text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
��
o
q ·
0
0
0
o
q
0
0 + q2
y − q2
x − q2
q2
z
2(qyqx + q0qz)
2(qzqx − q0qy)
0
2(qxqy − q0qz)
y − q2
x + q2
0 − q2
q2
z
2(qzqy + q0qz)
0
2(qxqz + q0qy)
2(qyqz − q0qx)
y + q2
x − q2
0 − q2
q2
z
The matrix ¯QT Q has skew-symmetric components and symmetric components. This is useful for conversions. Given a
quat... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
the angle (we can solve for θ through arccos of the expression above), it is not the best way to do
instead.
so: we will encounter problems near θ ≈ 0, π. Instead, we can use the off-diagonal elements, which depend on sin
(cid:17)
(cid:16) θ
2
Note that this works because at angles θ where cos
and vice versa.
(cid:17)
(... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
= 4q0qx
r21 + r12 = 4qxqy
r32 + r23 = 4qyqz
r13 + r31 = 4qzqz
Adding/subtracting off-diagonals give us 6 relations, of which we only need 3 (since we have 1 relation from the diagonals). For
instance, if we have qi = qy, then we pick off-diagonal relations involving qy, and we solve the four equations given by:
1 − r11 +... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
. sl
∆= (cid:80)n
i=1 ||r(cid:48)
l,i||2
2
Then we can write this objective for the optimal scaling factor s∗ as:
s∗ = arg min
s
{J(s) ∆= sr − 2sD + s2sl}
Since this is an unconstrained optimization problem, we can solve this by taking the derivative w.r.t. s and setting it equal to 0:
dJ(s)
ds
=
(cid:16)
d
ds
sr − 2sD... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
we can write our objective and optimization problem over scale as:
||
1
√
s
r(cid:48)
r,i −
√
sR(r(cid:48)
l,i)||2
2
(cid:19)
− 2
n
(cid:88)
(cid:16)
||r(cid:48)
r,i||2
2
r,iR(r(cid:48)
r(cid:48)
l,i)
n
(cid:88)
(cid:17)
+ s
||R(r(cid:48)
l,i)||2
2
n
(cid:88)
i=1
n
(cid:88)
i=1
(cid:18) 1
s
s∗ = arg min
s
= arg min
s
=... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
scaling factor s∗ as:
s∗ = arg min
s
{J(s) ∆=
1
s
sr − 2D + ssl}
4
Since this is an unconstrained optimization problem, we can solve this by taking the derivative w.r.t. s and setting it equal to 0:
dJ(s)
ds
=
(cid:19)
(cid:18) 1
s
sr − 2D + ssl
d
ds
1
s2 sr + sl = 0 =⇒ s2 =
= 0
= −
sl
sr
Therefore, we can see that go... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
the derivative of this objective w.r.t. our quaternion
o
q and setting it equal to zero. Note the following helpful identities with matrix and vector calculus:
1.
2.
d
da (a · b) = b
d
da (aT M b) = 2M b
However, since we are working with quaternions, we must take this constraint into account. We saw in lecture 18 that... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
= 0
N
(
o
q
2
T o
q)2
o
q
2N
T o
o
q
q
−
o
q
(
5
= 0
From here, we can write this first order condition result as:
o
q
o
q =
N
o
q
o
q
T
o
q
N
T o
q
Note that
T
o
q
o
N
q
T o
q
o
q
∈ R (this is our objective). Therefore, we are searching for a vector of quaternion coefficients such applying the
rotation matrix to this ve... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
.1 How Many Correspondences Do We Need?
Recall that we are looking for 6 parameters (for translation and rotation) or 7 parameters (for translation, rotation, and scaling).
Since each correspondence provides three constraints (since we equate the 3-dimensional coordinates of two 3D points in space),
assuming non-redund... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
note, this is often not needed, especially for finding the absolute orientation between two cameras, because
oftentimes the only transformations that need to be considered due to the design constraints of the system (e.g. an autonomous
car with two lidar systems, one on each side) are translation and rotation.
1.4.2 Whe... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
ar.
1.4.3 What Happens When Points are Coplanar?
When points are coplanar, we have that the matrix N , composed of the sum of dyadic products between the correspondences in
the two point clouds, will be singular.
To describe this plane in space, we need only find a normal vector ˆn that is orthogonal to all points in th... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
at least Span({ˆn}), and therefore
M is singular. Recall that a matrix M ∈ Rn×d is singular if ∃ x ∈ Rd, x (cid:54)= 0 such that M x = 0, i.e. the matrix has a non-trivial
null space.
1.4.4 What Happens When Both Coordinate Systems Are Coplanar
Visually, when two point clouds are coplanar, we have:
Figure 3: Two coplan... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
can
orient two planes on top of each other, and then just solve a 2D least-squares problem to solve for our in-place rotation.
With these steps, we have a rotation between the two point clouds!
1.5 Robustness
In many methods in this course, we have looked at the use of Least Squares methods to solve for estimates in th... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
just immediately terminating once you
have a good fit, you can run this many times, and then take the best fit from that.
Furthermore, for step 3, we threshold the band from the fitted line/hyperplane to determine which points of the dataset are
inliers, and which are outliers (see figure below). This band is usually given... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
this space? Many orientation problems we have studied so far do not have a closed-form
solution and may require sampling. How do we sample from the space of rotations?
1.6.1
Initial Procedure: Sampling from a Sphere
Let us start by sampling from a unit sphere (we will start in 3D, aiming eventually for 4D, but our fram... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
the sphere.
Generalization to 4D: As we mentioned above, our goal is to generalize this from 3D to 4D. Cubes and spheres simply
become 4-dimensional - enabling us to sample quaternions.
1.6.3 Sampling From Spheres Using Regular and Semi-Regular Polyhedra
We saw the approach above requires discarding samples, which is c... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
:0)− π
4
o
q = (cos (cid:0)− π
4
(cid:1) , sin (cid:0)− π
4
(cid:1) ˆx) = 1√
2
(1, −ˆx)
(cid:1) , sin (cid:0)− π
4
(cid:1) ˆy) = 1√
2
(1, −ˆy)
(cid:1) , sin (cid:0)− π
4
(cid:1) ˆz) = 1√
2
(1, −ˆz)
These 10 rotations by themselves give us 10 ways to sample the rotation space. How can we construct more samples? We can
d... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
1)
(cid:17)
(cid:16) θ
2
= 1
2 =⇒ θ
2 = π
3 =⇒ θ = 2π
3
Therefore, we have produced a new rotation that we can sample from!
These are just a few of the pairwise quaternion products we can compute. It turns out that these pairwise quaternion products
produce a total of 24 new rotations from the original 10 rotations. Th... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
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I... | https://ocw.mit.edu/courses/8-322-quantum-theory-ii-spring-2003/36b3cccc5336969c0304126613a3121e_83223Lecture2.pdf |
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in
meunrcc... | https://ocw.mit.edu/courses/8-322-quantum-theory-ii-spring-2003/36b3cccc5336969c0304126613a3121e_83223Lecture2.pdf |
Lecture 2
8.321 Quantum Theory I, Fall 2017
5
Lecture 2 (Sep. 11, 2017)
2.1 More Relevant Math
2.1.1 Inner Products
Last time, we discussed the concept of a maximally linearly independent set, which is a set {|αj(cid:105)}
of vectors that are linearly independent, and such that there exists no |β(cid:105) such that {|α... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
α(cid:105), |β(cid:105))∗ = (|β(cid:105), |α(cid:105)) .
(|α(cid:105), |β(cid:105) + |β(cid:48)(cid:105)) = (|α(cid:105), |β(cid:105)) + (|α(cid:105), |β(cid:48)(cid:105)) .
4. For all |α(cid:105) ∈ V ,
and if (|α(cid:105), |α(cid:105)) = 0, then |α(cid:105) is the zero ket, |α(cid:105) = 0.
(|α(cid:105), |α(cid:105)) ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
8)
Lecture 2
8.321 Quantum Theory I, Fall 2017
as we discussed in the last lecture, which can be endowed with the inner product
ˆ
(f, g) =
0
1
dx f ∗(x)g(x) .
6
(2.9)
Now we introduce a bit of terminology. The vectors we have been discussing are referred to as
kets. Two kets are orthogonal if (|α(cid:105), |β(cid:105)... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
104)γ|, such
that for any basis {|αj(cid:105)}, the statement
γ : |αj(cid:105) (cid:55)→ cj
(2.14)
implies that
(cid:104)γ|αj(cid:105) = cj .
We see that the dual space is just the space of linear functionals on V . The {γ} form an n-
dimensional vector space V ∗, where n = dim V . We then have a duality V ↔ V ∗, given... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
017
7
2.1.4 Operators
Recall the second postulate of quantum mechanics: observables are represented in quantum me-
chanics as Hermitian operators acting on the Hilbert space H (from here on out, we will consider
states in the Hilbert space H rather than an arbitrary vector space V ). In order to understand this
stateme... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
105) + c∗
βX|β(cid:105)
(2.22)
for all cα, cβ ∈ F and |α(cid:105), |β(cid:105) ∈ H.
We can also define an action of these same operators on the dual space, which will be a right
action of the form (cid:104)β|X for (cid:104)β| ∈ H∗. We define this object via its inner product with kets, by
requiring that it satisfy the pr... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
2.27)
for all |α(cid:105) ∈ H. Given an orthonormal basis |{a(cid:48)}(cid:105) (notation from Sakurai), the identity operator
can be expressed in the form
1 =
(cid:88)(cid:12)
(cid:12) .
(cid:12)a(cid:48)(cid:11)(cid:10)a(cid:48)(cid:12)
We can use Eq. (2.28) to write any operator X in the form
a(cid:48)
X = 1X1
(cid:... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
(cid:48)|X|a(cid:48)(cid:48)(cid:105), given any orthonormal basis |{a(cid:48)}(cid:105). This defines an
n × n matrix with complex entries corresponding to each operator X. For this reason, we will often
use the words “operator” and “matrix” interchangeably if the chosen basis is clear, even though
the concept of an op... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
|(XY )† = (cid:104)γ| ,
(2.31)
(2.32)
(2.33)
(2.34)
which we can compare with the second equation in (2.33) to yield (XY )† = Y †X †.
This same concept has a well-defined meaning for matrices. As we have seen, any operator has
the same information content as an n × n matrix, where n = dim H. If M is a matrix correspondi... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
(2.37)
(2.38)
As a second example, consider the Hilbert space H of square-integrable functions on (−∞, ∞),
with inner product
(cid:104)f1|f2(cid:105) =
ˆ
∞
−∞
dx f1
∗(x)f2(x)
(2.39)
for |f1(cid:105), |f
d
2(cid:105) ∈ H. Now consider the operator A = . Using integration by parts, we see that
dx
ˆ
∞
ˆ
dx f1
∗(x)
d
dx
f2... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
which we have already seen.
For some operators, we can define an inverse operator: for an operator A, its inverse operator
A−1 (if it exists) satisfies
A−1A = AA−1 = 1 .
(2.44)
Inverse operators are not guaranteed to exist.
Of special interest to us are unitary operators: a unitary operator U is one that satisfies
U −1 = ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
of A with distinct eigenvalues are orthogonal.
Proof. Suppose that for some Hermitian operator A and kets |a(cid:48)(cid:105), |a(cid:48)(cid:48)(cid:105) ∈ H we have
(cid:12)a(cid:48)(cid:11) = a(cid:48)(cid:12)
A(cid:12)
a
(cid:12)
(cid:12)
(cid:12)
(cid:10)
(cid:10)
(cid:12)A = a(cid:48)(cid:48)
a(cid:48)(cid:48)
(c... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
Eqs. (2.48a) and (2.48b), we see that
(cid:2)(cid:0)a(cid:48)(cid:48)(cid:1)∗ − a(cid:48)(cid:3)(cid:10)a(cid:48)(cid:48)(cid:12)
(cid:11)
(cid:12)a(cid:48) = 0 .
(2.48a)
(2.48b)
(2.49)
|a(cid:48)(cid:48)(cid:105) and we have
(cid:54)= a(cid:48)(cid:48), then this proves that (cid:104)a(cid:48)(cid:48)|a(cid:48)(cid:10... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
5)(cid:104)a| .
We can check that for some eigenket |b(cid:105), we have
A|b(cid:105) =
(cid:88)
a
a|a a
(cid:105)(cid:104) |b(cid:105) =
(cid:88)
a
a
|a(cid:105)δab = b|b(cid:105) .
Note that the equation
is simply a special case of Eq. (2.51).
(cid:88)
1 =
|a(cid:105)(cid:104)a|
a
(2.50)
(2.51)
(2.52)
(2.53)
MIT Ope... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
Queueing Systems: Lecture 5
Amedeo R. Odoni
October 30, 2006
Lecture Outline
• A fundamental result for queueing networks
• State transition diagrams for Markovian
queueing systems and networks: examples
• Examples
• Dynamic queueing systems and viable
approaches
• Qualitative discussion of behavior
Reference:... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/36d90014e175b04466290678ce09bfbf_lec9.pdf |
Poisson
arrivals; λ
One server;
neg. exp’l; μ 1
One server;
neg. exp’l; μ 2
No queuing
space
No queuing
space
Note: The queuing system on the right may “block” the one on
the left.
Example 2: M/Ek/1 System, with
system capacity for total of N users
See distributed notes.
Example 3: Two Types of Users
a... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/36d90014e175b04466290678ce09bfbf_lec9.pdf |
imate Methods
Problems with Approach 1:
•
1. For cases in which demand varies significantly (e.g.,
>10% from overall average value) the delay estimates
can be VERY poor
2. Will underestimate overall average delay, possibly by a
lot
Problems with Approach 2:
•
1. May not have ρ < 1, for some intervals; then wha... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/36d90014e175b04466290678ce09bfbf_lec9.pdf |
15
1:00
3:00
5:00
7:00
9:00
11:00
13:00
15:00
17:00
19:00
21:00
23:00
Dem
R1
R2
R3
R4
(R1= capacity is 80 movements per hour; R2 = 90; R3 = 100; R4 = 110)
Two Recent References on Numerical
Methods for Dynamic Queueing Systems
• Escobar, M., A. R. Odoni and E. Roth, “Approximate
Solutions for Mult... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/36d90014e175b04466290678ce09bfbf_lec9.pdf |
“Medicine and the Computer:
The Promise and Problems of Change”
(cid:216) Perceived problems
—W.B. Schwartz, NEJM 1970
(cid:216) Physician shortage and maldistribution
(cid:216) Ever-expanding body of knowledge, so that the physician
cannot keep up
(cid:216) Exploit the computer as an “intellectual”, “deductive”
i... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
) use some representation of disease and a diagnostic
algorithm
(cid:216)disease/symptom associations (since 1960’s)
(cid:216)probabilistic version (since 1960’s)
(cid:216)causal models (since 1980’s)
5
Flowcharts contain all of…
Diagnostic
Approach
Domain
Knowledge
Inference
Engine
9
Flowcharts
(cid:216) ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
organism is
organism is
streptococcus.
streptococcus.
streptococcus.
streptococcus.
11
Mycin consult
Mycin consult
Davis, et al., Artificial Intelligence 8: 15-45 (1977)
12
12
How Mycin Works
(cid:216) To find out a fact
(cid:216) If there are rules that can conclude it, try them
(cid:216) Ask the user
(ci... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
Representation
s1
s2
s3
s4
s5
s6
s7
s8
s9
s10
s...
Disease
s1
s2
s3
s4
s5
s6
s7
s8
s9
s10
s...
Disease
16
Diagnosis by Card Selection
s1
s2
s3
s4
s5
s6
s7
s8
s9
s10
s...
Disease
17
Diagnosis by Edge-Punched
Cards
(cid:216) Dx is intersection of sets of diseases that may
cause al... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
NEPHROTIC SYNDROME, a clinical state
FINDINGS:
1* Low serum albumin concentration
2. Heavy proteinuria
3* >5 gm/day proteinuria
4* Massive symmetrical edema
5* Facial or peri-orbital symmetric edema
6. High serum cholesterol
7. Urine lipids present
IS-SUFFICIENT: Massive pedal edema & >5 gm/day proteinuria
MUST-NOT-HAV... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
SYMMETRICAL. …
DIAGNOSES THAT HAVE BEEN ACCEPTED ARE: NEPHROTIC SYNDROME AND
SODIUM RETENTION.
THE LEADING HYPOTHESIS IS IDIOPATHIC NEPHROTIC SYNDROME.
IDIOPATHIC NEPHROTIC SYNDROME
ACUTE GLOMERULONEPHRITIS
HENOCH SCHOENLEIN PURPURA
fit
0.80
0.22
0.07
explained score
0.58
0.37
0.24
0.27
0.09
0.10
24
Other “Frame-bas... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
4
M5
M6
H2
29
Competitors
H1
M1
M2
M3
M4
M5
M6
H2
30
Still Competitors
H1
M1
M2
M3
M4
M5
M6
H2
31
Probably Complementary
H1
M1
M2
M3
M4
M5
M6
H2
32
Multi-Hypothesis Diagnosis
(cid:216) Set aside complementary hypotheses
(cid:216) … and manifestations predicted by them
(cid:216) Solve diagnostic problem among comp... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
loss
(cid:216) Vomiting causes acid loss
(cid:216) Therefore, normal pH is a manifestation of
{diarrhea + vomiting}!
39
Temporal Reasoning
(cid:216)Keeping track of multiple forms of temporal relations
(Kahn '75)
(cid:216)The time line
(cid:216)“On Dec. 12 last year . . .”
(cid:216)Special reference events
(cid:216... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
to anticipate them all
(cid:216) Strawman: Fit parameters to a physiological
model, then predict consequences to suggest
(cid:216) other expected findings
(cid:216) reasonable interventions
(cid:216) Qualitative models
(cid:216) Combining associational and model-based
reasoning
44
Guyton's Model of
Cardiovascular D... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
216) A few sophisticated, modern, probability-based systems
are now being built
(cid:216) HIS's really are being developed (slowly, but surely) and
will provide a critical opportunity for experimentation
(cid:216) No large, broad-domain, deep systems are being
tackled today
(cid:216) Research advances are laying the... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
Lecture #1 Instructor Notes
First off, welcome. I hope that these notes are interesting and helpful to you.
Also, please note that there is a set of “Comments” on each lecture, that go along with the
readings and the Instructor’s Notes here.
Let us start with the first question you should always ask in a course….. ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/3719d6726de02bbc57b61939f0cd14c5_MIT2_682S12_lec01.pdf |
the
water comes from an equation of state relating the soundspeed to temperature T, salinity S, and
depth z. In a very simplified form, the soundspeed as a function of depth z (its main
dependence) is
This is the first equation in the Computational Ocean Acoustics book, and shows that the
soundspeed is very sensit... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/3719d6726de02bbc57b61939f0cd14c5_MIT2_682S12_lec01.pdf |
8
Continuous-Time
Fourier Transform
In this lecture, we extend the Fourier series representation for continuous-
time periodic signals to a representation of aperiodic signals. The basic ap-
proach is to construct a periodic signal from the aperiodic one by periodically
replicating it, that is, by adding it to itself s... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
represents the sig-
nal. This envelope is defined as the Fourier transform of the aperiodic signal
remaining when the period goes to infinity.
Although the Fourier transform is developed in this lecture beginning
with the Fourier series, the Fourier transform in fact becomes a framework
that can be used to encompass bo... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
Aperiodic Signals: The Continuous-Time Four-
ier Transform, pages 186-195
Section 4.5, Periodic Signals and the Continuous-Time Fourier Transform,
pages 196-202
Continuous-Time Fourier Transform
MARKERBOARD
8.1(a)
Cw%-nwnsA -TWAe
Ferio. SAsDg
Nce pevb~e.t.
r.
T . 4,)
.4
FOURIER REPRESENTATION OF APERIODIC SIGNALS
(""N... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
dt
Then:
Toak X(W) I=kcoo
X(w) is the envelope of To ak
e jkoot =
+00
k=-co
k=-00
+00
x(t) = 27
X(kco)
k
k=-Oo
X(kwO) ejkwot
Continuous-Time Fourier Transform
+00
X(o) =
x(t) ei jWt dt
00
Fourier transform
- analysis
x(t) =
E X(kwo) ejkwo tWO
k=-0O
As To-- oo,
coo -* 0 1x t) - x (t), we doE -
+a(
x(t) = 2
X(o) e j... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
4T1
x(t)1
-T1
0
T,
JT
F7-1 *..
To
TRANSPARENCY
8.7
±(t) and its Fourier
series coefficients
with To = 8T1. [As the
figure is drawn, To and
T, are not to scale.]
T,1X(W)
t. e I
- S T
t V
T
1
/
'it/I
Continuous-Time Fourier Transform
MARKERBOARD
8.1(b)
ExCapkt (Td' 4.'1)
4S
e.e
-i
-
e.
I
~ c*w-+ a4-30&
I~
V~V
~-~-... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
-20
-40
-601
-r/4
-~ /2
-37r/4
-
_
~
0.1a
_
~
I
10a
I
100a
frequency (CO )
Fourier Series coefficients equal
times samples of Fourier
1
To
transform of one period
7(t)
r __
.
.
27>
T /2
x(t) = one period of x(t)
x(t) +
k
x(t) -)
X((I)
Continuous-Time Fourier Transform
TRANSPARENCY
8.11
The Fourier transform
... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
PERIODIC SIGNAL R(t)
TRANSPARENCY
8.14
Representation of the
Fourier series in terms
of the Fourier
transform.
(t) +
N(t)
ak
X
Fourier series
coefficients
Fourier transform
+00
i(t)
-
21r _ .,
X(o) eict dw
+00
2r k=- 00
27r ak
+00
f
(w-kco0 ) e- jet dw
Continuous-Time Fourier Transform
8-11
TRANSPARENCY
8.15
Illu... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
series for
a periodic signal.
TRANSPARENCY
8.18
Summary example
illustrating some of
the relationships
between the Fourier
series and Fourier
transform.
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Signals and Systems
Professor Alan V. Oppenheim
The following may not correspond to a particular course on MIT Ope... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
Observers, state feedback
6.011, Spring 2018
Lec 10
1
Observers
2 System (“plant”)
x[n]
w[n]
q[n]
A, b, cT, d
y[n]
+
1[n]
3
A good model
x[n]
w[n]
[n
q[n]
A, b, cT, d
b
y[n]
+
y[n]
b
1[n]
4
Observer configuration
x[n]
w[n]
q[n]
A, b, cT
Plant
y[n]
+
Z[n]
y[n]
q[n]
[n... | https://ocw.mit.edu/courses/6-011-signals-systems-and-inference-spring-2018/374940065e1339ea37dd2aa3cf8c54bd_MIT6_011S18lec10.pdf |
1 2 3 4 5 6 7 8 MIT OpenCourseWare
http://ocw.mit.edu
6.890 Algorithmic Lower Bounds: Fun with Hardness Proofs
Fall 2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/378d4376b42f8454d18bbaa761b01fe2_MIT6_890F14_L03.pdf |
18.336 spring 2009
lecture 13
03/19/09
Initial Value Problems (IVP)
⎧
⎨
⎩
in Ω×]0, T [
on Ω × {0}
on ∂Ω×]0, T [
ut = Lu
u = u0
u = g
where L differential operator.
←
←
←
PDE
initial condition
boundary condition
⎫
⎬
⎭
Ex.: • L = �
2
→
Poisson equation
•Lu = b · �u
advection equation
•Lu = −�2(�2
biharm... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/379860bd755f63b2873eda4e2fc5a337_MIT18_336S09_lec13.pdf |
Δt) − u(x, t)
Δt
Approximate time derivative by step:
d
dt
→
unew(x) = u(x) + ΔtLu(x) = (I + ΔtL)u(x)
Need to know about ODE solvers.
Stationary problem:
[explicit Euler]
1
Numerical Methods for ODE
�
�
y(t) ∈ Rd
y˙(t) = f (y(t))
y(0) = ˚y
yn ≈ y(t), yn+1 ≈ y(t + Δt)
y
Linear approximation: y˙ ≈
Explicit ... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/379860bd755f63b2873eda4e2fc5a337_MIT18_336S09_lec13.pdf |
stable: Δt <
IE unconditionally stable
2
ρ(A)
Message: One step implicit is more costly than one step explicit.
, then implicit pays!
But: If ρ(A) large
�
�
��
stiffness
Ex.: Different time scales
−50
49
49 −50
A =
�
Solution: y(t) = e−t
�
, ˚y =
�
·
��
1
1
behavior
�
�
�
2
0
�
�
+ e−99t −1
·
1
��
... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/379860bd755f63b2873eda4e2fc5a337_MIT18_336S09_lec13.pdf |
6.241 Dynamic Systems and Control
Lecture 1: Introduction, linear algebra review
Readings: DDV, Chapter 1
Emilio Frazzoli
Aeronautics and Astronautics
Massachusetts Institute of Technology
February 2, 2011
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
1 / 22
Outline
1
Syllabus review
2
Linear Algeb... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
algebra, and of least squares problems.
Representation, structure, and behavior of multi-input, multi-output (MIMO)
linear time-invariant (LTI) systems.
Robust Stability and Performance. Approaches to optimal and robust control
design.
Hopefully, the material learned in this course will form a valuable foundation ... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
Vaccaro, Digital Control: A State-Space Approach, McGraw-Hill, 1995.
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
7 / 22
Tentative schedule
#
1
2
3
4
5
6
7
Date Topic
Feb 2, 2011
Introduction to dynamic systems and control.
Matrix algebra.
Feb 7, 2011 Least Squares error solut... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
6
18
19
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
8 / 22
Tentative schedule
Date Topic
#
14
15
16
17
18
Mar 28, 2011
Mar 30, 2011
Apr 4, 2011
Apr 6, 2011
Apr 11, 2011
19
Apr 13, 2011
20
Apr 20, 2011
21
22
23
24
25
26
Apr 25, 2011
Apr 27, 2011
May 2, 2011
May 4, 2011
May 9, ... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
(MIT)
Lecture 1: Introduction
Feb 2, 2011
10 / 22
Vector Spaces
A vector space is defined as a set V over a (scalar) field F , together with two
binary operations, i.e., vector addition (+) and scalar multiplication ( ), satisfying
the following axioms:
·
Commutativity of +: u + v = v + u, ∀u, v , ∈ V ;
Associat... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
y (t) of the LTI ODE dy (t)/dt + 3y (t) = 0;
The set of points (x1, x2, x3) ∈ R3 satisfying x1
2 + x2
2 + x 2 = 1.
3
The set of solutions y (t) of the LTI ODE dy (t)/dt + 3y (t) = 0.
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
12 / 22
Subspaces
A subspace of a vector space is a subset of vectors that i... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
vectors that are independent.
Any set of n independent vectors is also called a basis for the space.
if a space contains a set of n independent vectors for any n ∈ N, then the
space is infinite-dimensional.
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
15 / 22
Norms
Norms measure the ‘length” of a vecto... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
17 / 22
�
Inner product
An inner product on a vector space V (with scalar field F ) is a binary
operation �·, ·� : V × V
F , with the following properties:
→
1
2
3
Symmetry: �x, y � = �y , x��, ∀x, y ∈ V ;
Linearity: �x, ay + bz� = a�x, y � + b�x, z�;
Positivit... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
norm is that induced by the inner product)
Proof
0 ≤ �x + αy , x + αy � = x �x + α�y �x + αx �y + |α|2 y �y
Choose α = −x �y /�y , y � :
0 ≤ �x, x��y , y � − �x, y �2
.
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
19 / 22
The Projection Theorem
Let M be a subspace of an inner product space V . Given s... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
.
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
21 / 22
�
Linear Systems of equations
Consider the following system of (real or complex) linear equations:
Ax = y ,
A ∈ Rm×n , x ∈ Rn , y ∈ Rm .
Given A and y , is there a solution x?
∃ a solution x ⇔
y ∈ A ⇔ R([A|y ]) = R(A).
There are three cases:
... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
this case it is
desired to find the solution than minimizes some cost criterion.
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
22 / 22
�
MIT OpenCourseWare
http://ocw.mit.edu
6.241J / 16.338J Dynamic Systems and Control
Spring 2011
For information about citing these materials or our Terms of Use, visi... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
Lectures 11 and 12
Air Pollution and SI Engine Emissions
Atmospheric Pollution
• SMOG
O
||
O3
NO2
– Ozone Nitrogen dioxide
R-C-OONO2
PAN(Peroxyacyl Nitrate)
• TOXICS
– CO, Benzene, 1-3 butadiene, POM (Polycyclic organic Matters),
Aldehydes
Primary Pollutants:
Direct emissions from vehicles
CO, HC, NOx... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
1994 TLEV
Euro 4
Euro 5
1997 TLEV
1997-2003 ULEV
)
e
l
i
/
m
g
(
x
O
N
0.1
1975
1977
1
1981
1994 TLEV
Euro 3
1997-2003 ULEV
Euro 4
Euro 5
PZEV
PZEV
1975 1980 1985 1990 1995 2000 2005 2010
1975 1980 1985 1990 1995 2000 2005 2010
0.01
Starting year of implementation
Starting year of implementation
H... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
rication oil contribution
• Hydrocarbon emissions
– Fuel hydrocarbons escape oxidation (or only partially oxidized) via
various pathways
3
Typical steady state SI engine-out
emissions
• NOx is a few thousand parts per million
• CO is around 0.5-1% for stoichiometric
operation
• HC is 500-2000 ppm for fully... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
�� Engine-out [NO2]/[NOx] 2%
5
dxNO
dt
(s-1)
x 0
NO
SI Engine NO formation
Fig. 11-4
Adiabatic flame temperature,
Kerosene combustion with
700K, 15 bar air
Dash line is
adiabatic flame
temperature for
kerosene
combustion
with 700K 15 bar
air
d[NO]
dt
(Zeldovich)
2k1
O
1... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
Very temperature sensitive
2000
2200
2400
2600
T (oK)
© McGraw-Hill Education. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use.
)
a
p
M
(
P
)
K
(
u
T
,
b
T
)
m
p
p
(
O
N
Thermodynamic
state of charge
x
b
Fig. 9-5... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
© McGraw-Hill Education. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use.
7
NO control by EGR
• EGR is a dilution effect
– Reduce burned gas temperature via increase in
thermal inertia
1600 rpm; v=0.5;
MBT tim... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
800 using O3 as indicator. Maximum O3 formation occurs at about 1500-1700 hr.
Carter Index for
Ozone Forming
Potential
(CARB July, 1992)
Table from SAE
Paper 932718
(Tauchida et.al)
Methodology explained in SAE Paper 900710
(Lowi and Carter)
© Society of Automotive Engineers. All rights reserved. This content... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
l
i
O
Compression stroke
Desorption of
fuel vapor
Burned
gas
m
l
i
f
l
i
O
Expansion stroke
Ishizawa and Takagi (Nissan)
JSME Int. Jnl. 1987 Vol. 30 No. 260 pp. 310-317
© JSME. All rights reserved. This content is excluded from our Creative Commons
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(1.6%)
Fully Burned Exhaust
Tailpipe- out HC (0.1-0.4%)
Catalyst
HC Sources: Magnitudes and
Percent of Total Engine-out Emissions*
(SAE Paper 932708)
Source
Crevices
Quench
Oil Layers
Deposits
Liquid Fuel
Valve Leakage
Total
% Fuel Escaping
Normal Combustion
5.2
Fraction Emitted % Fuel as HC % of Tota... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
ustion Engines
Spring 2017
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
6.825 Techniques in Artificial Intelligence
What is Artificial Intelligence (AI)?
Lecture 1 • 1
If you're going to teach or take an AI course, it's useful to ask: "What's AI?"
It's a lot of different things to a lot of different people. Let's go through a few
things that AI is thought to be and situate them within... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
area called
cognitive neuroscience. The research strategy is to affiliate with someone
who does experiments that reveal something about what goes on inside
people's heads and then build computational models that mirror those kind
of processes.
A crucial question is to decide at what level to mirror what goes on in... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
systems drawn from this larger class. But then you get into terrible trouble
because you have to say what it means to behave intelligently. We might
feel that although we can't define what it is to be intelligent, we can
recognize it when we see it. We'll give up on trying to decide what
intelligence is and spend o... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
. Some of these applications you might not want to call "intelligent" or "rational"
but it is work that has traditionally been done in the field of AI. Usually, they are problems in
computer science that don't feel well specified enough for the rest of the computer science
community to want to work on. For instance,... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
could get that very badly wrong and so you want a system that's good and
reliable. There are long lists of examples; AI applications are very viable.
We're going to spend most of our time thinking, or at least feeling motivated, by
computational systems that behave rationally. But a lot of the techniques that we wil... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
an
account of how an agent works by separating it from the world it works in,
because the interface is so big and so complicated. And that may be right.
That I can't get exactly right a description of how the agent needs to operate
in the world by separating it from the world. But, it gives me a kind of
leverage i... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
environment
actions
Lecture 1 • 10
And, so, here's another view of the world. We're going to be thinking about
the agent as the software that runs some big hardware system. That is not
to make light of or say that it's easy to design the hardware part, and
depending on how the hardware part has been designed your... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
World Model
• A – the action space
• P – the percept space
Lecture 1 • 13
Next, we need a percept space: what are all the things that the agent can
perceive in the world? These spaces can be continuous; you can imagine
that the agent can perceive how high its arm is raised or the temperature in
some reaction ves... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
Model
• A – the action space
• P – the percept space
•
E – the environment:
A*
•
Alternatively, define
P
→
•
•
•
S – internal state [may not be visible to agent]
Perception function: S
A
World dynamics: S
P
S
→
→
×
s
p
Perception
Function
World
Dynamics
a
Lecture 1 • 15
Usually we'll think... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
We need a utility function. It’s typically thought of as a mapping from states
in the world to real values, or maybe sequences of states into real values. It
is meant to say, "Agent, these are the states of the world and this is how
valuable they are from your perspective." So that indirectly tells the agent
what y... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
its goals. This is all in high-level pseudo-psychological talk that
makes some people nervous. We can cache it out into something more
concrete in a minute but the idea is that you're rational if you take actions
that are consistent with what you are trying to achieve in the grand scheme
of things.
Let's say that ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
to have
watched the weather forecast the night before. But, given what I knew it was
ok to ride my bike, even though it turned out be dumb at some level,
because I didn't know what was happening.
19
Rationality
• A rationa
l agent takes actions
it believes will
achieve its goals
.
• Assume I don’t like to get ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
big problem with our definition of
rationality…
Lecture 1 • 21
This is still not a good enough notion to decide what should go in the head of
our agent or our robot. Do you see any potential problem with this as a
criterion for behavior in real systems?
21
Limited Rationality
•
•
There is a big problem with o... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
, we might be able to make a robot that's the best possible
chess player, but it might not be able to cross the street. If our robot needs
to be able to cross the street, its aggregate behavior is not rational. So,
when we think about rationality we may we want to think about it in a much
broader context: given all... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
that someone was given the job of writing a specification of the
environment that we want our agent to work in. You could say: "Oh, but you
can't do that. This whole approach seems pretty silly because how is it that
anyone could specify the domain that the agent is going to work in?" It does
seem hard to write dow... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
those constraints. The problem is that, although
information theoretically this is a specification for the correct program, it is
not an effective specification. It's not a specification that the computer can
use. There is a huge gap between the specification for what you want the
agent to do and what you can write... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
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