text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
what it means.
31
Semantics
• Meaning of a sentence is truth value {t, f}
• Interpretation is an assignment of truth values to
the propositional variables
[
• ² i φ Sentence φ is t in interpretation i ]
• 2i φ [Sentence φ is f in interpretation i ]
• ² i true
• 2i false
interpret.]
• ² i ¬
φ
• ² i φ Æ ψ
• ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
conjunction]
if and only if ² i φ or ² i ψ [disjunction]
iff
i(P) = t
Lecture 3 • 33
Now we have one more clause. I’m going to do it by example. Imagine that
we have a sentence P. P is one of our propositional variables. How do we
know whether it is true in interpretation I?
Well, since I is a mapping from vari... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
al, equivalence]
→
→
(φ
Lecture 3 • 36
Finally, the double arrow just means that we have single arrows going both
ways. This is sometimes called a “bi-conditional” or “equivalence”
statement. It means that in every interpretation, Phi and Psi have the same
truth value.
36
Some important shorthand
• φ
ψ
... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
ψ
φ Ç ψ [conditional, implication]
→
¬
antecedent
≡
• φ
ψ
↔
≡
→
(φ
→
consequent
→
Truth Tables
P
P Æ Q
P Ç Q
P
f
f
t
t
Q
f
t
f
t
¬
t
t
f
f
f
f
f
t
f
t
t
t
Q
Q
P
→
t
t
f
t
P
→
t
f
t
t
P
Q
↔
t
f
f
t
Q is t
Note that implication is not “causality”, if P is f then P
→
Lecture 3 • 38
Mo... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
is
assigned to P by the interpretation, “P or not P” is true.
40
Terminology
• A sentence is valid iff its truth value is t in all
interpretations (²φ)
Valid sentences: true,
false, P Ç
P
¬
• A sentence is satisfiable iff its truth value is t in at
¬
least one interpretation
Satisfiable sentences: P, true,... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
wanted to know if a particular sentence was valid, or if I wanted to know if it
was satisfiable or unsatisfiable. I could just make a truth table. Write down
all the interpretations, figure out the value of the sentence in each
interpretation, and if they're all true, it's valid. If they're all false, it's
unsatisf... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
KB is also a model of \Phi.
Lecture 3 • 46
46
Models and Entailment
• An interpretation i is a model
of a sentence φ iff ² i φ
• A set of sentences KB entails φ
iff every model of KB is also a
model of φ
Sentences
Sentences
Lecture 3 • 47
So, here’s the picture. If we consider two groups of sentences, we mi... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
e
s
Sentences
Sentences
Interpretations
Interpretations
subset
Now, we can ask whether the first set of interpretations is a subset of the
second set.
Lecture 3 • 50
50
Models and Entailment
• An interpretation i is a model
of a sentence φ iff ² i φ
• A set of sentences KB entails φ
iff every model of ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
φ = B
U
Lecture 3 • 53
Now, we can use a Venn diagram to think about the interpretations. Let U
be the set of all possible interpretations.
53
Models and Entailment
• An interpretation i is a model
of a sentence φ iff ² i φ
• A set of sentences KB entails φ
iff every model of KB is also a
model of φ
s
c... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
m
e
s
Sentences
entails
Sentences
Interpretations
Interpretations
subset
KB = A Æ B
φ = B
A Æ B ² B
A Æ B
B
U
Lecture 3 • 56
So, we find that A and B entails B. If we know that A and B are true, then B
has to be true.
56
Models and Entailment
• An interpretation i is a model
of a sentence φ iff ² ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
. That lets you draw valid
conclusions from assumptions.
57
Examples
Let’s work through some examples. We can think about whether they're
valid or unsatisfiable or satisfiable.
Lecture 3 • 58
58
Examples
Interpretation that make
sentence’s truth value = f
Valid?
valid
smoke
Sentence
smoke
→
smoke Ç ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
We could show that by drawing out the truth table (and you should
do it as an exercise if the answer is not obvious to you). Another way to
show that a sentence is not valid is to give an interpretation that makes the
sentence have the truth value F. In this case, if we give “smoke” the truth
value F and and “fire”... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
smoke = t, fire = f
s = f, f = t
s
f = t,
→
¬
s
→
¬
f = f
Lecture 3 • 63
What about “b or d or (b implies d)”? We can rewrite that (using the definition
of implication) into “b or d or not b or d”, which is valid, because in every
interpretation either b or not b must be true.
63
Recitation Exerci... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
Programs with Flexible Time
When?
Contributions:
Brian Williams
Patrick Conrad
Simon Fang
Paul Morris
Nicola Muscettola
Pedro Santana
Julie Shah
John Stedl
Andrew Wang
courtesy of JPL
Steve Levine
Tuesday, Feb 16th
(cid:2)
Assignments
Problems Sets:
• Pset 1 due tomorrow (Wednesday) at 11:59pm
• Pset 2 released tomorr... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
(cid:29)CD or Y+ < X-
(cid:2)(cid:5)
[Villain & Kautz; Simmons]
Temporal Relations Described by a
Simple Temporal Network (STN)
• Simple Temporal Network
• Tuple <X, C> where:
• variables X1,…Xn, represent time points
(real-valued domains)
• binary constraints C of the form:
)
∈
X
X
−
(
[
ba
,
ik
ik
].
k
i
[Dechter... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
)(cid:8)+(cid:29)(cid:17)(cid:13)(cid:29)(cid:21)(cid:27)(cid:26)(cid:13)(cid:22)(cid:21)(cid:27)(cid:16)(cid:28)(cid:29)(cid:27)(cid:8)(cid:16)(cid:23)(cid:17)(cid:21)(cid:14)(cid:8)(cid:29)(cid:23)(cid:9)(cid:26)(cid:15)
[Dechter, Meiri, Pearl 91]
(cid:3)(cid:3)
Algorithm: offline scheduling
Initialize execution win... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
cid:29),(cid:27) (cid:21)(cid:27)(cid:8)
1. Describe Temporal Plan
1. Describe Temporal Plan
2. Test Consistency
2. Test Consistency
3. Schedule Plan
3. Reformulate Plan
4. Execute Plan
4. Dynamically Schedule Plan
(cid:13))) (cid:21)(cid:27)(cid:8)
(cid:13)(cid:27) (cid:21)(cid:27)(cid:8)
(cid:4)(cid:4)
Algorithm: of... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
]
-1
9
1
-1
-1
2
1
-2
10
0
[-∞, ∞]
11
-2
Assign the first event
(cid:5)$
[-∞, ∞]
Computing a schedule
outgoing edges to neighbor: u’ = min(u, ti + wu)
incoming edges from neighbor: l’ = max(l, ti
- w )
l
[-∞, 10]
t = 0
10
0
-1
9
1
-1
-1
2
1
-2
[-∞, ∞]
11
-2
[-∞, ∞]
Propagate updated time bounds to neighbors
(cid:5)(ci... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
[2, 11]
Arbitrarily pick another time point and assign it...
(cid:5)(cid:10)
t = 0
Computing a schedule
t = 3
-1
9
1
-1
-1
2
1
-2
10
0
[2, 2]
11
-2
[2, 11]
Propagate updated time bounds to neighbors
(cid:5)(cid:11)
t = 0
Computing a schedule
t = 3
-1
9
1
-1
-1
2
1
-2
10
0
[2, 2]
11
-2
[4, 4]
Propagate updated time bo... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
Schedule events “on the fly,” after observing past event times.
– Increases robustness to many unanticipated fluctuations.
– Flexible temporal constraints allow this!
(cid:10)(cid:2)
To Execute a Temporal Plan
(cid:25)(cid:26)*(cid:8)+" (cid:8)(cid:29),)) (cid:21)(cid:27)(cid:8)(cid:29)
(cid:25)(cid:26)*(cid:8)+" (cid... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
1
-2
10
0
[0, 9]
11
-2
Arbitrarily picking next event…
(cid:10)(cid:11)
Naïve (wrong!) online scheduling
t = 0
10
0
t = 3
-1
9
1
-1
-1
2
1
-2
t = 2
11
-2
...but wait! We just assigned a past time!
(cid:10)(
[4, 4]
Enablement conditions dictate the
ordering of dispatched events
• Online: must assign monotonically inc... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
ti
Propagate to all xi’s neighbors & update their windows
Add xi to S
Add to E any now-enabled events
(cid:11)(cid:3)
Running online dispatcher
Compute dispatchable form (i.e., APSP)
Initialize execution windows to [-∞, ∞]
E (cid:3) {events with no predecessors}
S (cid:3) {}
while unexecuted events:
Wait until some e... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
-enabled events
t = 0
10
0
Dispatch A and propagate
E = {C}
S = {A}
[1, 10]
-1
9
1
-1
-1
2
[0, 9]
11
-2
[2, 11]
1
-2
(cid:11)5
Running online dispatcher
Compute dispatchable form (i.e., APSP)
Initialize execution windows to [-∞, ∞]
E (cid:3) {events with no predecessors}
S (cid:3) {}
while unexecuted events:
Wait unt... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
Propagate to xi’s neighbors
Add xi to S
Add to E any now-enabled events
E = {B}
S = {A, C}
[3, 3]
t = 0
10
0
-1
9
1
-1
-1
2
[4, 4]
1
-2
B is now enabled (but still not D).
t = 2
11
-2
68
Running online dispatcher
Compute dispatchable form (i.e., APSP)
Initialize execution windows to [-∞, ∞]
E (cid:3) {events with no ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
in E is active
ti = now
Propagate to xi’s neighbors
Add xi to S
Add to E any now-enabled events
E = {}
S = {A, C, B, D}
t = 3
t = 0
10
0
-1
9
1
-1
-1
2
Finish up by dispatching D!
(1
t = 2
11
-2
t = 4
1
-2
73
Online dispatching algorithm remarks
• By considering predecessors, we guarantee that events
assigned monoton... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
events:
Wait until some event xi in E is active
ti = now
Propagate to xi’s neighbors
Add xi to S
Add to E any now-enabled events
(5
Online dispatcher efficiency
• Consider an STN with n edges.
• How many edges in APSP distance graph? n2.
• How many neighbors to propagate to each step? n.
Compute dispatchable form (i.e... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
8)
7-
79
You don’t need all those edges!
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
-0
You don’t need all those edges!
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
1
-1
1
9
-1
0
Equivalent minimal dispatchable network
-1
You don’t need all those edges!
Let’s consider a specific triangle
of edges.
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
-2
You don’... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
B could slide freely! Not the
same behavior. Collectively, AB and BD entail AD (but AD
does not entail both AB and AD).
.0
Upper dominating edges - detection
from APSP
dBC
dAC
If dAC, dBC ≥ 0 and
dAB + dBC = dAC
then BC dominates AC
(Proof omitted - based on triangle rule property of
APSP. Please see notes / readi... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
(cid:2)$0
Dominance example
Lower dominated!
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
(cid:2)$1
Dominance example
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
Lower dominated!
(cid:2)$2
Dominance example
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
Lower dominated!
(cid:2)$3
Dominance example
10
0
-1
9
1
-1
-1
2
1
-2
11
-2
(cid:2)$4
Lower dominated!
D... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
:2)09
Avoiding Intermediate Graph
Explosion
> Problem:
K All pairs shortest path table computation consumed O(n2) space
K Only used as an intermediate - not needed after minimal dispatchable graph
obtained.
> Solution:
K Interleave process of APSP construction with edge elimination.
> Never have to build whole APSP ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/5a40e2ff010a7721fdf6c5513ce5e26a_MIT16_412JS16_L4.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.005 Elements of Software Construction
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
6.005
elements of
software
construction
basics of mutable types
Daniel Jackson
heap semantics of Java
pop quiz
what ha... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
null, or get/set field
© Daniel Jackson 2008
6
the operator ==
the operator ==
‣ returns true when its arguments denote the same object
(or both evaluate to null)
for mutable objects
‣ if x == y is false, objects x and y are observably different
‣ mutation through x is not visible through y
for immutable obj... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
mutable datatypes
mutable vs. immutable
String is an immutable datatype
‣ computation creates new objects with producers
class String {
String concat (String s);
...}
StringBuffer is a mutable datatype
‣ computation gives new values to existing objects with mutators
class StringBuffer {
void append (St... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
value
a simple theorem
‣ if we define a ≈ b when f(a) = f(b) for some function f
‣ then the predicate ≈ will be an equivalence
an equivalence relation is one that is
‣ reflexive: a ≈ a
‣ symmetric: a ≈ b ⇒ b ≈ a
‣ transitive: a ≈ b ∧ b ≈ c ⇒ a ≈ c
© Daniel Jackson 2008
16
a running example
a duration class
‣... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
Duration d2 = new Duration(1,2);
System.out.println(d1.equals(d2)); // prints true
Object d1 = new Duration(1,2);
Object d2 = new Duration(1,2);
System.out.println(d1.equals(d2)); // prints false!
© Daniel Jackson 2008
20
explaining bug #2
what’s going on?
‣ we’ve failed to override Object.equals
‣ meth... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
new Duration(1,2);
System.out.println(d1.equals(d2)); // false
System.out.println(d2.equals(d1)); // true
© Daniel Jackson 2008
24
bug #4
yet another attempt
‣ this time not transitive
@Override public boolean equals(Object o) {
if (! (o instanceof Duration)) return false;
if (! (o instanceof ShortDuratio... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
get(k): to get value associated with key k
examine all entries in table[i] as for insertion
if find one with key equal to k, return val
else return null
resizing
‣ if map gets too big, create new array of twice the size and rehash
© Daniel Jackson 2008
29
hashing principle
why does hashing work?
e: table[i].... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
}
@Override public int hashCode () {return i;}
}
public static void main (String[] args) {
Set m = new HashSet <Counter> ();
Counter c = new Counter();
m.add(c);
System.out.println ("m contains c: " + (m.contains(c) ? "yes" : "no"));
c.incr();
System.out.println ("m contains c: " + (m.contains(c) ? "yes" : "no"))... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
principles
object heap is a graph
‣ to understand mutation & aliasing, can’t think in terms of values
equality is user-defined but constrained
‣ must be consistent and an equivalence
abstract aliasing complicates
‣ may even break rep invariant (eg, mutating hash key)
© Daniel Jackson 2008
36 | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/5a6830a94eaa166e2db4b5e820498609_MIT6_005f08_lec16.pdf |
operator ξ S†ΓµW hv which is not equivalent to (10.6) because S and W do not commute. This apparent
problem is solved by considering the remaining two diagrams of the same order as this one
n,p
11 SCETII APPLICATIONS
Diagrams.
These diagrams both yield the current
Fig() = Fig() =
2
if abcT c
g
2
ν
µ
n¯
n
n · qs... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5a73e8bb8f7f637a9b3eb58c5e733e6b_MIT8_851S13_SCETIIApplicat.pdf |
W (0))ΓµY †hv
3. Matching SCETI onto SCETII by taking Yn → Sn.
JII = (ξ
(0)
n
W (0))ΓµS†hv
(10.10)
(10.11)
(10.12)
11 SCETII Applications
(ROUGH) In this section we will apply the SCETII formalism developed in previous sections to various
processes to illustrate the formalism
• γ∗γ → π0
• B → Dπ
• The Massive Ga... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5a73e8bb8f7f637a9b3eb58c5e733e6b_MIT8_851S13_SCETIIApplicat.pdf |
ize the matrix element (Dπ| O0,8 |B). We can represent this factorization diagrammat
ically as (INSERT FIG) where there are no gluons between π quarks and B/D quarks. For this process
we expect a B → D form factor (Isgur-Wise form factor) and a pion wavefunction/distribution. This fac
torization will be possible bec... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5a73e8bb8f7f637a9b3eb58c5e733e6b_MIT8_851S13_SCETIIApplicat.pdf |
,5 . They simply take
[ξ W ΓlC0(P +)W †ξ(u)] → [ξ
n,p
(d)
n,p
(d)(0)
n,p
W (0)ΓlC0(P +)W (0)†ξ(u)(0)
n,p
]
(11.6)
where we used the fact that Y commutes with the wilson coefficient C0(P +). This argument cannot be
applied to Q1,5 because Y , containing generators of its own, does not commute with T a . However, by ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5a73e8bb8f7f637a9b3eb58c5e733e6b_MIT8_851S13_SCETIIApplicat.pdf |
Programming Languages
Copyright
c
Nancy Leveson, Sept. 1999
As difficult to discuss rationally as religion or politics.
Prone to extreme statements devoid of data.
Examples:
"It is practically impossible to teach good programming to
students that have had a prior exposure to BASIC; as
potential programmers the... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
defined (not
dependent on compiler.
Relationship between PL and Correctness (3)
Copyright
c
Nancy Leveson, Sept. 1999
Understandability
"The primary goal of a programming language is accurate
communication among humans."
Readability more important than writeability.
Well "punctuated" (easy to directly det... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
restrict programmer to decisions
that really matter.
Decisions should be recorded in program independent
of external documentation.
Simplicity of language less important than ability to
write conceptually simple programs.
Can programming language influence correctness?
Languages affect the way we think about... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
thesis 2: Every notation highlights some type of
information at the expense of others; the better notation
for a given task is theone that highlights the information
that given task needs.
More generally, the comprehensibility of a notation may depend
on the number and complexity of mental operations required to
... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
purposes for each part
Visible or easily inferred relationships between
each part and the larger structure.
Important to alleviate mismatches between programmer’s task
and program structure.
Hypothesis that role expressiveness tends to detract from reusability
of program fragments.
"When a program fragment makes... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
The Invention Machine
The Invention Machine
Computational adaptation of TRIZ,
Computational adaptation of TRIZ,
Value Engineering and
Value Engineering and
the Semantic Web
the Semantic Web
Thanks to Invention Machine and
Dr. Mikhail Verbitsky for materials and
consultation and
SDM04 students who participated
Cou... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
about Noisy Fan
4) Management pressure to reduce costs
© Speller 2007, Engineering Systems Division, Massachusetts Institute of Technology
7
Step 1: Value Equation Development
Problem-Statement
Value Engineering, TRIZ
Alternative Architecture
List of Problems
Value = F/(P+C)
© Speller 2007, Engineering Systems Divisio... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
: Effects
© Speller 2007, Engineering Systems Division, Massachusetts Institute of Technology
16
Courtesy of Invention Machine Corporation. Used with permission.
Traditional TRIZ: Effects
© Speller 2007, Engineering Systems Division, Massachusetts Institute of Technology
17
Courtesy of Invention Machine Corporation. U... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
NL
?
Process…
Semantic
Knowledge
Base
DOC
RTF
HTM(L)
PDF
TXT
© Speller 2007, Engineering Systems Division, Massachusetts Institute of Technology
24
Access to External and Internal Intellectual Assets
Invention Machine Content
Scientific
Effects
Patent
Collections
9,000
10,000,000
…and via Knowledge Producer
Includes ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
of Technology
32
-Theory (TRIZ, a theory of invention),
-Method (Altshuller’s step-by-step
creative process)
-Tool (ex. Goldfire1), SDM’rs evaluation
A theory of invention, as stated by Altshuller the
inventor of TRIZ should:
• Be systematic, a step-by-step procedure
• Be inclusive to a broad solution space hoping t... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
with different concepts, patents,
scientific effects, and technology trends.
• The fact that the intent is expressed in a high level allows thinking in
a broad range of solutions.
• A deep web search on patents classified through the use of the
semantic engine, again here the patents help to clarify the idea, but ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
and with a
steady flow. The process is therefore conceptualizing, and the operand will be the solution that changes
from non-existent to existent, delivering value to the Corporation (the beneficiary)
In the case of the user, the benefit is slightly different. Because of the exposure to many different ideas, the
sem... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
3.044 MATERIALS PROCESSING
LECTURE 5
General Heat Conduction Solutions:
∂T = ∇ · k∇T, T (¯x, t)
∂t
Trick one: steady state ∇2T = 0, T (x)
Trick two: low Biot number ∂T = α h(Ts
∂t
− Tf ), T (t)
Transient:
- semi-infinite
- infinite series: book, analytical
- graphical solutions
- computers, numerical, finite elements
Exam... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/5b02d3e73eaf3afa6ec9564595714beb_MIT3_044S13_Lec05.pdf |
K
T ≈ 750K
Solve for h (the only changable variable): h ≈ 260 W
m2 K
- oil bath w/ standoff/air gap
- big fans
- other gases
Example 2: Thermal Spray Coatings / Plasma Spray
Specific Example:
oxyacetylene torch: T = 2700◦C
powder: Ni alloy MAR-M200, r = 2 − 50μm
Problem Statement: need a particle to melt in flight (T = Tm... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/5b02d3e73eaf3afa6ec9564595714beb_MIT3_044S13_Lec05.pdf |
http://ocw.mit.edu
3.044 Materials Processing
Spring 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/5b02d3e73eaf3afa6ec9564595714beb_MIT3_044S13_Lec05.pdf |
6.801/6.866: Machine Vision, Lecture 11
Professor Berthold Horn, Ryan Sander, Tadayuki Yoshitake
MIT Department of Electrical Engineering and Computer Science
Fall 2020
These lecture summaries are designed to be a review of the lecture. Though I do my best to include all main topics from the
lecture, the lectures ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
the patent examiner, rather than the patent
authors
• Most patents end with something along the lines of “this is why our invention was necessary” or “this is the technical gap
our invention fills”
• Software is not patentable - companies and engineers get around this by putting code on hardware and patenting the
“appar... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
→ Ky =
0
0 −1
0
−1
1
0
0 −1
∂E
∂x0
∂E
∂y0
• Sobel Operator: This computational molecule requires more computation and it is not as high-resolution. It is also more
robust to noise than the computational molecules used above:
→ Kx =
⎡
−1
2
−1
⎣
0
0
0
⎤
1
2⎦
1
⎡
−1
→ Ky = ⎣ 0
1
⎤
0 ⎦
1... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
IC algorithm, we can estimate the
brightness gradient magnitude and direction. The CORDIC algorithm does this iteratively through a corrective feedback
mechanism (see reference), but computationally, only uses SHIFT, ADD, SUBTRACT, and ABS operations.
3. Choose Neighbors and Detect Peaks: This is achieved using brig... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
x|, |Gy|) (octant quantization)
4. S0 = min(|Gx|, |Gy |) (octant quantization)
At a high level, the apparatus discussed in this patent is composed of:
3
• Gradient estimator
• Peak detector
• Sub-pixel interpolator
Next, let us dive in more to the general edge detection problem.
1.3 Edges & Edge Detection
Let us... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
artificially high
(infinite) frequencies that prevent us from sampling without aliasing effects. Let us instead try a “soft” step function, i.e. a
“sigmoid” function: σ(x) =
. Then our u(x) takes the form:
1
1+e
−x
A “Soft” Unit Step Function, u(x)
)
x
(
u
1
0.8
0.6
0.4
0.2
0
−6
−4
−2
2
4
6
0
x
The gradient... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
1. u(x) = 1+exp (−x)
1
2. ru(x) =
du
dx
=
d
dx 1+exp −x
1
=
exp(−x)
(1+exp(−x))2
2
3. r u(x) =
d2 u
dx2
d
= (
dx (1+exp(−x))2
exp(−x)
) =
− exp(−x)(1+exp(−x))2+2 exp(−x)(1+exp(−x)) exp(−x)
(1+exp(−x))4
Building on top of this framework above, let us now move on to brightness gradient estimation.
... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
Let us first consider the simple two-pixel difference operator (in the x-direction/in the one-dimensional case),
i.e. → Kx = (−1 1). Let us look at the forward difference and backward difference when this operator is applied:
dE
dx
1
δ
1. Forward Difference:
∂E ≈ f (x+δx)−f (x)
∂x
δx
= f 0(x) + δx f 00(x) +
2
(δx)2
6... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
x − δ x x + ). There is no pixel at x but we can still compute
the derivative here. This yields an error that is 0.25 the error above due to the fact that our pixels are
apart, as opposed to δ
apart:
1
2δ
δ
2
δ
2
dx
2
error =
( xδ )2
2
6
f 000(x)
This makes sense intuitively, because the closer together a set ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
operator (in each dimension) as being the dis-
crete convolution of a 2-by-2 horizontal or vertical highpass/edge filer with a smoothing or averaging filter:
1. x-direction: 1
−1
2δx −1
1
1
∗
1 1
1
1
⎡
−1
= −2
⎣
−1
0
0
0
⎤
1
2
⎦
1
2. y-direction: 1
2δy
−1 −1
1
1
∗
1 1
1 1
⎡
= ⎣ 0
1
−1 −2 −1
0
... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
G0: (1) G−, (2) G0, and (3) G+. Let us look at the results for these two types
of functions:
1. Quadratic: s =
G+−G−
4(G0− 1 (G+−G−))
2
, this results in s ∈ [− 1 , ].
1
2 2
2. Triangular: s = 2(G0−min(G+ ,G−))
G+−G−
A few notes about these approaches:
• Note that in each case, we only want to keep if the magnit... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
Optics Overview
MIT 2.71/2.710
Review Lecture p-1
What is light?
• Light is a form of electromagnetic energy – detected through its
effects, e.g. heating of illuminated objects, conversion of light to
current, mechanical pressure (“Maxwell force”) etc.
• Light energy is conveyed through particles: “photons”
– ba... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
c/n
n : refractive index
(or index of refraction)
Absorption coefficient 0 Absorption coefficient α
energy decay coefficient,
after distance L : e–2αL
E.g. vacuum n=1, air n ≈ 1;
glass n≈1.5; glass fiber has α ≈0.25dB/km=0.0288/km
MIT 2.71/2.710
Review Lecture p-6
Materials classification
• Dielectrics
typic... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
71/2.710
Review Lecture p-9
The concept of a monochromatic
“ray”
t=0
(frozen)
λ
z
direction of
energy propagation:
light ray
wavefronts
In homogeneous media,
light propagates in rectilinear paths
MIT 2.71/2.710
Review Lecture p-10
The concept of a monochromatic
“ray”
t=∆t
(advanced)
λ
z
direction of... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
1.00
n
=1.51
TIR
n
=1.5105
n
=1.51
TIR
• Planar version: integrated optics
• Cylindrically symmetric version: fiber optics
• Permit the creation of “light chips” and “light cables,” respectively, where
light is guided around with few restrictions
• Materials research has yielded glasses with very low losses (... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
another point in space
(the “image.”)
• The latter function is the topic of the discipline of Optical Imaging.
MIT 2.71/2.710
Review Lecture p-23
Imaging condition: ray-tracing
thin lens (+)
2nd FP
chiefray
image
(real)
1st FP
object
• Image point is located at the common intersection of all rays which
em... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
Photographic camera
• Magnifier
• Microscope
• Telescope
MIT 2.71/2.710
Review Lecture p-28
The human eye
Remote object (unaccommodated eye)
Near object (accommodated eye)
MIT 2.71/2.710
Review Lecture p-29
The photographic camera
meniscus
lens
or (nowadays
zoom lens
)
“digital imaging”
Film
or
detect... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
degree that the “inverse
problem” is solvable (i.e. its condition)
2.717 sp02 for details
– electronic noise (thermal, Poisson) in cameras
– multiplicative noise in photographic film
– stray light
– speckle noise (coherent imaging systems only)
• Sampling at the image plane
– camera pixel size
– photographic f... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
�ΛΛΛ
incident
plane
wave
Λ
m=3
m=2
m=1
m=0
m=–1
m=–2
m=–3
…
“straight-through” order or DC term
Condition for constructive interference:
(
m
integer)
π
2
Λ π
=
2
m
λ
⇔ sinθ= m
λ
Λ
diffraction order
MIT 2.71/2.710
Review Lecture p-43
Grating dispersion
Anomalous
(or negative)
dispersion
poly... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
as a linear, shift-invariant system
Thin transparency
(
t
)
,
y
x
(
g
1
)
y
x ,
=
( ,
x h
=
)
y
1
i λ
z
exp i
2π
z
exp
λ
i π
2x + y 2
λz
x
,(
y
g
)
2
)
(
x
,
t y
=
x
,(
y
)
g
1
impulse response
convolution
=
(
′
′
x
y
g
,
3
∗
x
y
,(
)
g
2
o... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
Machine learning for Pathology
Andrew H Beck MD PhD
CEO @ PathAI
6.S897/HST.956: Machine Learning for Healthcare. MIT.
March 19, 2019
© source unknown. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
1
Pat... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
Even today, the anatomic path lab has been
largely unchanged for routine diagnostics
© sources unknown. All rights reserved.
This content is excluded from our Creative
Commons license. For more information,
see https://ocw.mit.edu/help/faq-fair-use/
10
And core technology
breakthroughs in rout... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
on skin biopsies
• 187 pathologists interpreted skin lesion biopsies, resulting in an overall discordance of 45%
• 118 pathologists read the same samples 8 months apart, and had an intraobserver
discordance of 33%
Courtesy of Elmore, et al. Used under CC BY-NC.
BMJ 2017;357:j2813 | doi: 10.1136/bmj.j2813
13
... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
et al. Relevant impact of central pathology review on nodal classification in individual breast
cancer patients. Ann Oncol. 2012;23(10):2561-2566.
18
The data - CAMELYON
• H & E stained, Formalin-Fixe
d
Paraffin-Embedded (FFPE)
• 270 training slid... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
12;318(22):2199-2210.
26
22
Deep learning model outperforms human pathologists
in the diagnosis of metastatic cancer
Pathologists in competition
Pathologists in clinical practice1
Pathologists on micro-metastasis2
Deep learning model
Error Rate (1-AUC)
3.5%
13 ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
unknown. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
28
Pathology Report
Pathology Report
Confirm
Patient: John Doe
Diagnosis:
Size:
pTNM staging:
# of Pos LN:
# of Neg LN:
Time per slide:
1 –... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
32
Manual interpretation of PD-L1 IHC is highly
variable
PDL1 manual IHC scores on immune cells are unreliable
© American Medical Association. All rights reserved. This content is excluded from our Cre... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
further
From quantitative assay to patient prediction
● PD-L1 scoring alone reduces billions of pixels to 1-2 numbers.
● Can we identify additional relevant information?
○ Using data from randomized controlled clinical trials
● However: Millions of patches, hundreds of patients
Proprietary & Confidential
41
... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
Total number of macrophages in invasive margin (250um)
Total number of lymphocytes in epithelial/stromal interface on H&E stain
Total number of plasma cells in epithelium on H&E stain
Total number of plasma cells in stroma on H&E stain
Tumor (epithelium + stroma) area on H&E stain
Total number of plasma cells in e... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
Map
© source unknown. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
TCGA-EE-A2GL, Malignant Melanoma
Tumor Stroma
Tumor Epithelium
51
Melanoma Cell Map
© source unknown. All rights reserved. This content... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
Stromal plasma cell area RNA signature
strongly enriched for immune genes
Gene Set Name
REACTOME_IMMUNE_SYSTEM
REACTOME_ADAPTIVE_IMMUNE_SYSTEM
PID_TCR_PATHWAY
REACTOME_IMMUNOREGULATORY_INTERACTIONS_BETWEEN_A_
LYMPHOID_AND_A_NON_LYMPHOID_CELL
KEGG_PRIMARY_IMMUNODEFICIENCY
PID_IL12_2PATHWAY
PID_CD8_TCR_PATHWAY ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
type
• Contrast to traditional approach:
hand-crafted algorithms.
Proprietary & Confidential
56
Extensive Slide Search & Data Standardization
Proprietary & Confidential
57
Automated quality control
Blurred areas
Folded /
damaged
tissue
© source unknown. All rights reserved. Thi... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
AI in medicine
Some closing thoughts
• ML in the real world:
• Building the right dataset is
75% of the challenge
• Modern ML: engineering and
empirical science
• Rigorous validation is key
• Ideas and algorithms vs.
teams and infrastructure
Proprietary & Confidential
63
... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
7.91 / 20.490 / 6.874 / HST.506
7.36 / 20.390 / 6.802
C. Burge Lecture #9
Mar. 6, 2014
Modeling & Discovery
of Sequence Motifs
1
Modeling & Discovery of Sequence Motifs
• Motif Discovery with Gibbs Sampling Algorithm
• Information Content of a Motif
• Parameter Estimation for Motif Models (+ others) ... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
Member of the Trypanosoma Cruzi CCHC Zinc Finger Protein
Family with Nuclear Localization." Genetics and Molecular
Research 5, no. 3 (2006): 553-63.
CX2CX4HX4C
Zinc finger (DNA binding)
Ericsson et al. Genet. Mol. Res. 2006
Courtesy of the authors. License: CC-BY-NC.
Source: Bentem, Van, Sergio de la Fuente, et al.... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
7 S8 S9
Odds Ratio: R =
P(S|+) = P-3(S1)P-2(S2)P-1(S3) ••• P5(S8)P6(S9)
P(S|-) = Pbg(S1)Pbg(S2)Pbg(S3) ••• Pbg(S8)Pbg(S9)
Background model homogenous, assumes independence
7
Ways to describe a motif
Common motif adjectives:
exact/precise versus degenerate
stron... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
knows what entropy really is, so in a debate you will always have
the advantage.”
source: Wikipedia
10
Information Content of a DNA Motif
... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
stochastic search of the space of possible PSPMs
• Deterministic Optimization (e.g., MEME)
- deterministic search of space of possible PSPMs
13
What the motif landscape might look like
14
14
Monte Carlo Algorithms
The Gibbs motif sampler is a
Mont... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
seq 1).
3) Make a weight matrix model of width W from the sites
in all sequences except the one chosen in step 2.
4) Assign a probability to each position in seq 1 using the
weight matrix model constructed in step 3:
p = { p1, p2, p3, …, pL-W+1 }
Lawrence et al. Science 1993
17
... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
Gibbs Sampling Algorithm V
5. Sample a new site proportional to likelihood and update motif
instances
Likelihood
(probability)
Θ
23
Gibbs Sampling Algorithm VI
6. Update weight matrix
Likelihood
(probability)
Θ
24
Gibbs Sampling Algorithm VII
7. Iterate until convergence ( ... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
matrix
)
4
x
(
y
c
n
e
u
q
e
r
F
.
m
u
C
Nucleotide:
Information content
Motif
strength
)
s
t
i
b
(
n
o
i
t
a
m
r
o
f
n
I
Iteration
Courtesy of Mehdi Yahyanejad. Used with permission.
Position in seq
Current
Position
in seq
.
o
N
e
c
n
e
u
q
e
S
Position in seq
Probability
density
e
c
n
e
u
q
e
S
Gibbs sam... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
acgaaacgagggcgatcaatgcccgataggactaataag
tagtacaaacccgctcacccgaaaggagggcaaatacctt
atatacagccaggggagacctataactcagcaaggttcag
cgtatgtactaattgtggagagcaaatcattgtccacgtg
...
Features that affect motif finding
No. of sequences
Length of sequences
Information content of motif
Match between expected length and
actual l... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
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