text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
8 Keys to DfM in
Developing Countries
• Understand manufacturing
capabilities
• Incorporate the most
accessible, affordable
manufacturing techniques into
your detailed design
19 Design for Assembly
"a process for improving product
design for easy and low-cost
assembly, focusing on functionality
and on assem... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/31bc1da7c60661b5848601aa2f9eba07_MITEC_720JS10_lec12.pdf |
Provide alignment features
• Eliminate fasteners
• Don’t put fasteners in places where you
can’t get access to
27 MIT OpenCourseWare
http://ocw.mit.edu
EC.720J / 2.722J D-Lab II: Design
Spring 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/31bc1da7c60661b5848601aa2f9eba07_MITEC_720JS10_lec12.pdf |
III. The Scaling Hypothesis
III.A The Homogeneity Assumption
In the previous chapters, the singular behavior in the vicinity of a continuous transi
tion was characterized by a set of critical exponents {α, β, γ, δ, ν, η, · · ·}. The saddle–point
estimates of these exponents were found to be unreliable due to the imp... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
= 0
.
(III.1)
The singularities in the free energy can in fact be described by a single homogeneous
function† in t and h, as
f (t, h) = |t|2 gf
h/|t|Δ
.
(III.2)
The function gf only depends on the combination x ≡ h/|t|Δ, where Δ is known as the gap
exponent. The asymptotic behavior of gf is easily obtai... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
going beyond the saddle–point approxima
(III.3)
Δ =
.
tion, the singular form of the free energy (and any other thermodynamic quantity) retains
the homogeneous form
fsing.(t, h) = |t|2−α gf
h/|t|Δ
.
(III.4)
The actual exponents α and Δ depend on the critical point being considered. The depen
(cid:0)
(cid:1) ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
→ 0.
(cid:0)
(cid:1)
It may appear that we have the freedom to postulate a more general form
C±(t, h) = |t|−α±g±
h/|t|Δ±
,
(III.7)
with different functions and exponents for t > 0 and t < 0, that match at t = 0. However,
(cid:0)
(cid:1)
this is ruled out by the condition that the free energy is analytic every... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
for large arguments,
and {A±, B±} are the corresponding pre-factors. Matching to the taylor series in eq.(III.8)
requires p±Δ± = −α± and q±Δ± = −(1 + α±), and leads to
C±
t ≪ hΔ
= A±h−α±/Δ± + B±h−(1+α±)/Δ± |t| + · · · .
(III.10)
(cid:0)
(cid:1)
Continuity at t = 0 now forces α+/Δ+ = α−/Δ−, and (1 + α+)/Δ+ = (1 ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
∼ |t|2−α−Δ gm
h/|t|Δ
.
(III.12)
In the limit x → 0, gm(x) is a constant, and
(cid:0)
(cid:1)
m(t, h = 0) ∼ |t|2−α−Δ , =⇒
β = 2 − α − Δ.
(III.13)
On the other hand, if x → ∞, gm(x) ∼ xp, and
m(t = 0, h) ∼ |t|2−α−Δ
h
|Δ
|t
(cid:18)
p
.
(cid:19)
(III.14)
Since this limit is independent of t, we must have ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
odynamic derivatives, the same gap expo
nent Δ, occurs for all such quantities.
(3) All (bulk) critical exponents can be obtained from only two independent ones, e.g. α
and Δ.
(4) As a result of the above, there are a number of exponent identities. For example,
eqs.(III.13), (III.15), and (III.16) imply
α + 2β + ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
0.67
0.70
1
η
0.04
0.04
0.04
1/4
III.B Divergence of the Correlation Length
The homogeneity assumption relates to the free energy and quantities derived from
it. It says nothing about the behavior of correlation functions. An important property of
a critical point is the divergence of the correlation length, wh... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
9)
where gs and ga are non-singular functions of dimensionless parameters (a is an appropriate
microscopic length). The singular part of the free energy comes from the first term, and
behaves as
fsing.(t, h) ∼
Z
ln
Ld
∼ ξ−d ∼ |t|dν gf
h/|t|Δ
.
(III.20)
(cid:0)
(cid:1)
A simple interpretation of the above res... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
of critical
behavior must thus account for the validity of this relation in low dimension, and its
breakdown for d > 4.
39
III.C Critical Correlation Functions and Self-Similarity
One exponent that has not so far been accounted for is η, describing the decay of
correlation functions at criticality. Exactly at th... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
x
|x|d−2+η ∼ ξ2−η ∼ |t|−ν(2−η), =⇒
γ = (2 − η)ν
. (III.24)
C ∼
dd xGc
EE(x) ∼
Z
Z
ξ
ddx
|x|d−2+η′ ∼ ξ2−η ′
∼ |t|−ν(2−η ′ ) , =⇒
α = (2 − η ′ )ν
.
(III.25)
As before, two independent exponents are sufficient to describe all singular critical behavior.
An important consequence of these scaling ideas is that the... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
. Unfortunately, it is not possible to
directly see how such a requirement constrains the effective Hamiltonian. One notable
exception is in d = 2, where dilation symmetry implies conformal invariance, and a lot
of information can be obtained by constructing conformally invariant theories. We shall
instead prescribe... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
2.161 Signal Processing: Continuous and Discrete
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Massachusetts Institute of Technology
Department of Mechanical Engineering
2.161 Signal Processing - Continuous and Di... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
(n =
m) so that
lim |H(jΩ)| = 1.
Ω→∞
Low Frequency Behavior: There are a pair of zeros at the origin so that
lim |H(jΩ)| = 0
Ω→0
and the low frequency asymptotic slope is +40dB/decade.
Mid Frequency Behavior: The response in the region Ω ≈ a0 is determined by
the systems damping ratio ζ, and will exhibit a res... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
High Frequency Behavior: The number of poles equals the number of zeros (n =
m = 2) so that
lim |H(jΩ)| = 1.
Ω→∞
Low Frequency Behavior: There are no zeros at the origin and
lim |H(jΩ)| = 1
Ω→0
Mid Frequency Behavior: There are a pair of imaginary zeros at s = ±j a0 forcing
√
√
the response magnitude to zero at ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
-pass and band-stop
filters it is convenient to define a center frequency Ωo as the geometric mean of the pass-pand
8–4
(cid:6)
(cid:5)
(cid:10)
(cid:14)
(cid:6)
(cid:5)
(cid:10)
(cid:14)
(cid:6)
(cid:5)
(cid:10)
(cid:14)
(cid:2)
(cid:3)
(cid:8)
(cid:2)
(cid:5)
(cid:3)
(cid:11)
(cid:21)
(cid:22)
(cid:14)
(cid:17)
(cid:... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
c
s
s2 + Ω2
o
s
sΩ2
c
s2 + Ω2
o
The band-pass and band-stop transformations both double the order of the filter, since s2
is involved it the transformation. The low-pass filter is designed to have a cut-off frequency
Ωc = Ωcu − Ωcl.
The above transformations will create an ideal gain characteristic from an ide... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
and bandwidth
ΔΩ.
Step 1: Design a first-order prototype low-pass filter with cut-off frequency
ΔΩ:
Hlp(s) =
ΔΩ
s + ΔΩ
Step 2: Transform the prototype using
g(s) =
s2 + Ω2
o
s
so that
H(s) = �
ΔΩ
�
2+Ωs
2
o
s
+ ΔΩ
ΔΩs
= 2 + ΔΩs + Ω2
s
o
Example 4
Design a second-order band-stop filter with center freque... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
example the third-order
Butterworth high-pass filter
would be implemented as
as shown below:
(cid:18) (cid:17)
(cid:18) (cid:17)
(cid:14) (cid:21)
(cid:3)
(cid:18) (cid:17)
H(s) =
s3
s3 + 40s2 + 800s + 8000
H(s) =
s2
s2 + 20s + 400
×
s
s + 20
(cid:12)
(cid:14)
(cid:6)
(cid:3)
The design of each low-ord... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
8–7
(cid:14)
(cid:17)
(cid:17)
(cid:18)
(cid:17)
(cid:12)
(cid:6)
(cid:14)
(cid:17)
(cid:18)
(cid:17)
(cid:19)
(cid:6)
(cid:12)
(cid:14)
(cid:14)
(cid:17)
(cid:18)
(cid:17)
(cid:12)
(cid:6)
(cid:14)
(cid:17)
(cid:21)
(cid:17)
(cid:19)
(cid:6)
(cid:14)
(cid:17)
(cid:12)
(cid:17)
(cid:20)
(cid:6)
(cid:6)
(cid:14)
(cid:1... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
) (cid:9)
(cid:6)
This basic structure may be used to realize the four basic filter types by appropriate
choice of the numerator.
Hlp(s) =
Hbp(s) =
Hhp(s) =
Hbs(s) =
Y1(s)
U (s)
Y2(s)
U (s)
Y3(s)
U (s)
Y4(s)
U (s)
=
=
=
=
a0
s2 + a1s + a0
a1s
s2 + a1s + a0
s2
s2 + a1s + a0
s2 + a0
s2 + a1s + a0
... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
Introduction to Engineering
Introduction to Engineering
Systems, ESD.00
Lecture 1
Lecturers:
Professor Joseph Sussman
Dr Afreen Siddiqi
Dr. Afreen Siddiqi
TA: Regina Clewlow
Motivation I
Motivation I
Society faces many large-scale problems-- we
call them critical contemporary issues (CCIs)
They aren’t simply ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/321cd9d9f5c380b76f1939b678e8b9c9_MITESD_00S11_lec01.pdf |
Connections
Lady Bird Johnson found that in dealing with
highway beautification it was like "picking up a
tangled skein of wool; all the threads are
interwoven - recreation and pollution and
interwoven recreation and pollution and
mental health. and the crime rate, and rapid
transit and the war on poverty, and
par... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/321cd9d9f5c380b76f1939b678e8b9c9_MITESD_00S11_lec01.pdf |
simply observers
CSS Example
CSS Example
We now introduce a number of CSS
concepts. We will use the
Boston/Cambridge transportation system
as a CSS example to illustrate some of the
concepts.
Systems Parallel to our CSS I
Systems Parallel to our CSS I
Our CSS has various
parallel systems. These
parallel sy... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/321cd9d9f5c380b76f1939b678e8b9c9_MITESD_00S11_lec01.pdf |
the rail
to
ate hich incl des the ail
network, the highway network on which
the MBTA’s buses operate, fare
t e
a e
collection systems, operating staffs
ope ate,
s buses
ope
Subsystems of our CSS III
Subsystems of our CSS III
Travelers who will make a choice of what
mode to use: driving, taking public
transpor... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/321cd9d9f5c380b76f1939b678e8b9c9_MITESD_00S11_lec01.pdf |
Partial derivatives
Partial derivatives
Let w = f (x, y) be a function of two variables. Its graph is a surface in xyz-space, as
pictured.
Fix a value y = y0 and just let x vary. You get a function of one variable,
(1)
w = f (x, y0),
the partial function for y = y0.
Its graph is a curve in the vertical plane y ... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/3299372c0b77500ada4ea558fea4db80_MIT18_02SC_MNotes_ta1.pdf |
specific point; the second is common in science
and engineering, where you are just dealing with relations between variables and don’t
mention the function explicitly; the third and fourth indicate the point by just using a
single subscript.
Analogously, fixing x = x0 and letting y vary, we get the partial function w... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/3299372c0b77500ada4ea558fea4db80_MIT18_02SC_MNotes_ta1.pdf |
same: to define the partial
derivative with respect to x, for instance, hold all the other variables constant and take the
ordinary derivative with respect to x; the notations are the same as above:
d
dx
f (x, y0, z0, . . . ) = f x(x0, y0, z0, . . . ),
∂f
∂x
�
0
,
�
�
1
∂w
∂x
.
0
�
... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/3299372c0b77500ada4ea558fea4db80_MIT18_02SC_MNotes_ta1.pdf |
6.172
Performance
Engineering
of Software
Systems
LECTURE 14
Caching and Cache-
Efficient Algorithms
Julian Shun
© 2008-2018 by the MIT 6.172 Lecturers
1
!"##$*
%&'&(!"#)*+)$#)*+,*-./01*
!"##$*
%&'&(!"#)*+)$#)*+,*-./01*
CACHE HARDWARE
© 2008-2018 by the MIT 6.172 Lecturers
2
Multicore Cache Hierarchy
DRAM... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
48
Cache size M = 32.
Line/block size B = 4.
0x0040
0x0024
0x0014
0x003C
0x0030
0x0008
tag
A cache block can reside
anywhere in the cache.
To find a block in the cache, the entire cache must be
searched for the tag. When the cache becomes full, a
block must be evicted to make room for a new block.
The re... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
single
location in the cache
need be searched.
Set-Associative Cache
Cache size M = 32.
Line/block size B = 4.
k=2-way associativity.
0x0040
0x0030
0x0014
0x0024
0x0008
0x003C
tag
A cache block’s set determines
k possible cache locations.
w-bit
address
space
0x0000
0x0004
0x0008
0x000C
0x0010
0x0014
0x0018
0... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
the MIT 6.172 Lecturers
7
Conflict Misses for Submatrices
4096 columns
of doubles
= 215 bytes
Conflict misses can be
problematic for caches with
limited associativity.
4096
rows
32
A
address
tag
tag
w – l... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
Q cache misses on an ideal cache of size M. Then on
a fully associative cache of size 2M that uses the
least-recently used (LRU) replacement policy, it
incurs at most 2Q cache misses. ∎
Implication
For asymptotic analyses, one can assume optimal or
LRU replacement, as convenient.
Software Engineering
∙ Design a t... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
.172 Lecturers
(cid:20)
(cid:48)
(cid:48)
(cid:16)(cid:3553)
(cid:48)
(cid:16)(cid:3553)
(cid:21)(cid:48)
(cid:16)(cid:3553)
(cid:16)(cid:3553) (cid:15)
(cid:30)
(cid:30)
(cid:2862)
(cid:30)
12
(cid:20)(cid:84)
(cid:20)(cid:84)
(cid:20)(cid:48)
!
(cid:3553)(cid:10)(cid:16)(cid:3553)
(cid:16)(cid:3553)
Tall Caches
me... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
the number of misses to read all
A’s elements is at most 3n2/B.
Proof. We have N = n2, n = r = si, B ≤ n = N/r, and N
< M /3. Thus, the Cache-Miss Lemma applies. ∎
© 2008-2018 by the MIT 6.172 Lecturers
15
!"##$*
%&'&(!"#)*+)$#)*+,*-./01*
CACHE ANALYSIS OF ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
008-2018 by the MIT 6.172 Lecturers
B
18
Analysis of Cache Misses
void Mult(double *C, double *A, double *B, int64_t n) {
for (int64_t i=0; i < n; i++)
for (int64_t j=0; j < n; j++)
for (int64_t k=0; k < n; k++)
C[i*n+j] += A[i*n+k] * B[k*n+j];
}
Assume row major and tall cache
A
© 2008-2018 by the MIT 6.172... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
0; k < n; k++)
for (int64_t j=0; j < n; j++)
C[i*n+j] += A[i*n+k] * B[k*n+j];
}
Assume row major and tall cache
C
© 2008-2018 by the MIT 6.172 Lecturers
B
21
Analyze matrix B.
Assume LRU.
Q(n) = n·!(n2/B) =
!(n3/B), since
matrix B can exploit
spatial locality.
!"##$*
%&'&(!"#)*+)$#)*+,*-./01*
TILING
© 2008... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
0; i1<n; i1+=s)
for (int64_t j1=0; j1<n; j1+=s)
for (int64_t k1=0; k1<n; k1+=s)
for (int64_t i=i1; i<i1+s && i<n; i++)
for (int64_t j=j1; j<j1+s && j<n; j++)
for (int64_t k=k1; k<k1+s && k<n; k++)
C[i*n+j] += A[i*n+k] * B[k*n+j];
}
s
s
n
Analysis of cache misses
! Tune s so that the submatrices
just fit int... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
C[i*n+j] += A[i*n+k] * B[k*n+j];
C[i*n+j] += A[i*n+k] * B[k*n+j];
}
s
s
n
Analysis of cache misses
! Tune s so that the submatrices
just fit into cache ! s = "(M1/2).
! Submatrix Caching Lemma implies
"(s2/B) misses per submatrix.
"((n/s)3(s2/B))
Q(n) =
= "(n3/(BM1/2)).
!
Remember
this!
© 2008-2018 by t... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
&& i<i2+t && i<n; i++)
for (int64_t j=j1; j<j1+s && j<j2+t && j<n; j++)
tuning optimization
cannot be done with
for (int64_t k=k1; k1<k1+s && k<k2+t && k<n; k++)
binary search.
C[i*n+j] += A[i*n+k] * B[k*n+j];
n
}
© 2008-2018 by the MIT 6.172 Lecturers
27
s
t
uut
s
n
Three-Level Cache
n
∙ Three “voodo... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
that n is an exact power of 2.
void Rec_Mult(double *C, double *A, double *B,
int64_t n, int64_t rowsize) {
if (n == 1)
C[0] += A[0] * B[0];
else {
int64_t d11 = 0;
int64_t d12 = n/2;
int64_t d21 = (n/2) * rowsize;
int64_t d22 = (n/2) * (rowsize+1);
Coarsen base case to
overcome function-
call overheads.
Re... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
else {
int64_t d11 = 0;
int64_t d12 = n/2;
int64_t d21 = (n/2) * rowsize;
int64_t d22 = (n/2) * (rowsize+1);
Rec_Mult(C+d11, A+d11, B+d11, n/2, rowsize);
Rec_Mult(C+d11, A+d12, B+d21, n/2, rowsize);
Rec_Mult(C+d12, A+d11, B+d12, n/2, rowsize);
Rec_Mult(C+d12, A+d12, B+d22, n/2, rowsize);
Rec_Mult(C+d21, A+d21,... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
ize+1);
Rec_Mult(C+d11, A+d11, B+d11, n/2, rowsize);
Rec_Mult(C+d11, A+d12, B+d21, n/2, rowsize);
Rec_Mult(C+d12, A+d11, B+d12, n/2, rowsize);
Rec_Mult(C+d12, A+d12, B+d22, n/2, rowsize);
Rec_Mult(C+d21, A+d21, B+d11, n/2, rowsize);
Rec_Mult(C+d21, A+d22, B+d21, n/2, rowsize);
Rec_Mult(C+d22, A+d21, B+d12, n/2, ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
W(n/2)
1
W(n/4) W(n/4) ⋯
W(n/4)
© 2008-2018 by the MIT 6.172 Lecturers
36
Analysis of Work
W(n) = 8W(n/2) + #(1)
recursion tree
1
8
W(n/2)
1
W(n/2)
1
8
!
W(n/2)
1
lg n
W(n/4) W(n/4)
1
1
!
W(n/4)
1
#leaves = 8lg n = nlg 8 = n3
"
#(1)
Note: Same work as looping versions.
© 2008-2018 by ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
n/2, rowsize);
Rec_Mult(C+d12, A+d12, B+d22, n/2, rowsize);
Rec_Mult(C+d21, A+d21, B+d11, n/2, rowsize);
Rec_Mult(C+d21, A+d22, B+d21, n/2, rowsize);
Rec_Mult(C+d22, A+d21, B+d12, n/2, rowsize);
Rec_Mult(C+d22, A+d22, B+d22, n/2, rowsize);
} }
Submatrix
Caching
Lemma
Q(n) =
!(n2/B) if n2<cM for suff. small c... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
/4)
8
#
1
© 2008-2018 by the MIT 6.172 Lecturers
41
Analysis of Cache Misses
Q(n) =
#(n2/B) if n2<cM for suff. small const c&1,
8Q(n/2) + #(1) otherwise.
recursion tree
1
8
lg n-!lg(cM)
1
Q(n/2) Q(n/2)
8
1
!
1
Q(n/2)
Q(n/4) Q(n/4)
1
1
!
Q(n/4)
1
$
#leaves = 8lg n – %lg(cM)
= #(n3/M3/2).
G
e
o
m
e
... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
== 1)
C[0] += A[0] * B[0];
else {
int64_t d11 = 0;
int64_t d12 = n/2;
int64_t d21 = (n/2) * rowsize;
int64_t d22 = (n/2) * (rowsize+1);
cilk_spawn Rec_Mult(C+d11, A+d11, B+d11, n/2, rowsize);
cilk_spawn Rec_Mult(C+d21, A+d22, B+d21, n/2, rowsize);
cilk_spawn Rec_Mult(C+d12, A+d11, B+d12, n/2, rowsize);
Rec_Mu... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
worker steals a continuation, its cache is
completely cold in the worst case. But after M/B (cold) cache
misses, its cache is identical to that in the serial execution. The
same is true when a worker resumes a stolen subcomputation
after a cilk_sync. The number of times these two situations can
occur is at most 2S... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
2, A+d22, B+d22, n/2, rowsize);
cilk_sync;
cilk_spawn Rec_Mult(C+d11, A+d12, B+d21, n/2, rowsize);
cilk_spawn Rec_Mult(C+d21, A+d21, B+d11, n/2, rowsize);
cilk_spawn Rec_Mult(C+d12, A+d12, B+d22, n/2, rowsize);
Rec_Mult(C+d22, A+d21, B+d12, n/2, rowsize);
cilk_sync;
} }
Cache misses: Qp = '1 + O(SPM(B)
= "(n3/B... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
LECTURE 13
Homotopy Coinvariants, Abelianization, and Tate
Cohomology
Recall that last time we explicitly constructed the homotopy invariants X hG of a
qis
−−→ Z,
G is a canonical complex of free G-modules in non-positive degrees. Then
complex X of G-modules. To do this, we constructed the bar resolution P can
where P ... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
, ϕ
is a 1-coboundary if there exists some m ∈ M such that ϕ(g) = g · m − m for all
g ∈ G. The upshot is that
H 1(G, M ) := H 1(M hG) = {1-cocycles}/{1-coboundaries}.
As a corollary, if G acts trivially on M , then H 1(G, M ) = HomGroup(G, M ), since
the 1-coboundaries are all trivial, and the 1-cocycles are just ordin... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
(cid:88)
ni,
i
i
We claim that IG is Z-spanned by {g − 1 : g ∈ G} (which we leave as an exercise).
A corollary is that
is exact, since Z[G]⊕G (cid:16) IG via 1 (cid:55)→ g − 1 on the gth coordinate.
Z[G]⊕G → Z[G] → Z → 0
Remark 13.1. The correct algorithm for computing tensor products is as fol-
lows: recall that tenso... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
, and tensor with that instead: XhG := X ⊗Z[G] PG.
Definition 13.3. A complex F of left A-modules is flat is for every acyclic
complex Y of right A-modules, Y ⊗A F is also acyclic, that is, − ⊗A F preserves
injections.
We now ask if PG is flat. In fact:
Claim 13.4. Any projective complex is flat.
An easier claim is the fo... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
so F • = hCoker(F −1 → F 0). Then since tensor products commute with
homotopy cokernels, we obtain
Y ⊗A F = hCoker(Y ⊗A F −1 → Y ⊗A F 0),
so by Case 1, if Y is acyclic, then Y ⊗A F 0 and Y ⊗A F −1 are as well, hence
Y ⊗A F is as well by the long exact sequence on cohomology. A similar (inductive)
argument gives the cas... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
(it’s turtles
all the way down!).
Definition 13.7. The ith torsion group (of Y against X) is TorA
i (Y, X) :=
H −i(Y ⊗der
A X).
Definition 13.8. The homotopy coinvariants of a chain complex X is the
Z (cid:39) X ⊗Z[G] PG (which we note is only well-defined up
complex XhG := X ⊗der
Z[G]
to quasi-isomorphism).
60
13. HOM... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
0(G, Z) = H 0(Z)G = Z, and H1(G, Z[G]) = 0 as
Z[G]hG := Z[G] ⊗Z[G] PG = PG (cid:39) Z
is a quasi-isomorphism. Thus, our exact sequence is really
0 → H1(G, Z) ∼−→ H0(G, IG) → Z ∼−→ Z,
which gives the noted isomorphism. The upshot is that
H1(G, Z) = (IG)G = IG/I 2
G
since MG = M/IG · M .
Claim 13.12. The map
is an isomor... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
] − 1) + ([h] − 1) ≡ [gh] − 1 mod I 2
G,
(¯g, 1) (cid:55)→ [g] + 1 − 1 = [g] and [g] + n − 1 (cid:55)→ (¯g, 1)(1, n − 1) = (¯g, n),
as desired.
This proves the claim.
(cid:3)
(cid:3)
Finally, we define the norm map XhG
N−→ X hG to be the composition
XhG = X ⊗Z[G] PG → X ⊗Z[G] Z → HomZ[G](Z, X) → HomZ[G](PG, X) = X hG,
w... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
N(L×) (cid:39) Gab.
MIT OpenCourseWare
https://ocw.mit.edu
18.786 Number Theory II: Class Field Theory
Spring 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
18.600: Lecture 33
Entropy
Scott Sheffield
MIT
1 Outline
Entropy
Noiseless coding theory
Conditional entropy
2 Outline
Entropy
Noiseless coding theory
Conditional entropy
3 I Familiar on some level to everyone who has studied chemistry
or statistical physics.
I Kind of means amount of randomness or disorder.
... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
P{X = x} = 2−k .
I In information theory it’s quite common to use log to mean
log2 instead of loge. We follow that convention in this lecture.
In particular, this means that
log P{X = x} = −k
I Since there are 2k values in S, it takes k “bits” to describe an
for each x ∈ S.
element x ∈ S.
I Intuitively, could say that ... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
a fair coin k times.
I Then the state space S is the set of 2k possible heads-tails
sequences.
I If X is the random sequence (so X is a random variable), then
for each x ∈ S we have P{X = x} = 2−k .
10 I Since there are 2k values in S, it takes k “bits” to describe an
element x ∈ S.
I Intuitively, could say that w... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
for each x ∈ S.
I Since there are 2k values in S, it takes k “bits” to describe an
element x ∈ S.
12 Information
I Suppose we toss a fair coin k times.
I Then the state space S is the set of 2k possible heads-tails
sequences.
I If X is the random sequence (so X is a random variable), then
for each x ∈ S we have... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
948.
14 I If a random variable X takes values x1, x2, . . . , xn with positive
probabilities p1, p2, . . . , pn then we define the entropy of X by
H(X ) =
pi (− log pi ) = −
pi log pi .
n
X
i=1
n
X
i=1
I This can be interpreted as the expectation of (− log pi ). The
value (− log pi ) is the “amount of surprise” when we... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
1948.
I Goal is to define a notion of how much we “expect to learn”
from a random variable or “how many bits of information a
random variable contains” that makes sense for general
experiments (which may not have anything to do with coins).
I If a random variable X takes values x1, x2, . . . , xn with positive
pro... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
= x}
x
Dog
Cat
Cow
Pig
Squirrel
Mouse
Owl
Sloth
Hippo
Yak
Zebra
Rhino
1/4
1/4
1/8
1/16
1/16
1/16
1/16
1/32
1/32
1/32
1/64
1/64
2
2
3
4
4
4
4
5
5
5
6
6
I Can learn animal with H(X ) questions on average.
19 Twenty questions with Harry
I Harry always thinks of one of the following ... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
) =
n
X
i=1
pi (− log pi ) = −
pi log pi .
n
X
i=1
21 I If X takes k values with equal probability, what is H(X )?
I What is H(X ) if X is a geometric random variable with
parameter p = 1/2?
Other examples
I Again, if a random variable X takes the values x1, x2, . . . , xn
with positive probabilities p1, p2, .... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
)?
I If X takes k values with equal probability, what is H(X )?
I What is H(X ) if X is a geometric random variable with
parameter p = 1/2?
24 I Then we write
H(X , Y ) = −
p(xi , yj ) log p(xi , yi ).
X
X
i
j
I H(X , Y ) is just the entropy of the pair (X , Y ) (viewed as a
random variable itself).
I Claim: if X a... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
itself).
27 Entropy for a pair of random variables
I Consider random variables X , Y with joint mass function
p(xi , yj ) = P{X = xi , Y = yj }.
I Then we write
H(X , Y ) = −
p(xi , yj ) log p(xi , yi ).
X X
i
j
I H(X , Y ) is just the entropy of the pair (X , Y ) (viewed as a
random variable itself).
I Clai... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
↔ 00
B ↔ 01
C ↔ 10
D ↔ 11
A ↔ 0
B ↔ 10
C ↔ 110
D ↔ 111
32 I What does 100111110010 spell?
I A coding scheme is equivalent to a twenty questions strategy.
Coding values by bit sequences
I David Huffman (as MIT student) published in “A Method for
the Construction of Minimum-Redundancy Code” in 1952.
I If X take... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
we can code them by:
I Or by
A ↔ 00
B ↔ 01
C ↔ 10
D ↔ 11
A ↔ 0
B ↔ 10
C ↔ 110
D ↔ 111
I No sequence in code is an extension of another.
I What does 100111110010 spell?
I A coding scheme is equivalent to a twenty questions strategy.
35 I Note: The expected number of questions is the entropy if
each question... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
n be i.i.d. instances of X .
Do there exist encoding schemes such that the expected
number of bits required to encode the entire sequence is
about H(X )n (assuming n is sufficiently large)?
I Yes. Consider space of N n possibilities. Use “rounding to 2
power” trick, Expect to need at most H(x)n + 1 bits.
Twenty questions... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
.
I Note: The expected number of questions is the entropy if
each question divides the space of possibilities exactly in half
(measured by probability).
I In this case, let X take values x1, . . . , xN with probabilities
p(x1), . . . , p(xN ). Then if a valid coding of X assigns ni bits
to xi , we have
N
X
i=1
... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
40 Outline
Entropy
Noiseless coding theory
Conditional entropy
41 Outline
Entropy
Noiseless coding theory
Conditional entropy
42 I But now let’s not assume they are independent.
I We can define a conditional entropy of X given Y = yj by
HY =yj (X ) = −
p(xi |yj ) log p(xi |yj ).
X
i
I This is just the entropy of... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
I This is just the entropy of the conditional distribution. Recall
that p(xi |yj ) = P{X = xi |Y = yj }.
I We similarly define HY (X ) = P
j HY =yj (X )pY (yj ). This is
the expected amount of conditional entropy that there will be
in Y after we have observed X .
Conditional entropy
I Let’s again consider random variab... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
Y with joint mass
function p(xi , yj ) = P{X = xi , Y = yj } and write
H(X , Y ) = −
p(xi , yj ) log p(xi , yi ).
X X
i
j
I But now let’s not assume they are independent.
I We can define a conditional entropy of X given Y = yj by
X
HY =yj (X ) = −
p(xi |yj ) log p(xi |yj ).
I This is just the entropy of the c... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
X ) =
(X ) = −
P
(X )pY (yj ).
P
=yj
j HY =yj
i p(xi |yj ) log p(xi |yj ) and
48 I In words, the expected amount of information we learn when
discovering (X , Y ) is equal to expected amount we learn when
discovering Y plus expected amount when we subsequently
discover X (given our knowledge of Y ).
I To prove ... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
j pY (yj ) log pY (yj ) P
i p(xi |yj ) −
i p(xi |yj ) log p(xi |yj ) = H(Y ) + HY (X ).
Properties of conditional entropy
I Definitions: HY
HY (X ) =
P
(X ) = −
HY =yj (X )pY (yj ).
=yj
P
j
i p(xi |yj ) log p(xi |yj ) and
I Important property one: H(X , Y ) = H(Y ) + HY (X ).
I In words, the expected amount o... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
given our knowledge of Y ).
I To prove this property, recall that p(xi , yj ) = pY (yj )p(xi |yj ).
51 Properties of conditional entropy
(X ) = −
HY (X ) =
I Definitions: HY
P
(X )pY (yj ).
I Important property one: H(X , Y ) = H(Y ) + HY (X ).
I In words, the expected amount of information we learn when
i p(x... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
X (x1), pX (x2), . . . , pX (xn)} is a weighted
average of vectors vj := {pX (x1|yj ), pX (x2|yj ), . . . , pX (xn|yj )}
as j ranges over possible values. By (vector version of)
Jensen’s inequality,
H(X ) = E(v ) = E(P pY (yj )vj ) ≥ P pY (yj )E(vj ) = HY (X ).
Properties of conditional entropy
I Definitions: HY
HY (X... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
) with equality if
and only if X and Y are independent.
54 I Proof: note that E(p1, p2, . . . , pn) := − P pi log pi is concave.
I The vector v = {pX (x1), pX (x2), . . . , pX (xn)} is a weighted
average of vectors vj := {pX (x1|yj ), pX (x2|yj ), . . . , pX (xn|yj )}
as j ranges over possible values. By (vector vers... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
(X ) =
(X ) = −
P
(X )pY (yj ).
P
j
=yj
HY
=yj
i p(xi |yj ) log p(xi |yj ) and
I Important property two: HY (X ) ≤ H(X ) with equality if
and only if X and Y are independent.
I In words, the expected amount of information we learn when
discovering X after having discovered Y can’t be more than
the expecte... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
pY (yj )vj ) ≥
pY (yj )E(vj ) = HY (X ).
P
P
P
57 MIT OpenCourseWare
https://ocw.mit.edu
18.600 Probability and Random Variables
Fall 2019
For information about citing these materials or our Terms of Use, visit:
https://ocw.mit.edu/terms.
58 | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
6.S897/HST.956 Machine Learning for Healthcare
Lecture 10: Application of Machine Learning to Cardiac Imaging
Instructors: David Sontag, Peter Szolovits
1 Background
This lecture was a guest lecture by Rahul Deo, the lead investigator of the One Brave Idea project at Brigham
and Women’s Hospital. Rahul is also Adj... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
that can grow
to as much as 35 liters per minute during intense exercise. One crucial aspect of cardiac function is that
the body must maintain extremely rhythmic beating of the heart, a not inconsequential task given that the
average human heart generates a total of more than 2 billion heartbeats over a lifetime.
... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
(EKG). A Wiggers Diagram can be used to demonstrate the interconnectedness of these electrical and
6.S897/HST.956 Machine Learning for Healthcare — Lec10 — 1
Courtesy of OpenStax. Used under CC BY.
Figure 1: The major chambers, valves, and blood vessels of the human heart.
mechanical systems, allowing one to see ho... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
various imaging techniques that each play critical roles in diagnosis. Here is a brief
overview of some of the most important ones:
• EKG - An extremely cheap technique based on measuring voltage differences in the heart over time.
Can be used, for example, to diagnose myocardial infarction.
6.S897/HST.956 Machine Lea... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
often stuck with the data that is already out
there because someone decided it was worth paying for. The available data often controls the risk model
and decision analysis you can undertake. Finally, while imaging data can be found for patients with diseases
for which the imaging process is seen as an essential part of... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
cost of the study increases and/or the perceived
utility of the data decreases, the availability of data goes down. As an example, an imaging technique like
PET that is very expensive has only 8000 studies available at Brigham and Women’s Hospital, whereas over
30 million EKGs can be accessed.
4.3 Characteristics o... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
of great interest to cardiac imaging as a result.
5.1
Image Classification
In image classification, the goal is to assign a label to a given image or video. This is a ripe candidate for
applying supervised machine learning techniques to cardiology. There are many simple disease recognition
tasks in medicine such as ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
involved with medical
image classification, with radiologist being the most sued profession in medicine. This liability means that
radiologists don’t feel sufficiently convinced to pass the task off to a black box computer system.
Nevertheless, applications of automated image classification still exist - just not in the ide... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
iency map to
identify the pixels that maximally activate the given class.
5.2 Semantic Segmentation
In semantic segmentation, the goal is to assign each pixel of the image with a class label. For example, one
common task in cardiology is delineating the boundaries of the heart in an image, and radiology reports
of... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
such as PET, in which case one must average images across many cardiac cycles (i.e. gating). Conditional
variational autoencoders have shown to be a particularly well suited model for this task by learning geometric
transformations between pairs of images.
6 A fully automated pipeline for echocardiogram interpretati... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
3 Focus of machine learning in cardiac diseases
We can use machine learning to:
1. enable much greater of volumes of data to be interpreted, so that we reduce costs of acquisition and
interpretation, as well as augment interpretations of simple data.
2. augment surveillance within a hospital system, e.g. patient identi... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
it. We see that both niches fulfill high reward, but the left is low risk and
the right is high risk. For early stages, it would be beneficial to use the automated pipeline for low liability,
low cost, and quick decisions for whether further analysis is needed. For late stages, it would be better to
use a more trusted sk... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
and
emerging algorithms.
4. Physicians are only interested in classifications or risk models that will change and improve practice,
thus evidence is required to justify a shift.
5. There is the question of how more data will be obtained and dispersed for research.
8
Biology
There are some goals in biology that can be ac... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
year all
with expressive phenotyping and full medical records
2. Use of cell morphology/cell coutner data to massively expand phenotypic space at low cost using
perturbations and diverse readouts
3. Overlapping of multiple phenotypic scales in dfiferent cohorts to convert costly, tissue-localized pheno-
types (e.g. ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
priors for recon-
struction, are these geometric priors being reintroduced in some way in modern times?
Answer 3: This is not something that is widely-used anymore, and the data to do this is also unavail-
able.
References
[Org17] World Health Organization. Cardiovascular diseases (cvds), May 2017.
6.S897/HST.956 M... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/32e3387bbd9f6ea7df9f9195337ed5b4_MIT6_S897S19_lec10note.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.641 Electromagnetic Fields, Forces, and Motion, Spring 2005
Please use the following citation format:
Markus Zahn, 6.641 Electromagnetic Fields, Forces, and Motion, Spring
2005. (Massachusetts Institute of Technology: MIT OpenCourseWare).
http://ocw.mit.edu (accessed MM DD, ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/32f92e8162acbc3041e3ac32861394bb_lecture8.pdf |
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