generalized
nets
IFSs and
Clustering
IFDFC
Model
Parvathi Rangasamy
[email protected]
Peter Vassilev
[email protected]
Krassimir Atanassov
[email protected]
Stefan Hadjitodorov
[email protected]
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
Brief introduction to
Generalized nets
BIOMATH 2011, Bulgaria
Generalized nets
generalized
nets
IFSs and
Clustering
IFCC
model
Generalized nets (GNs) are extensions of Petri
nets. They are characterized by their:
• static structure
• dynamic elements
• global temporal components
• memory
Let us look at some of these net components.
BIOMATH 2011, Bulgaria
Static structure
generalized
nets
IFSs and
Clustering
IFDFC
Model
•
sets of transitions
• functions determining the priorities of
the places and transitions
•
functions giving the places’ capacities
BIOMATH 2011, Bulgaria
Dynamic elements
generalized
nets
IFSs and
Clustering
IFDFC
Model
• sets of tokens
• functions giving the tokens’ priorities
• moment of time, when a token enters
the net
BIOMATH 2011, Bulgaria
Global temporal components
generalized
nets
IFSs and
Clustering
•
moment of firing (starting) the net
•
elementary time-step
• duration of the net’s active state
multiple
knapsack
problem
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Memory
generalized
nets
IFSs and
Clustering
IFDFC
Model
•
•
set of initial characteristics of tokens
functions giving the next characteristics
of tokens (during their moving into the
net)
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Transitions
generalized
nets
IFSs and
Clustering
IFDFC
Model
set of input places
set of output places
moment of firing
duration of active state
index matrix
Figure 1: Components of the generalized net
BIOMATH 2011, Bulgaria
Index matrix
generalized
nets
IFSs and
Clustering
It contains the net’s
inputs and
and ouputs
ouputs in
order to join them
with predicate a R.
In1
…
IFDFC
Model
… Outn
Out1
a In ,Out  R
i
Inm
Figure 2: Index matrix
j
BIOMATH 2011, Bulgaria
Token's movement into the net
generalized
nets
IFSs and
Clustering
IFDFC
Model
Figure 3:
A token moves, splits
into two tokens, each
of them obtains new
characteristics, loops
and at the end both
tokens merges.
BIOMATH 2011, Bulgaria
Algebraic operations
generalized
nets
IFSs and
Clustering
IFDFC
Model
Different algebraic operations like:
• union
• intersection
• composition
• iteration
can be defined over the set of generalized
nets.
BIOMATH 2011, Bulgaria
Relations
generalized
nets
IFSs and
Clustering
IFDFC
Model
There are also relations applicable over the set
of generalized nets:
• inclusion
• equality
• graphical (structural) inclusion
• graphical (structural) equality
• inclusion according to the work done
• equality according to the work done
BIOMATH 2011, Bulgaria
Comments
generalized
nets
IFSs and
Clustering
IFDFC
Model
25
All analytical functions in a GN model can be
described by the tokens’ characteristics.
All logical conditions that exist in a GN can
be defined by the transitions’ predicates.
Thus, we are able to make model of any
arbitrary real process.
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
Intuitionistic Fuzzy Sets
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Intuitionistic Fuzzy Sets
generalized
nets
Membership function
A : X  [0,1]
IFSs and
Clustering
Non-membership function
 A : X  [0,1]
IFDFC
Model
so that
0  A ( x)  A ( x)  1
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•Cluster analysis or clustering is the assignment
of a set of observations into subsets (called
clusters) so that observations in the same cluster
are similar in some sense.
generalized
nets
IFSs and
Clustering
IFDFC
Model
•Data is divided into distinct clusters, where
each data element belongs to exactly one cluster.
•Distance measure - An important step in most
clustering is to select a distance measure, which
will determine how the similarity of two
elements is calculated. This will influence the
shape of the clusters, as some elements may be
close to one another according to one distance
and farther away according to another.
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
In fuzzy clustering, each point has a degree of
belonging to clusters, as in fuzzy logic, rather than
belonging completely to just one cluster. Thus,
points on the edge of a cluster, may be in the
cluster to a lesser degree than points in the center
of cluster.
Fuzzy c-means (FCM), the centroid of a cluster
is the mean of all points, weighted by their
degree of belonging to the cluster.
BIOMATH 2011, Bulgaria
The degree of belonging is related to the inverse
of the distance to the cluster center.
generalized
nets
IFSs and
Clustering
IFDFC
Model
Then the coefficients are normalized and
fuzzified with a real parameter m > 1 so that
their sum is 1. So
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Modified FCM with Intuitionistic Fuzzy values
generalized
nets
IFSs and
Clustering
IFDFC
Model
Step 1: Initially, select c number of centroids
with membership and non-membership values.
Step 2: Compute the membership degree of
each object to each cluster U ij using IF
similarity measure.
Step 3: Update the centroids matrix Vi
Step 4: Compute membership and
nonmembership degrees of Vi .
Step 5: Repeat the Steps 2 to 4 until converges.
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
GN-model of
Intuitionistic Fuzzy
based Distributed Fuzzy
Clustering
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
Traditional clustering methods require all data to be
located at one place and are practically inapplicable
in the case of multiple distributed datasets. It is not
always possible to transmit all data from the local
sites to single location and then perform global
clustering. Distributed clustering algorithms reduce
the communication overhead, central storage
requirements and computation times. Recently,
intuitionistic fuzzy based clustering for centralized
environment has been proposed. There is evidence
that intuitionistic fuzzy based FCM clustering can
perform better than the well established FCM
algorithm.
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
The Generalized Net Model
of IFDFC
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
The GN contains 7 transitions, 19 places and
three types of tokens: p in number α-tokens, one
β token and p in number γ-tokens. Let i-th
characteristic of token ω be noted as xiω.Each
one of the α-tokens (let it be i-th α -token)
enters place l1 with initial characteristic
x α i0 = “i-th input dataset”
and the initial characteristic of β -token is
xβ i0 = “values of constant c”
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Let i-th dataset have the form
generalized
nets
where i = 1, 2, …, p and ni is the number of sites and
a's are numerical values of the feature vectors to be
clustered.
The transitions have the form:
IFSs and
Clustering
IFDFC
Model
The token αi enters place l4 with a charactersitic
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On the next time-moment this token splits into
two tokens: token ®i that enters place l3 without
a new characteristic and token °i that enters
place l2 . The first ° -token (we will mark it
without index i) enters l2 with characteristic
generalized
nets
IFSs and
Clustering
IFDFC
Model
°
®
x1 = x1 i
while the next (let it be k-th token for 2 ≤ k ≤ p)
° token unites with the first one and it obtains the
characteristic
°
°
°
®k
k
xk = \ < max(pr1xk¡1; pr1x®
);
min(pr
x
;
pr
x
¡
2
2
1
1 )>"
k 1
where for every natural numbers s, v, s ≤ v
BIOMATH 2011, Bulgaria
prs is a projective function defined by:
prs < u1 ; u2 ; :::; uv >= us :
generalized
nets
IFSs and
Clustering
IFDFC
Model
where W6,5 =“places l1, l3 and l4 are already
empty”
W6,6 is the negation of the predicate W6,5.
All α tokens are collected in place l6 in
stepwise manner without acquiring
characteristic there, and further they go to l5
where they obtain characteristic:
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
where
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
We must mention that here we use one special
capability of the GN-models: token β enters place
l7 with the above mentioned characteristic. Its
value is necessary for the further GN-functioning,
but this token is not necessary for the future
process. Therefore, it stays only in its (input for the
GN) place. Each one of the -tokens obtains in place
l the characteristic
x®i = ¹i (x9j ; ¸) = 1 ¡ (1 ¡ ¹i (xj ))¸ &ºi (xj ; ¸) = 1 ¡ (1 ¡ ¹i (xj ))¸(¸+1)
3
BIOMATH 2011, Bulgaria
(¸ = 0:95)
where λ is in the interval [0,1]
The same token obtains
in ®place l8 the characteristic
®
x
4
i
= pr1 x i :
3
generalized
nets
IFSs and
Clustering
W11,10 = “places l8 and l9 are already empty”
W11,11 is the negation of W11,10
IFDFC
Model
All α tokens are collected in place l11 and after that
enter place l10 where they obtain a characteristic
x®i = updated centroid with membership and non-membership values
5
BIOMATH 2011, Bulgaria
generalized
nets
where W14,12 = “places l8 and l9 are already empty and
the centroid is not changed”
IFSs and
Clustering
W14,13 = “places l8 and l9 are already empty and the
centroid is changed”
All α tokens are collected in place l14 and obtain a new
characteristic
IFDFC
Model
x®i = ¹i (xj ; ¸) = 1 ¡ (1 ¡ ¹i (xj ))¸ &ºi (xj ; ¸) = 1 ¡ (1 ¡ ¹i (xj ))¸(¸+1)
6
BIOMATH 2011, Bulgaria
After this, again step-by-step they enter place l13
without a new characteristic or place l12 with a
characteristic
x®i = 00 updated centroid 00
7
generalized
nets
IFSs and
Clustering
IFDFC
Model
where W15,15 = “places l12 and l14 are not empty”
W15,17 is the negation of the predicate W15,15
All α-tokens step-by-step are collected in place
l15 as follows. The first α-token enters place l15.
Any subsequent token entering l15 is united with
the first.
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After the union with the last α-token with the rst
one,xthis
token
thecentroids
characteristic
®i =united
list of the
c inobtain
number
8
generalized
nets
IFSs and
Clustering
IFDFC
Model
On the next time-step the α-token enters place l17 and
x®i = ¹iobtains
(xj ; ¸) =the
1 ¡characterisitc
(1 ¡ ¹i (xj ))¸ &ºi (xj ; ¸) = 1 ¡ (1 ¡ ¹i (xj ))¸(¸+1)
9
Finally, on the next time-step the -token enters place
l16 and obtains the characterisitc
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
where W19,18 = “places l16 and l17 are already
empty”,
W19,19 is the negation of W19,18
All α-tokens step-by-step are collected in place l19
where they are united with the first α-token entered
there, that had not obtain any characterisitc and
after this, it enters place l19, where it obtains a
characteristic
x®i = ¯nal centroids of the global clusters
11
BIOMATH 2011, Bulgaria
generalized
nets
IFSs and
Clustering
IFDFC
Model
Thank you!