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Objective

Minimize instantaneous squared error

e k w 2 y k x k w 2

Lms algorithm

w k w k 1 μ x k e k
Where w k is the new weight vector, w k 1 is the old weight vector, and μ x k e k is a small step in the instantaneous error gradient direction.

Interpretation in terms of weight error vector

Define

v k w k w opt
Where w opt is the optimal weight vector and
ε k y k x k w opt
where ε k is the minimum error. The stochastic difference equation is:
v k I μ x k x k v k 1 μ x k ε k

Convergence/stability analysis

Show that (tightness)

B v k B 0
With probability 1, the weight error vector is bounded for all k .

Chebyshev's inequality is

v k B v k 2 B 2
and
v k B 1 B 2 v k 2 v k
where v k 2 is the squared bias. If v k 2 v k is finite for all k , then B v k B 0 for all k .

Also,

v k tr v k v k
Therefore v k is finite if the diagonal elements of Γ k v k v k are bounded.

Convergence in mean

v k 0 as k . Take expectation of using smoothing property to simplify the calculation. We haveconvergence in mean if

  • R xx is positive definite (invertible).
  • μ 2 λ max R xx .

Bounded variance

Show that Γ k v k v k , the weight vector error covariance is bounded for all k .

We could have v k 0 , but v k ; in which case the algorithm would not be stable.
Recall that it is fairly straightforward to show that the diagonal elements of the transformed covariance C k U Γ k U tend to zero if μ 1 λ max R xx ( U is the eigenvector matrix of R xx ; R xx U D U ). The diagonal elements of C k were denoted by γ k , i i i 1 p .
v k tr Γ k tr U C k U tr C k i 1 p γ k , i
Thus, to guarantee boundedness of v k we need to show that the "steady-state" values γ k , i γ i .

We showed that

γ i μ α σ ε 2 2 1 μ λ i
where σ ε 2 ε k 2 , λ i is the i th eigenvalue of R xx ( R xx U λ 1 0 0 λ p U ), and α c σ ε 2 1 c .
0 c 1 2 i 1 p μ λ i 1 μ λ i 1
We found a sufficient condition for μ that guaranteed that the steady-state γ i 's (and hence v k ) are bounded: μ 2 3 i 1 p λ i Where i 1 p λ i tr R xx is the input vector energy.

With this choice of μ we have:

  • convergence in mean
  • bounded steady-state variance
This implies
B v k B 0
In other words, the LMS algorithm is stable about the optimumweight vector w opt .

Learning curve

Recall that

e k y k x k w k 1
and . These imply
e k ε k x k v k 1
where v k w k w opt . So the MSE
e k 2 σ ε 2 v k 1 x k x k v k 1 σ ε 2 x n ε n n n k v k 1 x k x k v k 1 σ ε 2 v k 1 R xx v k 1 σ ε 2 tr R xx v k 1 v k 1 σ ε 2 tr R xx Γ k 1
Where tr R xx Γ k 1 α k 1 α c σ ε 2 1 c . So the limiting MSE is
ε k e k 2 σ ε 2 c σ ε 2 1 c σ ε 2 1 c
Since 0 c 1 was required for convergence, ε σ ε 2 so that we see noisy adaptation leads to an MSE larger than the optimal
ε k 2 y k x k w opt 2 σ ε 2
To quantify the increase in the MSE, define the so-called misadjustment :
M ε σ ε 2 σ ε 2 ε σ ε 2 1 α σ ε 2 c 1 c
We would of course like to keep M as small as possible.

Learning speed and misadjustment trade-off

Fast adaptation and quick convergence require that we take steps as large as possible. In other words,learning speed is proportional to μ ; larger μ means faster convergence. How does μ affect the misadjustment?

To guarantee convergence/stability we require μ 2 3 i 1 p λ i R xx Let's assume that in fact μ 1 i 1 p λ i so that there is no problem with convergence. This condition implies μ 1 λ i or μ λ i 1 i i 1 p . From here we see that

c 1 2 i 1 p μ λ i 1 μ λ i 1 2 μ i 1 p λ i 1
This misadjustment
M c 1 c c 1 2 μ i 1 p λ i
This shows that larger step size μ leads to larger misadjustment.

Since we still have convergence in mean, this essentially means that with a larger step size we "converge"faster but have a larger variance (rattling) about w opt .

Summary

small μ implies

  • small misadjustment in steady-state
  • slow adaptation/tracking
large μ implies
  • large misadjustment in steady-state
  • fast adaptation/tracking

w opt 1 1 x k 0 1 0 0 1 y k x k w opt ε k ε k 0 0.01

Lms algorithm

initialization w 0 0 0 and w k w k 1 μ x k e k k k 1 , where e k y k x k w k 1

Learning curve

μ 0.05

Lms learning curve

μ 0.3

Comparison of learning curves

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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