Mod-06 Lec-41 FCM and Soft-Computing Techniques

Mod-06 Lec-41 FCM and Soft-Computing Techniques


Today I will be talking about fuzzy C means
algorithm I will be talking about fuzzy C means algorithm before that let me just say
a little bit about soft computing techniques rare fuzziness is one of the components of
soft computing techniques soft computing techniques were first introduced by someone named lottery
today in 1965 actually he introduced fuzzy sets in his famous paper trust published information
Sciences fuzzy sets is a variation from the ordinary sets. Note that in ordinary sets if you have a set
A and if you have a point x we know that either x belongs to A or x does not belong to only
one of them is true another way of saying it is you have an indicator function of the
set a where this IX=if x does not belong to A and 1 if X belongs to A I is said to
be indicated function of the set indicator function so they defined fuzzy sets as. You have a set A with the membership function
value membership function which is µ for a fuzzy set A it takes values not only 0 and
1 but it can also take values in between 0 and 1 that is this is known as membership
function membership function and the µAx takes values
in the interval 0 to 1 not only 0 A1 it may take a value in between.
There are many examples one example is suppose you want to look at the set of all let us
say tall persons and then if someone’s height is let us just say five feet, five inches
and would you just call him tall probably all the persons whose height is more than
5 feet 7 inches you may call them tall and probably all the persons whose height is less
than let us just say 5 feet 2 inches you may call them not tall.
But from 5 feet 2 inches to 5 it is 7 inches you have you may not have a clear-cut idea
of what exactly is tall probably. Let us say this is 155 centimeters and then say it is
150, 164 let us just say 166 centimeters yes I have taken some values so and here I am
writing membership function say this is zero then says it is one if the height is 155cm
probably you may not want to call the person as tall if it is 166cm surely you would like
to call the person as tall and anything greater than 166 you would like to call the person
as tall anything less than or equal to 155 it is surely not tall this is this one and
between 155 to 166 probably you have like this and this is called membership function.
The membership first the attribute tall is will be increasing and it will go to one this
many times in reality we use adjectives without clear-cut mathematical formulations without
clear-cut mathematical formulation exact and precise mathematical formulation is out there
and by using those adjectives when I speak to you or when you speak to me when use those
adjectives are when I use those adjectives. We understand each other when we are able
to understand each other and we would like to have we would like to have our computer
also to understand these words we would like to have the computer also to understand the
verse that is the basis for soft computing I mean that is the basis for fuzzy set theoretic
that is the basis for fuzzy sets. It is also the basis for what is known as
fuzzy logic what is known as fuzzy logic in the usual binary valued logic that is 0 or
1 a statement is either true or it is not true but in fuzzy logic that is a statement
is true with some membership value rather just say 0.5, 0.6, 0.7 and it is not true
with some membership value find again with some membership.
And like that you can have I mean remember ship functions in many problems which is actually
the case in reality in reality we do not always make mathematically precise statements in
fact many times most of the times we do not make mathematically precise statements even
then we understand each other then we would like to have our computer also to understand
the language which we are speaking then come then the analysis should be done in such a
way that this understanding is possible. So one of the ways that the suggested eth
2 x 2 use these number shift values in the analysis so there are very many developments
after fuzzy sets neural networks is considered to be a part of soft computing multi-layer
perceptions and radial basis function networks you have principal component analysis networks
these are considered to be part of soft computing genetic algorithms are considered to be part
of soft computing rough sets are considered to be part of soft computing soft computing
actually means. That we can have input imprecise we can have
ambiguous input and sometimes even in character C when we discuss many things sometimes we
make incorrect statements but even then we understand each other always whenever I mean
the basic significance of all these things is we would like to make our computer to understand
and to behave in such a way as a human being since we sometimes make in correct statements
still we understand each other then how do you make the computer also understand these
sort of things. Basically with this brief introduction to
soft computing like fuzzy sets neural network genetic algorithms and rough search okay I
am not going to discuss all these things in this class I will have only talked about fuzzy
sets because I will be using them in the fuzzy seeming algorithm Fuzzy sets means you basically
have some membership values you basically have some membership values in the membership
values are used in several applications and that is the reason why I introduced fuzzy
sets A set a is said to be a fuzzy set. If the membership values they lie between
0 and 1 n and there is at least one point for which the membership value is strictly
in between 0 and 1 if all the membership values are either 0’s or 1’s then it is going
to be called as a crisp set if all the membership values are either 0’s or 1’s and nothing
in between 0 and 1 then the set is called as a crisp set position is there exists at
least one membership value which lies strictly in between 0 and 1 it is neither 0 or 1 at
least 1x for which µix is strictly in between 0 and 1 so that is fuzzy sets. Now using these fuzzy sets we can also discuss
what is known as fuzzy partition fuzzy C partition fuzzy C partition first let me tell you the
meaning of ordinary partition that we all know but I will repeat it again then we will
go for fuzzy C partition now the usual C partition C is=2 and C is an integer
usual C partition of a set S is let us just say x1 to xm let us say is x1 to xn is what
it is like this usual C partition of a set s represented as let me just call it P is
for partition A1 A2 Ac you need to give partition of S x C subsets partition of S X C subset
C is the number of subsets is what represented as this.
And the definition is what is the definition is first each Ai ? 5 right and Ai intersection
Aj=right and Union right partition of S in to see subsets that
is the urban air to AC if I represent the C subsets as a 180 AC then the definition
is that each Ai ? ? and Ai intersection Aj is ? and Union Ai 1 to c=S so this is a
usual C partition now what is fuzzy partition. Now fuzzy C partition of s basically you need
to give membership values basically you need to give membership values fuzzy C partition
of S represented by
represented by U is U S is the set okay where U is n / c matrix where U is and n/c matrix
U is equal to so ith row this column element is okay where
uij denotes membership value membership value of ith point to the jth set
jth fuzzy set. jth fuzzy set naturally the number of points
is n so I lies between 1 to n and C1=j=c fuzzy C partition of S represented by US where
U is an n / c matrix Uij denotes membership value of ith point with it fuzzy sets okay
satisfying the following properties satisfying the properties
stated I do not I do not want to write below because I need to write there so satisfying
the properties stated let me just try it state it here just one minute I will erase this
portion. The first property is naturally 0=uij=1
for all ij membership values, two ? uij, j=1 to c this means you have taken diet point
you fix the ith appoint its membership for that for the first set is ui1 a 1 no matter
for the second set is ui2 number three for the jet-set is uic membership at the safe
set is you I see, so this ?it is=1 okay its ?is=1 in the usual partition if your point
X belongs to 1 of the sets it is not going to belong to any other set, so for that point
if you sum up all the membership values 1 place you will get 1 and at all other places
you are going to get 0 so ?is 1. Here the ? is 1 but it is not necessarily
true that exactly at 1 place here you have 1 and at other places we have 0 you may get
some fractional values here 3 is ?I is=1 to n uIJ that is you are fixing the set now
and you are looking at the sum of all the membership values for a particular set, it
should be strictly>0 it should be strictly1 and for a given positive definite matrix A the
objective function value is given by this expression, we are supposed to find the best
U given S, A and r one needs to find the optimal U are the best U best from the point of view
of that you which minimizes Jr, U, S, A. Now how does 1 do it before I go into the
algorithm part I would like to mention a few things this algorithm was first proposed by
James Bedeck, a very famous figure and a proof was also given to the algorithm that it converges
and then the proof was modified and then a modified proof was given but then ultimately
after a I should say a few iterations ultimately the correct proof was found the algorithm
was same throughout in all these things the proof needs knowledge quite a bit of knowledge
of mathematics including topology and other such fields I will not going to the proof
of the algorithm I will basically tell you the steps of the algorithm. So this is f c m algorithm F for fuzzy c means
fuzzy c means algorithm what are the steps first one is we are given the set S which
has smaller number of points and it is a subset of RM m dimensional space we are given the
positive definite matrix a and then exponent are I am saying that we are given.
Okay usually people choose the value of R some R greater than 1usually people choose
the value of R something that is greater than 1 and people choose it as something which
is very close to 1 from one point zero one some people make use to also but this value
is generally chosen by users and of course the number of clusters see this C is same
as the same we will start with a fuzzy c partition U of S. So once you is there you can always calculate
the means V J’s what is the expression for me the expression for mean is this, this is
the expression for me so you can always calculate this you can always calculate this expression
so you can always calculate V and once you calculate VGS you can calculate you I guess
how the expression for u IJ is this summation K=1 to C of D IJ square by D I K2 K=1 this
K is 1 to C this thing to the power 1 by r – 1.
And hold to the power of minus 1 here R

5 thoughts to “Mod-06 Lec-41 FCM and Soft-Computing Techniques”

  1. Thank you sir for this wonderful explanation. Can you give me link or some material of FCM proof ? I would like to go through it for my curiosity. sir I have a question for you though : How can I came to know the value of matrix A with respect to my data-set and what are different methods or technique of comparison for optimal value of U matrix, r and A matrix. Can gain chart be an effective technique for these comparison ?

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