(cid:1)
6
5.5. Application to the Gaussian sequence model
117
For s = log(1 + d ). Putting everything together, we get
2k
ωj = ωk : ρ(ωj, ωk) < k
≤
exp
log M + k log 2
IP
∃
(cid:0)
sk
− 2
k
2
k
2
−
−
(cid:1... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
(ωj, ωk) log(1 + )
(ii)
θj−
|
for β =
2
d
ρ(ωj, ωk) log(1+ )
2k
θk|2 =
α . Applying now Theorem 5.11 yields
8
≤
d
2k ≥
4
2 2
β σ
8n
2kβ2σ2
n
2 2
β σ
n
β2σ2
n
k log(1 + )
d
2k
log(1+ )
d
2k ≤
2ασ2
n
log(M ) ,
p
inf sup IPθ |
ˆθ θ
IRd
∈
(cid:0)
k
θ
|0≤
|
ˆ
θ
θ
2
|2 ≥
−
α2σ2
64n
It implies the following corollary.
k log(1... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
that
for the lower bound.
|1 =
n . We can essentially do the same
log d
θ′
θ
σ
|
|
q
(0, 1) be a parameter to be chosen later
Assume that d √
n and let β
and define k to be the smallest integer such that
≥
∈
k
R
≥ βσ
r
n
log(ed/√
n)
.
Let ω1, . . . , ωM be obtained from the sparse Varshamov-Gilbert Lemma 5.14
with this ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
. Recall that
that
ε > 0, the minimax rate of estimation over
model is
IRd such
n1/2+ε,
B1(R) in the Gaussian sequence
|1 ≤
∈
≥
(R)
⊂
B
1
|
0(k)) = min(R , Rσ
2
φ(
B
log d
n
) .
Moreover, it is attained by the constrained least squares estimator θls
R
and by the trivial estimator θ = 0 otherwise.
σ
ˆ
ˆ
B1(R)
if
log d
n... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
j=0,1
≥
1
4
e−
KL(IP0,IP1)
.
Problem 5.3. For any R > 0, θ
B2(θ, R) the (Euclidean)
ball of radius R and centered at θ. For any ε > 0 let N = N (ε) be the largest
integer such that there exist θ1, . . . , θN ∈ B2(0, 1) for which
IRd, denote by
∈
for all i = j. We call the set
an ε-packing of
{
(a) Show that there exist... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
2β+1
.
fj =
C
√
n
N
ωjiϕi
i=1
X
N for some appropriately chosen
where C is a constant, ωj ∈ {
N and
1 is the trigonometric basis.]
0, 1
}
ϕj}j
{
≥
6
6
Bibliography
[AS08]
[Ber09]
[Bil95]
[Bir83]
[BLM13]
[BRT09]
[BT09]
[Cav11]
[CT06]
Noga Alon and Joel H. Spencer. The probabilistic method. Wiley-
Interscience Series in... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
. Inverse problems in statistics. In Inverse prob-
lems and high-dimensional estimation, volume 203 of Lect. Notes
Stat. Proc., pages 3–96. Springer, Heidelberg, 2011.
Thomas M. Cover and Joy A. Thomas. Elements of information
theory. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ,
second edition, 2006.
121
Bibli... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
pringer, New York, 2013.
Branko Grunbaum. 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 Studie... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
Gabriel Peyr´e.
Harry Markowitz. Portfolio selection. The journal of finance,
7(1):77–91, 1952.
Arkadi Nemirovski. Topics in non-parametric statistics. In Lec-
tures on probability theory and statistics (Saint-Flour, 1998),
volume 1738 of Lecture Notes in Math., pages 85–277. Springer,
Berlin, 2000.
G. Pisier. Remarques... | 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 |
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 |
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 inhibit the reproduction
process, and clear the virus cells from their body.
However, some of t... | 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 |
, in %
float resistance;
static const float defaultReproductionRate = 0.1;
// resistance against drugs, in %
public:
constructors
Virus(float newResistance);
Virus(float newReproductionRate, float newResistance);
Virus* reproduce(float immunity);
bool survive(float immunity);
};
method
don’t forget the semi-c... | 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 |
.g. virus must be able to reproduce)
implementation: parts that may change frequently
(e.g. representation of resistance inside virus)
26
Protect your private parts!
Client
Interface
Implementation
Why is this bad?
X
27
Access control: public vs. private
class Virus {
float reproductionRate;
float resistance;... | 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 |
this virus cell survives, given the patient's immunity
bool Virus::survive(float immunity) {
// If the patient's immunity is too strong, then this cell cannot survive
if (immunity > resistance)
return false;
return true;
}
const float Virus::defaultReproductionRate;
31
Header inclusion
#include <stdlib.h>
#... | 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 |
void takeDrug();
bool simulateStep();
};
37
Patient class declaration
#include “Virus.h”
#define MAX_VIRUS_POP 1000 Array of pointers to objects
class Patient {
Virus* virusPop[MAX_VIRUS_POP];
int numVirusCells;
float immunity;
// degree of immunity, in %
Constructor
public:
Patient(float initImmunity, int 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 |
rand()/RAND_MAX;
virusPop[i] = new Virus(resistance);
}
numVirusCells = initNumVirusCells;
}
43
Using dynamically allocated objects
bool Patient::simulateStep() {
Virus* virus;
bool survived = false;
...
for (int i = 0; i < numVirusCells; i++){
virus = virusPop[i];
survived = virus->survive(immunity);
if (... | 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 |
{
...
bool checkRep();
public:
...
};
void Patient::takeDrug() {
assert(checkRep());
...
assert(checkRep());
}
Patient::Patient(float initImmunity, int initNumViruses) {
...
assert(checkRep());
}
51
Preserving rep. invariant
Will calling checkRep() slow down my program?
Yes, but you can take them out... | 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 |
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 measurable.”
Basic properties of measurable funct... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/509c71b048eab7f1971e8f1cea40392e_18125_lec1.pdf |
�. 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 measurable,
u−1(I1) ∈ M, and since v is measurable, v−1(I2)... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/509c71b048eab7f1971e8f1cea40392e_18125_lec1.pdf |
(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 of subsets of X. Then there
exists a smallest σalgebra M∗ such that F ⊂ M∗. We call M ∗ the
σalgebra g... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/509c71b048eab7f1971e8f1cea40392e_18125_lec1.pdf |
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 measurable. In particular, continuous functions are Borel
measurable.
Terminology:
• Fσ (“Fsigma”) = countable union of closed se... | 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 |
model, had many more innovative
features
– tagged data
– virtual memory
– multiple processors and memories
September 14, 2005
6.823 L3- 6
Arvind
A Stack Machine
Processor
stack
:
A Stack machine has a stack as
a part of the processor state
Main
Store
typical operations:
push, pop, +, *, ...
Instruction... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
Size and Memory References
6.823 L3- 10
Arvind
a b c * + a d c * + e - /
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
R0 R1
R0 R1 R2
R0 R1
R0
memory refs
a
b
c, ss(a)
sf(a)
a
d, ss(a+... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
1 R2
R1
R0
R0 R1 R2
R1
R0
R0
a and c are
“loaded” twice
⇒
not the best
use of registers!
September 14, 2005
Register Usage in a GPR Machine
(a + b * c) / (a + d * c - e)
6.823 L3- 13
Arvind
a
c
b
R1
R0
d
R1
R0
e
R0
R3
More control over register usage
since registers can be na... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
any element in
the data area
– jump to any instruction
in the code area
– move any element in
the stack frame to the
top
machinery to
carry out
+, -, etc.
⇔
SP
DP
PC
stack
a
b
c
.
.
.
data
push a
push b
push c
*
+
push e
/
code
September 14, 2005
Stack versus GPR Organization
Amdahl, Blaa... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
language
– Fixed-height stack design simplified implementation
– Stack trashed on context swap (fast context switches)
– Inmos T800 was world’s fastest microprocessor in late 80’s
• Forth machines
– Direct support for Forth execution in small embedded real-
time environments
– Several manufacturers (Rockwell, Pat... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
Control Store Read only 1µsec
256K - 512 KB
64-bit
5 nsec/level
Transistor Registers
Conventional circuits
IBM 360 instruction set architecture completely hid
the underlying technological differences between
various models.
With minor modifications it survives till today
September 14, 2005
IBM S/390 z900 Mi... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
2
SS format: store to store instructions
M[(B1) + D1] ← M[(B1) + D1] op M[(B2) + D2]
iterate “length” times
Most operations on decimal and character strings
use this format
MVC move characters
MP multiply two packed decimal strings
CLC
compare two character strings
...
Multiple memory operations per instruct... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
addr
result
addr
September 14, 2005
CDC 6600:
A Load/Store Architecture
6.823 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
... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/50a585c85a21b1fbbb3f9d11e86ba850_l03_sixties.pdf |
” 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-oriented, …)
– Applications change (e.g., multimedia, ....)
– Bad ne... | 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 |
(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(D+)
FP=p(T+|D-)
D-
Bayes’ Rule:
TN=p(T-|D-)
T-
Pi(D j)P(S| D j)
Pi+1( D j) = n
� Pi(Dk)P(S|D k)
k=1
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
• value may depend on how we got there (see below)
• therefore, value of a treatment can be determined by
expectation
• Test: lead to few results, revise probability
distribution of diseases, and impose disutility
• Questions: lead to few results, revise probability
distribution
Treatment Outcome
(not as in ARF... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
.01
0.01
0.8
0.4
0.1
0.1
0.001
0.01
0.1
0.001
0.001
0.8
0.2
0.3
0.2
0.8
0.2
0.6
0.8
0.7
0.1
0.2
0.4
0.2
0.4
0.1
0.001
0.001
0.8
0.2
0.001
0.001
0.1
0.2
0.9
0.8
0.5
0.8
0.6
Questions
casts in urine sediment
• Blood pressure at onset
• proteinuria
•
• hematuria
• history of pr... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
Pi(Dk)P(S|Dk)
k=1
Bayes’ rule
Value of treatment
• Three results: improved, unchanged,
worsened
– each has an innate value, modified by “tolls”
paid on the way
• Probabilities depend on underlying
disease probability distribution
I
V(I)
Ip
Tx
Up
U
V(U)
W
V(W)
Modeling treatment
Steroids
improved unchan... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
vasc
scl
cgae
mh
How large is the tree?
• Infinite, or at least (27+3+8)^(27+3+8), ~10^60
• What can we do?
– Assume any action is done only once
– Order:
• questions
• tests
• treatments
• 27! x 4 x 3 x 2 x 8, ~10^30
• Search, with a myopic evaluation function
– like game-tree search; what’s the static ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ae225657d5179de807b5039fb54eb6_lecture2.pdf |
the patient's sex?
1 Male
2 Pregnant Female
3 Non-pregnant Female
Reply: 1
. . .
Local Sensitivity Analysis
Case-specific Likelihood Ratios
Therapy Planning Based on
Utilities
Global Sensitivity Analysis
• When asking questions, “how bad could it
get for the leading hypothesis?”
– Assume all future answers are w... | 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 |
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
the molecular
conformation, spatial arrangement, structural dimensions, rate dependent
processes, etc.)
information on
-Basic ... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/50b9921af4b6e9f6f039b4489ab10c2a_lec6.pdf |
m
δt(max)
m
PROBE TIP SHARPNESS
Sheng, et al. J. Microscopy 1999, 196, 1.
Image removed due to
copyright restrictions.
Image removed due to copyright restrictions.
3-D model of sharp probe tip on a protein,
from Lieber et al, 2000 (http://cnst.rice.edu)
3
3.052 Nanomechanics of Materials and Biomaterial... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/50b9921af4b6e9f6f039b4489ab10c2a_lec6.pdf |
750 nm scan
courtesy C. Tolksdorf, Digital Instruments/Veeco,
Santa Barbara, USA, and R. Schneider and G.
Muskhelishvili, Istitut für Genetik und
Mikrobiologie, Germany.
Courtesy of Veeco Instruments and
G. Muskhelishvili. Used with permission.
Courtesy of Zhifeng Shao. Used with permission.
http://people.virginia... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/50b9921af4b6e9f6f039b4489ab10c2a_lec6.pdf |
Biophys. J. 2006 91, 2532.
DOPC
Courtesy of the Biophysical Society. Used with permission.
7
3.052 Nanomechanics of Materials and Biomaterials Tuesday 02/27/07
NANOMECHANICS OF SUPPORTED LIPID BILAYERS
Prof. C. Ortiz, MIT-DMSE
Courtesy of the Biophysical Society... | 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 |
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, if the smallest subgroup of G that contains S is G itself.
Definition 4.5. Let... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
(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.
Now we know that every group of order 1, 2, 3 and 5 must be cyclic.
Suppose that G has order 4. There are two cases. If G has an element
a of o... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
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 the whole table
∗
e e
a a
b b
c c
e
a b c
a b c
e c b
c e a
b a e
So now we have a complete classification of all fini... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
subgroup (g) generated by g, we might as
well assume that G is cyclic, generated by g.
Suppose that gl = e. I claim that in this case
G = { e, g, g , g , g , . . . , g l−1 }.
3
Indeed it suffices to show that the set is closed under multiplication
4
2
and taking inverses.
Suppose that gi and gj are in the set. Then... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
a = b between 0 and k − 1. Suppose that a < b.
Then gb−a = e. But this contradicts the fact that k is the smallest
D
integer such that gk = e.
Lemma 4.10. Let G be a finite group of order n and let g be an element
of G.
Then gn = e.
Proof. We know that gk = e where k is the order of g. But k divides
n. So n = km... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
n. The
group of rotations of an n-gon forms a cyclic group of order n. Indeed
any rotation may be expressed as a power of a rotation R through
2π/n. On the other hand, Rn = 1.
However there is another way to write down a cyclic group of order
n. Suppose that one takes the integers Z. Look at the subgroup nZ.
Then... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
a + pn
b' = b + qn,
and
where p and q are integers.
Then
'a + b' = (a + pn) + (b + qn) = (a + b) + (p + q)n.
So we are okay
[a + b] = [a + b'],
'
5
MIT OCW: 18.703 Modern AlgebraProf. James McKernan
and addition is well-defined. The set of left cosets with this law of
addition is denote Z/nZ, the integers mod... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
0].
So if you throw away [0] then you have to throw away [2]. In fact
given n, you should throw away all those integers that are not coprime
to n, at the very least. In fact this is enough.
Definition-Lemma 4.12. Let n be a positive integer.
The group of units, Un, for the integers modulo n is the subset of
Z/nZ o... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
inverse of [a]. We
want an integer b such that
This means that
[ab] = 1.
ab + mn = 1,
for some integer m. But a and n are coprime. So by Euclid’s algorithm,
D
such integers exist.
Definition 4.13. The Euler φ function is the function ϕ(n) which
assigns the order of Un to n.
Lemma 4.14. Let a be any integer... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
if a is coprime to p.
Proof. Follows from (4.14).
D
How about ϕ(pk)? Let us do an easy example.
Suppose we take p = 3, k = 2. Then of the eight numbers between
1 and 8, two are multiples of 3, 3 and 6 = 2 · 3. More generally, if a
number between 1 and pk − 1 is not coprime to p, then it is a multiple
of p. But t... | https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/50c134275caf32dbf4430ab097185157_MIT18_703S13_pra_l_4.pdf |
.
52 = 25 = 1 mod 6.
How about U8? Well
ϕ(8) = 4.
So either U8 is either cyclic of order 4, or every element has order 2.
1, 3, 5 and 7 are the numbers coprime to 2. Now
32 = 9 = 1 mod 8,
8
MIT OCW: 18.703 Modern AlgebraProf. James McKernan
and
52 = 25 = 1 mod 8,
72 = 49 = 1 mod 8.
So
[3]2 = [5]2 = [7]2 = ... | 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 |
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 problem
3
... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
-.
Give a polynomial-time reduction / mapping ! to $%&'(↑.
.
/ 0 = 23, 25
0 ∈ ! iff 6 23 = 6 25
all strings except a rejecting computation history for + on 0.
Construct 23 so that 6 23
Construct 25 = Δ∗ ( Δ is the alphabet for computation histories, i.e., Δ = Γ ∪ % ∪ # ) •
=
23 construction: 23 = 2<=>?@A=BA ∪ 2<=... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
H2 = Hstart
2 AL
⋯
H?
abababa #
⋯
#
2 AL
⋯
= reject ⋯
Hreject
*+,-./0,10 generates all strings that do not start with Hstart = =>424? ⋯ 4A ˽ … ˽
*+,-./0,10 = M> ∪ M2 ∪ M? ∪ ⋯ ∪ MA ∪ Mblanks ∪ M#
Remember: Δ is the alphabet for computation histories, i.e., Δ = Γ ∪ % ∪ # )
Notation: Δd = Δ ∪ {f}
Δ.+ = Δ wit... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
:.
*8 = *+,-.<7,47 ∪ *+,-./012 ∪ *+,-.425267
Rejecting computation history for 9 on ::
2 BM
>?:8:@ ⋯ :B
˽ … ˽
#
2 BM
ababa ⋯ abababa
#
⋯
#
I8 = Istart
I@
2 BM
⋯ >reject ⋯
Ireject
267 generates all strings that do not contain >re
ject
*+,-.425
*+,-.425267 = Δ.O
∗
reject
*+,-./012 generates all strings th... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
to solve the coNP problem: 7 and 9 are equivalent
3. Accept if 7 and 9 are equivalent. Reject if not.”
8
... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/50cb369d1be3c7fbe0886e318aea13c2_MIT18_404f20_lec22.pdf |
ed !"#$%↑ is EXPSPACE-complete and thus !"#$%↑ ∉ PSPACE
4. Defined oracle TMs
5. Showed P( = NP( for some oracle *
6. Discussed relevance to the P vs NP question
10
!"REX ∈ PSPACE
Theorem: !"REX ∈ PSPACE
Proof: Show !"RE' ∈ NPSPACE
“On input (), (... | 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 |
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 concrete of Zakim Bridge
– Planning maps missing the Fleet Center
– "Based on anecdotal evi... | 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 |
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 model given
unseen data distributed like D:
R(α ) = 2 ∫ | y(α, x) − t | dP( x, t)
1
Stat... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
• 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 bound
f1(α)
f2(α)
f3(α)
f4(α)
f5(α)
best
Model selection
• Cross-validation: use test sets to
estimate error
• Penalize model complexity:
– Akaike info... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
(w • xi + w0) –1 ≥ 0
Optimal hyperplanes
• Optimal hyperplane has largest margin
(“large margin classifiers”)
• Parameter estimation problem turned into
constrained optimization problem
• Unique solution w = Σαixi over all inputs xi
on the margin (“support vectors”)
• Decision function g(x) = sign(Σαixi ·x + w0)... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
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:
K(xi ,xj) = Φ(xi)·Φ(xj)
• Advantage: no need to calculate Φ
• Advantage:... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/50ff51d9bebe9dbba735b715c65824e5_lecture12.pdf |
VM examples
Cubic polynomial
Degree 4 poly.
SVM examples
Gaussian, σ = 1
Gaussian, σ = 3
Performance comparison
• Log. regression ⇔ ANN ⇔ SVM
• Real-world data set
• 1619 lesion images
• 107 morphometric features:
– Global (size, shape)
• size
• shape
– Local (color distributions)
• Use ROC analysis
C... | 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 |
2007, Lecture 5 - 3
Sample process: multilayer SU-8 microfluidics
> Spin coat, prebake, expose, and
postbake first layer
> Spin coat, prebake, expose, and
postbake second layer
> Develop both layers
> Cap with SU-8 coated transparent plate
> Expose to crosslink SU-8 “glue”, final
Described in Jackman, J. Micromech.... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
Ware (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 6
SU-8 Removal
> When using SU-8 as a resist and not a structural material, it must
be removed!
• Enduring challenge – best option is not to strip the SU-8
> Postbaked material ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
CH3
CH3
CH3
> Upon treatment in oxygen
CH3 Si O Si O
Si
CH3
CH3
CH3
CH3
n
plasma, PDMS seals to itself,
glass, silicon, silicon nitride,
and some plastic materials.
Plasma oxidation
Air (~ 10 min)
contact PDMS
surfaces
irreversible seal:
formation of
covalent bonds
Courtesy of Hang Lu and Rebecca Jackman. Used with ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
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 - 11
Microcontact printing
Stamp
Stamp
Stamp
> Apply ink to an elastomer stamp
> Bring stamp i... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
deformable
Master
Master
> Imprint in a thermoplastic material
by heating and applying pressure
> Typical material: PMMA
(polymethyl-methacrylate)
> Or imprint in UV-curable fluid, like
polyurethane
> Process usually leaves trace
material in “clear” areas, which may
be removed by dry etch
> Can replicate nanoscale... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
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 - 16
Outline
> Soft Lithography
• Materials and processes
• Patt... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
. "Protein Patterning.“
Biomaterials 19, nos. 7-9 (April 1998): 595-609.
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].
... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
Ware (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 21
Biomaterials processing by stencils
> Stencils
• Use PDMS stamps as dry resists
• Physically pattern biomaterials
• Can use with most any substrate
• Potential damage to cells... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
many MEMS devices is in
coupling one domain to another, and this coupling is
typically described by material properties
• Mechanical to electrical
• Electrical to thermal
• Thermal to fluids
> The failure modes of many MEMS devices are in
coupling one domain to another
• For example, package stress interacting with ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 26
Planning around material properties
> Well-controlled material properties can pose a design challenge
• Many factors, may point design in opposite directions
> Poorly-controlled material properties are worse
• Every device has specifications, which must be met by either
g... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
Inverse of viscosity
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 - 29
Scala... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
J/2.372J Spring 2007, Lecture 5 - 31
Outline
> Soft Lithography
• Materials and processes
• Patterning biomaterials
> 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... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
> Characterize by four-point measurement of test structures with
known geometry.
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 Mont... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
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 - 36
Residual film stress
> Stress in a film deposited on a Si wafer, in the absence of
external loading.
> Two flavor... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
TE mismatch plus high T processing or operation
creates stress (and deformation and/or destruction)
> Examples:
• Bonding glass (quartz or Pyrex) to Si
• Thermal stress in a film that is deposited at high T
> CTE is tabulated, and one of the less variable material
properties.
Cite as: Carol Livermore, course materi... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
is described by a fourth-rank
tensor
] J
⋅σ⋅Π+ρ=E
[
e
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 ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
.777J/2.372J Spring 2007, Lecture 5 - 42
Piezoresistivity in Silicon
> Coefficients depend on doping, and decrease rapidly above
about 1019 cm-3
> Coefficients are functions of temperature
> Typical values
Type
Units
n-type
p-type
Resistivity
Ω-cm
11.7
7.8
π11
10-11 Pa-1
-102.2
6.6
π12
10-11 Pa-1
53.4
-1.1
π44
10-11 ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
perpendicular to the applied electric field
(actuation)
• Straining a piezoelectric creates an electric field both parallel
and perpendicular to the imposed strain (sensing)
> Interaction between stored mechanical energy and stored
electrostatic energy
• Permits both sensing and actuation
• Unlike piezoresistivity, ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
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