III the other fireflies to move toward the

III HYBRIDFIREFLY ALGORITHMFeature extraction creates new featuresfrom functions of original features, whereas feature selection returns a subsetof the features. The goal of the subset selection process is to obtain a subsetof features that allows better rates of correct identification than with theentire set of features.

Feature subset selection is an optimization problem,since the aim is to obtain any subsets that minimize the particular measure. Anoptimization problem can be solved through stochastic algorithms. In thisstudy, for better accuracy, hybrid firefly algorithm is proposed for feature selection18,19. A. Firefly algorithmFirefly is an insect that mostlyproduces short and rhythmic flashes that produced by a process ofbioluminescence. The function of the flashing light is to attract partners(communication) or attract potential prey and as a protective warning towardthe predator. Thus, this intensity of light is the factor of the other firefliesto move toward the other firefly.The light intensity is varied at thedistance from the eyes of the beholder.

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It is safe to say that the lightintensity is decreased as the distance increase. The light intensity also theinfluence of the air absorbs by the surroundings, thus the intensity becomesless appealing as the distance increase.B. Firefly RulesHybrid Firefly algorithm is based onidealizing the flashing characteristic of fireflies 13-17. The idealizedthree rules are:       i.

        Fireflies areattracted toward each other regardless of gender.     ii.        The attractiveness isproportional to the brightness, and they both decrease as their distanceincreases. Thus for any two flashing fireflies, the less bright one will movetowards the brighter one.

If there is no brighter one than a particularfirefly, it will move randomly.    iii.        The brightness of afirefly is determined by the landscape of the objective function.

For amaximization problem, the brightness is proportional to the objectivefunction’s value.C. Structure of firefly algorithmIn firefly algorithm, there are twoimportant variables; the light intensity and attractiveness. A firefly isattracted towards the other firefly having brighter flash than itself. Theattractiveness is depended with the light intensity.In the simplest case, for maximizationproblems, the brightness I of a firefly at a particular location x can bechosen as:  I (r) ? f (x).

However, theattractiveness ? is relative; it should be seen in the eyes of the beholder orjudged by the other fireflies. In the simplest form, the light intensity I(r)varies with the distance r monotonically and exponentially (Equation-4). .

(4) Sometimes, we may need a function whichdecreases monotonically at a slower rate. In this case, we can use theapproximation as follows (Equation-5): . (5) As firefly’sattractiveness is proportional to the light intensity seen by adjacentfireflies (6), we can now define the variation of attractiveness ? with thedistance r by .

(6) where ?0 is theattractiveness at r = 0. It is worth pointing out that the exponent ?(r*r) canbe replaced by other functions such as ?rm when m > 0.The distance r between any twofireflies i and j at xi and xj , respectively, is theCartesian distance defined by (Equation-7): rij = ||xi – xj || =  . (7)  The movement of a firefly i attractedto another more attractive (brighter) firefly j is determined by(Equation-8)  xit+1 = xit +?0 e-? ( r(i,j)* r(i,j) ) (xjt -xit) +?t ?it . (8) where the second term is due to theattraction, the third term is randomization with ?t being therandomization parameter, and ?it is a vector of randomnumbers drawn from a Gaussian distribution or uniform distribution at time t.If ?0 = 0, it becomes a simple random walk. On the other hand, if ?=0, it reduces to a variant of particle swarm optimization. Furthermore, therandomization ?it can easily be extended to otherdistributions.

The hybrid Firefly Algorithm is described as follows:D. The standard Firefly AlgorithmObjective Function, which is to bemaximized, f(x), x = (x1, x2,…, xd)TGenerate initial population offireflies xi ( i = 1, 2,…, m)Light intensity Ii at xiis determined by f(xi)if Ij < IithenMove firefly i towardsfirefly j in d-dimension;end ifAlternativeness varieswith distance r via e–?rfor i = 1: n dofor j =1: i do  Define light absorption coefficient ?  while t < Max Generation do  Evaluate new solutions and update lightintensityend for j end for iRank the fireflies and findthe current bestend while The initial population is considered asmean value of the duration for individual users and objective function is thefeature subsets obtained by using FA. Each mean value is chosen and comparedwith every other mean value in population and then the largest difference areneglected (i.e. the light intensity) to find the subset. The similar meanvalues are considered as single in a set.

Hence the irrelevant and repeateddata are removed using FA in order to find subset. E. StandardDifferential EvolutionDifferential evolution (DE) was proposed byStorn and Price in 1996, which uses a vectorized mutation operator and twoforms of crossover (either exponential or binomial) to evolve from the randomly generated, initial starting points to thepotentially optimal solution. There are many DE variants. In this paper, we usethe so-called DE/rand/1/bin scheme/variant.This variant is probably the most widely used in practice, which can be brieflydescribed as follows 12.

For a given D-dimensional minimization problem, apopulation consists of n individualsolution vectors. This means that the updateis accepted only if a better objective is achieved. Algorithm2 summarizes thebasic steps of the simple differential evolution algorithm.Algorithm 2 Pseudo code for the standard DE algorithmStep 1: The first step is the random initialization of the parentpopulation. Randomly generate a population of (say) NP vectors, each of ndimensions: xi,j= xmin,j + rand(0, 1)(xmax,j-xmin,j), where xmin,j and xmax arelower and upper bounds for jth component respectively, rand(0,1) isa uniform random number between 0 and 1.Step 2: Calculate the objective function value f(Xi) for all Xi. Step 3: Select three points from population and generate perturbedindividual Vi using equation (1a).Step 4: Recombine each target vector xi with perturbed individual generatedin step 3 to generate a trial vector                Ui.

Step 5: Check whether each variable of the trial vector is within range. Ifyes, then go to step 6 else make it                    within range using ui,j =2*xmin,j – ui,j ,if ui,j < xmin,j and ui,j =2* xmax,j - ui,j , if ui,j>xmax,j, and go to step 6. Step 6: Calculate the objective function value for vector Ui.

Step 7: Choose better of the two (function value at target and trial point)using equation (3) for next generation. Step8: Check whether convergence criterion is met if yes then stop; otherwisego to step 3.F. Hybrid Firefly Algorithm(HDFA):Both the firefly algorithm and Differential Evolution algorithmshave their own advantages and they both work well for a wide range ofoptimization problems.

The following algorithm describes the HDFA algorithm,which combines the advantages of Firefly and Diffuse Evolution algorithms.Algorithm 3 Pseudo-code for the HDFA algorithmBeginDividethe whole group into two groups: G1 and G2Initializethe populationsG1 and G2Evaluatethe fitness value of each particleRepeat   Do in parallel   Perform FA operation on G1   Perform DE operation on G2 End Do inparallel   Update the global best in the wholepopulation   Mix the two groups and regroup them randomlyinto new groups: G1 and G2   Evaluate the fitness value of each particleUntila terminate-condition is metEndIV. RESULTSExperiments were carried out with thekey stroke data collected from 25 users, with 10 valid samples from each user,in a span of one month. Duration, latency and digraph timing were measured foreach sample and their mean, median and standard deviation were stored in thereference file.

A system using Visual Basic was created, to organize thesemeasurements. The system registered each typing entry and stored thecorresponding data in samples’ file. Table 1 shows the measured keystrokefeature values of duration timing of user-1 for the password “welcome” and thecorresponding each key value obtained in milliseconds. The computations asshown in Table.1 serve as the initial reference signature or template for featurestring. Extracted features are optimized using HDFA. Firefly was implementedusing the algorithm as explained in section 3.

It was observed, from Figure 2, that the convergencerate of PSO was 58 ms and took 425 generations to reach an optimal solution andGA took 68 ms and 440 generations to converge and produced only near optimalsolution. ABC was very competitive, performed relatively well and showed lowercomputational overhead than PSO and GA and also fewer parameters to set. However,HDFA has proved to be most efficient algorithm in this case, with convergencerate as 40 ms and 360 generations to reach the optimum solution; which is infact 41 % and 18 % less respectively as compared to those of maximum valuestaken by other algorithms. 

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