Abstract
The Swine Influenza Model Based Optimization (SIMBO) family is a newly introduced speedy optimization technique having the adaptive features in its mechanism. In this paper, the authors modified the SIMBO to make the algorithm further quicker. As the SIMBO family is faster, it is a better option for searching the basin. Thus, it is utilized in local searches in developing the proposed memetic algorithms (MAs). The MA has a faster speed compared to SIMBO with the balance in exploration and exploitation. So, MAs have small tradeoffs in convergence velocity for comprehensively optimizing the numerical standard benchmark test bed having functions with different properties. The utilization of SIMBO in the local searching is inherently the exploitation of better characteristics of the algorithms employed for the hybridization. The developed MA is applied to eliminate the power line interference (PLI) from the biomedical signal ECG with the use of adaptive filter whose weights are optimized by the MA. The inference signal required for adaptive filter is obtained using the selective reconstruction of ECG from the intrinsic mode functions (IMFs) of empirical mode decomposition (EMD).
1. Introduction
Genetic algorithm, particle swarm optimization, bacterial foraging optimization, differential evolution, evolutionary programming, and so forth are the stochastic optimizers that have drawn the attentions in recent time [1–7]. In these, a population of the solutions is utilized in the search process. These algorithms are capable of exploring and exploiting the promising regions in the search space but take relatively longer time [1]. Hence, algorithms are combined for required properties like faster exploration and exploitation capabilities making the combination faster and accurate [8–11].
Since 1960, genetic algorithm (GA) is a potential optimizer in the field of optimization [2]. In GA, chromosomes are used to encode problem parameters and to search for solution with iterations or generations in a natural selection process using genetic operations such as selection, crossover, and mutation. GA offers a robust search mechanism in complex spaces applicable to problems in various fields like science, engineering, commerce, and so forth.
The hybrid algorithm tries to keep a balance in processes of exploration and exploitation achieving global optimization. Hybrid methods like least-square-fuzzy BFO [8], GA-BFO [9], and BFO-Nelder-Mead [10] are developed for taking advantage of such combinations. In developing some of the memetic algorithms, GA is used, in which gradient-based information is utilized for local search. But gradient-based methods fail in the cases of multimodal and nondifferentiable functions [12]. Population-based search algorithms have advantages over the gradient-type searches as the former is capable of overcoming the local optima issue. But some of the population-based search algorithms like particle swarm optimization (PSO) sometimes have tendency of premature convergence. So, to overcome that, stochastic local search is utilized to take out the solution from local trapping [13]. This property of heuristic search methods having the capability to overcome premature convergence makes them suitable for local refinement in the memetic algorithm [14]. One such combination is known as memetic algorithm (MA). It is inspired by both Darwinian’s principle of natural evolution and Dawkins’ notion of a meme as a unit of information or idea transmission from brain to brain or cultural evolution capable of individual learning [1, 15].
This paper proposal is in two ways, that is, modification of the newly introduced algorithms, Swine Influenza Model-Based Optimization (SIMBO) [16], and development of the new variants of memetic algorithm (MA) having genetic algorithm (GA) as the main algorithm with modified SIMBOs as local search mechanisms.
The paper is organized in seven folds. Section 2 briefs the SIMBO. In Section 3, modifications of SIMBOs are discussed, and in Section 4, development of memetic algorithms with the use of modified SIMBOs as local searches is discussed. The experimental results of the proposed modified SIMBO and the proposed MAs on benchmark functions are presented in Section 5. The application of ECG signal processing is discussed in Section 6, and Section 7 is the conclusion.
2. Swine Influenza Model-Based Optimization (SIMBO)
The Swine Influenza Model-Based Optimization (SIMBO) developed by Pattnaik et al. [16] introduces a novel optimization scheme. The SIMBO is imitated the concept in the models of swine flu that is, Susceptible-Infectious-Recovered (SIR) [17]. This is similar to the development of algorithm family based on natural phenomena in physics [18]. The initial population is divided into three classes: susceptible (S), infectious (I), and recovered (R) [17]. The population with single infectious individual goes susceptible (S). The infectious individual (I) transmits influenza towards all susceptible individuals (S). Afterwards, researchers modified the SIR model by incorporating seasonality, vaccination, treatment, quarantine, isolation, and so forth [17]. The susceptible individuals (S) can be vaccinated, and infected individuals (I) can be treated with antiviral drugs (T) [17]. SIR model was further extended by adding two behavioural modifications: quarantine and isolation measures (Q) [17]. The expansion of SIMBO is with the help of treatment (SIMBO-T), vaccination (SIMBO-V), and quarantine (SIMBO-Q) depending on the probabilities. These SIMBO variants are employed in optimization of the multimodal functions and this shows the enhancement in convergence as well as accuracy. In the process, first, swine flu test depending on the dynamic threshold is recognizing confirmed cases of swine flu. Then the susceptible cases in the community population are recommended the vaccination for gaining immunity against swine flu. The confirmed cases of swine flu in the population are quarantined. Also, the suspected cases are administered the antiviral drugs and are treated. The antiviral drugs quantity administered to the individual is based on the current health of the individual. The states of the individuals are updated with direct vaccination/quarantine and indirect treatment in SIMBO-V and SIMBO-Q. Details on SIMBO can be obtained from [16].
2.1. Comparison with Artificial Immune System
Artificial immune system (AIS) is inspired by the immunology of the body’s immune system that categorizes all cells (or molecules) within the body as self-cells or nonself-cells [19, 20]. In implementing a preliminary artificial immune system, four steps are present, namely, encoding, similarity measure, selection, and mutation [19, 20]. There are a variety of the AIS techniques developed like negative selection algorithms, artificial immune networks, clonal selection algorithms, danger theory, and dendritic cell algorithms [19, 20]. These are based on harmful entity suppression mechanism inside the body, whereas the SIMBO is based on the mechanism that suppresses the viral spread in the human populations by adopting some techniques like treatment, vaccination, and quarantine. The biological immune system is highly distributed, highly adaptive, self-organising in features, preserving a memory of earlier period came across, and has the ability of continuous learning about new experiences [19, 20] that is tried to adopt in AIS. In the SIMBO, also the adaptability and memory in terms of best health information are present [16]. The AIS is made for optimization and learning, whereas SIMBO is purely an optimization technique [16, 19, 20].
3. Modifications in Swine Influenza Model-Based Optimization (SIMBO)
The modifications in the SIMBO are incorporated in the proposed work as (i) constraining the individual range in the process of optimization, and (ii) new process of state change, having population diversity in the vicinity of the current individual, in SIMBO-V and SIMBO-Q.
3.1. Modified Swine Influenza Model-Based Optimization with Treatment (Modified SIMBO-T)
The key terms and definitions used in SIMBO have been used in the modified SIMBO. These are listed below.
Key Terms and Definitions Day (D): current generation or iteration. TD: total number of days or generations. State (S): individual position. Health (H): fitness. Pandemic state (PS): pandemic (global) best state in all the individuals. Pandemic health (PH): fitness value corresponding to pandemic state. Primary symptoms of swine flu: swine flu symptoms are fever, runny nose, cough, sore throat, headache, body aches, fatigue, and chills. Swine flu test: the laboratory test-based verification of an infected individual with swine flu. Vaccination (V): it vaccinates against the swine flu to the susceptible/unexposed population. Amount of vaccine (Vc): the vaccine dose supplied to the individual. Dose: antiviral drugs supplied to individual for curing the swine flu. μ: vaccination probability. α: recovery probability. : quarantine probability. Momentum factor of dose (Md): it is for controlling the individual dose. Momentum factor of state (MS): it is for controlling the individual state.
The list of the parameters in SIMBO used in mSIMBO is shown in Table 1 with their values and ranges. The same tuning of the parameters is used in cases of all problems and parameters are not tuned differently for every problem.
The modified SIMBO-T, which is named as mSIMBO-T, also does optimization by treatment mechanism using probability of treatment similar to that of basic SIMBO-T. Initially, all individuals are susceptible due to the infected individual and the treatment is given to all susceptible cases by antiviral drug dose. The amount of dose is dependent on primary and secondary symptoms as well as current health and pandemic health.
The mSIMBO-T is also doing optimization through two steps as given below.
Steps of mSIMBO-T Step 1: evaluate health. Step 2: treatment within limits.
3.1.1. Evaluate Health
The health of the individual depends upon given fitness function. Initially, the health of all individuals is evaluated for checking susceptible patient to swine flu as diagnostic confirmation.
3.1.2. Treatment
Treatment is based on symptoms and is often based on a trial and error. A physician generally begins the treatment with a typical dose and monitors for a response as well as side effects. The standardization of a dose is based on population dose response characteristics and not on individual optimal outcomes [16]. When the dose is accustomed eventually depending on the individual feedback, then optimal result is reached faster.
Similar to the basic SIMBO algorithms (i.e., SIMBO-T, SIMBO-V, and SIMBO-Q), the modified SIMBO algorithms, which are named as mSIMBO-T, mSIMBO-V, and mSIMBO-Q, also have the amount of dose based on primary symptoms, secondary symptoms, current health, and pandemic health, where the primary symptoms and secondary symptoms are shown in (1) and (2). where Fe: fever, Co: cough, fathead: fatigue and headache, NV: Nausea and Vomiting, and Dai: diarrhea. The first term in (1) is total influence of primary symptom and second term increases the primary symptom per day: The dose administered to the individual is given by (3). After the progress in the health of the individual, the dose is reduced due to the last two terms in (3) as the current health of the individual is closer to the pandemic health. The momentum factors are utilized in controlling the dose and state of individual for current iteration those are found based on their previous iteration status. For applying the treatment, the probability of recovery used is α = 0.2. As the part of proposed work, the modification is incorporated as: after calculating the dose it is checked for its value with the help of range and limited it to the feasible range. With this controlled dose, the individual state is changed as indicated in (4): Here, one of the modifications is incorporated in the SIMBO family of the algorithms as applying boundary constrain on the dose keeps it within the search range. In the modified SIMBO algorithms, for such constrain, the dose is checked for its value and it is put at one of the extremum of range that is nearer if it is out of range. As the dose is within the range, the individual will also eventually return to the search range. When the feasible range is (range1, range2) then repair is incorporated with application of this feasibility constrain on the dose as shown in (5):
3.2. Modified Swine Influenza Model-Based Optimization with Vaccination (mSIMBO-V)
It does the optimization by utilizing the vaccination and treatment mechanism. The susceptible individuals go through the swine flu test. The susceptible class is presented for swine flu vaccination and acquiring immunity. The SIMBO-V mechanism of optimization has four steps as shown below and mSIMBO-V also utilizes these.
Steps of mSIMBO-V Step 1: evaluate health. Step 2: swine flu test. Step 3: modified Vaccination. Step 4: treatment within limits.
The explanation of Steps 2 and 3 is as follows.
3.2.1. Swine Flu Test
Dynamic threshold is used for evaluation of the patient states due to affect of swine flu virus. When current health values of individuals are more than the dynamic threshold (D_Threshold), they are susceptible and the remaining other are recovered cases. Dynamic threshold is calculated by utilizing the health information of better half of population, quarantine or vaccination probability (β or μ), and primary symptoms as given in (6):
The following is the algorithm showing the application of swine flu test: for k = 1 : TI if current_health (k) > D_Threshold Susceptible case else Recovered case end end
3.2.2. Vaccination
With the application of a vaccine, the vaccination is carried out for the protection from disease with the help of immunity [21]. In the basic SIMBO-V, the vaccination is done by the amount of vaccine by multiplying the state of individual with random value in the range of 0 to 1 [16].
In mSIMBO-V, this process of the state change of the individual is also modified and multiplicand Vc modified as in (7) that randomly perturbs the original state value in the range from 100% to −50% of original state value to get all directional diversities for changing the health of individual: The probability of vaccination (μ) is selected high (0.8) for having higher exploitation in the neighbourhood of the individual by direct state change like the local refinement.
The following is the algorithm showing the application of modified vaccination: for k = 1 : TI if rand > μ if current_health (k) > D_Threshold if flag (k)= =0 flag (k) = 1 end end end end
3.3. Modified Swine Influenza Model-Based Optimization with Quarantine (mSIMBO-Q)
It does the optimization with quarantine and treatment mechanism. Out of all susceptible individuals who undergo the swine flu test, the confirmed cases have quarantine from population. The treatment is applied to all individuals by dose amount based on current health. Similar to the basic SIMBO-Q, mSIMBO-Q also does optimization in four steps as given below. Steps 1 and 4 of mSIMBO-Q are the same as those of mSIMBO-T. Step 2 is the same as the above explanation, whereas Step 3 is explained below. Steps 1, 2, and 4 of mSIMBO-Q are similar to those of mSIMBO-V.
Steps of mSIMBO-Q Step 1: evaluate health. Step 2: swine flu test. Step 3: modified Quarantine. Step 4: treatment within limits.
3.3.1. Quarantine
Quarantine is the separation and isolation of individuals or restriction forced on unbound reposition to prevent contagious disease spread. The confirmed cases in swine flu tests are isolated or quarantined. The original algorithm quarantines the individual by multiplying the best state of the individual with that is a random number in the range from 0 to 1.
The modification introduced in calculating the value of is used for quarantining the individuals. This is able to generate new individual in the vicinity to the original value of the individual by perturbation ranging from 100% to −50%. This introduces the diversity in all directions in the population and also keeps the individual in the vicinity to the original cluster of the population. This is achieved with the help of (8) shown below: The best state is chosen because quarantine generates the individual with probably having immunity and best health for faster convergence. If β is greater than rand and current health of individual is greater than D_Threshold, then individual is quarantined. The probability of quarantine (β) is kept high (0.8) for having more exploitation in the neighbourhood of the best individual having local refinement.
The following is the algorithm that shows the application of modified quarantine: for k = 1 : TI if rand > β if current_health (k) > D_Threshold end end end
4. Memetic Algorithm Using Local Search as Modified SIMBO Algorithms
Hybridization of genetic algorithm (GA) with local search algorithm is memetic algorithm (MA) [1, 2, 14, 15, 22, 23]. In general, MA is a synergy of evolution and individual learning [1, 22], which improves the capability of evolutionary algorithms like GA to find optimal solutions accurately in function optimization problems with higher speed of convergence. Genetic algorithm (GA) is nature-inspired optimization technique that represents an intelligent exploration having a random search confined within a defined search space for solving a problem optimally with the help of population [24]. Elitism helps prevent the loss of good solutions once they are found. This also can speed up the performance of the GA significantly [25, 26]. Elitism is used to keep the best solution intact without changing it due to the evolution in the evolutionary algorithms like genetic algorithm [24–26]. These best solutions are used for the reproduction, which helps the algorithm to converge fast with better accuracy.
The unique aspect of the MA is that the chromosomes and offspring are facilitated to gain some experience with a local search process in between regular evolutionary process [15, 27]. The local search process is inspired by individual learning using meme, which is a unit of information or idea that gets transmitted. Similar to the GA, the MA also creates an initial population randomly and searches the fitness landscape. The local search process drags solutions in the direction of local optima. These improvements go on accumulating over all generations, resulting in a larger improvement in the total performance [27]. The meta-heuristic search mechanism in the memetic algorithms offers it with the speed and quality of convergence [28].
Evolutionary algorithms follow heuristic search procedures that incorporate random selection and variation. Therefore, Swine Influenza Model-based Optimization Algorithms (SIMBOs) are modified and used as local searches in the main algorithm GA to make MAs. In this work, the variants of MAs are named like memetic algorithm with BLX- and SBX crossover operators, having mSIMBO-T for local search, as MA-BLX-T and MA-SBX-T, respectively. Similarly, MAs with mSIMBO-V as local search are named as MA-BLX-V and MA-SBX-V; also with mSIMBO-Q as local search they are named as MA-BLX-Q and MA-SBX-Q. The modified SIMBOs are population-based heuristic algorithms derived from SIR model of swine influenza and are utilized for local search in development of the new variants of memetic algorithm that converges in cases of unimodal, multimodal, shifted, rotated, and hybrid composite functions.
The GA is used for the purpose of exploration and the modified SIMBOs as local search for exploitation in the meta-heuristic combination of the proposed MAs. So, this gives a better converging capability to the algorithm for global convergence with better accuracy in convergence besides speed along with higher success rate. The inherent properties of basic SIMBO family are present in mSIMBO-T, mSIMBO-V, and mSIMBO-Q and they are having the dynamic adaptation for every individual with the help of learning from its neighbours facilitating the mSIMBO-T, mSIMBO-V, and mSIMBO-Q in dealing with complex, multimodal search landscapes efficiently. When these mechanisms are used in searching the search landscapes under limited exposure of local areas of landscapes with local boundary constraints in memetic framework, these give the capability of finding solutions in the complex local areas also. The exploration-exploitation balancing adaptation in the evolutionary algorithm like modified SIMBO family inherently has its local refining characteristics and again it is utilized in refining the smaller area of landscape in MA framework making it sort of meta-meme in nature. This is observed in the memetic framework under consideration.
The GAs and the MAs based on real number representation are called real-coded genetic algorithms (RCGAs) and real-coded memetic algorithms (RCMAs), respectively [2, 23]. The population is real coded for both evolving algorithm and local search algorithm. The evolving algorithm has the population that is obtained by generation process. The generation process generates the uniformly distributed random population in the specified range in [7] for particular function shown in Table 2. The fitness of the population is determined and current best individual is opted for the elite preservation. Then this population goes under the GA operations such as crossover and mutation.
The crossovers used are BLX-αcrossover [30] and SBX crossover [31]. The offsprings and are generated from parents and with “” dimension. In BLX-α crossover, is chosen randomly between the interval [(), ()] with the condition < and = max(, ) − min(, ). In SBX crossover, the effect of the one-point crossover of the binary representation is tried to emulate. The crossover generates two offspring and , where is a sample from random generator having density function as shown in (9):
This distribution is obtained by transforming with uniform random number source (0, 1) in (10) which is expressed as where is taken for better results. Exploitation capacity of operator increases for higher values of . The mutation mutates the individual in the specified range of the variables to change it randomly under uniform normal distribution. Elitism speeds up the performance of the GA largely, which also prevents the loss of good solutions once they are found [25, 26]. The elitism is also used in the MA for enhancing its convergence process.
The local search algorithm is evoked after a decided number of iterations or generations of evolving algorithm (known as Glocal) which is selected as 2 for better result. In the MA, local searching is of the search space around best candidate solution obtained due to the previous process of exploration. The smaller population for the refinement is generated around the current best individual by little perturbations.
Then the individuals undergo treatment, quarantine, and vaccination processes according to the threshold, state of the individual, quarantine and vaccination probabilities, and so forth. The calculated dose used for treatment depends on primary and secondary symptoms, current health and best health of individual, and previous dose for the same individual. Then the controlled dose is applied to the individual for recovery. The best individual is found out in all iterations during the refinement process. After the local refinement, the better result due to the refinement process is taken back in the evolving algorithmic process. It strengthens the algorithm for convergence of shifted and/or rotated unimodal/multimodal single/composite well-known standard benchmark functions [7] and the functions in the CEC 2005 [29], which are listed in Table 2. The performance of the algorithms is checked with 38 standard benchmark functions shown in Table 2. The proposed algorithms are proved on the standard benchmark functions used and the performance is noted by determining accuracy with the help of error and convergence speed by the number of fitness function evaluations (NFFEs).
Pseudo-Code of the Proposed MA Initialization: Generate a random initial population Number of generation: = 0, Glocal = 0 while Stopping conditions are not satisfied do Evaluate all individuals in the population Find current best individual If Glocal = 2 for best individual do (refinement by modified SIMBO) Obtain the local population around the current best individual by small perturbation While stopping conditions are not satisfied do Evaluate and sort the population Classify the individuals in classes S, I and R Update current_best_individual, current_best_health Calculate dynamic threshold (DT) Classify and tag individuals into S, I, R classes Quarantine or vaccination S & I individuals based probability (β or μ), DT, health Calculate Dose for applying to individual Provide treatment to S based on probability α, health and PH end while Glocal = 0 Return best individual to evolving algorithm end if Select individuals for crossover based on crossover probability Pc Crossover the parents by BLX-αcrossover or SBX crossover operators Correct the feasibility of the produced individuals Mutate some of the descendant population based on mutation probability Pm Replace the old population by new preserving the elite ++, Glocal++ end whilewhere is generations, Glocal is counter for application of local refinement, current_best_individual is current best individual from local population, current_best_health is current best health of the individual which is among the local population, Pc is crossover probability, and Pm is mutation probability.
5. Experimentation
The experimental results on benchmark functions using modified SIMBO variants and modified SIMBO-based memetic algorithms are presented here in this section.
5.1. Parameter Setting and Experimentation
In the genetic as well as in memetic algorithms, a population of 100 individuals of real-valued representation is used. The crossover operator used is the BLX-αcrossover or SBX crossover with crossover probability set to 0.8. For BLX-αcrossover, α = 0.3 and for SBX crossover, are used. The mutation probability is set to 0.05. The algorithm uses generational replacement of individuals preserving one elite. Two stopping criteria are used either the number of fitness function evaluations (NFFE as in Table 2 for basic functions; and 20000 or 100000 for 02 or 10 dimensions, resp., in case of CEC 2005 functions) or an error value of 10−8. In case of basic SIMBO family, modified SIMBO family for its individual performance validity and for combining modified SIMBO variants as local search with GA for making MA, all similar parameters are used, like real-coded population size 100, the treatment probability of recovery α = 0.2, probability for quarantine β = 0.8, and probability for vaccination μ = 0.8. The ensemble strategy to tune parameters and operators can be useful to develop further the more adaptive techniques that collectively may take care of parameters [32]. All algorithms with various dimensions, namely, 02, 10, 30, and 50 are executed for 25 independent trials and the best errors in trails (runs) are used for averaging and calculating standard deviation (Std Dev) presented in Tables 3, 4, and 5 in cases of basic standard benchmark functions and in Tables 7 and 8 in cases of CEC 2005 functions. Also, the algorithms are compared with statistical testing (Student’s -test) for statistical validity of the obtained results in Table 6 in cases of basic standard benchmark functions and in Tables 9 and 10 in cases of CEC 2005 functions [33].
5.2. Discussion
The algorithms are tested using two sets of standard benchmark functions, namely, well-known basic benchmark functions [7] and benchmark problems in CEC 2005 [29].
5.2.1. Basic Standard Benchmark Functions [7]
The algorithms are tested on well-known basic standard benchmark functions having unimodal, multimodal properties with noise and discontinuity. Functions Fsph (sphere model), Fs2.22 (Schwefel’s problem 2.22), Fs1.2 (Schwefel’s problem 1.2), Fs2.21 (Schwefel’s problem 2.21), and Fros (generalized Rosenbrock’s function) are functions having unimodal property. Function Fste (step function) is step function, which has one minimum and which is discontinuous function. Function Fqua (quartic function) is a noisy quartic function comprising of random [0, 1) which is a uniformly distributed random variable in [0, 1). Functions Fs2.26 (generalized Schwefel’s problem 2.26), Fras (generalized Rastrigin problem), Fack (Ackley’s function), Fgri (generalized Griewank function), Fpen1 (generalized penalized function no. 1), and Fpen2 (generalized penalized function no. 2) are multimodal functions in which the number of local minima increases exponentially with the problem dimension [7]. They are the most difficult class of problems for many optimization algorithms [7]. For unimodal functions, the convergence rates are more interesting than the final results of optimization. For unimodal functions, there are other methods like gradient based methods which are specifically designed to optimize these. In case of multimodal functions, the final results are important since they reflect an algorithm’s ability to escape from poor local optima and locating a good near-global optimum.
From Tables 3, 4 and 5, it is clearly seen that the proposed algorithms all modified SIMBOs and all MAs with the modified SIMBOs as local searches generate better results compared to all the competitor algorithms used for comparison. The modified SIMBOs as well as MAs based on modified SIMBOs generally demonstrate the superiority than all GAs and basic SIMBOs in solving the standard benchmark test problems. In case of all unimodal functions (Fsph, Fs2.22, Fs1.2, Fs2.21, and Fros), all the MAs converge successfully for all the dimensions of 10, 30, and 50, whereas basic SIMBO family and modified SIMBO family converge successfully in all unimodal functions for all dimensions except Rosenbrock’s function, that is, Fros. The basic as well as modified SIMBO families not at all successfully converge for any of the dimensions in case of Rosenbrock’s function, that is, Fros. The GAs do not reach the required accuracy in stipulated NFFEs for any of the run and for any of the dimensions in cases of unimodal functions (Fsph, Fs2.22, Fs1.2, Fs2.21, and Fros). All algorithms with all dimensions under test in this experiment successfully converge for the discontinuous step function (Fste). In case of noisy quartic function (Fqua), all basic as well as modified SIMBOs successfully converge in all runs for all dimensions, whereas all GAs do not converge successfully and all MAs converge successfully in very few runs. For generalized Schwefel’s problem 2.26 (Fs2.26) test all GAs as well as all MAs show successful convergence in many runs, whereas the basic as well as modified SIMBOs do not at all converge. In cases of multimodal functions, namely, generalized Rastrigin problem, Ackley’s function, generalized Griewank function, that is, Fras, Fack, and Fgri, all the algorithms except the GAs successfully converge to set accuracy. For the generalized penalized functions nos. 1 and 2, that is, Fpen1 and Fpen2, the GAs as well as MAs are more promising than the basic and modified SIMBOs for convergence characteristics.
As seen from Figures 1(a) and 1(b), the SIMBO has the better characteristics of convergence and mSIMBO shows still better converging characteristic in general in almost all the types of numerical standard benchmark functions for all 10, 30, and 50 dimensions. The mSIMBOs have quite faster speed of convergence but when the mSIMBOs are embedded in the evolutionary algorithm GA, the developed MAs show slightly lesser speed of convergence than the mSIMBOs as seen from Figures 2(a), 2(b), 2(c), and 2(d). The very small tradeoffs are repaid by solving the larger number of problem functions of a variety of domains for all MAs as compared to all SIMBOs. Along with SIMBOs, the MAs also successfully converge in cases of unimodal (Fsph, Fs2.22, Fs1.2, and Fs2.21) and multimodal (Fras, Fack, and Fgri). Additionally, the success in convergence is attained by MAs in cases of Fs2.26, Fpen1, and Fpen2 along with promising characteristics of GAs in the same cases. The SIMBOs have quite large drag in origin centric solutions and fail in other cases which are overcome by MAs with heavy explorations along with the exploitations by mSIMBOs. Among the SIMBOs and mSIMBOs, treatment based are the slowest and quarantine based are the fastest; similarly, the MAs based on mSIMBOs have the same characteristics, that is, treatment based are the slowest and quarantine based,the are fastest.
(a)
(b)
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A statistical test, Student’s -test, is performed for statistical validity of the obtained results [33]. Here, N.A. denotes “not applicable.” From Table 6, it is seen that the proposed algorithm modified SIMBOs (mSIMBO_T, mSIMBO_V, and mSIMBO_Q) show better performance than basic SIMBOs (SIMBO_T, SIMBO_V, and SIMBO_Q) in their respective comparison. Both basic as well as proposed algorithms show better performances as per statistical comparison for many cases of basic standard benchmark functions of 10 dimensions and considering a significance level of 0.05. As the significance level becomes 0.01, the superiority of the modified SIMBOs becomes clearly distinct. The modified SIMBOs also show the consistent performance for higher dimensions of 30 and 50, and therefore, the proposed algorithm wins in the statistical tests in these cases. Overall, it is seen that the proposed SIMBOs are better algorithms than basic SIMBOs.
5.2.2. CEC 2005 Standard Benchmark Problems [29]
There are various types of optimization algorithms available and used to solve real-parameter function optimization problems. There is a variety of mechanisms having variations in their operators and working principles. Catering to the need of evaluating the algorithms in a more systematic manner by specifying a common termination criterion, size of problems, initialization scheme, linkages/rotation, and so forth, the CEC 2005 having 25 test problems is designed [29].
All functions are nonseparable and scalable except F1 and F9, which are separable, whereas F15 is separable near the global optimum (Rastrigin). The functions in the test suit are shifted and rotated. Functions F1–F5 are unimodal and functions F6–F25 are multimodal in which F6–F12 are single functions, F13-F14, are expanded functions and F15–F25 are hybrid composition functions. Functions F9, F10, and F15–F25 have a huge number of local optima in their fitness landscape. In functions F15–F25, different function’s properties are mixed together to make hybrid composition functions. Functions F4 and F17 have Gaussian noise in fitness and they are similar to functions F2 and F16, respectively. Functions F15–F20 have effect due to sphere functions giving two flat areas for the function and functions F24-F25 have effect due to unimodal functions giving flat areas for the function. Functions F18–F20 have a local optimum and are set on the origin. Functions F8, F20, F22-F23, and F25 have Global optimum on the bound. For the performance evaluation, the function error, NFFEs, and convergence graphs for the problems are used [29]. Function error value is , where global optimum. Minimum error is recorded when the maximum number of fitness function evaluation is reached or NFFEs is recorded when error of 10−8 is attained, each in all 25 runs [29]. Mean and standard deviation of the error are also calculated. Convergence graphs (or run-length distribution graphs) are the semilog graphs that should show log10 versus NFFEs for each problem [29].
From Tables 7 and 8, the convergence properties GABLX, GASBX, and MAs with various modified SIMBOs as local searches are tested for convergence on CEC 2005 functions. All the SBX crossover algorithms, that is, GASBX and MASBX, with all types of mSIMBOs as local search and with 2 dimensions are 100% successfully converging in all of the runs for function F5. All the algorithms (GABLX, GASBX, MABLX, and MASBX with all mSIMBOs as local searches) are successfully attaining optimum with error less than 10−8 in all of the runs or some of the runs with 2 dimensions for functions F1, F2, F9–F11, F13, F15, and F16. All the algorithms (GABLX, GASBX, MABLX, and MASBX with all mSIMBOs as local searches) show promising characteristics in convergence for function F14 with 2 dimensions and for functions F1, F9, and F13 with 10 dimensions. All the SBX crossover algorithms that is, GASBX and MASBX, with all types of mSIMBOs as local search and with 2 dimensions show achievement of success with error less than 10−8 with more than 92% successful runs for functions F1, F2, F4, and F15; 52% to 79% successful runs for function F3; 20% to 24% successful runs for functions F6, and F8; 32% to 48% successful runs for functions F12-F13, F16–F18; 4% to 12% successful runs for functions F20, F21, and F23. Also, all SBX algorithms promisingly converge in case of the function F7.
As per Figures 3(a), 3(b), 3(c), and 3(d), MA with mSIMBO_Q as local search shows more promising average convergence property, and after that, MAs with mSIMBO_V and mSIMBO_T as local searches follow in the convergence performance. For some functions, MAs with mSIMBO_V and mSIMBO_T also show better average performance characteristics than MA with mSIMBO_Q. In general, among the MAs with treatment based mSIMBOs as local searches are the slowest and those of quarantine based are the fastest converging characteristics for CEC 2005 functions. The averaged error and convergence characteristics show the success in a variety of functions with unimodal, multimodal, shifted, rotated, and composite properties.
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When the statistical test, Student’s -test, is performed in comparison of the results of MAs and GAs for CEC 2005 problems, it is seen that MAs behave better in generality for convergences with 2 dimensions and also show promising performance in many cases with dimensions 10. As per Tables 9 and 10 for 2 dimensions, both GA and MAs promisingly perform and also perform similar in some cases, but for 10 dimensions MAs show consistency in performance. Therefore, with higher dimensions, MAs win in the statistical tests in generality.
6. The LMS Filter Weight Optimization for PLI Removal from ECG
The LMS-based adaptive filter having a length is depicted in Figure 4. It generally uses the gradient-based technique to update the weights with, , where is the weight vector. For th index, , error = , having as the desired response available in starting training period and μ indicating step-size parameter, and is reference or inference signal necessary for the adaptive filter [34].
The stochastic optimization with proposed MAs is used to search for the optimum filter weight vector giving error. In this case to remove the noise from the ECG signal, the ECG signal contaminated by the PLI signal is used in case of the desired response in the adaptive filter shown in Figure 4. The required inference signal is obtained with the help of selective usage of the intrinsic mode functions (IMFs) of empirical mode decomposition (EMD) in generating the signal and is used in the adaptive filter [34]. Where the inference signal obtained from the EMD process is named as that is in the correlation with and is applied to the filter, that is, . The filter error becomes , where, is the filter output given by .
As the signal and noise are uncorrelated, then mean squared error (MSE) becomes: In this method of MA-based optimization, the filter weights are generated with minimization of the MSE results in a filter output which is the best least-squares estimate of the signal .
The ECG contaminated by the artificial PLI with 50 Hz frequency is put forward for the PLI removal, to the LMS adaptive filter for its weights optimization by the MA using fitness function articulated in (11). The MSE is used to measure the difference between the original “clean” ECG and the reconstructed ECG. The MSE between original and recovered ECG indicates that MA optimizes the filter weights effectively with better mean square error (MSE) as in Table 11. Frequency spectrum of ECG having PLI and after filtering with LMS adaptive filter are shown in Figures 5(a) and 5(b), respectively, which indicates the absence of only power line inference (PLI) with other frequency spectrum remaining unaffected after filtering. In Figure 6(a), the corrupted ECG and in Figure 6(b) the recovered ECG are shown. The ECG is used from the benchmark MIT-BIH database [35, 36].
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7. Conclusion
The Swine Influenza Model-Based Optimization (SIMBO) is a faster optimization technique having the adaptive nature in its process. With the help of the three-fold modifications, the algorithm is made still faster in this work. The SIMBO family is very good in searching the basin under consideration, so it is utilized in local searches to develop the memetic algorithms (MAs). The SIMBO family has high pressure in the direction of origin and hence demonstrates better results in noisy function also. But this makes it weak in case of an other type of the solution that requires more explorations. This is achieved with help of MA having balance in exploration and exploitation. So, MAs have little tradeoffs in convergence speed but are more comprehensive in solving the numerical standard benchmark test bed of functions having the different properties. The modifications in SIMBO and application of it as the local searchers in making MAs as a process exploit the properties of both the algorithms to enhance the efficiency. The ensemble strategy to tune parameters and operators can be useful to develop further the more adaptive techniques.
The developed MA is also applied to remove the power line interference (PLI) from the biomedical signal ECG with the help of adaptive filter whose weights are optimized by the MA. In this, the adaptive filter inference signal is generated with the selective reconstruction of ECG from the intrinsic mode functions (IMFs) of empirical mode decomposition (EMD). The adaptive filter with the help of the optimized weights filters the PLI from ECG effectively with reduced error (mean square error (MSE)).