# 3.2 Blind source separation via ica: math behind method

 Page 1 / 1
This module describes the math behind how our project on blind source separation using ICA works.

#### The math of ica

The Independent Components Analysis algorithm allows two source signals to be separated from two mixed signals using statistical principles of independence and nongaussianity.

#### Defining the problem

ICA assumes that the value of each source at any given time is a random variable. It also assumes that each source is statistically independent, meaning that the values of one source cannot be correlated to values in any of the other sources.

With these assumptions, ICA allows us to separate source signals from mixtures of these source signals. The algorithm requires that there be as many sensors as input signals. For example, with three independent sources and three mixtures being recorded, the problem could be modeled as:

$\begin{array}{l}{x}_{1}\left(t\right)=a{s}_{1}\left(t\right)+b{s}_{2}\left(t\right)\\ {x}_{2}\left(t\right)=c{s}_{1}\left(t\right)+d{s}_{2}\left(t\right)\end{array}$

Using matrix notation, the problem can be generalized to any number of mixtures. For some number of sources n to be identified, n mixtures would need to be recorded.

$x=As$

The goal of blind source separation using ICA is to invert this procedure; that is, given the mixtures x as inputs, ICA finds s . Because the mixing matrix A is square, we can write the reverse procedure as

$x={A}^{-1}s$

or, if we define W to be equal to the inverse of A ,

$x=Ws$

is an equivalent expression of the problem at hand.

#### Isolating the independent sources

The Central Limit Theorem provides the key to unlocking the mystery matrix W . The central limit theorem says that a sum of independent random variables can be approximated by a normal curve. The greater the number of variables summed, the more normal, or gaussian, the distribution. Since each of the mixtures being received by the sensors represents a linear combination of samples from each source in s , the distribution of the mixed signals is more gaussian than either of the two independent sources (by the central limit theorem).

In order for the ICA algorithm to use this principle, the algorithm needs a way of determining how gaussian a particular signal is. There are two main quantitative measures of nongaussianity. The first of these measures is kurtosis, which measures the “spikiness” of a signal. Kurtosis is 0 for gaussian random variables, positive for random variables that are more spiky than gaussian variables, and negative for random variables that are flatter than gaussian variables. The second measure is negentropy, which measures the “simplicity” of a signal. Negentropy is also 0 for gaussian random variables.

The FastICA package that we used for our project uses both of these measures of nongaussianity to identify independent source signals. It begins by guessing a row of the matrix W , which we can call w . This row represents the weighting coefficients for finding one of the original source signals. It then measures the nongaussianity of the proposed independent source defined by its guess of w , and finds the gradient of nongaussianity in an n-dimensional space to determine how the coefficients in w should change. It then uses a projection of the gradient to create a new guess of the coefficients in w , and continues in a cycle until the coefficients converge on certain values. Once this occurs, the resulting independent source is as nongaussian as it can be. This in turn means that it is the furthest the algorithm could get from the source being a sum, which means that one of the independent sources has been isolated.

The algorithm repeats this process for finding all the rest of the independent sources, taking care not to find the same source twice.

#### Ambiguities and limitations

Just by examining the statement of the problem at the beginning of this module, two significant ambiguities arise in the ICA algorithm.

#### Indeterminate energy

Because a scalar multiplier could be pulled out of s and multiplied to A with no change in the above equations, the ICA algorithm cannot determine the energy contained in any of the independent sources it finds. The amplitudes it gives the output components are arbitrary, and the true source signal could be one the isolated sources multiplied by any scalar multiple. This includes a negative multiple, which means that often, the output signals are also inversions of the original signals.

The seriousness of this ambiguity depends on the application. For sound signals, inversions are irrelevant because the only important part of the signal is the different between voltages, not the polarity. Gain can also be added on to sound systems to deal with the amplitude ambiguity. In other applications, such as image processing, the inability to distinguish energy is much more significant.

#### Order ambiguity

Because the algorithm chooses coefficients of w at random when it searches for the sources, the isolated sources that the algorithm finds can come out in any order. So, it would take some additional processing to determine which independent sources is the one of interest to you.

#### Under-determination

There must be as many sensors as there are sources in order to properly isolate the sources. If there are not enough sensors, the resulting signals will not match any of the sources, but rather will still be mixtures of multiple sources.

#### Under-determination

ICA can only handle linear mixtures that can be represented in the form x = As . The algorithm cannot accurately guess the independent sources if the sources are out of phase in the mixtures or if the mixtures have other nonlinear features.

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
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
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!