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6.776
High Speed Communication Circuits and Systems
Lecture 9
Enhancement Techniques for Broadband Amplifiers,
Narrowband Amplifiers
Massachusetts Institute of Technology
March 3, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H.
Perrott
Shunt-Series Peaking
(cid:131) Series inductors isolate load capacitance fro... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
"Circuit Techniques for a 40 Gb/s Transmitter in 0.13um CMOS",
J. Kim, et. al. ISSCC 2005, Paper 8.1
H.-S. Lee & M.H. Perrott
MIT OCW
Bandwidth Enhancement With ft Doublers
I1
I2
M1
M2
vin
Vbias
2Ibias
(cid:131) A MOS transistor has ft calculated as
(cid:131) ft doubler amplifiers attempt to
increase the ratio of
... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
2 will ideally match that of M1
H.-S. Lee & M.H. Perrott
MIT OCW
Problems of ft Doubler in Modern CMOS RF Circuits
(cid:131) Problems:
- Works if Cgs dominates capacitance , but in modern
CMOS, this is not the case (for example, Cgd=0.45Cgs in
0.18 µ CMOS)
- achievable bias voltage across M1 (and M2) is severely
r... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
.H. Perrott
-80
1/100
1/10
1
Normalized Frequency (Hz)
Choosing the Optimal Number of Stages
(cid:131) To first order, there is a constant gain-bandwidth
product for each stage
lower its gain
- Increasing the bandwidth of each stage requires that we
- Can make up for lost gain by cascading more stages
(cid:131) ... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
transmission lines have (ideally) infinite
bandwidth, but can be modeled as LC networks
- Can we lump device capacitances into transmission line?
H.-S. Lee & M.H. Perrott
MIT OCW
Distributing the Input Capacitance
Rs=Z0
delay
Zo
vin
Cout
Cout
Cout
M1
Zo
M2
Zo
M3
Zo
RL=Z0
vout
RL=Z0
(cid:131) Lump input capacitance in... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
(cid:131) Can we take advantage of this fact when designing
the amplifier?
H.-S. Lee & M.H. Perrott
MIT OCW
Tuned Amplifiers
Vdd
LT
RL
vout
CL
M1
Rs
vin
Vbias
(cid:131) Put inductor in parallel across RL to create bandpass
filter
- It will turn out that the gain-bandwidth product is
roughly conserved regardless of t... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
looks like an open circuit at resonance, so C doesn’t
load the amplifier)
(cid:131) This is often called low-pass to band-pass transform
in filter design:
- Replace C with parallel LC tank
- Replace L with series LC tank
H.-S. Lee & M.H. Perrott
MIT OCW
Gain-Bandwidth Product for Tuned Amplifiers
vout
vin
gmRp
Cp
Lp... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
LT
RL
Zin
Cgd
M1
Cgs
vout
CL
Rs
vin
Vbias
(cid:131) At frequencies below resonance, tank looks inductive
H.-S. Lee & M.H. Perrott
Negative
Resistance!
MIT OCW
Use Cascode Device to Remove Impact of Cgd
Vdd
LT
RL
Vbias2
Zin
Cgd
Cgs
M2
M1
vout
CL
Rs
vin
Vbias
(cid:131) At frequencies above and below resonance
H.-S. Lee ... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
1: Tuning LC
C2 : positive feedback
LE: RF choke (large inductance)
•When the RF amplitude becomes large, it is rectified at the emitter of Q1
•This raises the DC potential at the emitter Q1 eventually turning it off
•The RF oscillation dies (quenched), and the DC potential at emitter of Q1
returns
•Amplitude of oscil... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
ed load behave much like
a parallel LC circuit, while with open load it behaves like a series LC
- The problem is the dimension. For 900MHz mobile phone frequency, λ/4 in
- With high permittivity dielectric material (ceramic), the size can be reduced
to a reasonable dimensions. With εr=10, the length of waveguide is... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
[
s
s
o
L
n
o
i
t
r
e
s
n
I
0
5
10
15
20
25
30
35
40
45
50
2200
2450
Frequency [MHz]
0
1
2
3
4
5
6
7
8
9
10
2700
Figure by MIT OCW.
JRC NSVS754 2.4 GHz RF SAW Filter Charactersitic
H.-S. Lee & M.H. Perrott
Adapted from Japan Radio Co.
MIT OCW | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
6.776
High Speed Communication Circuits
Lecture 7
High Freqeuncy, Broadband Amplifiers
Massachusetts Institute of Technology
February 24, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H.
Perrott
High Frequency, Broadband Amplifiers
(cid:131) The first thing that you typically do to the input signal
is amplify i... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
consumption must be also considered
H.-S. Lee & M.H. Perrott
MIT OCW
Gain-bandwidth Observations
(cid:131) Constant gain-bandwidth is simply the result of single-
pole role off – it’s not fundamental!
(cid:131) It implies a single-pole frequency response may not be
the best for obtaining gain and bandwidth
simultane... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
Zout
Vout
ZL
(cid:131) Output impedance:
This makes sense because the input of the amplifier is
‘virtual ground’ if gain is large
H.-S. Lee & M.H. Perrott
MIT OCW
Amplifier Example – CMOS Inverter
(cid:131) The Miller effect gives a quick way to estimate the
bandwidth of an amplifer without solving node equations:
... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
tot
1
2πCtotRf
f
Rf
vin
Vbias
H.-S. Lee & M.H. Perrott
MIT OCW
Ctot = Cdb1+Cdb2 + Cgs3+Cgs4 + K(Cov3+Cov4) + CRf /2 + Cfixed
(+Cov1+Cov2)
Miller multiplication factor
We Can Still Do Better
(cid:131) We are fundamentally looking for high gm to capacitance ratio
to get the highest bandwidth
- PMOS degrades this ratio
... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
in
Rout
Rs
Rf
RL
vout
M1
R1
vin
Vbias
(cid:131) Use resistors to control the bias, gain, and
input/output impedances
- Improves accuracy over process and temp variations
(cid:131) Issues
- Degeneration of M1 lowers slew rate for large signal
- There are better high speed approaches – the advantage
applications (such... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
of NMOS Load Amplifier
(cid:131) Gain is well controlled despite process variations
(cid:131) NMOS is not a low parasitic load
- Cgs of M2 loads the output
(cid:131) Biasing Problem: VT of M2 lowers the gate bias voltage of
the next stage (thus lowering its achievable ft)
- Severely hampers performance when amplifier ... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
fixed
Ibias
M6
M3 M4
M7
αIbias
M5
(cid:131) Benefits
- Bias stability without feedback
- Common-mode rejection
(cid:131) Negative
- More power than single-ended version
H.-S. Lee & M.H. Perrott
MIT OCW
Open-Circuit Time Constants
(cid:131) The Miller capacitance analysis is a reasonably good
method, but is somewhat l... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
τ
n
=
1
n
∑
τ
i
1
i
=
H.-S. Lee & M.H. Perrott
MIT OCW
OCT Method, Continued
It can be shown
n
n
τ
τ
∑=∑
i
i
1
1
=
=
jo
j
Thus
ω
h
≈
1
n
∑
τ
j
1
=
jo
where
CR=τ
jo
jo
:open-circuit time constants
j
The open-circuit time constants can be found without node
equations, often by inspection
H.-S. Lee & M.H. Perrott
MIT OC... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
a
v
1−=
Rg
Lm
Cascode to Improve Bandwidth
VCC
RL
vo
R
S
,
eff
=
r
o
1
R
effL
,
=
1
g
m
,2
eff
=
g
m
2
1
+
g
mb
2
M2
M1
RS
vi
R =1
o R
S
R
2
o
=
R
o
1
=
R
S
=
R
S
1
+
⎛
⎜⎜
⎝
(
1
++
1
g
+
+
m
2
Rg
1
effLm
,
+
R
effL
,
R
S
⎞
⎟⎟
⎠
effL
,
)
RRg
Sm
1
Rg
Sm
1
g
+
mb
2
R
3
o
≈
R
S
,
eff
1
g
m
,2
eff
≈
g
m
R
4
o
=
r
b
⎛
⎜
⎜
⎜
... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
18.336 spring 2009
lecture 18
04/14/09
Intermezzo:
Boundary Conditions for Advection
Linear advection
�
ut + ux = 0
x ∈ [0, 1]
Image by MIT OpenCourseWare.
Upwind (first order) treats boundary conditions naturally correctly.
LF, LW: Need artificial/numerical boundary conditions at x = 1,
that acts as close to “do ... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/3ed0b03979b2d5868bbf5f44fdc5f3a2_MIT18_336S09_lec18.pdf |
−1)
+1 + Uj
Δt
+
n
n
+1) − f (Uj
−1)
f (Uj
2Δx
= 0
Image by MIT OpenCourseWare.
(no straightforward LW, since based on linear Taylor expansion)
Numerical Flux Function
Uj
n+1 − Uj
n
Δt
Fj
+
n − Fj−1
n
Δx
= 0
Upwind: Fj
n =
�
f (Uj
n)
f (Uj+1
n)
if
n
f (U +1
j
n
U +1−U
j
)−f (U n
j )
n
j
�
≥ 0
< 0
... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/3ed0b03979b2d5868bbf5f44fdc5f3a2_MIT18_336S09_lec18.pdf |
Op#miza#on Problems,
John Gu7ag
MIT Department of Electrical Engineering and
Computer Science
6.0002 LECTURE 2
1
Relevant Reading for Today’s Lecture
§ Chapter 13
6.0002 LECTURE 2
2
The Pros and Cons of Greedy
§ Easy to implement
§ Computa<onally efficient
§ But does not always yield the best solu<on
◦... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
’tTake
Val = 170
Cal = 766
Val = 120
Cal = 766
Val = 140
Cal = 508
Val = 90
Cal = 145
Val = 80
Cal = 612
6.0002 LECTURE 2
Val = 30
Cal = 258
Val = 50
Cal = 354
Val = 0
Cal = 0
6
Image © source unknown. All rights reserved. This content is excluded from our Creative
Commons license. For more informat... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
have not yet considered
avail. The amount of space still available
6.0002 LECTURE 2
9
Body of maxVal (without comments)
if toConsider == [] or avail == 0:
result = (0, ())
elif toConsider[0].getUnits() > avail:
result = maxVal(toConsider[1:], avail)
else:
nextItem = toConsider[0]
withVal, withToTake... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
),
random.randint(1, maxVal),
random.randint(1, maxCost)))
return items
for numItems in (5,10,15,20,25,30,35,40,45,50,55,60):
items = buildLargeMenu(numItems, 90, 250)
testMaxVal(items, 750, False)
6.0002 LECTURE 2
13
Is It Hopeless?
§ In theory, yes
§ In prac<ce, no!
§ Dynamic programming to the rescue
... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
��b(6)
fib(5)
fib(4)
fib(4)
fib(3)
fib(3)
fib(2)
fib(3)
fib(2)
fib(2)
fib(1)
fib(2)
fib(1)
fib(1)
fib(0)
fib(2)
fib(1)
fib(1)
fib(0)
fib(1)
fib(0)
fib(1)
fib(0)
fib(1)
fib(0)
6.0002 LECTURE 2
17
Clearly a Bad Idea to Repeat Work
§ Trade a <me for space
§ Create a table to record what we’ve done
◦ Before compu<ng... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
1, fib(x) = fib(x - 1) + fib(x – 2)
§ Overlapping subproblems: finding an op<mal solu<on
involves solving the same problem mul<ple <mes
◦ Compute fib(x) or many <mes
6.0002 LECTURE 2
20
What About 0/1 Knapsack Problem?
§ Do these condi<ons hold?
Ques<ons 2 and 3
6.0002 LECTURE 2
21
Search Tree
Op<mal substruct... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
problems
6.0002 LECTURE 2
27
Modify maxVal to Use a Memo
§ Add memo as a third argument
◦ def fastMaxVal(toConsider, avail, memo = {}):
§ Key of memo is a tuple
◦ (items leS to be considered, available weight)
◦ Items leS to be considered represented by
len(toConsider)
§ First thing body of func<on does is che... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
6.0002 LECTURE 2
30
Summary of Lectures 1-2
§ Many problems of prac<cal importance can be
formulated as op<miza<on problems
§ Greedy algorithms oSen provide adequate (though not
necessarily op<mal) solu<ons
§ Finding an op<mal solu<on is usually exponen<ally
hard
§ But dynamic programming oSen yields good
pe... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
Tracking Indoors
1
Pervasive Computing MIT 6.883 SMA 5508 Spring 2006 Larry
Rudolph
Location of what?
• Objects
• Static, Moveable, or Mobile
• Frequency of movement: door, desk,
laptop
• Dumb or Networked
• People
• Waldo asks “Where am i?”
• System asks “where’s Waldo?”
• Services
• applications, resourc... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/3ef14f365936b016b8aedc689b1e8843_l7_batscrickets.pdf |
330 m/s + .6*temp; >2 receivers ==>
location
More on BATs
• Deployment
• 50 staff members, 200 BATS, 750
Receivers, 3 Radio cells, 10,000 sq ft
office space
• 20 ms per bat enables 50 BATs / sec
• Smart scheduling reduces BAT’s power
• while at rest, reduce frequency of query
• detect activity at PC to deduce “re... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/3ef14f365936b016b8aedc689b1e8843_l7_batscrickets.pdf |
e
c
a
f
r
e
t
n
I
C
e
l
c
a
r
O
ORACLE 7
Relational Database
THREE-TIER ARCHITECTURE
Figure by MIT OCW.
Better Trackers
Bayesian filtering on sensory data
Predict where person will be in future.
position and speed over near past
behavior (avg speed) over long
term
Uses
Filter bad sensory data
Likely plac... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/3ef14f365936b016b8aedc689b1e8843_l7_batscrickets.pdf |
6.034 Artificial Intelligence. Copyright © 2004 by Massachusetts Institute of Technology.
6.034 Notes: Section 10.1
Slide 10.1.1
So far, we've only talked about binary features. But real problems are typically characterized by
much more complex features.
Slide 10.1.2
Some features can take on values in a discrete... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
Slide 10.1.6
We'll use the example of predicting whether someone is going to go bankrupt. It only has two
features, to make it easy to visualize.
One feature, L, is the number of late payments they have made on their credit card this year. This is
a discrete value that we're treating as a real.
The other feature, ... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
11
If we just use Euclidean distance in this space, the number of cylinders will have essentially no
influence on nearness. A difference of 4 pounds in a car's weight will swamp a difference between 4
and 8 cylinders.
6.034 Artificial Intelligence. Copyright © 2004 by Massachusetts Institute of Technology.
Slide ... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
say we've thought
about the domain and decided that the R feature (ratio between expenses and income) needs to be
scaled up by 5 in order to be appropriately balanced against the L feature (number of late payments).
So we'll use Euclidian distance, but with the R values multiplied by 5 first. We've scaled the axes o... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
being represented by the edges in the Voronoi partition that
separate a region associated with a positive point from a region associated with a negative one. In
our example, that generates this bold boundary.
It's important to note that we never explicitly compute this boundary; it just arises out of the "nearest
n... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
this one go bankrupt in general. The other is
to say that this example is an "outlier". It represents an unusual case that we would prefer largely to
ignore, and not to incorporate it into our hypothesis.
Slide 10.1.31
So, what happens in nearest neighbor if we get a query point next to this point?
6.034 Artifici... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
often the method of choice) for problems in relatively low-
dimensional real-valued spaces.
But as the dimensionality of a space increases, its geometry gets weird. Here are some suprising (to
me, at least) facts about high-dimensional spaces.
Slide 10.1.37
In high dimensions, almost all points are far away from on... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
Slide 10.1.43
Here's a graph of the cross-validation accuracy of nearest neighbor on the heart disease data, shown
as a function of k. Looking at the data, we can see that the performance is relatively insensitive to
the choice of k, though it seems like maybe it's useful to have k be greater than about 5.
6.034 A... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
This class of splits allows us to divide our feature-space into a set of exhaustive and mutually
exclusive hyper-rectangles (that is, rectangles of potentially high dimension), with one rectangle for
each leaf of the tree. So, each rectangle will have an output value (1 or 0) associated with it. The set
of rectangle... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
tree, we'll choose the split
that minimizes the average entropy of the resulting child nodes.
Slide 10.2.11
Let's see what actually happens with this algorithm in our bankruptcy domain.
We consider all the possible splits in each dimension, and compute their average entropies.
Slide 10.2.12
Splitting in the L dim... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
The best performance of this algorithm is about .77, which is slightly worse than the performance of
nearest neighbor.
Slide 10.2.20
But performance isn't everything. One of the nice things about the decision tree algorithm is that we
can interpret the hypothesis we get out. Here is an example decision tree resulti... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
. Whew!
Slide 10.2.27
If they're older than 57.5, then we examine some technical feature of the cardiogram, and let that
determine the output.
Hypotheses like this are very important in real domains. A hospital would be much more likely to
base or change their policy for admitting emergency-room patients who seem ... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
data.
6.034 Artificial Intelligence. Copyright © 2004 by Massachusetts Institute of Technology.
Slide 10.3.3
When you get a new query point x, you find the k nearest points.
Slide 10.3.4
Then average their y values and return that as your answer.
Of course, I'm showing this picture with a one-dimensional x, but ... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
Then, to compute the predicted y value, we just add up all of the y values of the points used in the
prediction, multiplied by their weights, and divide by the sum of the weights.
Slide 10.3.10
Here is one popular kernel, which is called the Epanechnikov kernel (I like to say that word!). You
don't have to care too... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
value for the leaf. In the numeric case,
we'll use the average output value. It makes sense, and besides there's a hairy statistical argument in
favor of it, as well.
6.034 Artificial Intelligence. Copyright © 2004 by Massachusetts Institute of Technology.
Slide 10.3.15
So, if we're going to use the average value... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
split.
Slide 10.3.22
Doing so, we can see that the average variance of splitting on feature 3 is much lower than of
splitting on f7, and so we'd choose to split on f3.
Just looking at the data in the leaves, f3 seems to have done a much better job of dividing the values
into similar groups.
6.034 Artificial Inte... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
L8a Markov Decision Processes:
Reactive Planning to Maximize Reward
Brian C. Williams
16.410 / 13
October 5th, 2015
Slides adapted from:
Manuela Veloso,
Reid Simmons, &
Tom Mitchell, CMU
Assignments
• Reading:
• Today: Markov Decision Processes: AIMA 17.1-3.
• Wednesday: Hidden Markov Models: AIMA 15.... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
0 for all other states
• Trained by playing 1.5 million games against self.
è Became roughly equal to best human player.
16.410/13 F15: Markov Decision Processes
5
MDP Examples: Aerial Robotics [Frazzoli & Feron]
Computing a Solution ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
,a)
1. Observe state st in S.
2. Choose action at in A.
3. Receive immediate reward rt.
4. State changes to st+1.
Example:
s1 a1
a0
s0
a1
s1
a2
s2
s3
r2
r1
r0
10
G
10
10
• Legal transitions shown.
• Reward on unlabeled
transition is 0.
16.410/13 F15: Markov Decision Processes
10
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
environment model as a MDP, create a policy for acting
that maximizes lifetime reward.
16.410/13 F15: Markov Decision Processes
13
MDP Problem: Lifetime Reward
Agent
State
Reward
Action
Environment
a0
s0
a1
s1
a2
s2
s3
r2
r1
r0
Given an environment model as a... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
Environment
a0
s0
a1
s1
a2
s2
s3
r2
r1
r0
Given an environment model as a MDP, create a policy for acting
that maximizes lifetime reward.
V = r0 + γ r1 + γ 2 r2 . . .
16.410/13 F15: Markov Decision Processes
17
Assume deterministic world
Policy π : S à... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
Value Function Vπ for a Given Policy π
• Vπ(st) is the accumulated lifetime reward resulting from
starting in state st and repeatedly executing policy π:
Vπ(st) = rt + γ rt+1 + γ 2 rt+2 . . .
Vπ(st) = ∑i γ i rt+I
where rt, rt+1 , rt+2 . . . are generated by following π,
starting at st .
π
Vπ
Assume γ = .9
9
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
γV(δ(s, a)]
γ = 0.9
Model + V:
100
90
100
G
0
100
81
90
100
16.410/13 F15: Markov Decision Processes
24
Example: Mapping Value Function to Policy
• Agent selects optimal action from V:
π(s) = argmaxa [r(s,a) + γV(δ(s, a)]
γ = 0.9
• a: 0 + 0.9 x 100 = 90
• b: 0... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
• b: ?
• c: ?
Ø select ?
16.410/13 F15: Markov Decision Processes
27
Markov Decision Processes
• Motivation
• Markov Decision Processes
• Computing Policies From a Model
• Value Functions
• Mapping Value Functions to Policies
• Computing Value Functions... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
Value Function V* for an optimal policy π*
Example
RA
A
B
SA
RB
SB
RA
A
RB
B
• Optimal value function for a one step horizon:
V*1(s) = maxai [r(s,ai)]
• Optimal value function for a two step horizon:
V*2(s) = maxai [r(s,ai) + γV 1
*(δ(s, ai))]
• Optimal value function for an n step horizon:
V*n(s) = ma... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
using Bellman’s Equation:
V*t+1(s) ← maxa [r(s,a) + γV*
t(δ(s, a))]
• Terminate when values are “close enough”
|V*t+1(s) - V*
t (s) | < ε
• Agent selects optimal action by one step lookahead on V* :
π*(s) = argmaxa [r(s,a) + γV*(δ(s, a)]
16.410/13 F15: Markov Decision Processes
33
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
6.092: Thursday
Lecture
Lucy Mendel
MIT EECS
MIT 6.092
IAP 2006
1
Topics
z Interfaces, abstract classes
z Exceptions
z Inner classes
MIT 6.092
IAP 2006
2
Abstract Classes
z Use when subclasses have some code in common
abstract class Person {
private String name = “”;
public String getName() { return name; ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
public int width;
}
class Rectangle {
Should:
Square extend Rectangle?
Rectangle extend Square?
public int width,height;
}
…
int calculateArea (Square x) {
return (x.width)*(x.width); }
int calculateCircumference (Rectangle x) {
return 2*(x.width+x.height); }
MIT 6.092
IAP 2006
8
Rectangle extends Square
... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
it
class ListSet { // might want to implement Set
private List myList = new ArrayList();
void add(Object o) {
if (!myList.contains(o)) myList.add(o);
}
…
MIT 6.092
IAP 2006
13
Exceptions
z Goal: help programmers report and handle errors
z What happens when an exception is thrown?
z Normal program control flow... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
Class { … }
}
MIT 6.092
IAP 2006
20
Inner Classes
(non-static nested classes)
z Associated with an instance of the enclosing
class
z Cannot declare static members
z Can only be instantiated within context of
enclosing class
MIT 6.092
IAP 2006
21
Local Anonymous Inner Class
public class Stack {
private Arr... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
class Paperboy {
public Money payment;
public void getPaid( Man m ) {
payment += m.myWallet.money;
}
}
MIT 6.092
IAP 2006
27
Keep field references unique
z Copy parameters before assigning them to fields
z Copy fields before returning them
public final class Period {
private final Date start;
private final ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
Lecture 6: Quantitative Aspects Of
Networks III: Outline
• Network Analysis Terminology notated
• Connectivity
• Some Social Network Concepts-intuition and calculation
transitivity (clustering)
•
• centrality
• degree, closeness, betweenness, information, eigenvector
• prestige and acquaintance
• degree distributions
... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
the “sparseness” or normalized
interconnection “density”
• Path length, l
1
(
nn
−
)1
l
=
1
2
∑
i
≥
j
ijd
Professor C. Magee, 2006
Page 3
Connectivity
• Fraction of nodes connected in a network
• Of interest in resilience/robustness which we will cover in a
later lecture and then an inverse path length definition is... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
Analysis (1994) The following slides cover a few
selected examples in one area from that book. The purpose is
to give some feel for the application of such metrics which
attempt to measure structural properties of direct interest for
analysis
• We should also note that transitivity (clustering) and almost all
other... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
• Higher order clusters (groups of n related nodes) also of
•
interest but no clean way (yet) to separate lower order and
higher order tendencies
In directed graphs, n=2 effects (the proportion of nodes that
point at each other) can be of interest and is labeled
reciprocity.
Professor C. Magee, 2006
Page 9
Central... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
n
−
2/()1
n
−
)3
Professor C. Magee, 2006
Page 14
Florentine Families Centrality Metrics II
Closeness
)
B nC′
( i
)
I nC′
( i
)
Acciaiuoli
Ablizzi
Barbadori
Bischeri
Castellani
Genori
Guadagni
Lamberteschi
Medici
Pazzi
Peruzzi
Pucci
Ridolfi
Salvati
Strozzi
Tornabuoni
)
D nC′
( i
0.071
0.214
0.143
0.214
0.214
0.071
0.... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
Not covered in ESD Terms and Definitions but..
• Hierarchy: A description of a group of elements (system?)
where each element is graded or ranked and then arranged in a
structure that separates elements according to rank which each
descending rank being in some way subordinate to the next
higher rank (this leads to... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
rentine Families:
15th Century Marriage Relations
n5:Castellani
n11:Peruzzi
n12:Pucci
n15:Strozzi
n4:Bischeri
n3:Barbadori
n13:Ridolfi
n16:Tornabuoni
n9:Medici
n14:Salvati
n10:Pazzi
n7:Guadagni
n8:Lamberteschi
n2:Ablizzi
n1:Acciaiuoli
n6:Genori
Professor C. Magee, 2006
Page 19
Betweeness Centrality II
• Actor
• Power ... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
∑
nC
(
i
I
i
)
Professor C. Magee, 2006
Page 21
Florentine Families Centrality Metrics
Acciaiuoli
Ablizzi
Barbadori
Bischeri
Castellani
Genori
Guadagni
Lamberteschi
Medici
Pazzi
Peruzzi
Pucci
Ridolfi
Salvati
Strozzi
Tornabuoni
)
D nC′
( i
0.071
0.214
0.143
0.214
0.214
0.071
0.286
0.071
0.429
0.071
0.214
---
0.214
0.14... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
. The centrality of each vertex is therefore
determined by the centrality of the vertices it is connected to. The parameter á is
required to give the equations a non-trivial solution and is therefore the
reciprocal of an eigenvalue. It follows that the centralities will be the elements
of the corresponding eigenvec... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
.214
0.143
0.214
0.214
0.071
0.286
0.071
0.429
0.071
0.214
---
0.214
0.143
0.286
0.214
)
C nC′
( i
0.368
0.483
0.438
0.400
0.389
0.333
0.467
0.326
0.560
0.286
0.368
---
0.500
0.389
0.438
0.483
)
B nC′
( i
0.000
0.212
0.093
0.104
0.055
0.000
0.255
0.000
0.522
0.000
0.022
---
0.114
0.143
0.103
0.092
)
I nC′
( i
0.049
0.0... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
in search, navigation
and community structure models but otherwise the “Network
Science” Community does not utilize these measures. CM bias is
that they are probably useful in social and other networks.
• Hidden Hierarchy, robustness –communication and other
meanings are all dependent on effects such as those defin... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
Some Social Network Concepts-intuition and calculation
• clustering (transitivity)
• centrality
• degree, closeness, betweenness, information, eigen
• prestige and acquaintance
• degree distributions
• skew (and non-skew) distributions
• fitting power laws to observed data
• the normality of power laws
• truncation
• S... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
105
Population of City
10-2
10-4
10-6
10-8
104
105
106
107
Population of City
Figure by MIT OCW.
See
Newman, M. E. J. cond-mat/0412004v2
Professor C. Magee, 2006
Page 33
Degree Distributions II
kp
• Define as the fraction of nodes in a network with degree k. This is
equivalent to the probability of randomly picki... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
10-4
10-6
10-8
C
B
100
10-1
10-2
10-3
10-4
100
104
101
k
101
102
103
100
k
102
k
Figure by MIT OCW.
Barabasi and Albert(1999) A is actor collaboration, B is www
and C is the Western Power Grid (incorrectly identified as power law)
Professor C. Magee, 2006
Page 36
Courtesy of National Academy of Sciences, U. S. A. Use... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
of a large n.
From Systems engineering lecture by Dan Frey
Professor C. Magee, 2006
Page 39
Marginal and Markov process defined
• Marginal probability- In a multivariate distribution, the
probability of one variable, or function of several of these
variables, taking a specific value (or falling in a range)
• Me... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
2006
Page 42
Degree Distributions III
kp
• Define as the fraction of nodes in a network with degree k. This is
equivalent to the probability of randomly picking a node of degree k
• A plot of can be formed by making a histogram of the degrees of
the vertices. This is the degree distribution of the network. Some
dis... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
j
• Connectivity
• Clustering (2 definitions)
• Centrality (5 definitions + prestige and acquaintance)
• Degree Distribution
• Compare some systems (see handout for assignment # 3)
Professor C. Magee, 2006
Page 45
Networks structural characteristics:
Preliminary summary of results
• Most measures-even simple ones- sh... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
qu, S., Stable Non-Gaussian Random
Processes: Stochastic Processes with Infinite Variance,
Chapman and Hall, London, (1994)
• A Barabasi and R. Albert, “The Emergence of Scaling Laws in Random
Networks”, Science 286, pp 509-512 (1999)
• Amaral, L. A. N., Scala, A., Bertelemy, M. and Stanley, H. E. “Classes
of Smal... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
18.034 ;
Feb 6, 2004
Lecture 2
1. Set-up model for a mixing problem
Rate of mass of chemical in
=(conc. in)× (rate of flow of liquid)= a.c(t)
Rate of mass out = (conc. out) × (rate of flow) =
q ⋅
So
y
='
( )
tca
⋅
−
a
v
y
'
y
+
a
v
y
=
( )tca
.
where a,
0>V
are constants
( )
ty
v
.
2. Discussed method of ... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/3f60608b2696971c5f2fea7198a1b0e5_lec2.pdf |
ts. on (
defined on all of (
( )tq
)ba, ⊂ IR , then there exists a
)ba,
, the solution is unique,
=
( )
ty
=
e
( )
tp
−
t
∫
0
e
sp
( ) ( )
dssq
+
ey
0
−
tp
)(
,
where
3. Used this method to solve the mixing problem:
( )tp
( )
'
tp
=
( ) 0
0 =
p
( )
ty
=
e
α
v
t
ae
−
α
v
t
t
∫
0
( )
dssc
+
ey
0
−
α
v
t
(a) If
( )
... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/3f60608b2696971c5f2fea7198a1b0e5_lec2.pdf |
φ
=
ω
λ
. Didn’t have time to really
( )
ty
=
aA
2
2
ωλ
+
sin
(
)
φω
−
t
+
be
−
t
λ
for some b
4. Particular solution method. To find the general solution of
( )
(i) Find general solution of undriven/ homo system
ytp
(ii) Find a particular solution py of original equation.
(iii) General solution is
py
y +0
.
'... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/3f60608b2696971c5f2fea7198a1b0e5_lec2.pdf |
Key Concepts for section IV (Electrokinetics and Forces)
1: Debye layer, Zeta potential, Electrokinetics
2: Electrophoresis, Electroosmosis
3: Dielectrophoresis
4: Inter-Debye layer force, Van-Der Waals forces
5: Coupled systems, Scaling, Dimensionless Number
Goals of Part IV:
(1) Understand electrokinetic phenomena... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3f79682d8f684ab43e9dbd4208737945_electrokin_lec2.pdf |
m
μ
(
y
t
i
c
o
e
V
l
40
20
0
-20
-40
-60
Velocity of DNA in obstacle-free channel
10X TBE
0
0.25
0.5
0.75
1
0.5X TBE
T7
T2
Buffer concentration(M)
Electroosmosis
• The oxide or glass surface
become unprotonated (pK ~ 2)
when they are in contact with
water, forming electrical
double layer.
• When applied an elec... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3f79682d8f684ab43e9dbd4208737945_electrokin_lec2.pdf |
particle
motion (v)
generates
E-field (ΔΦ)
Electrophoresis
Sedimentation potential
Figure by MIT OCW.
Fick’s law of diffusion
Maxwell’s equation
Concentration(c)
(ρ)
Electrophoresis
ρ, J : source
E and B field
Convection
Osmosis
Electroosmosis
Streaming
potential
(aqueous) medium,
Flow velocity (vm)
Navier-Stokes’ ... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3f79682d8f684ab43e9dbd4208737945_electrokin_lec2.pdf |
iseuille flow
⎛
− ⎜
⎝
electroosmotic flow
R
−
r
μ
Δ
4
2
Δ ≠
P
0,
E
z
=
0 :
Poiseuille flow
parabolic flow profile
Δ =
P
0,
E
z
≠
0 :
Electroosmotic flow
flat (plug-like) profile
v
EEO
= −
εζ
μ
=
E
z
μ
EEO
E (outside of the Debye layer)
z
μ
EEO
:
electroosmotic 'mobility'
Paul’s experiment
Figure by MIT OCW. After Paul... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3f79682d8f684ab43e9dbd4208737945_electrokin_lec2.pdf |
18.465 March 29, 2005, revised May 2
M-estimators and their consistency
This handout is adapted from Section 3.3 of 18.466 lecture notes on mathematical
statistics, available on OCW.
A sequence of estimators Tn , one for each sample size n, possibly only defined for
n large enough, is called consistent if for X1, X... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
, A∞, P ∞) of copies of (X, A, P )
(RAP, Sec. 8.2). A statistic Tn = Tn(X1, ..., Xn) with values in Θ will be called an M-
estimator if
�n
1
n
h(Tn , Xi) = inf θ∈Θ
1
n
i=1
h(θ, Xi).
i=1
�n
Thus, in the log likelihood case, an M-estimator is a maximum likelihood estimator.
The outer probability P (C) of a not n... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
that there is a set A ⊂ X with
P (A) = 0 and a countable subset S ⊂ Θ such that for every open set U ⊂ Θ and every
closed set J ⊂ [−∞, ∞],
{x : h(θ, x) ∈ J for all θ ∈ S ∩ U } ⊂ A ∪ {x : h(θ, x) ∈ J for all θ ∈ U }.
1
This will be true with A empty if each function h(·, x) is continuous on Θ and S is dense
in Θ,... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
x in X, the function h(·, x) is lower semicontinuous on Θ, meaning that
h(θ, x) ≤ lim inf φ→θ h(φ, x) for all θ.
Often, but not always, the functions h(·, x) will be continuous on Θ. Consider for
:=
example the uniform distributions U [θ, θ + 1] on R for θ ∈ R. The density f (θ, x)
1[θ,θ+1](x) is not continuous in ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
(θ, ·) is well-defined (possibly +∞)
and not −∞ for all θ, and for some θ, also Eh(θ, ·) < +∞, so it is some finite real number.
If a(·) is a measurable real-valued function on X such that h(θ, x) − a(x) is adjusted
for P , then h(·, ·) will be called adjustable for P and a(·) will be called an adjustment
function fo... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
is defined by minimizing or approximately minimizing
�n h(θ, Xi). If ∫ h(θ, x)dP (x) is finite, it is the limit of the sample averages by the
1
n
strong law of large numbers. But if it isn’t finite, it may be made finite by subtracting
an adjustment function a(x) from h. Since a(·) doesn’t depend on θ, this change doe... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
h(θ, x) − a1(x)]
for P -almost all x. To check this we need to take account that h can have values ±∞. For
any θ, h(θ, x) > −∞ for P -almost all x since h is adjustable. We have h(θ1, x) < +∞ and
h(θ2, x) < +∞ for P -almost all x. Thus the given expression for (a1 −a2)(x) is well-defined
for P -almost all x and θ = ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
a probability measure Pθ. The distribution P of
the observations may not be in the parametric family of laws Pθ, and if not, no true value
of θ exists, but often a pseudo-true value exists.
By Proposition 3.3.4, θ0 does not depend on the choice of adjustment function. After
some more assumptions, it will be shown t... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
is any law on R with a
unique median. Consistency, to be proved below, will imply that sample medians converge
to the true median in this case.
Some consequences of the assumptions will be developed. The first one follows directly
from Proposition 3.3.4 and the definitions:
3.3.8 Lemma. For any adjustable h(·, ·) an... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
is monotone. Since b(·) is
continuous and positive, it is bounded on any neighborhood Uk with compact closure, say
0 < b(φ) ≤ M for all φ ∈ Uk . Then by (3.3.5), h(φ, x) − a(x) ≥ −M u(x) for all φ ∈ Uk
and all x. Thus the stated convergence holds by monotone convergence (RAP, 4.3.2) for
a fixed sequence of neighborh... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
.10 Lemma. If (A-1), (A-3), (A-4), and (A-5) hold, then there is a compact set C ⊂ Θ
such that for every sequence Tn of approximate M-estimators, almost surely there will be
some n0 such that Tn ∈ C for all n ≥ n0, in the sense that
(3.3.11)
1{Tn∈C} → 1 almost uniformly as n → ∞.
Proof. If Θ is compact there is no... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/3f81a744094897380591c3d2b21201e8_m_estimates.pdf |
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