So, we were discussing about the fuzzy neural
network, fuzzy min, max neutral network for
pattern recognition. And in the last class
we have said that how the fuzzy, how the hyper
box classifiers can be extended to fuzzy min
max neural network, where every middle layer
node in the fuzzy min max neural network will
compute the fuzzy membership function of an
unknown point to a particular hyper box. Then
outputs of all the hyper boxes belonging to
the same class they are gathered together
in one of the output layer nodes or class
nodes. But the class node decides about what
is the maximum membership given by the different
hyper boxes belonging to different classes.
And the membership function to that class
is this maximum membership of different hyper
boxes belonging to a same class. So, if I
have c number of output neurons pertaining
to c different classes, then every c neuron
will give me a membership function and this
membership function is the membership of the
unknown feature vector to the corresponding
class, okay. And we said that this itself
can be taken as fuzzy classification because
this is nothing but a membership vector or
if we want this classification or hard classification
then whichever neutron gives the maximum membership
value we can put the unknown feature vector
into the corresponding class. So, we can have
a max operator which will give us a binary
vector where only one of the components of
a binary vector will be equal to 1. And that
obviously corresponds to a maximum membership
and all other components of the binary vector
will be equal to 0. And that is equivalent
to our hard classification, okay. So, today
what we will do is we will take the same example
that we have taken for designing the hyper
boxes, okay. And we will take the same example
and we will re draw the hyper boxes and simultaneously
we will try to design a fuzzy min max neutral
network, okay. So, this is nothing but learning
or training of the fuzzy min max neutral network,
okay. So, the set of points that we had taken
in the last class is something
like this. We had taken a point having the
components 0 point 7 0 point 7 and we said
that this belongs to class 1, okay. And let
us simultaneously draw the hyper boxes that
we will get out of these points that we have
taken, okay? So, we will do it like this.
The lines are very light 
so I will put it like this is 0 point 1, 0
point 2, 0 point 3, 0 point 4, okay. So, when
we said that this point, this feature vector
0 point 7 0 point 7 this belongs to class
1. So, immediately we said that there will
be a hyper box for which the min point and
max points are same, okay. So, I will have
a hyper box somewhere over here 0 point 7
0 point 7. So, at
this location I will have a point hyper box
for which min point and max point is same,
that is 0 point 7 0 point 7. And for this
hyper box let us draw, try to generate this
fuzzy min max neutral network. So, as we said
that I assume that there are two classes,
class 1 and class 2. So, obviously in the
output layer I will have two nodes, one corresponding
to class 1and other one corresponding to class
2. So the first node corresponds to class
1 and the second node corresponds to class
2. Similarly, at the input layer also I we
will have two nodes because I am considering
feature vectors of dimension 2. So, at the
input layer also we have two nodes, okay.
So, I have node 1 and input node, okay. And
this will have the input feature vectors.
And while designing this fuzzy min max neutral
network what we have to do is we have to generate
three matrices because from the middle layer
to the output layer, the connections are actually
binary connections, either 0 or 1. 1 means
there is a connection from a middle layer
node to that corresponding output layer node.
0 means there is no connection from the corresponding
middle layer node to the corresponding output
layer node. And I will have two matrices,
which represents the connections from the
input layer nodes to the middle layer nodes.
And those two matrices actually represented
the min point and max point. So, for min point
I will have a matrix U. For max point I will
have a matrix W and for the middle layer node
to the output layer node I will have, sorry
I will have matrix V and
W, not even W. V corresponding to the min
points and W corresponding to the max points.
And I will have a matrix U which represents
the connections from the middle layer nodes
to the output layer nodes, okay. So, if I
use this convention that for every node in
the input layer I will have one row in matrix
U and one row in matrix W, okay, so this matrix,
sorry matrix V and matrix W. So, matrix V
will have two rows corresponding to the min
points and the number of columns will actually
be variable because till now I do not know
what the number of nodes in the middle layer
is. So, the number of nodes in the middle
layer that will actually represent that how
many columns I need in this matrices V and
W, okay. So, matrix V will have two rows.
Similarly, matrix W will also have two rows.
Similarly, for matrix U also let us assume
that we will have two nodes. It depends upon
which way I represent the matrix. So, here
in matrix U our row will correspond to one
of the nodes in the output layer node. The
other row we correspond to the other node
in the output layer node, okay. So, from input
layer to middle layer a connection V i j will
represent a connection from ith node in the
input layer to the jth node in the middle
layer, okay. Whereas, in matrix U the node,
the entry u j i will represent the connection
from the jth node in the middle layer to the
ith node sorry, a connection from u j i will
represent a connection from the ith node in
the middle layer to the jth node in the output
layer, okay. I just said the reverse, jth
node in the middle layer to the ith node in
the output layer. So accordingly for every
node in output layer I will have a row. So,
in this matrices what I have to go on adding
is the number of columns, okay. So, for the
first point that you considered, this point
is 0 point 7 0 point 7, okay. So, min point
and max point that is there same, right? So,
accordingly in matrix V, I will have to introduce
0 point 7 0 point 7 because that is the min
point, okay. Similarly, matrix W, I have to
introduce 0 point 7 0 point 7 because that
is also the max point, okay. And I have to
introduce one node in the middle layer corresponding
to this particular feature vector 0 point
7 0 point 7. So, the corresponding entries
in matrix V and W, I already established.
It is known that this feature vector belongs
to class 1. So, as this node belongs to class
1 so the corresponding column in matrix U
will be 1 0 indicating that
this node is connected to c 1 only. This node
is not connected to c 2, okay. Similarly,
when I consider the second point. The second
point that I have is point B which is 0 point
75 0 point 75 and this belongs to again class
1, right? So, coming over here 0 point 75
0 point 75 is this point. This also belongs
to class 1. So, I generate a hyper box. Assuming
that this expansion is permitted, I generate
a single hyper box containing these two points,
okay. So now, find that the earlier min point
and max point that we have generated, this
min point and max point gets changed and I
am not introducing a new node in the middle
layer. Only the min and max points I have
to change and because I am not introducing
any node in the middle layer so the matrix
U will remain unchanged, okay? So, what I
have to do is over here, the min point is
now minimum of these two, okay. In the first
dimension the minimum of 0 point 7 and minimum
of 0 point 75, it remains 0 point 7. And similarly,
minimum of 0 point 7 and minimum of 0 point
75, it remains 0 point 7. Whereas, the max
matrix that will be maximum of these two.
So, it will be 0 point 75 0 point 75, right?
So, the matrix W though initially we had set
it 0 point 7 0 point 7. Now, it will be 0
point 75 and 0 point 75, okay. So, any time
the min point and max point gets changed.
I have to change the corresponding entries
in matrix V and matrix W. Now, comes the third
point.
The third point that we considered was point
C which is 0 point 9 0 point 9 and we said
that this point belongs to class 2, okay.
So, I will take 0 point 9, 0 point 9 over
here. So, 0 point 9 and 0 point 9 is this
one, okay. And this belongs to class c 2.
So, obviously I cannot expand any of them
to include this particular point. And this
is the first point, the first feature vector
that is considered from class 2. So, naturally
I have to introduce a new node in the middle
layer, right? So, I introduce a new node in
the middle layer. And as I know that this
hyper box will belong to class c 2. So, this
will only have a connection from here to c
2. It will not have any connection upto c
1. So, accordingly the corresponding entity
in matrix U will be 0 1, okay. Now, I have
to set the entries in the other two matrices,
that is V and W, okay. And you find that this
is the only feature vector considered so far
belonging to class c 2. So, naturally its
min point and max point will be the same.
And min point max point both of them will
be 0 point 9 0 point 9. Max point will also
be 0 point 9 0 point 9. Now, I consider the
third point where the third point is D and
it is 0 point 8 0 point 8 and this also belongs
to class 2, right? So, corresponding to this
I have 0 point 8 0 point 8 somewhere over
here. And suppose I can expand this hyper
box to include this also, okay. So the new
hyper box that I will get is this one. Now,
in this hyper box we find that the max point
will remain the same as the previous hyper
box whereas, the min point has to change,
okay. So, earlier the min point was 0 point
9 0 point 9 Now, the min point has to be minimum
of these two. So, it will now be 0 point 8
0
point 8. So, earlier this 0 point 9 0 point
9, which we had put, this min point is to
be 0 point 8 0 point 8, okay? So, now let
us take the fifth point which is E and the
vector is 0 point 1 0 point 1, and we said
that this feature vector belongs to class
1, okay? So, for this I will have the hyper
box at 0 point 1 0 point 1, right? And suppose
that this hyper box belonging to class 1,
I cannot expand to include this, okay. So,
I have to generate a new node for this hyper
box and that new node have to generate over
here.
And as I know that this belongs to class c
1. So, the corresponding entry in my matrix
U has to be 1 0 indicating that this node
belongs to class c 1, right? And then I have
to make an entry into the matrix V. I have
also have to make the entry into the matrix
W. And here again because this is a point
hyper box so the min point and max point they
will remain the same, that is 0 point 1 0
point 1. Here also it is 0 point 1 0 point
1. Is that okay? So, then let us consider
the other point that is point F. Whose coordinates
are 0 point 2 0 point 2 and this point also
belongs to class 1, right?
So, over here this corresponds to 0 point
2 0 point 2 a point somewhere over here. And
assuming that this 0 point 1 0 point 1 can
be expanded to include this point. I get a
single hyper box, and this single hyper box
is going to class c 2, okay. And because I
am not adding, I am not adding any new hyper
box, okay. So I do not have to add any node.
The previous node that I have only in that
node I have to change the min point max point.
So, accordingly I do not have to make any
entry into matrix U. In matrix V and W change
the min point and max point. So, the new min
point that remains minimum of these two, okay.
So, the min point does not change but max
point has to be maximum of these two.
So, the max point has to be changed to 0 point
2 0 point 2 from 0 point 1 0 point 1. So,
the new max point that I get, earlier it was
0 point 1 0 point 1 now I have to make it
0 point 2 0 point 2. The other matrices remain
the same. Then let us take the next point,
that is G where the vector is 0 point 3 0
point 3 and this belongs to class 2, okay.
So, 0 point 3 0 point 3 if I put over here
0 point 3 0 point 3 comes at this location.
So, you find that this belongs to class c
2. The earlier hyper box for c 2 was this.
Assuming that this cannot be expanded into
this, so I have to add a new node in the middle
layer, okay? So, the new node in the middle
layer comes somewhere over here. And I know
that this belongs to class c 2. So, it will
only have a connection to c 2. The classifying
neurons c 2 and accordingly the corresponding
entity in the U matrix will be 0 1, okay.
So, this is U matrix. Now, this is a new node
and a point hyper box. So, I will have 
the min point and max point which is same
as 0 point 3 0 point 3. Max point we will
also be 0 point 3 0 point 3, okay, the same
min point max point.
Then you come to the next point which is 0
point 15, 0 point 15 and this also belongs
to class 2, okay. coming over here 0 point
15 0 point 15 is this, right? And assuming
that I can expand this to include this one.
So, the new hyper box that I will generate
is this one. So, this becomes my new hyper
box. And now we find that this hyper, this
one is hyper box belonging to class c 2. This
is hyper box belonging to class c 1. So, whenever
I have such a situation, because there is
an overlap I have to break this two hyper
boxes. And to break these two hyper boxes
I
have to find out that in which dimension the
overlapping is minimum. So, here you find
that along the horizontal dimension the overlap
is, this is 0 point 15 and this is 0 point
2. So, the overlapping is 0.05. In this dimension
it is 0 point 15. This is 0 point 2. So, the
overlapping is 0.05. So, in both the dimensions
the amount of overlap is same. So, I can break
this hyper box in any of the two dimensions,
okay. Let us assume that we break the hyper
box along the middle of the first dimension
over here. So, when I break this I get two
hyper boxes. One hyper box is this one which
belongs to class c 1 and the other hyper boxes
is this one which belongs to class c 2, right?
Now, over here you find that when I am breaking
these two hyper boxes, I am not generating
any new hyper box. There was a hyper box corresponding
to this and there was a point hyper box corresponding
to this. So, it is the point hyper box which
has been expanded to include the second point.
And while it was expanded there was an overlap
because of this overlap I had to contact this
hyper box, okay. So, what I have to do is
I have to change the min point max point accordingly
for this hyper box which was a point hyper
box earlier. I have to change the min point
and max point for this which hyper box was
already existing, right? So, what will be
the min point max point? For this hyper box
the min point remains the same, right? What
is the max point? Max point is this coordinate,
right? And what is this coordinate? This coordinate
we be 0 point 15 and 0 point 2, sorry max
point, max point for this hyper box will be
0 point 15 0 point 2, right? So, this is 0
point 15 0 point 2 and that is the max hyper
box. Max point of this hyper box, the modified
max point, okay, similarly the min point of
this hyper box which is nothing but this location,
okay. This can be 0 point 15 and 0 point 15,
okay. Earlier it was 0 point 15 now it will
be 0 point 175, 0 point 175 and this will
remain same as 0 point 15, right? And this
is 0 point 175 and 0 point 2, 0 point 175
and 0 point 2. So, the max point of this hyper
box is going to change, okay and for this
hyper box earlier I had a point hyper box
for which the min point and max point was
same. Now, the new min point will be this.
Max point will remain unchanged. Is it okay?
So, coming to our V matrix and W matrix you
find that for the earlier one where the min
point was 0 point 1 0 point 1 and max point
was 0 point 2 0 point 2. Now, the new max
point is 0 point 175 0 point 2.
So, this 0 point 2 now it gets replaced by
0 point 175. The other point, the other dimensions
remains as 0 point 2, right? For this one
0 point 3, 0 point 3 which was a point hyper
box. For this point hyper box the max point
remains the same. But the min point becomes
0 point 175 and 0 point 15, is that okay?
So, the min point will be for this will be
changed to 0 point 175 and 0 point 15, is
it okay? Now, let us take the next point.
The next point is I for which the coordinate
is, okay, vector is 0 point 45 0 point 45
and this belongs to class 1, okay?
So, for this I have 0 point 45 and 0 point
45. This belongs to class 1, and suppose this
cannot be included in any of the hyper boxes
after expansion. So, I have to make a new
node for this particular point whose corresponding
to which vector is 0 point 45 0 point 45.
So, I generate a new node and I know that
this node belongs to class c 1. So, accordingly
I have to make a connection from this node
to class c 1 and correspondingly matrix U
will be modified as 1 0,
okay? And I have to enter the min point and
max points in this two matrices. And because
it is a point hyper box so the min point and
max point will remain the same. So, both of
them are 0 point 45 0 point 45. Here also
it is 0 point 45, here also it is 0 point
45, okay? So, when I consider the next point
that is point J corresponding to this the
feature vector is 0 point 5 0 point 5 and
this also belongs to class 1, okay. If I come
over here 0 point 5 0 point 5, that is this,
okay? And assuming that this point hyper box
can be expanded to include this. So, I generate
a new hyper box and because I am not adding
any new node because the previous point hyper
box has been expanded to include this. So,
U matrix will remain the same.
However, I have to look at the V matrix and
W matrix. And here you find that the max point
that is being changed. So, earlier max point
was 0 point 45 0 point 45. Now, the new max
point is 0 point 5 0 point 5, okay? So, I
have to change this corresponding entry in
matrix W. So, it will be 0 point 5 0 point
5, okay? Then let us consider the next point,
the next point is K. The feature vector is
0 point 42, 0 point 42 and this belongs to
class 2, okay. So, I come over here. This
is 0 point 45, 0 point 45, okay. 0 point 42,
0 point 42 will be somewhere over
here. Somewhere over here 0 point 42, 0 point
42 and assuming that this has to be point
hyper box I have to have a corresponding node
in the middle layer, okay? So, have to have
a node in the middle layer and as I know that
this node belongs to class c 2. So, I have
to change the corresponding U matrix, right?
So, I have to make a column, add a column
in the U matrix which will be 0 1 because
this belongs to class c 2, okay. And in the
U matrix, V matrix I have to generate, I have
to add a column corresponding to this particular
point, okay, so this 0 point 42, 0 point 42.
Here also it is 0 point 42, 0 point 42, okay.
Then I take the next point.
The next point is 0 point 55 0 point 55 and
this belongs to class 2. So, according to
this I have over here 0 point 55, 0 point
55 somewhere over here. And assuming that
this can be expanded to include this I have
this new hyper box, the modified hyper box
which comes like this and now you find that
there is a containment. So, because there
is a containment I had to contract this hyper
box. So, this new hyper box will be contracted.
Let us assume that we contract it from this
location. So, in this side I have one hyper
box belonging to class c 1 and
on this side I have another hyper box belonging
to class c 2, okay. So, over here you find
that this hyper box was already existing and
for this hyper box and min max point has not
been changed, okay. So, accordingly the earlier
entry in our matrix V and W matrix that remains
the same, that is 0 point 45 0 point 45, 0
point 5 0 point 5. This remains unchanged,
right? Whereas, for this new hyper box, okay.
I had earlier the min point, which is 0 point
42 0 point 42 that corresponds to this particular
point. And when I have expanded this to include
the second point and because of this expansion
there was a containment and due to this containment
I had to contract this hyper box. So, as I
contracted this hyper box the min point has
now changed from here to here, right? And
this is my max point. So, what is this new
min point? The new point is 0 point 5 0 point
42 that is the new min point. And the max
point is 0 point 55 0 point 55, 0 point 55
and 0 point 55. So, accordingly I had to make
the modification over here. Now, the min point
becomes 0 point 5 0 point 42, okay. So, this
will be 0 point 5. 0 point 42 as it is and
the max point which earlier was 0 point 42
0 point 42 because it was a point hyper box,
okay. Now the new max point will be 0 point
55 and 0 point 55, that is this one, okay.
So, this max point, new max point becomes
0 point 55 and 0 point 55. So, with this I
have included all the training points that
were given. So, at the end of the training,
the three matrices that I have generated are
like this, okay. So, I have this U matrix
which represents connections from the middle
layer nodes to the output layer nodes. And
I have two matrices V and W corresponding
to min points and max points of the nodes
which are generated in the middle layer. So,
I over here these are represented by the matrices
V and W. And over here the connection is presented
by matrix U and I have the final memberships
computed to the two different classes c 1
and c 2 which comes from the output layer
nodes, is that okay? So, we you find that
your network has been trained in a single
pass unlike in case of multi-layer perceptron
or single layer perceptron. What we have done
is, we have changed the weight vector or the
weight matrices iteratively every time a given
feature vector becomes misclassified then
following the gradient descent approach or
following the back propagation learning we
had to change the weight matrices for different
layers. And once it is changed I have to re-check
with all the samples all the feature vectors
which was used to train the neural network
before this. That means I have to come to
conclusion that the multilayer perceptron
or single layer perceptron has been trained
properly only when in a single pass all the
samples are correctly classified, all the
training samples are correctively classified
or the error that you get is less than certain
specified value or less than the tolerable
value so the training algorithm of the neural
network in such cases was an iterative algorithm.
Whereas, in this case the neural network is
trained in a single pass because every time,
okay. And what I am doing is if there is an
overlap immediately contacting the existing
hyper box, which takes care of the fact that
there is no overlap or no ambiguity during
classification or during testing, okay. However,
as we said before that the problem with this
fuzzy min max neutral network is that as we
are going for contraction, okay.
So, here to find that say this point which
we know that these belongs to class c 2. But
because of overlap and contraction finally
this point has been classified into class
c 1. Because now if I after this contraction
operation if I present the same point to this
neural network, the neural network will say
that this point belongs to class c 1, though
I know that this point belongs to class c
2. Similarly, for this I know that this point
belongs to class c 1. But after creation of
this hyper boxes and contraction of these
hyper boxes, if I present the same point to
the neural network it will say that this black
point belongs to c 2. Though I know that this
point belongs to class c 1, okay? So, that
is the disadvantages of this kind of neural
network. And such a kind of problem you get
both over here and over here. Because earlier
we know that this point belongs to class c
1, okay. But after creation of this hyper
boxes and contraction. Now, if I present this
point to the neural network, what it will
do? For this point, it will try to calculate
the fuzzy membership function to this hyper
box and also the fuzzy membership function
to this hyper box. And now you find that this
point is nearer to this hyper box, then this
hyper box. So, the fuzzy membership function
for this hyper box will be more than the fuzzy
membership function for this hyper box. So,
naturally if I go for hard classification.
This point will be classified to class c 1,
though you know that it actually belongs to
class c 2, okay. So, after this expansion
and contraction process you say that the training
samples which are used to train the
classified, train the neural network those
samples themselves will be misclassified or
some of those samples themselves will be misclassified.
So, naturally due to this contraction we are
introducing some amount of error in our learning
process or in training process. So we have
suggested some modification to this kind of
algorithm, that is by introducing the concept
of compensatory neurons. So, what we have
done is the basic structure or the classified
section of this neural network called reflex
neural network remains the same as what has
been suggested by Patrick Simson, that is
fuzzy min max neural network, okay? So, in
the new architecture I have as before a number
of classifying neurons in the output layer,
okay. A number of input layer neurons. Let
me draw in another paper.
So, I have a number of classification neurons.
I have a number of input layer neurons. Output
layer nodes is same as the number of classes
we are considering. The number of input layer
neurons is same as the dimensionality of the
feature vector that we are considering, okay.
So, the major part of this reflex fuzzy neural
network is same as the fuzzy min max neural
network which has been suggested by Simson.
And we have this middle layer nodes which
are basically hyper boxes, okay. And as we
have just seen that these number of nodes
in the output layer and the number of nodes
in the input layer they are fixed by the problem.
Whereas, number of nodes in the middle layer
that we create during the training process,
okay and accordingly we generate matrix U,
matrix V and matrix W. Now, what we have suggested
is, we have suggested two more sections in
this neural network which is called the compensation
section, and in compensation section again
as we have just seen that we can have two
types of situations. One is overlap situation
and the other one is containment situation
and the kind of compensation that you have
to impart is different in case of overlap
compensation we have to give some sort of
compensation. In case of contentment we have
to give some other type of compensation. So,
accordingly this compensation section that
we have generated, this compensation section
is again divided into two parts. One of them
we are calling as overlap compensation, the
other one is called containment compensation,
okay. So, I will have some additional nodes.
One in the overlap compensation section. This
we call as overlap compensation 
and the other one is containment compensation.
So, what is done in this overlap compensation
and containment compensation? Whenever there
is an overlap like here while training if
we find that there is overlap in the previous
case, what we have done is we have contacted
the hyper boxes to remove that problem. Now,
we do not go for contraction. But we go for
the gradation of the membership function within
that overlap region. So, whenever there is
overlap one new node is introduced in the
overlap sections, okay. So, over here in the
overlap section we will introduce a new node,
okay and this new node will give one compensation
to the classifying neurons. Corresponding
to this hyper box and another compensation
to the classifying neurons. Corresponding
to this hyper box because I cannot grade the
membership of only one hyper box. I have to
grade the membership of both the hyper boxes.
Because if a point falls within this region,
the overlap region normally our classifying
neurons. In this section, classification section
both of them will give membership function
to this class and membership function of one
to this class, okay? I have to modify both
of them or compensate both of them. So, this
new neuron that we are introducing in the
overlap compensation section that will actually
generate two outputs. One output is meant
for compensation of one class and the other
output is for compensation of the other class,
okay.
So, if I have class c i and class c j, the
hyper boxes belonging to c i and the hyper
boxes belonging to class c j there they overlap,
okay. Then output of the compensation neuron
that will give compensation to both c i and
c j and however the amount of compensation
to c i and the amount of compensation to c
j will be different. The reason is if this
point is nearer to this point then the membership
function to c i to c j should be more. And
if this point is towards this, then the membership
function to plus c i should be more, okay?
So, the amount of compensation that you give
to plus c i and the amount of compensation
that you give to plus c 2 or c j, they cannot
be same. They have to be different and while
calculating this compensation values you have
to take care that, to which of the points
your unknown point is nearer. So, what you
are doing is, in this section also I will
have a number of classifying neurons or compensation
neurons which is same as the number of classes
that I have, okay. This will get output from
the input. This will get input from the input
layer neurons and output of this will actually
go to two of the output layer neurons because
I have to compensate for ith class as well
as the jth, okay? Similarly, in the containment
section also I will also have a number of
output layer neurons which is same as the
number of classes I have, okay. Here again
whenever there is a containment like this
I will put a containment compensation neuron,
okay. Now, why I said that the amount that
the type of compensation that have to give
in this case is different from
the type of compensation that you have to
give in this case is that, here I said that
I have to compensate the outputs, the membership
functions of both class c i and class c j,
right? Whereas, in this case if any feature
point lies over here we are saying that it
should get a membership value of 1 to this
class. Whereas, the membership value to this
class has to be 0. And my hyper box in the
classifying section that itself will compute
membership value of 1 to this class. It will
also get a membership value 1 to this class.
I have to make the membership value to this
class equal to 0. And the membership value
to this class should remain unchanged. That
means I have to only compensate the membership
value of one class which is containing the
hyper box of another class, okay. So, the
hyper box that is contained, the membership
value of that should not be changed. However,
the hyper box which contains the other hyper
box, the membership value of the class of
the hyper box that contains that should be
made equal to 0. Is that okay? So, accordingly
output of a compensation neuron in the containment
section should go to only one class, that
is the class of this particular hyper box.
It should not go to the membership function.
It should not compensate the membership function
of this class, okay. So, here in the overlap
compensation section the output of this compensating
neuron goes to two classes. Whereas, in this
case the output of the containment compensation
neuron will go to only one class, right? So,
I get two types of compensation; one from
the overlap compensation neurons and the other
one for the containment compensation neurons.
Now, I have to combine these two compensations,
okay. So, I have to have a combiner, right?
So, over here, that belongs to the class of
smaller hyper box, because that defines its
class membership more vigorously than the
class membership of the other one, okay. So,
I will have to combine the outputs, combine
the outputs of this compensation neurons.
So, I have to have a number of combiner, okay?
So, I will combine the output of this and
the output of this. So, these are outputs
of the output layer neurons of the compensation
class belonging to the same class. So, this
is class c 1. This is class c 1 you combine
these two. Similarly, class c 2, class c 2
you combine these two, okay. Finally, after
combination combining these outputs I know
that what is the overall compensation that
I had to give to the compared to the membership
values
generated by these neurons. These neurons
are generating the membership value as if
there is no containment or there is no overlap,
okay. We are trying to modify these outputs,
these membership values as computed by this
compensation values, okay? So, what I have
to do is, I have to add these values to these
membership values. So, this will come over
here, this will come over here and then finally
I get the final membership value, okay. So,
we find that in case of this neural network
without any compensation section if the membership
value to a particular class say ith class
was mu i. In this case the membership value
after this compensation to ith class will
be mu i plus let as put it as o i, where o
i is the combination of the outputs of the
overlap compensation neuron and outputs of
the compensation neuron.
So, over here I will have this output as o
i, okay? So, the final membership value will
be mu i. mu i is what is computed by these
neurons. I have to modify this with the outputs
of the compensation neurons so the final membership
value becomes mu i plus o i. And that I can
take as the final membership value say C i.
Two class to the ith class for any of the
unknown feature vectors which have been put
up in the neuron, okay. So, when these compensation
section neurons whether it is overlap compensation
or containment compensation, they compute
the compensation values they will make use
of the same V and W matrices, the min points
and max points that we have generated for
these hyper boxes, okay. So, they will make
use of the same min point and max point to
compute what should be the compensation given.
And this composite compensation computed only
when an unknown feature vector falls within
either the overlap region or the contract
region. This will not be computed if the unknown
feature vector falls outside the overlap region
or falls outside the containment region. And
when it get, this the final membership value
your classification will depend upon this
final membership value. So, the classification
instead of doing based on mu i it will be
based on c i, okay? So, in the next class
we will see the details of the membership
computation or the compensation computation
as done by this overlap compensation neurons
and the complete containment compensation
neurons, okay. So, we will take the same
example to design our reflex for the min max
neural networks with, I will take the same
example because I have the case of overlap
I also have a case of containment, okay. So,
with that we will illustrate that how this
neural network with compensation neurons that
can also be generated, okay. So, we will stop
here today. Thank you.
