Apso Computational Statistics Final
Transcript of Apso Computational Statistics Final
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Association Rule Mining through Adaptive
Parameter Control in Genetic Algorithm and
Particle Swarm Optimization
K.Indira
Research Scholar
Department of Computer Science and Engineering
Pondicherry Engineering College
Puducherry, India
919585287662
S.Kanmani
Professor
Department of Information Technology
Pondicherry Engineering College
Puducherry, India
Abstract: Association rule mining is a data mining task on a great deal of academic research has
been done and many algorithms are proposed. Association rule mining is treated as a twofold
process by most of the methods. It increases the complexity of the system and takes up more
time and space. Evolutionary Algorithms (EA) are fast growing search based optimization method
for association rule mining. Among EAs Genetic Algorithm (GA) and Particle Swarm Optimization
(PSO) are more suited for mining association rules. The bottleneck in both GA and PSO is setting
the precise values for their control parameters. Setting values to the control parameter is done
either through parameter tuning or parameter control. This paper proposes an adaptive
methodology for the parameter control of 1) Mutation rate in GA based on the fitness values and
mutation rate history. 2) Acceleration coefficients and inertia weight in PSO based on Estimation
of Evolution State (EES) and fitness value respectively. Both of the proposed adaptive methodswhen tested on five datasets from University of California Irvine (UCI) repository proved to
generate association rules with better accuracy and rule measures compared to simple GA and
PSO.
Keywords: Association Rule Mining, Genetic Algorithm, Particle Swarm Optimization,
Adaptive GA, Adaptive PSO, Estimation of Evolution State.
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected] -
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1. Introduction
Classification rule discovery and association rules mining are two important data
mining tasks. Association rule mining discovers all those rules from the training
set that satisfies minimum support and confidence threshold while classification
rule mining discovers a set of rules for predicting the class of unseen data. Minedrules can be effective in uncovering unknown relationships, providing results that
can be the basis of forecast and decision [1]. They have proven to be very useful
tools for an enterprise as it strives to improve its competitiveness and profitability.
The application area of association rule mining and Classification rule discovery
varies from market analysis to business intelligence, and now extended to
epidemiology, clinical medicine, fluid dynamics, astrophysics, and crime
prevention. Hence the accuracy of the rules mined and the relationship between
attributes has become an important issue. The standard rule mining methods such
as Apriori [2,19], FP growth tree [19,20] scans the whole dataset for each attribute
match, increasing the input/output overhead of the system. The rules generated are
with single objective aiming at accuracy alone and the number of rules generatedis vast. Pruning and summarization are needed to filter the significant rules [6].
The efficiency of association rule mining can be enhanced by
Reducing the number of passes over the database
Making the process as multiobjective
Sustaining the search space efficiently
Evolutionary Algorithms (EA) such as Genetic Algorithm (GA) and Particle
Swarm Optimization (PSO), provides solution for the above three requirements
while mining association rules [13, 24, 30]. Both methods generate rules withbetter predictive accuracy and reduced complexity. However one laborious aspect
of all EAs, including PSO and GA, is setting appropriate values for the controlparameter [46]. Mining association rules for classification task is proposed in this
paper using Adaptive Genetic Algorithm (AGA) and Adaptive Particle Swarm
Optimization (APSO).
In solving problems with GA and PSO, the properties affect their performance.
The properties of both GA and PSO depend on the parameter setting and hence
users need to find the apt value for the parameters to optimize the performance.
The interactions between the parameters have a complex behavior and so each
parameter will give a different effect depending on the value set for others.Without prior knowledge of the problem, parameter setting is difficult and time
consuming. The two major ways of parameter setting are through parameter
tuning and parameter control [16]. Parameter tuning is the commonly practiced
approach that amounts to find appropriate values for the parameters before
running the algorithm. Parameter control steadily modifies the control parameter
values during the run. This could be achieved either through deterministic or
adaptive or self-adaptive techniques [15].
Deterministic parameter control is made using a deterministic rule that modifies
the strategy parameter without any feedback. This method becomes unreliable for
most problems because the parameter adjustments must rely on the status of theproblem at current time. In self-adaptive approach the parameters to be controlled
are encoded into the candidate solution which may result to deadlock, good
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solution depends on finding the good setting of parameters and obtaining the good
setting of parameters depends on finding the good solution. Moreover, extra bits
are required to store these strategy parameters, so the dimensionality of the search
space is increased. Thus the corresponding search space becomes larger and hence
the complexity of the problem is increased.
Therefore adaptive method is the solution. There have been several attempts in
adapting the parameters of GA and PSO. The crossover and mutation rate both are
adopted based on reproduction rate by Srinivas and Patnaik [45] aiming high
convergence speed and precision. An adaptive GA that determines mutation and
crossover rate of an individual by its location in a two dimensional lattice plane is
proposed in [26]. The algorithm keeps diversity of these parameters by limiting
the number of individuals in each lattice.The role of inertia weight is consideredcritical for the convergence of PSO. An adaptive inertia weight is proposed by
Eberhart and Kennedy [14] and Inertia weight is linearly decreased over the
generations in Arumugam et al. [4]. The acceleration coefficients are adapted in
Ratnaweera et al. [39] dynamically and in Arumugam et al. [4] by balancing thecognitive and the social components. The acceleration coefficients are adjusted
based on evolution state estimator and the inertia weight is adapted through
evolution factor by Zhan et al. [52] to perform global search over entire search
space with faster convergence. The adaptive methodologies adjust the parameters
based on strategies independent of the data used in the applications.
The literature on adaptive strategies in GA and PSO focuses on the mechanisms
of adaptation over generation either decreasing or increasing the parameter values,
independent of the data involved to enhance the performance. As association rule
mining is mainly based on relationship or correlation between items in a dataset,
data dependent strategy of adaptation could generate better association rules both
qualitatively and quantitatively. The performance of the adaptive strategy for
mining association rules could be enhanced if the parameter adjustments are done
depending on i) the data involved and ii) fitness values used for evolution through
generations. In this paper the parameter control mechanism based on EES is
proposed for adopting the acceleration coefficients in PSO and the inertia weight
is adopted during the evolutionary process based on the fitness values. In adaptive
GA the mutation rate is made adaptive relying on the fitness value and mutationrate history.
The rest of the paper is organized as follows. Section 2 introduces thepreliminaries on association rules, GA and PSO. Section 3 reviews the literature
related to proposed methodology. In section 4 the proposed methodology for
mining association rules with Adaptive Genetic Algorithm and Adaptive Particle
Swarm Optimization is given. Section 5 reports the experiment settings,
experimental results and discussions. Finally section 6 draws the conclusion.
2. Preliminaries
This section briefly discusses about the association rules and their related factors.
The basic features of multiobjective optimization, Genetic Algorithm and ParticleSwarm Optimization are also discussed.
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2.1 Association Rule
Association rules are a class of important regularities in data. Association rule
mining is commonly stated as follows [1]: Let I = {i1,i2, , in} be a set of n binaryattributes called items. Let D = {t1, t2, , tm} be a set of transactions called the
database. Each transaction in D has a unique transaction ID and contains a subsetof the items in I. A rule is defined as an implication of the form xy where
x, y I and x y = . The set of itemsets xand y are called antecedent (left-hand-side or LHS) and consequent (right-hand-side or RHS) of the rule. Often
rules are restricted to only a single item in the consequent. The same is adopted in
this paper also.
There are two basic measures for association rules, support and confidence.
Support of an association rule is defined as the percentage/fraction of records that
containxyto the total number of records in the database. Support is calculatedusing the following equation
(1)Confidence of an association rule is defined as the percentage/fraction of the
number of transactions that contain to the total number of records thatcontain where if the percentage exceeds the threshold of confidence aninteresting association rule can be generated. (2)Confidence is a measure of strength of the association rule.
is
the support of both antecedent and consequent occurring together in the dataset.
2.2 Multiobjective Optimization
A general minimization problem of M objectives can be mathematically stated as:
Given = [x1, x2, . . . , xd], where d is the dimension of the decision variablespace,
Maximize : [ ], subject to :gj() 0, j= 1, 2, . . . , J, andhk(
) = 0, k = 1, 2, . . .,K, where
is the ithobjective function, M is the
number of objectives, gj() is the jth inequality constraint, and hk() is the k
thequality constraint.
A solution is said to dominate another solution if it is not worse than that solution
in all the objectives and is strictly better than that in at least one objective. The
solutions over the entire solution space which are not dominated by any other
solution are called Pareto-optimal solutions.
Association rules generated are usually large in numbers and have redundant
information. Much work on association rule mining has been done focusing on
efficiency, effectiveness and redundancy. Focus is also needed on the quality of
rules mined. In this paper association rule mining using GA and PSO is treated as
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a multiobjective problem where rules are evaluated quantitatively and
qualitatively based on the following five measures.
Predictive Accuracy
Laplace
Conviction Leverage
Lift
i. Predictive Accuracy (also Confidence)measures the effectiveness of the
rules mined. The mined rules must have high predictive accuracy.
(3)wherexis the antecedent andythe consequent.
ii. Laplace is a confidence estimator that takes support into account,
becoming more pessimistic as the support of decreases [12,18].It rangesbetween [0, 1] and is defined as
(4)iii. Conviction is sensitive to rule direction and attempts to measure the
degree of implication of a rule [8]. It ranges between [0.5, ]. Values farfrom 1 indicate interesting rules.
(5)
iv. Leverage also known as Piatetski-Shapiro measure [36], measures how
much more counting is obtained by the rule from the co-occurrence of the
antecedent and consequent from the union. It ranges between [0.25, 0.25]and is defined as (6)
v. Lift (also called interest by Brin et al. [8]) measures how far from
independence are antecedent and the consequent. It ranges within [0,+].A lift value of 1 indicates that the items are co-occurring in the database as
expected under independence. Values greater than one indicate that theitems are associated.
(7)The objective function or fitness function is derived to meet the above objectives
based on the confidence and support values of the mined rules. Maximization of
the fitness function is the goal.
(8)
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2.3 Genetic Algorithm
Genetic algorithm (GA) is a search method for solving optimization problems and
modeling evolutionary systems. A typical GA works with a population of
individuals, each being a solution for the optimization problem with an associated
fitness value. Successive generations of the population are produced based on theprinciple of survival of the fittest. The evolution process is achieved through
genetic operations, such as crossover and mutation, on the population of
individuals.
The steps in Genetic Algorithm are given below.
Genetic algorithm ( )
{
Initialize population randomly;
Evaluate fitness of each individual in the population;While stopping condition not achieved
{
Perform selection;
Perform crossover and mutation;
Evaluate fitness of each individual in the population;
}
}
In simple GA, the three basic operators namely, selection, crossover and mutation
operators are fixed apriori. The speed and success of the GA depends greatly on
the correct mix of GA operators and the probabilities of crossover and mutationoperations. The optimum parameters for these operators depend on problem on
which the GA is applied and also on the fitness of the current population.
2.4 Particle Swarm Optimization
Swarm Intelligence (SI)[28] is an innovative distributed intelligent paradigm for
solving optimization problems that originally took its inspiration from the
biological examples by swarming, flocking and herding phenomena in
vertebrates. Particle Swarm Optimization (PSO)[27] incorporates swarming
behaviors observed in flocks of birds, schools of fish, or swarms of bees, and even
human social behavior.
PSO is initialized with a group of random particles (solutions) and then searches
for optima by updating generations. During all iterations, each particle is updated
by following the two best values. The first one is the best solution (fitness) ithas achieved so far. This value is called pBest . The other best value that istracked by the particle swarm optimizer is the best value obtained so far by any
particle in the population. This best value is a global best and is called gBest.After finding the two best values; each particle updates its corresponding velocity
and position.
Each particle p, at some iteration t, has a position x(t), and a displacement velocityv(t). The personal best (pBest) and global best (gBest) positions are stored in the
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associated memory. The velocity and position are updated using equations 9 and
10 respectively. (9)
(10)
Where is the inertia weight is particle velocity of the ithparticle before updation is particle velocity of the ithparticle after updation is the ith, or current particlei is the particles numberd is the dimension of search space
rand ( ) is a random number in the range (0, 1) is the individual factor
is the societal factor
is the particle bestis the global bestParticles velocities on each dimension are clamped to a maximum velocity Vmax.
If the sum of accelerations causes the velocity on that dimension to exceed Vmax,
which is a parameter specified by the user, then the velocity on that dimension is
limited to Vmax. This method is called Vmaxmethod [42].
Inertia weight controls the impact of the velocity history into the new velocity and
balances global and local searches. Suitable fine-tuning of cognitive parameter
and social parameter c1 and c2 result in faster convergence of the algorithm and
alleviate the risk of settling in one of the local minima . The Pseudo code for PSO
is given below.
For each particle
{
Initialize particle
}
END
Do
{For each particle
{
Calculate fitness value
If the fitness value is better than the best fitness value (pBest) in
history
set current value as the new pBest
}
End
Choose the particle with the best fitness value of all the particles as the
gBest
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For each particle
{
Calculate particle velocity according to equation (9)
Update particle position according to equation (10)
}
End}
While maximum iterations or minimum error criteria is not attained
3. Literature review
The issue of parameter setting has been relevant since the beginning of the EA
research and practice. Eiben et al. [15,16] emphasize that the parameter values
greatly determinate the success of EA in finding an optimal or near-optimal
solution to a given problem. Choosing appropriate parameter values is a
demanding task.There are many methods to control the parameter setting duringan EA run. Parameters are not independent but trying all combinations isimpossible and the found parameter values are appropriate only for the tested
problem instances. Nannen et al. [33] state that major problem of parameter
tuning is weak understanding of the effects of EA parameters on the algorithm
performance.
In view of the potential important applications of GAs, many improvement ideas
have been proposed.Marseguerra et al. [63]proposed a drop by drop idea forprogressive Monte Carlo simulation to reduce the total running time of the GA
process. Kwon et al. [64] suggested a successive zooming genetic algorithm to
identify global solutions using continuous zooming factors. Paszkowicz [65]
equipped the GA with a dynamic penalty function. Ilgi and Tunali[66] carried out
a factorial experiment and regression analysis to identify the best GA parameter
settings. Back [5] embedded the mutation rate into chromosomes in GA to
observe the performance of the self-adaptive approach in different functions.
Spears [43]added an extra bit in the schematic of chromosome to investigate the
relative probability of operating two-point crossover and uniform crossover in
GA.
Besides these methods, an abundance of literature has proved that adaptive GAs
(AGAs) is able to improve the GA performance. San Jose-Revuelta [67] showed
that adaptive GA requires a much smaller population size, in the range of 1020,yet could obtain a satisfactory solution. Many approaches have been proposed foradjusting GA parameters such as mutation probability, crossover probability
[68,69] and population size [70] in the course of a genetic algorithm. Hop and
Tabucanon [21] presented a new and original approach to solve the lot size
problem using an adaptive genetic algorithm with an automatic self-adjustment of
three operator rates namely, the rate of crossover operation, the rate of mutation
operation and the rate of reproduction operation. Adaptive directed mutation
operator in real coded genetic algorithm was applied to solve complex function
optimization problems [37]. It enhances the abilities of GAs in searching global
optima as well as in speeding convergence by integrating the local directional
search strategy and the adaptive random search strategies. A hybrid and adaptivefitness function [10], in which both filter and wrapper approaches were applied
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The parameters involved in GA and PSO are
Table 1. Parameters of GA and PSO
Parameter GA/PSO Parameter Role
Mutation (pm) GA Maintains the genetic diversity of the
population from one generation toother
Crossover (pc) GA Sharing Information between
chromosomes
Inertia weight ( PSO Controls the impact of the velocityhistory into the new velocity
Acceleration
Coefficient c1
PSO Maintains the diversity of swarm
Acceleration
Coefficient c2
PSO Convergence towards the global
optima
4.1 Adaptive Genetic Algorithm (AGA)
Genetic algorithms when applied for mining association rules performs global
search and copes better with attribute interaction.
The
roulette wheel selection is
adopted where the parents for crossover and mutations are selected based on their
fitness values, i.e. if a candidate is having a high fitness value then the chances of
selection is high. The crossover and mutation operations are manifested by the
crossover and mutation rate usually defined by the user.
The efficiency of the rules mined by genetic algorithm mainly depends on the
mutation rate [7] and the crossover rate which affects the convergence rate. In the1960s, L. Fogel et al. [59] illustrated how mutation and selection can be used toevolve finite state automatons for a variety of tasks. Sophisticated versions of
evolution strategies, with adaptive mutation rates, proved quite useful for function
optimization tasks [58,61]. Recent studies confirm this view, illustrating the
power of mutation [62]. D. Fogel has continued the earlier work of L. Fogel and
makes an even stronger claim - that crossover has no general advantage over
mutation [60]. It is possible to implement mutation with a parameter that is
adapted during genetic algorithm search.
Tuning the value of the mutation rate becomes an important criterion for mining
association rules with GA. Higher mutation rate generates chromosomes muchdeviated from original values resulting in higher exploration time. Lower
mutation rate results in crowding of chromosomes towards the global optima
thereby limiting the search space. The dataset from which association rules are
mined is considered for modifying the mutation rate during evolution process. So
mutation rate is made adaptive based on the history of mutation rate and the
fitness value.
The algorithm for adaptive genetic algorithm for mining association rules is as
given below
Initialize population randomly;
Evaluate fitness of each individual in the population;
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While the stopping condition is not met
{
Perform selection;
Perform crossover and mutation;
Evaluate fitness of each individual;
Change mutation operator.}
The Michigan encoding method is applied for the chromosomes representation.
The fitness function used for evaluating the chromosome is based on equation 8.
The mutation operator is made adaptive to maintain and introduce the diversity of
the population throughout the evolution process. The adaption of the mutation rate
is made dependent on fitness value, to introduce the diversity of the population in
the future generations. The initial mutation rate set along with an adjustment
factor maintains the diversity of the population. The proposed adaptive mutation
rate derived from the objective function is given in equation 11.
(11)pm
(n+1) is the mutation rate of (n+1)th generation. The first generation mutation
rate is pm0, fmean is the mean fitness of itemset. fmax
(n+1)is the highest fitness of the
(n+1)th individual stocks. fin is the fitness of the nthindividual i. is the adjustment
factor, which is set within the range [0,1].
The selection of individuals for reproduction (crossover and mutation) is based on
fitness value. Adapting the mutation rate based on the fitness values of the itemset
results in generation of offspring for the new population with higher fitness value
[45]. This leads towards the global optima easily thereby maintaining the search
space effectively. The main drawback of GA is its lack of memory. The mutation
rate adaption based on the history of mutation rates solves this problem, thus
enhancing the performance of GA.
4.2 Adaptive Particle Swarm Optimization (APSO)
A swarm [17] consists of a set of particles {
1,,xm}of size m, moving within
the search space, S Rd, each representing a potential solution of the problem as
Find subject to appropriate constraints;Where is the particle, S is the search space, d the dimension of the search space,Rd isthe set of potential solution and F is the fitness function associated with the
problem, which we consider to be a minimization problem without loss of
generality.
PSO is mainly conducted by three key parameters important for the speed,
convergence and efficiency of the algorithm [48]: the inertia weightand twopositive acceleration coefficients(c1and c2). Inertia weight controls the impact of
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the velocity history into the new velocity. Acceleration parameters are typically
two positive constants, called the cognitive parameter c1and social parameters c2.
Inertia Weight plays a key role in the process of providing balance between
exploration and exploitation process. The linearly decreasing inertia weight over
time enhances the efficiency and performance of PSO [47]. Suitable fine-tuning ofcognitive and social parameters c1and c2may result in faster convergence of the
algorithm and alleviate the risk of settling in one of the local minima. To adapt the
parameters of PSO in the proposed adaptive PSO methodology for association
rule mining the parameter tuning is performed based on the evolutionary state to
which the data fit. The acceleration coefficients are adapted based on the dataset
from which association rules are mined. The inertia weight is adjusted based on
the fitness value of the individuals. The pseudo code for the Adaptive PSO is
given below
/* Ns: size of the swarm, C: maximum number of iterations, O f: the final output*/
i. t = 0, randomly initialize S0,
Initialize xi, i, i {1, . . .,Ns} /* xi : the ithparticle */ Initialize vi, i, i {1, . . .,Ns} /* vi: the velocity of the ithparticle*/ Initialize 1, c11, c21 /*: Inertia weight, c1,c2 :
Acceleration
Coefficients */
Pbi xi,i, i {1, . . .,Ns} /* Pbi: the personal best of the ithparticle */
Gb xi /* Gb : the global best particle */
ii. for t = 1 to C, /* C : total no of iterations */
for i = 1 to NsF(xi) = confidence(xi ) log (support (xi) (length(x) + 1)
/* F(xi) : Fitness of xi */
If ( F(xi) < F(Pbi))
Pbi xi /* Update particle best */
Gb min(Pb1, Pb2, , PbNs) /* Update global best */
adjust parameters(t, c1t, c2t) /* Adaptive Adjustment */vi(t) =
tvi(t-1)+ c1
tr1(Pbi xi) + c2tr2(Gb xi )
/* Velocity Updationr1, r2 : random values*/
xi(t)= xi(t-1) + vi(t) /* Position Updation */
At non dominated(StAt) /* Updating the Archive*/iii. Of Atand stop /* Of: Output*/
adjust parameters (t, c1t, c2t) in the above pseudo code is achieved throughadaptive mechanism newly proposed. The proposed approach distributes the
evolution of data into four states based on estimation of evolutionary states:
Convergence, Exploration, Exploitation and Jumping out.
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4.2.1 Estimation of Evolutionary State (EES)
During the PSO process, the population distribution characteristics vary not only
with the generation number but also with the evolutionary state. For example, at
an early stage, the particles may be scattered in various areas, and, hence, the
population distribution is dispersive. As the evolutionary process goes on,however, particles would cluster together and converge to a locally or globally
optimal area. Hence, the population distribution information would be different
from that in the early stage. In order to detect the different population distribution
information and use this information to estimate the evolutionary state is carried
out.
Based on the search behaviors and the population distribution characteristics of
the PSO Estimation of Evolution State is done as follows
a. The distance between particles is calculated using the Euclidean distance
measure for each particle i using the following equation (12)where N is the population size, xiandxjare the i
thand jthparticle in the
population respectively.
b. Calculate the evolutionary state estimator e, defined as
(13)Where
is the distance measure of the gBest particle,
,
are
the maximum and minimum distance measures respectively from step a.c. Record the evolutionary e factor for 100 generations for each dataset
individually
d. Classify the estimator e, to the state it belongs: Exploration, Exploitation,
Convergence, Jumping out for the datasets based on the evolutionary
states.
The evolutionary state estimator e over hundred generations is plotted for the zoo
dataset as shown in Figure 1 below. The evolution states are estimated based on
the e values. The state transition is different for the five datasets and are nondeterministic and fuzzy. Based on the evolution state estimator values the
classification is done for determining the intervals at which class transition occurs.
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Figure 1. Evolutionary state information robustly revealed by e at run time for Zoo dataset
The intervals arrived for the five datasets based on EES is as shown in Table 2.
Table 2. Classification into States by Evolutionary Factor
Datasets
State
LensesCar
Evaluation
HabermansSurvival
Postoperative
PatientZoo
Convergence 0.00.3 0.0- 0.15 0.0 - 0.4 0.0 - 0.5 0.0- 0.15
Exploitation 0.1- 0.4 0.15- 0.25 0.3-0.7 0.2- 0.6 0.1- 0.35
Exploration 0.2- 0.7 0.1- 0.3 0.6- 0.9 0.4-0.8 0.2- 0.4
Jumping out 0.6- 1 0.3- 1 0.8- 1 0.7-1 0.3-1
The change of state reflected as per the PSO sequence is Convergence
Exploitation Jumping Out Convergence.The formulation for numerical implementation of the classification for the
Habermans Survival dataset is as follows.
Case (a) Exploration: A medium to large value of e represents exploration,whose membership function is defined as
(14a)
Case (b)Exploitation: A shrunk value of e represents exploitation whose
membership function is defined as
(14b)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1 6 11 16 21 26 31 36 41 46
Evaluatione
stimator(e)
Iteration No.
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Case (c)Convergence: A minimal value of e represents convergence, whose
membership function is defined as
(14c)
Case (d)Jumping Out: When PSO is jumping out of a local optimum, the
globally best particle is distinctively away from the swarming cluster. Hence, the
largest values off reflect S4, whose membership function is, thus, defined as
(14d)Therefore, at a transitional period, two memberships will be activated, and e can
be classified to either state. The singleton method of defuzzification is adopted.
The distribution of all particles in PSO based on Euclidean distance calculation is
illustrated in Figure 2.
, Exploration , Exploitation, Convergence , Jumping Out , - Best particle of the swarmFigure 2. Population Distribution in PSO based on Evolutionary factor e
Based on the estimation into states using evolutionary estimation factor e and
fitness values the adaptation of acceleration coefficients and inertia weight is
performed.
4.2.2 Adaptive Control of Acceleration Coefficients
The acceleration coefficients are made adaptive through the classification of
evolutionary states. Parameter c1 represents the self-cognition that pulls theparticle to its own historical best position, helping in exploration of local niches
and maintaining the diversity of the swarm. Parameter c2 represents the socialinfluence that pushes the swarm to converge to the current globally best region,helping with fast convergence. Both c1 and c2 are initially set to 2 [29, 31,42]
based on the literature works. The acceleration coefficients are adaptively altered
during evolution according to the evolutionary state, with strategies developed.
The strategy to adopt for the four states is as given in Table 3. The values for and are empirically arrived for the datasets taken up for study.
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Table 3. Control Strategies for c1and c2
State/Acceleration
Coefficient
c1 c2
Exploration Increase by Decrease by
Exploitation Increase by Decrease by
Convergence Increase by Increase by
Jumping out Decrease by Increase by
The increase or decrease in the values of c1and c2arrived in Table 3 is discussed
below.
Exploration: During exploration particles should be allowed to explore as many
optimal regions as possible. This avoids crowding over single optima, probably
the local optima and explores the target thoroughly. Increase in the value of c1anddecrease in c2facilitate this process.
Exploitation: In this state based on the historical best positions of each particle
they group towards those points. The local information of the particle aids this
process. A slight increase in c1 advances the search around particle best (pBest)
positions. At the same time the slight decrease in c2avoids the deception of local
optima as the final global position has yet to be explored.
Convergence: In this state the swarm identifies the global optima. All the other
particles in the swarm should be lead towards the global optima region. The slight
increase in the value of c2 helps this process. To fasten up the process ofconvergence a slight increase in the value of c1 is adopted.
Jumping Out:The global best (gBest) particle moves away from the local optima
towards global optima, taking it away from the crowding cluster. Once any
particle in the swarm reaches this region then all particles are to follow the same
pattern rapidly. A large c2along with a relatively small c1value helps to obtain
this goal.
4.2.3 Acceleration Coefficient Bounds
The adjustments on the acceleration coefficients should be minimum so as to
maintain the balance between values c1and c2. Hence, the maximum increase or
decrease between two generations is bounded by the range of [0.02, 0.1]. The
value of is set as 0.02 based on trials and the value for is set as 0.06 similarly.c1 and c2 values are clamped in the interval [1.5, 2.5] in order to maintain the
diversity of the swarm and fast convergence [11]. The sum of the acceleration
coefficients is limited to 4.0[51], when the sum exceeds this limit then both c1 and
c2are normalized based on the equation 15.
(15)
The adaptation behavior of the acceleration coefficients c1 and c2through change
of states in estimation of evolution state for Zoo dataset is shown in Figure 3.
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Figure 3. Adaptation of Acceleration Coefficients through EES for Zoo Dataset
4.2.4 Inertia Weight Adaptation
The inertia weight controls the impact of previous flying experience, which is
utilized to keep the balance between exploration and exploitation. The particle
adjusts its trajectory according to its best experience and enjoys the information of
its neighbors. In addition, the inertia weight is also an important convergence
factor; the smaller the inertia weight, the faster the convergence of PSO. A linear
decrease in inertia weight gradually may swerve the particle from their global
optima [53]. Hence a nonlinear adaptation of inertia weight as proposed in the
given equation 14 is the solution. The global best particle is derived based on the
fitness value of the particles in the swarm. The proposed methodology for
adopting the inertia weight is based on the fitness values exhibited by the
particles.
{
( ) (16)
where f, fmin and favg are the fitness values, which are decided by the current
particles in the swarm. The minimum fitness value of all particles in swarm is
defined as fmin, while the mean value is selected as favg. As a result, all the
information searched, whatever is now or past, is utilized to update the future
velocity. It is also a way to deal with the problem of falling into the localoptimum. The adaptation of inertia weight for zoo dataset over generations is
given in Figure 4 as follows.
1.51.61.71.81.9
22.12.22.32.4
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
CoefficientValue
Generation Number
C1
C2
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Figure 4. Inertia Weight Adaptation for Zoo dataset
From Fig. 2, it can be concluded that, in general, the change in inertia weight
decreases linearly. In detail, it is adapted based on the fitness information
dynamically. The value of inertia weight swings around 0.461 frequently at later
generations. In this area, the algorithm can balance the searching and convergence
ability in the search space to avoid the risk of more likely to be trapped into local
minima, and improve the diversity of PSO.
5. Experimental Results and Discussions
The objective of the adaptive GA and adaptive PSO is to enhance the performance
of association rule mining by making the adaption dependent on dataset used. Tovalidate the performance of the proposed methodology, Nine datasets from
University of California Irvine (UCI) repository [32]: Lenses, HabermansSurvival, Car Evaluation, Postoperative Patient, Zoo, Iris, Nursery, Tic Tac Toe
and Wisconsin breast cancer have been used for the study.
The experiments were developed using Java and run in windows environment.
Each methodology was run 10 times on the dataset chosen. The number of
generations was fixed to 100. The datasets considered for the experiments is listed
in Table 4.
Table 4. Datasets Description
Dataset Attributes Instances Attribute
characteristics
Lenses 4 24 Categorical
Car Evaluation 6 1728 Categorical, Integer
Habermans Survival 3 310 IntegerPost-operative Patient Care 8 87 Categorical, Integer
Zoo 16 101 Categorical, Binary,
Integer
Iris 4 150 Real
Nursery 8 12960 CategoricalTic Tac Toe 9 958 Categorical
Wisconsin Breast Cancer 32 569 Real
0
0.1
0.2
0.3
0.40.5
0.6
0.7
0.8
0.9
1
1 713
19
25
31
37
43
49
55
61
67
73
79
85
91
97
InertiaW
eight
Iterations
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The first five datasets ( Lenses, Car Evaluation , Habermas Survival, PostOperative Patient and Zoo) were used for analyzing the performance of AGA and
APSO in comparison with simple GA and Simple PSO.
In adaptive GA mutation rate was made self adaptive, based on the analysis of
control parameters [22], where the mutation operation was found to influence theaccuracy of the system while the crossover operation affects the convergence rate
alone. Hence the crossover rate is kept at fixed value. For the mutation rate, in
addition to the feedback from earlier mutation rate, the fitness value is also
considered while adaption. This enhances the accuracy of the association rules
mined and makes the adaption data dependent.
The parameter setting for both the AGA and APSO methodology is given in the
Table 5.
Table 5. Parameter Setting for AGA and APSO
Parameter Name Parameter Value
Population Size Lenses : 20
Habermans Survival : 300Car Evaluation : 1000
Postoperative Care : 50
Zoo : 100
Number of Generations 100 0.3pc 0.3
0.25
c1, c2 2 0.9 0.4The scope of this study on association rule mining using adaptive GA and
adaptive PSO is to
Mine association rules with adaptive GA and PSO with multiobjectives
namely accuracy and significance of the rules generated
Compare the performance with simple GA and simple PSO (non adaptive
methodologies)
Analyze the performance over generations
Convergence analysis of AGA and APSO in terms of execution time
The AGA and APSO methodology are applied on the five datasets to mine
association rules and the predictive accuracy is recorded. The results are
compared with the simple GA and PSO performance [23] implemented on the
same platform with same datasets.
5.1 Performance of AGA
The predictive accuracy of the association rules mined by this method is
compared with the results of Simple GA as plotted in Figure 5. The predictiveaccuracy of the rules mined by the AGA is improved by 8% to 12.5% for the
datasets in comparison with the simple GA. The iteration at which maximum
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coefficients accordingly. Based on the fitness values the inertia weight is adapted.
Then the velocity modification is based on the state in which the particle lies. This
balances between exploration and exploitation thus escaping from premature
convergence. The predictive accuracy of the association rules mined is improved
through the evolution process.
The predictive accuracy for the adaptive PSO methodology over 100 generations
is shown in Figure 6.
Figure 6. Predictive Accuracy of Adaptive PSO over hundred Generations
The adaptive PSOs performance on Car Evaluation dataset and Zoo data set isconsistent as seen from the figure above. The performance on Habermanssurvival dataset and Post operative patient dataset is maximum at the end of
evolution. This avoids premature convergence at initial stages. For the Lensesdataset where the dataset size is small, global search space is maintained
effectively making the convergence possible even at earlier iterations.
The Laplace, Conviction, Leverage and Lift measures of the five datasets for the
adaptive PSO is recorded as in Table 7.
Table 7. Multiobjective Measures of APSO
LensesHaberman's
Survival
Car
Evaluation
Postoperative
PatientZoo
Laplace 0.52941 0.501608 0.502488 0.5028 0.5024
Conviction Infinity Infinity Infinity Infinity Infinity
Leverage 0.026 0.003039 0.002394 0.0301 0.0249
Lift 1.548 2.0458 1.9875 2.2478 2.4558
When the Laplace measure is away from 1, indicates that the antecedent values
are dependent on the consequent values and hence the rules generated are of
significant. The conviction measure which is infinity for all datasets show that the
rules generated is interesting. The Leverage measure being far away from 1 again
insists on the interestingness of the rules generated. The measure of Lift for all the
datasets, greater than1 signifies the dependency of the consequent on the
antecedent. The rules generated by Adaptive PSO are of importance based on theLift measure.
9394
95
96
97
98
99
100
10 20 30 40 50 60 70 80 90 100
Pre
dictiveAccuracy(%)
Iteration No.
Car
Haberman
Lens
Postop
Zoo
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The predictive accuracy achieved from the adaptive PSO methodology is
compared with the simple PSO as in Figure 7. The Adaptive PSO performs better
than the simple PSO for all the five datasets when applied for association rule
mining.
Figure 7. Predictive Accuracy comparison of Adaptive PSO with PSO
The number of rules generated by APSO with predictive accuracy less than 0.05
percentage of the maximum predictive accuracy arrived is plotted for mining
association rules against the simple PSO methodology for the datasets in Figure 8.
In terms of number of rules generated adaptive PSO performs better for
Hagermans Survival, Zoo and car evaluation datasets. The number of rules
generated for lens and postoperative patient dataset is marginally better thansimple PSO. As the number of instances in both datasets is small, the increase in
the number of rules is minimum.
Figure 8. Comparison of No. of Rules Generated by APSO and PSO
The execution time of APSO for mining association rules from the datasets in
comparison with PSO methodology is given in Figure 9. This is the time taken by
the system to generate association rules (CPU time). The number of iteration
when the predictive accuracy is at maximum is taken as point at which theexecution time is recorded. The execution time of the APSO for mining
association rules is more when compared to PSO. The adaption mechanism of
65
70
75
80
85
90
95100
Lenses Po-opert
Care
Habermans
Survival
Zoo Car
Evaluation
PredictiveAccuracy(%)
Dataset
PSO
APSO
0
5
10
15
20
25
30
35
40
Lens Postop Haberman Zoo Car
No.
ofRulesGenerated
Datasets
PSO
APSO
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acceleration coefficients based on evolution factor, and inertia weight adaptation
increases the complexity of the algorithm resulting in higher execution time when
compared to PSO.
Figure 9. Comparison of Execution Time of APSO and PSO
When number of instances in the dataset is less then increase in execution time is
marginal. Thus the execution time difference for mining rules from the Lenses,
Postoperative Patient and Zoo datasets are minimum for APSO than PSO. For the
Habermans survival dataset with moderate number of instances there isnoticeable increase in execution time, whereas for the car Evaluation dataset the
difference in execution time is large. The consistent performance of the APSO in
term of predictive accuracy throughout evolution and the predictive accuracy
achieved from APSO for mining association rules balances for the execution timedifference.
The major drawback of the simple PSO is its premature convergence where the
particle fixes some local optima as the target (global search area), where all
particles converge locally. One of the objectives of Adaptive PSO is to avoid the
premature convergence, thereby balancing between exploration and exploitation.
The iteration number at which predictive accuracy is high is plotted for both the
APSO and PSO methodologies for mining association rules for the datasets used
in Figure 10.
Figure 10. Comparison of Convergence of APSO and PSO
84
1243
2552
3425
6123
156
1841
6532
4586
9634
0
1000
2000
3000
4000
5000
6000
7000
80009000
10000
Lens Postop Haberman Zoo Car
ExecutionTime(ms)
Datasets
PSO
APSO
0
20
40
60
80
100
120
Lenses Zoo Car
Evaluation
Habermans
Survival
Postop
IterationNumber
Datasets
PSO
APSO
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The analysis of the performance of the five datasets through the adaptive methods
of GA and PSO has been compared with the simple GA and simple PSO. The
Adaptive GA and Adaptive PSO generated association rules with better predictive
accuracy and qualitative measures. To test the significance of the proposed
adaptive techniques on GA and PSO, the AGA and APSO is compared with Ant
Colony Optimization (ACO) methods for mining rules. Four more datasets from
UCI repository for which association rule mining using ACO has been carried out
in literature have been chosen. The predictive accuracy of the mined rules using
AGA and APSO for these datasets was recorded. The comparative results
obtained by AGA, APSO and ACO are shown in Table 8.
Table 8. Predictive Accuracy Comparison of AGA and APSO
Dataset
Adaptive
GA
Adaptive
PSO
Taco-Miner
zbakr etal [55]
Ant Colony
Parpinelliet al [57]
Ant
Colony
Liu et al
[54]
Ant
Colony
zbakret al [56]
Lenses 87.5 98.1 - - - -
Habermans
Survival68 99.6 - - - -
Car
Evaluation96 99.95 - - - -
Postoperative
Care92 99.54 - - - -
Zoo 91 99.72 - - - -
Iris 96 98.86 98 - - 98.67
Nursery 97.30 99.12 97.24 - - 97.16
Tic Tac Toe 88.36 98.76 - 75.57 76.58 97.59
Wisconsin
Breast Cancer92.54 98.75 - 96.97 94.32 -
The adaptive PSO methodology generates association rules with better predictive
accuracy when compared to other methodologies analyzed. Adaptive GA
generates association rules with predictive accuracies closer to the compared
methodologies.
The adaptive particle Swarm Optimization methodology when applied for mining
association rules generates rules with increase in predictive accuracy, increase in
the number of rules generated along with better rule set measures. The increase in
the execution time of APSO when compared to simple PSO methodology is also
minimum taking into account the complexity of the methodology and the increase
in accuracy achieved over PSO.
The balance in setting the global search space between exploration and
exploitation is also attained while mining association rules with APSO for all the
datasets used. This is noted from the shift in iteration number of APSO when
compared to simple PSO where highest predictive accuracy is noted. Adopting theacceleration coefficients c1 (cognitive factor) and c2(social factor) through EES
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methodology makes the adjustments in fixing up the global search space
effectively with the help of local optimas. The fitness value of the particles isused in adapting inertia weight. This helps in better balance between exploration
and exploitation when particles fly through the search space.
6. Conclusion
Genetic algorithm and particle Swarm Optimization are the fast and efficient
search methods. Both simple GA and simple PSO are problem dependent and
setting of the control parameters is a trivial task. This paper proposes adaptive
methods of parameter control for association rule mining using GA and PSO. The
fitness value plays a vital role in passing data over generations and adaptation
mainly depends on the fitness values of the chromosomes or particles. In Adaptive
GA the mutation operator was made self adaptive based on feedback of previous
generations and fitness function. The crossover rate was kept constant. In
Adaptive PSO the acceleration coefficients were adapted through estimation of
evolutionary state. In each generation, particles were set into one of the fourevolutionary states and based on the states the acceleration coefficients were
altered. The inertia weight parameter adaptation relied on the fitness values and
maximum and minimum values set for inertia weight.
When tested on five datasets from UCI repository both AGA and APSO generated
association rules with better predictive accuracy and good rule set measures. The
execution time of both the methods for mining association rules is found to be
optimum taking into account the accuracy achieved.
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Annexure 1
UCI Datasets
The datasets that have been used in this research work have been taken from the
UCI machine learning repository. The UCI Machine Learning Repository is a
collection of databases, domain theories, and data generators that are used by themachine learning community for the empirical analysis of machine learning
algorithms. The archive was created as an ftp archive in 1987 by David Aha and
fellow graduate students at UC Irvine. The details of the datasets used in this
study fromhttp://archive.ics.uci.edu/ml/about.html are presented below.
Lenses Dataset
Number of Instances : 24Number of Attributes : 4 (all nominal)
Attribute Information : 3 Classes
1 : the patient should be fitted with hard contact lenses,
2 : the patient should be fitted with soft contact lenses,
3 : the patient should not be fitted with contact lenses.
age of the patient: (1) young, (2) pre-presbyopic, (3) presbyopic
spectacle prescription: (1) myope, (2) hypermetrope
astigmatic: (1) no, (2) yes
tear production rate: (1) reduced, (2) normal
Number of Missing Attribute Values: 0
Habermans Survival Dataset
Number of Instances : 306
Number of Attributes : 4 (including the class attribute)
Attribute Information :
1. Age of patient at time of operation (numerical)
2. Patient's year of operation (year - 1900, numerical)
3. Number of positive axillary nodes detected (numerical)
4. Survival status (class attribute)
http://archive.ics.uci.edu/ml/about.htmlhttp://archive.ics.uci.edu/ml/about.htmlhttp://archive.ics.uci.edu/ml/about.htmlhttp://archive.ics.uci.edu/ml/about.html -
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1 = the patient survived 5 years or longer
2 = the patient died within 5 year
Missing Attribute Values : None
Car Evaluation Dataset
Number of Instances : 1728
(instances completely cover the attribute space)
Number of Attributes : 6
Attribute Values :
buying v-high, high, med, low
maint v-high, high, med, low
doors 2, 3, 4, 5-more
persons 2, 4, more
lug_boot small, med, bigsafety low, med, high
Missing Attribute Values : none
Post Operative Patient Dataset
Number of Instances : 90
Number of Attributes : 9 including the decision (class attribute)
Attribute Information :
1. L-CORE (patient's internal temperature in C):
high (> 37), mid (>= 36 and
-
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Missing Attribute Values : Attribute 8 has 3 missing values
Zoo Dataset
Number of Instances : 101
Number of Attributes : 18 (animal name, 15 Boolean attributes, 2 numerics)
Attribute Information : (name of attribute and type of value domain)
1. animal name Unique for each instance
2. hair Boolean
3. feathers Boolean
4. eggs Boolean
5. milk Boolean
6. airborne Boolean
7. aquatic Boolean
8. predator Boolean
9. toothed Boolean
10. backbone Boolean
11. breathes Boolean
12. venomous Boolean
13. fins Boolean
14. legs Numeric (set of values: {0,2,4,5,6,8})
15. tail Boolean16. domestic Boolean
17. catsize Boolean
18. type Numeric (integer values in range [1,7])
Missing Attribute Values : None
Iris Dataset
Number of Instances : 150 (50 in each of three classes)
Number of Attributes : 4 numeric, predictive attributes and the class
Attribute Information :Sepal length, Sepal width, Petal length and Petal width in cm
Class Types : Iris Setosa, Iris Versicolour and Iris Virginica.
Missing Attribute Values : None
Nursery Dataset
Number of Instances : 12960 (instances completely cover the attribute
space)
Number of Attributes : 8
Attribute Values :
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parents usual, pretentious, great_pret
has_nurs proper, less_proper, improper, critical, very_crit
form complete, completed, incomplete, foster
children 1, 2, 3, more
housing convenient, less_conv, critical
finance convenient, inconv
social non-prob, slightly_prob, problematic
health recommended, priority, not_recom
Missing Attribute Values : none
Tic Tac Toe Dataset
Number of Instances : 958 (legal tic-tac-toe endgame boards)
Number of Attributes : 9, each corresponding to one tic-tac-toe square
Attribute Information : (x=player x has taken, o=player o has taken,
b=blank)
1. top-left-square: {x,o,b}
2. top-middle-square: {x,o,b}
3. top-right-square: {x,o,b}
4. middle-left-square: {x,o,b}
5. middle-middle-square: {x,o,b}
6. middle-right-square: {x,o,b}7. bottom-left-square: {x,o,b}
8. bottom-middle-square: {x,o,b}
9. bottom-right-square: {x,o,b}
10. Class: {positive,negative}
Missing Attribute Values : None
Wisconsin Breast Cancer Dataset
Number of Instances : 699
Number of Attributes : 10 plus the class attribute
Attribute Information :
Sample code number, Clump Thickness, Uniformity of Cell Size,
Uniformity of Cell Shape, Marginal Adhesion, Single Epithelial Cell
Size, Bare Nuclei, Bland Chromatin, Normal Nucleoli, Mitoses and
Class (2 for benign, 4 for malignant).
Missing attribute values : 16