Backtracking search heuristics for identification of electrical muscle stimulation models using Hammerstein structure

Applied Soft Computing Journal 84 (2019) 105705

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Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc

Backtracking search heuristics for identification of electrical muscle stimulation models using Hammerstein structure Ammara Mehmood a , Aneela Zameer b , Naveed Ishtiaq Chaudhary c , ∗ Muhammad Asif Zahoor Raja d , a

Department Department c Department d Department b

of of of of

Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, Pakistan Computer and Information Sciences, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, Pakistan Electrical Engineering, International Islamic University, Islamabad, Pakistan Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock 43600, Pakistan

article

info

Article history: Received 18 June 2018 Received in revised form 7 July 2019 Accepted 2 August 2019 Available online 19 August 2019 Keywords: Electrical muscle stimulation Parameter estimation Hammerstein systems Backtracking search optimization Heuristic computing

a b s t r a c t The electrical muscle stimulation models (EMSMs) are effectively described through Hammerstein structure and are used to restore the functionality of paralyzed muscles after spinal cord injury (SCI). In the present study, global search efficacy of evolutionary computing paradigm through backtracking search algorithm (BSA) is exploited for parameter estimation of EMSMs. The approximation theory in mean squared error sense is used for the construction of a merit function for EMSMs based on deviation between optimal and approximated parameters. Variants of BSA are designed based on memory size and population dynamics for the minimization problem of EMSMs having cubic spline as well as sigmoidal nonlinearities. Comparative studies by means of rigorous statistics establish the worth of scheme for effective, accurate, reliable, robust and stable identification of EMSMs in rehabilitation scenarios of SCI. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The volitional activity of muscle is affected from spinal cord injury (SCI) and causes the reduction in the cross-sectional area of the muscle by almost 45% in the first few weeks of SCI [1]. The persons suffering from SCI experience twice more lifetime fracture risk than non-SCI individuals [2]. In order to prevent post SCI atrophy, i.e., to stop the gradual decline in the effectiveness of muscle after SCI, rehabilitation interventions are required. After SCI, electrical muscle stimulation (EMS) is an effective scheme that induces muscle hypertrophy, i.e., an increase in muscle mass and cross-sectional area, improves torque output and combats fatigue [3–5]. EMS is also useful for restoration of functional tasks such as standing and reaching. The countless applications of electrical stimulation for rehabilitation of SCI individuals [3–6] motivated the researchers to design control systems with real-time adjustment of the stimulus parameters for accommodating the changes in the muscle output. In order to effectively adapt the stimulus parameters in real-time situation, the mathematical models are required to identify the muscle torque output. The research community has ∗ Corresponding author. E-mail addresses: [email protected] (A. Mehmood), [email protected] (A. Zameer), [email protected] (N.I. Chaudhary), [email protected] (M.A.Z. Raja). https://doi.org/10.1016/j.asoc.2019.105705 1568-4946/© 2019 Elsevier B.V. All rights reserved.

proposed different models for muscle [7–9] and investigation has been carried out to model muscle dynamics with blockoriented Hammerstein–Wiener structures [10–12]. In a detailed study conducted at the rehabilitation center of the Southampton university, it was concluded that Hammerstein control autoregressive (H-CAR) structure is an effective model for paralyzed muscle dynamics under electrical stimulus conditions [12–14]. Hammerstein model consists of two blocks, first block corresponds to nonlinearity of the system and followed by a second block that represents the linear characteristics. In case of EMS, the nonlinear block represents the isometric recruitment curve, i.e., the static gain relation between the activation level of stimulus and the output torque, when the muscle is at fixed length. While the linear block represents EMS dynamic response [14]. After muscle modeling, the next step is to estimate the parameters of Hammerstein structure representing EMS [12]. Research community has growing interest to develop efficient methodologies for parameter estimation of nonlinear systems [15–17] in particular represented with Hammerstein structure. Ding contributed significantly for Hammerstein system identification and developed various stochastic gradient and least squares strategies based on hierarchical principle, multi innovation theory, auxiliary model, parameter separation idea and decomposition technique [18–24]. Recently, fractional calculus based adaptive algorithms were successfully applied to parameter

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A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

Fig. 1. Workflow diagram of proposed study.

identification of Hammerstein systems [25–30]. All these deterministic paradigms have their own perks, applicability and limitations while the stochastic procedures based on global search strength of genetic algorithms, particle swarm optimization, differential evolution, cuckoo search, and gravitational search were also exploited for parameter estimation of Hammerstein systems [31–38] to overcome the limitations of deterministic solver for getting stuck in a local minimum rather more frequently. Stochastic optimization solvers based on artificial intelligence looks promising to explore for accurate reliable and robust solution of the problems arising in nonlinear optics, nanotechnology, astrophysics, plasma physics, atomic physics, electrical conducting solids, bioinformatics, power, and economics [39–47] and reference cited therein. As per our investigations, the recently

proposed stochastics paradigm based on backtracking search optimization algorithm (BSA) has yet not exploited for parameter estimation of nonlinear Hammerstein systems since its introduction by Civicioglu in 2013 [48]. The brilliant performance of BSA is well established over other counterparts for number of applications arising in spectrum of fields such as parameter estimation of photovoltaic models [49], flow shop scheduling problem [50], community detection in complex networks [51], active noise control [52], optimization of controller for induction motors [53], non-convex economic dispatch problems [54], optimal design of power system stabilizers [55], economic emission dispatch problems [56], automatic generation control systems [57], optimization of model for active earth pressure on retaining wall

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

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Fig. 2. Process flow diagram of standard backtracking search optimization algorithm.

supporting c-Φ backfill [58], stiff constrained optimization problem [59], optimal control problems [60], fed-batch fermentation processes [61] and hydroelectric generators [62]. These are the motivating factors for the authors to explore in estimation of Hammerstein parameters representing the scenarios of EMS through global search efficacy of BSA. The highlights of the contributions in terms of salient features are briefly described as follows:

• Novel application of recently introduced evolutionary computing paradigms based on variants of BSA for accurate, reliable and robust parameter estimation of electrical muscle stimulation model represented through Hammerstein structure. • The population dynamics and memory size of BSA is exploited for design of the variants and the eminence of the proposed scheme is to provide reasonably correct results without estimation of the redundant parameters and equally robust for the low and high noise scenarios of the EMS system. • The performance of the designed methodologies is verified and validated for equally virtuous outcomes for different forms of nonlinearity in first block of the Hammerstein structure based on polynomial, cubic spline and sigmoid type functions. • The consistency of the worthy performance for the variants of BSA is endorsed through multiple trials based statistics in terms of local and global performance measures on accuracy and complexity. Remaining paper is organized as follows: the proposed design methodology for mathematical formulation of EMS system and its

optimization mechanism based on BSA is presented in Section 2; in Section 3, performance metrics to evaluate the performance of the proposed method are presented; in Section 4, discussion on the simulation results for single best run as well as on statistics through large dataset are provided through single and supporting objective based fitness functions; while conclusions are listed in the last section. 2. Design methodology Proposed methodology consists of two parts; in the first part, an overview of EMS system along with development of the cost function is presented, while in the second part, optimization procedure of BSA is described for parameter estimation of EMSM. The process workflow diagram of the proposed methodology is shown graphically in Fig. 1. 2.1. Electrical muscle stimulation modeling The dynamics of EMS are modeled through H-CAR structure and mathematically it is written as [12–14] y (t ) =

R (z ) S (z )

f (u (t )) +

1 S (z )

n (t ) ,

(1)

where u(t) and y(t) represent model input and output, respectively and n(t) is disturbance noise. R(z) and S(z) are the polynomials defined as R (z ) = r1 z −1 + r2 z −2 + · · · + rnr z −nr ,

(2)

S (z ) = 1 + s1 z −1 + s2 z −2 + · · · + sns z −ns

(3)

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Fig. 3. Results for Iterative adaptation of merit function by the variants of BSAs along with analysis through different performance indices for the EMS system in Example 4.1.

m−2 ∑

Let f (u) is the nonlinear function of system input, representing the nonlinear activation dynamics of a muscle. In case of polynomial type nonlinearity, f (u) is described as

f (u) =

f (u) = β1 u + β2 u2 + β3 u3 + · · · + βm um ,

Assuming one knot in the cubic spline function at 150, Eq. (6) becomes

(4)

while the nonlinear function represented by sigmoid and cubic spline functions are respectively defined as [12]: f (u) = β1 .

eβ2 u − 1 eβ2 u + β3

βi |u − ui+1 |3 + βm−1 + βm u + βm+1 u2 + βm+2 u3 , (6)

i=1

f (u) = β1 |u − 150|3 + β2 + β3 u + β4 u2 + β5 u3 . The true parameter vector of EMS system is

,

(5)

θ = [θ l , θ n ] ,

(7)

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Fig. 4. Comparison of the results based fitness against independent trials of BSAs through sorted, and zoomed illustrations for all three case studies of EMS systems.

where θ l and θ n are the parameters corresponding to linear and nonlinear characteristics of EMS respectively. The linear parameter vector θ l is defined as:

[ ] θ l = s1 , s2 , . . . , sns , r1 , r2 , . . . , rnr ,

(8)

and θ n corresponding to input nonlinear functions and in case of polynomial, sigmoid and spline type functions, θ n is respectively defined as [12]: [β1 , β2 , . . . , βm ] , θ n = [β1 , β2 , β3 ] , [β1 , β2 , . . . , β5 ]

{

polynomial type sigmoid type spline type.

functions are given, respectively, as:

( y (t ) = −

ns ∑

Using Eqs. (2)–(9) in (1), then the output of EMS model based on polynomial, sigmoid and cubic spline type input nonlinear

si z

i=1

y (t ) +

(n r ∑

) ri z

−i

( β1 u (t ) + β2 u2 (t )

i=1

) + · · · + βm u (t ) + n (t ) , (10) ( n ) (n )( ) s r β2 u(t) ∑ ∑ e −1 y (t ) = − si z −i y (t ) + ri z −i β1 . β u(t) e 2 + β3 m

i=1

(9)

) −i

i=1

+ n (t ) , ( n ) (n ) s r ∑ ∑ −i −i y (t ) = − si z y (t ) + ri z i=1

i=1

(11)

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Fig. 5. Comparison of the results on the basis of MAEs for EMS system in Example 4.1.

×

) m−2 ( ∑ βi |u (t ) − ui+1 (t )|3 + βm−1 + + n (t ) . (12) βm u (t ) + βm+1 u2 (t ) + βm+2 u3 (t )

functions are updated as:

i=1

The finite output responses of EMS for K time instances based on polynomial, sigmoid and cubic spline type input nonlinear

( y (tk ) = −

ns ∑ i=1

) si z

−i

( y (tk ) +

nr ∑ i=1

) ri z

−i

( β1 u (tk ) + β2 u2 (tk )

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

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Fig. 6. Comparison of the results on the basis of RMSE and TIC values through bars, histograms, and Stack bar illustrations for EMS system in Example 4.1.

) + · · · + βm um (tk ) + n (tk ) , ( n ) s ∑ y (tk ) = − si z −i y (tk ) i=1 nr

( +

∑ (

y (tk ) = −

ri z −i

)(

i=1 ns

∑ i=1

β1 .

y (tk ) +

nr ∑

K )2 1 ∑( y (tk ) − yˆ (tk ) , K

(17)

k=1

)

eβ2 u(tk ) + β3

(

mathematically it is written as

ε1 =

eβ2 u(tk ) − 1

) si z −i

(13)

+ n (tk ) ,

(14)

) ri z −i

i=1

) m−2 ( ∑ βi |u (tk ) − ui+1 (tk )|3 + βm−1 + × + n (tk ) , βm u (tk ) + βm+1 u2 (tk ) + βm+2 u3 (tk ) i=1

where K represents total number of time instances, y (tk ) is the kth output of EMS model as given in Eqs. (13)–(15), while the yˆ (tk ) is the kth output of estimated response based on approximate parameter vector θˆ with polynomial function based input as:

] [ ] [ θˆ = θˆ l , θˆ n = sˆ1 , sˆ2 , . . . , sˆns , rˆ1 , rˆ2 , . . . , rˆnr , βˆ 1 , βˆ 2 , . . . , βˆ m ,

(15) for k = 1, 2, . . ., K . Now intention is to derive the fitness function of EMS systems by exploiting the approximation theory in mean squared error sense. The fitness function ε for EMS model is constructed using the concept of supporting objective and is defined as the sum of two functions, one is dependent on the response of EMS model and second is based on the parameters of EMS system:

ε = ε1 + ε2 ,

(16)

The first error function ε1 is defined as a mean square difference between the estimated response and the desired output and

(18)

( yˆ (tk ) = −

)

ns

∑

sˆi z −i

( yˆ (tk ) +

i=1

)

nr

∑

rˆi z −i

( βˆ 1 u (tk ) + βˆ 2 u2 (tk )

i=1

) + · · · + βˆ m um (tk ) .

(19)

Similarly, the estimated response yˆ (tk ) with approximate parameter vector θˆ for sigmoid function based input is given as:

[ ] [ ] θˆ = θˆ l , θˆ n = sˆ1 , sˆ2 , . . . , sˆns , rˆ1 , rˆ2 , . . . , rˆnr , βˆ 1 , βˆ 2 , βˆ 3 , (20)

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Fig. 7. Comparison of the results through adaptation of single objective based merit function for all six BSA variants of the EMS Example 4.1.

Fig. 8. Comparison of the results through adaptation of single objective based merit function for all six BSA variants of the EMS Example 4.2.

( yˆ (tk ) = −

ns ∑ i=1

) sˆi z

−i

( yˆ (tk ) +

nr ∑ i=1

)( rˆi z

−i

eβ2 u(tk ) − 1 ˆ

βˆ 1 .

eβˆ 2 u(tk ) + βˆ 3

) ,

Accordingly, the estimated response yˆ (tk ) with approximate parameter vector θˆ for spline function based input is given as:

(21)

[ ] [ ] θˆ = θˆ l , θˆ n = sˆ1 , sˆ2 , . . . , sˆns , rˆ1 , rˆ2 , . . . , rˆnr , βˆ 1 , βˆ 2 , . . . , βˆ m , (22)

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Fig. 9. Comparison of the results through adaptation of single objective based merit function for all six BSA variants of the EMS Example 4.3. Table 1 Variants of backtracking search optimization algorithm. Method

Example 4.1

BSA-1 BSA-2 BSA-3 BSA-4 BSA-5 BSA-6

( yˆ (tk ) = −

ns ∑

Individuals

Generations

Individuals

Generations

40 40 40 20 40 60

200 400 600 200 200 200

40 40 40 10 40 60

1000 2000 3000 1000 1000 1000

40 40 40 20 40 60

500 750 1000 200 200 200

−i

( yˆ (tk ) +

nr ∑

) rˆi z

approximate parameter vector approaches its optimal, i.e., θˆ → θ . If only ε1 given in (17) is used for optimization of parameters in EMS model, then the desired response of EMS can be achieved but accuracy to obtain the desired parameters is vulnerable.

−i

i=1

i=1

) m−2 ( ∑ βˆ i |u (tk ) − ui+1 (tk )|3 + βˆ m−1 . × +βˆ m u (tk ) + βˆ m+1 u2 (tk ) + βˆ m+2 u3 (tk )

(23) 2.2. Optimization of parameters for EMSMs

i=1

The second error function ε2 is defined as a mean square deviation between the estimated and actual parameters of the system and mathematically it is expressed as:

ε2 =

N 1 ∑(

N

θn − θˆn

)2

,

(24)

n=1

where N is the total elements in the parameter vector, θn and θˆn be the nth entry of desired and estimated weights, respectively. In case of polynomial, sigmoid and cubic spline type input nonlinear functions, the parameter N = ns + nr + m, = ns + nr + 3 and = ns + nr + m, respectively. So, the fitness function (16) using Eqs. (17) and (24) is given as:

ε=

K N ( )2 )2 1 ∑ 1 ∑( y (tk ) − yˆ (tk ) + θn − θˆn . K N k=1

Example 4.3

Generations

) sˆi z

Example 4.2

Individuals

(25)

n=1

Now the requirements is to use the optimization techniques to solve the minimization problem (25) and find the appropriate parameters θˆ such that ε → 0, then the estimated response approach the actual response, i.e., yˆ → y, so accordingly, the

A brief introduction for training designed parameters through optimization of merit function (14) of EMS model is given here with the help of a metaheuristic optimization mechanism based on BSA. BSA belong to the class of evolutionary computational algorithms developed by Civicioglu [48] in 2013. BSA is modern stochastic global search technique, broadly employed to find the solution of nondifferentiable, nonlinear, and complex optimization problems. It has single control parameter and additional quality of possessing memory for providing search direction matrix. Trial population is generated with the help of three operators based on mutation, crossover and selection procedures. BSA has been widely applied to optimize real world problems in engineering and technology, i.e., antenna, robotics, wind speed prediction and load dispatch problem [63–66] and references cited therein. Effectiveness of BSA motivates authors to utilize the potentials of said evolutionary optimization procedure and its variants to find optimization variables of EMSMs. Generic flow graph of BSA is shown in Fig. 2, while the detailed stepwise procedural steps of BSA for EMSMs are illustrated in the pseudocode described in Algorithm 1.

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A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

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Table 2 Performance comparison through results of statistics for EMS system in Example 4.1. Example

Variant

BSA-1

BSA-2

BSA-3

Noise

Model

Approximate parameter vector i=1

i=2

i=3

i=4

i=5

i=6

0.0012

Best Mean Worst

−2.0000 −1.9956 −1.9999

1.0002 0.9985 1.0153

0.0001 0.0028 −0.0264

6.8988 6.9047 6.9807

0.0372 0.0398 −0.0078

2389.70 2389.70 2389.59

Best Mean Worst

−1.9987 −1.9980 −1.9994

0.9945 1.0000 0.9908

−0.0013

0.0102

0.0023 −0.0562

6.8982 6.9010 6.9965

0.0423 0.0419 −0.0373

2389.70 2389.70 2389.63

0.1002

Best Mean Worst

−2.0000 −1.9979 −2.0000

0.9968 0.9210 0.7912

−0.0101 −0.0573 −0.1288

6.8976 6.9037 6.9741

0.0429 0.0839 0.1829

2389.70 2389.70 2389.64

0.0012

Best Mean Worst

−1.9985 −1.9985 −1.9981

0.9985 0.9985 0.9999

0.0022 0.0022 0.0025

6.8994 6.8994 6.8991

0.0410 0.0410 0.0408

2389.70 2389.70 2389.70

0.0102

Best Mean Worst

−1.9985 −1.9985 −2.0000

0.9985 0.9985 0.9986

0.0022 0.0022 0.0021

6.8994 6.8994 6.8995

0.0410 0.0410 0.0410

2389.70 2389.70 2389.70

0.1002

Best Mean Worst

−1.9985 −1.9998 −2.0000

0.9984 0.8162 0.7992

−0.0009 −0.1341 −0.1686

6.8995 6.9027 6.9023

0.0411 0.1477 0.1751

2389.70 2389.70 2389.71

0.0012

Best Mean Worst

−1.9985 −1.9985 −2.0000

0.9985 0.9985 0.9985

0.0022 0.0022 0.0022

6.8994 6.8994 6.8994

0.0410 0.0410 0.0410

2389.70 2389.70 2389.70

0.0102

Best Mean Worst

−1.9985 −1.9985 −2.0000

0.9985 0.9985 0.9985

0.0022 0.0022 0.0022

6.8994 6.8994 6.8994

0.0410 0.0410 0.0410

2389.70 2389.70 2389.70

0.1002

Best Mean Worst

−1.9986 −1.9999 −2.0000

0.9984 0.8037 0.8178

−0.0010 −0.1422 −0.1625

6.8994 6.9024 6.9038

0.0411 0.1558 0.2247

2389.70 2389.70 2389.70

0.0012

Best Mean Worst

−1.9993 −1.9945 −1.9165

0.9920 0.9986 1.0039

0.0013 0.0037 0.0795

6.8969 6.9023 7.0000

0.0446 0.0358 0.0546

2389.69 2389.70 2389.72

Best Mean Worst

−1.9990 −1.9925 −1.7931

0.9996 0.9955 0.8542

0.0102

0.0102

−0.0016 −0.0141

6.9000 6.9043 7.0000

0.0409 0.0429 0.0724

2389.70 2389.70 2389.70

0.1002

Best Mean Worst

−2.0000 −1.9979 −2.0000

0.9981 0.9460 0.8528

−0.0030 −0.0363 −0.2430

6.8963 6.9074 6.9526

0.0437 0.0727 0.1813

2389.70 2389.70 2389.61

0.0012

Best Mean Worst

−1.9996 −1.9980 −1.9892

0.9981 0.9987 0.9378

0.0012 0.0043 0.0456

6.9035 6.8998 6.9294

0.0400 0.0373 −0.0320

2389.70 2389.70 2389.68

Best Mean Worst

−2.0000 −1.9986 −1.9976

0.9973 0.9972 1.0141

−0.0002

0.0102

6.9007 6.9024 7.0000

0.0398 0.0443 0.0071

2389.70 2389.70 2389.71

0.1002

Best Mean Worst

−2.0000 −1.9980 −2.0000

0.9961 0.9084 0.7937

−0.0596 0.1231

6.9000 6.9013 6.9734

0.0441 0.0893 −0.1438

2389.71 2389.70 2389.80

0.0012

Best Mean Worst

−1.9996 −1.9981 −2.0000

0.9989 0.9964 1.0373

0.0033 0.0024 −0.0056

6.8983 6.8979 6.8792

0.0393 0.0449 0.1256

2389.69 2389.70 2389.72

0.0102

Best Mean Worst

−1.9992 −1.9979 −2.0000

0.9993 0.9971 0.9972

0.0043 0.0014 0.0915

6.8988 6.9014 6.8425

0.0415 0.0413 −0.0054

2389.69 2389.70 2389.71

Best Mean Worst

−1.9990 −1.9987 −2.0000

0.9959 0.9030 0.8221

0.0071

0.1002

−0.0711 −0.1961

6.8947 6.8997 6.8864

0.0385 0.0952 0.1779

2389.71 2389.70 2389.76

−1.9985

0.9985

0.0022

6.8994

0.0410

2389.70

4.1

BSA-4

BSA-5

BSA-6

True values

3. Performance indices The four performance measuring operators for the parameter vector θ on the basis of mean absolute error (MAEθ ), normalizing error function, δθ , root of mean squared error (RMSEθ ), and Thiel’s inequality coefficient (TICθ ) are used to evaluate performance the designed algorithms for parameter estimation of EMS systems. The formulae of these indices are defined in this section.

0.0014 −0.0692 0.0048

The parameter MAEθ is mathematically written as:

MAEθ =

m ⏐ ⏐ 1 ∑⏐ ⏐ ⏐θi − θˆi ⏐ , m

(26)

i=1

where θi is the ith entry of true parameter vector θ while θˆi is its estimation.

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The parameter RMSEθ is mathematically defined as:

  m 1 ∑ (θi − θˆi )2 . RMSEθ = √ m

(27)

i=1

Normalize error function based on mean weigh deviation (MWDθ ), δθ is given as:

δθ =

  ˆ  θ − θ 

,

∥θ∥

(28)

where ∥ ∥ is the standard L2 norm. Thiel’s Inequality Coefficient (TICθ ) is formulized as:

  ⎛ ⎞   m ( m m )2 /  1 ∑ 1 ∑ 1 ∑ ⎝√ TICθ = √ θi − θˆi θi2 + √ θˆi2 ⎠. m

m

i=1

m

i=1

Accordingly, the global version of MAEθ (GMAEθ ) is formulated as:

GMAEθ =

1 ∑ IR

IR

(MAEθ )r =

r =1

1 ∑ IR

(

r =1

i=1

(30)

GRMSE θ

IR

IR

r =1

m

r =1

i=1

GTIC θ

r

IR

IR

r =1

r =1

m

i=1

m

i=1

εθ =

=

IR

IR

r =1

r

K N ( )2 )2 1 ∑ 1 ∑( y (tk ) − yˆ (tk ) + θn − θˆn K N k=1

e0.0410u(t −1) −1

e0.0410u(t −1) +2389.7

]

,

(34)

+ n (t ) ,

The supporting objective based fitness function for the case study is developed using (25) as:

ε=

(εθ )r

(

e0.0410u(t ) − 1

θ = [s1 , s2 , r1 , β1 , β2 , β3 ] , θ = [−1.9985, 0.9985, 0.0022, 6.8994, 0.0410, 2389.70] .

(32)

r =1

IR 1 ∑

= 6.8994.

20 6 ( )2 )2 1 ∑ 1 ∑( y (tk ) − yˆ (tk ) + θ k − θˆ k . 20 6

(35)

k=1

The fitness function without using supporting objective for the case study is developed using (17) as:

i=1

The mean of fitness ε θ based on total number of runs IR is formulized as: IR 1 ∑

eβ2 u(t ) − 1

k=1

 ⎞⎞ ⎛   m m 1 ∑ 1 ∑ ⎝√ θi2 + √ θˆi2 ⎠⎠ . m

n (t ) ,

eβ2 u(t ) + β3 e0.0410u(t ) + 2389.70 y(t) = − [−1.9985y (t − 1) + 0.9985y (t − 2)]

ε=

⎛  m ( )2 / ∑ ∑  ∑ 1 1 ⎝√ 1 = θi − θˆi (TICθ )r = IR

1 S (z )

R (z ) = r1 z −1 = 0.0022z −1 ,

Global TICθ (GTIC θ ) is formulated as: IR

f (u (t )) +

[ + 0.0022 6.8994 ×

⎞ ⎛  m ∑ 1 ∑ 1 1 ∑  ⎝√ = (θi − θˆi )2 ⎠ . (31) (RMSEθ )r = IR

S (z )

S (z ) = 1 + s1 z −1 + s2 z −2 = 1 − 1.9985z −1 + 0.9985z −2 ,

r

where, symbol IR represents the total independent runs. The global RMSEθ (GRMSE θ ) is mathematically written as: IR

R (z )

f (u (t )) = β1 .

) m ⏐ ⏐ 1 ∑⏐ ⏐ ⏐θi − θˆi ⏐ ,

m

Example 4.1 (EMS System with Sigmoid Function Based Input Nonlinearity). The parameters considered for the problem are taken from the data of experimental study conducted in [12], where the linear dynamics of EMS are model through transfer function based on one zero and two poles, while nonlinear characteristics are given by sigmoid function as per following details: y (t ) =

i=1

(29)

IR

individuals as per settings listed in Table 1 while the fixed or default parameters of each algorithm are given Algorithm 1 for BSA. Repeat the procedure given in Algorithm 1 for all six variants of BSA. Similarly, repeat Algorithm 1 for hundred times in order to conduct statistical analyses. In all three EMSMs, the performance of each variant of BSA is evaluated for three noise variances, i.e., σ 2 = 0.0012 , 0.012 , and 0.12 , for effective inferences.

n=1

) . r

(33)

20 )2 1 ∑( y (tk ) − yˆ (tk ) . 20

(36)

k=1

Example 4.2 (EMS System with Polynomial Function Based Input Nonlinearity). The linear dynamics of EMS in the said system are given through transfer function with two zeros and two poles, while the nonlinear characteristics are model with polynomial type function as given below [12]: y (t ) =

R (z ) S (z )

f (u (t )) +

1 S (z )

n (t ) ,

In the ideal scenarios of the modeling, magnitudes of all these indicators, i.e., MAEθ , δθ εθ , TICθ RMSEθ , and TICθ , along with global version should be zero.

S (z ) = 1 + s1 z −1 + s2 z −2 = 1 − z −1 + 0.8z −2 ,

4. Results and discussion

= 2.8u (t ) − 4.8u2 (t ) + 5.7u3 (t ) , y (t ) = − [−1.00y (t − 1) + 0.8y (t − 2)] + [(1) (2.8)] u (t − 1)

In this section, simulation results are presented for three problems of EMS model identification based on different forms of nonlinearity using six variants of BSA. The performance of BSA is studied using both single and supporting objective based fitness functions given in (17) and (25) respectively. 4.1. Experimental set up The variants of BSA-1 to BSA-6 are designed based on memory size, i.e., generations, and population dynamics, i.e., set of

R (z ) = r1 z −1 + r2 z −2 = z −1 + 0.6z −2 , f (u (t )) = β1 u (t ) + β2 u2 (t ) + β3 u3 (t )

+ [(1) (−4.8)] u2 (t − 1) + [(1) (5.7)] u3 (t − 1) + [(0.6) (2.8)] u (t − 2) + [(0.6) (−4.8)] u2 (t − 2) + [(0.6) (5.7)] u3 (t − 2) + n (t ) , θ = [s1 , s2 , r1 , r2 , β1 , β2 , β3 ] , θ = [−1.00, 0.80, 1.00, 0.60, 2.80, −4.80, 5.70] . (37)

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

The fitness function of the problem through supporting objective using (25) is given as:

ε=

20 7 ( )2 )2 1 ∑ 1 ∑( y (tk ) − yˆ (tk ) + θ k − θˆ k . 20 7 k=1

(38)

k=1

The fitness function of the problem without using supporting objective using (17) is given as:

ε=

20 )2 1 ∑( y (tk ) − yˆ (tk ) . 20

(39)

k=1

Example 4.3 (EMS System with Cubic Spline Function Based Input Nonlinearity). The parameters considered in the example are based on experimental studies reported in [12], where the linear dynamics of the system are given through two zeros and two poles based transfer function, while the nonlinear characteristics are model with cubic spline function having one knot at 150 as: y (t ) =

R(z ) f S (z )

(u (t )) +

S (z ) = 1 + s1 z

−1

1 n S (z ) −2

+ s2 z

(t ) ,

= 1 − 1.094z −1 + 0.109z −2 ,

R (z ) = r1 z −1 + r2 z −2 = z −1 + 0.249z −2 , f (u (t )) = β1 + β2 u (t ) + β3 u2 (t ) + β4 u3 (t ) + β5 ⏐u3 (t ) − 150⏐ , −6 2 f (u (t )) = −0.028 + 1.90 × 10−3 u (t ) − 7.83 × ⏐ 10 u (t ) ⏐ −8 3 −8 ⏐ 3 +1.78 × 10 u (t ) + 2.36 × 10 u (t ) − 150⏐ ,

⏐

⏐

(40) y (t ) = − [−1.094y (t − 1) + 0.109y (t − 2)] + [(1) (−0.028)]

[ ( )] + (1) 1.90 × 10−3 u (t − 1) [ ( )] + (1) −7.83 × 10−6 u2 (t − 1) [ ( )] + (1) 1.78 × 10−8 u3 (t − 1) ⏐ [ ( )] ⏐ + (1) 2.36 × 10−8 ⏐u3 (t − 1) − 150⏐ [ ( )] + [(0.249) (−0.028)] + (0.249) 1.90 × 10−3 u (t − 1) [ ( )] + (0.249) −7.83 × 10−6 u2 (t − 1) [ ( )] + (0.249) 1.78 × 10−8 u3 (t − 1) ⏐ [ ( )] ⏐ + (0.249) 2.36 × 10−8 ⏐u3 (t − 1) − 150⏐ + n (t ) , θ = [s1 , s2 , r1 , r2 , β1 , β2 , β3 , β4 , β5 ] , [ ] −1.094, 0.109, 1.00, −0.249, −0.028, θ= . 1.90 × 10−3 , −7.83 × 10−6 , 1.78 × 10−8 , 2.36 × 10−8 The fitness function for the case study using (25) is formulated as:

ε=

20 9 ( )2 )2 1 ∑ 1 ∑( y (tk ) − yˆ (tk ) + θ k − θˆ k . 20 9 k=1

(41)

k=1

The fitness function for the case study using (17) is formulated as:

ε=

20 )2 1 ∑( y (tk ) − yˆ (tk ) . 20

(42)

k=1

4.2. Analysis through multi objective for EMS The proposed heuristic algorithm based on BSA together with all six variants has been operated on 100 independent trials to optimize the fitness functions of Examples 4.1–4.3 given in (35), (38) and (41) respectively, in case of three different noise levels. The results of BSA-1 to 4 in term of learning curves, i.e., convergence plots of fitness with iterations, are shown graphically in Fig. 3(a–d) for Example 4.1, while in case of Examples 4.2 and 4.3, the respective results are given in Appendix Figures A1(a–d)

13

and Figures A2 (a–d), respectively, for all three noise variances σ 2 = 0.0012 , 0.012 , and 0.12 . In all these figures, it is found that for all three levels of noise variances, all the variants of BSA are convergent but the accuracy and convergence of the BSA3 is much superior than all other schemes for each EMSM. The magnitudes of performance operators of BSA-1 to BSA-4 in terms of MAEθ , RMSEθ and MWDθ are calculated using Eqs. (26), (27) and (28), respectively, and are illustrated graphically in Fig. 3(e– h), for Example 4.1, while in case of Examples 4.2 and 4.3, the respective results are presented in Appendix Figures A1(e–h), and A2(e–h), respectively. Generally, near-optimal value of all three performance measures achieved, but level of magnitudes of these metrics lie in the order of 10−02 to 10−12 , 10−04 to 10−14 and 10−04 to 10−12 , which further establish the consistency correctness of the proposed schemes. Besides this comparison, absolute errors (AEs) for all three EMS systems for each noise variation are shown graphically for BSAs in Fig. 3(i–l) for Example 4.1, while in case of Examples 4.2 and 4.3, the respective results are presented in Appendix Figures A1(i–l), and Figures A2(i–l), respectively. Generally, the AEs are found around 10−08 , 10−06 , and 10−04 using σ 2 = 0.0012 , 0.012 , and 0.12 , respectively, for Example 4.1, 10−08 , 10−05 , and 10−03 for σ 2 = 0.0012 , 0.012 , and 0.12 , respectively, for Example 4.2, and respective AEs are 10−07 , 10−06 , and 10−04 for σ 2 = 0.0012 , 0.012 , and 0.12 for Example 4.3 in case of BSA-3. The values of the performance indices in terms of accuracy measures of fitness ε , normalized error function δθ , MAEθ , RMSEθ , and TICθ , as defined in the last section, are calculated for the run of variants of BSA that gives the best fitness and results are shown in Table 2 together with the complexity measures based on time, generations and merit function evaluations. It is quite clear that the MWDθ , δθ values lie close to 10−09 to 10−10 , 10−07 to 10−08 , and 10−05 to 10−06 for noise variances σ 2 = 0.0012 , 0.012 , and 0.12 , respectively, in case of BSA-3 for EMS model in Example 4.1, while for other two case studies of EMS systems, the respective values lie around 10−09 to 10−08 , 10−05 to 10−04 , and 10−02 to 10−03 and 10−07 to 10−04 , 10−05 to 10−04 , and 10−02 to 10−03 , respectively. Generally, the reasonable magnitudes of performance metrics establish the consistency of the six variants of designed BSAs for each EMS systems, however, the BSA-3 outperforms by means of achieving the superior values of all performance indices from the rest of the algorithms for each scenario of all three case studies of EMSMs.(see Fig. 6). In order to establish efficacy, reliability and robustness of the schemes, the designed algorithms are evaluated for 100 trials in each case study of EMSM. The results of statistical observations through supporting objective based fitness are presented in the sorted and zoomed illustrations in Fig. 4. The comparative study based on the performance metrics of MAEθ , RMSEθ and TICθ are presented graphically in Fig. 5 for Example 4.1, while, in case of Examples 4.2 and 4.3, the respective results are given in Appendix Figure A3 and Figure A4 respectively. The MAEs for Examples 4.1–4.3, are illustrated graphically on semi-logarithmic scale to decipher small changes and are shown in Figures 5(a– d) in case of Example 4.1, while for Examples 4.2 and 4.3, the respective results are given in Appendix Figures A3(a–d), and Figures A4(a–d), respectively. The results reveal that the values of MAEs for all three case studies lie in the range of 10−01 to 10−02 , 10−01 to 10−02 and 10−02 to 10−04 in case of BSA-4. For further accuracy analysis of the designed variants of BSA, the histogram plots are presented and results on the basis of RMSEs for all three EMSMs are shown in Figures 6(a–h), Appendix Figures A3(e–l), and Figures A4(e–l), respectively. Moreover, TIC magnitudes of all six variants of BSA for all three problems are shown as stacked bar plots in Figures 6(i–j), Appendix Figures A3(m–n), and A4 10(m–n), respectively. All these graphical illustrations validate

14

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

Table 3 Performance comparison through results of statistics for EMS system in Example 4.2. Example

Variant

BSA-1

BSA-2

BSA-3

Noise

BSA-5

BSA-6

True values

Approximate parameter vector i=1

i=2

i=3

i=4

i=5

i=6

i=7

0.0012

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.79

−4.80 −4.80 −4.81

5.70 5.70 5.72

0.60 0.60 0.60

0.0102

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 0.99

2.80 2.80 2.81

−4.80 −4.80 −4.82

5.70 5.70 5.73

0.60 0.60 0.60

0.1002

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.79 2.79 2.79

−4.80 −4.80 −4.81

5.71 5.71 5.72

0.60 0.60 0.60

0.0012

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.80

−4.80 −4.80 −4.80

5.70 5.70 5.70

0.60 0.60 0.60

0.0102

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.80

−4.80 −4.80 −4.80

5.70 5.70 5.70

0.60 0.60 0.60

0.1002

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.79 2.79 2.79

−4.80 −4.80 −4.80

5.71 5.71 5.71

0.60 0.60 0.60

0.0012

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.80

−4.80 −4.80 −4.80

5.70 5.70 5.70

0.60 0.60 0.60

0.0102

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.80

−4.80 −4.80 −4.80

5.70 5.70 5.70

0.60 0.60 0.60

0.1002

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.79 2.79 2.79

−4.80 −4.80 −4.80

5.71 5.71 5.71

0.60 0.60 0.60

0.0012

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 0.96

2.80 2.81 2.85

−4.80 −4.82 −5.00

5.70 5.72 6.00

0.60 0.60 0.58

0.0102

Best Mean Worst

−1.00 −1.00 −0.99

0.80 0.80 0.81

1.00 1.00 1.00

2.80 2.80 2.82

−4.80 −4.79 −3.88

5.70 5.68 4.07

0.60 0.61 1.00

0.1002

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 0.96

2.80 2.80 2.84

−4.80 −4.81 −4.99

5.70 5.72 6.00

0.60 0.60 0.58

0.0012

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.81

−4.80 −4.80 −4.82

5.70 5.70 5.72

0.60 0.60 0.60

0.0102

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.80

−4.80 −4.80 −4.82

5.70 5.70 5.73

0.60 0.60 0.60

0.1002

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.79 2.79 2.79

−4.80 −4.80 −4.81

5.71 5.71 5.72

0.60 0.60 0.60

0.0012

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.81

−4.80 −4.80 −4.81

5.70 5.70 5.72

0.60 0.60 0.60

0.0102

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.80 2.80 2.80

−4.80 −4.80 −4.81

5.70 5.70 5.71

0.60 0.60 0.60

0.1002

Best Mean Worst

−1.00 −1.00 −1.00

0.80 0.80 0.80

1.00 1.00 1.00

2.79 2.79 2.79

−4.80 −4.80 −4.80

5.70 5.71 5.72

0.60 0.60 0.60

−1.00

0.80

1.00

2.80

−4.80

5.70

0.60

4.2

BSA-4

Model

the consistent accuracy of all the six variants of BSA for accurate, reliable and robust parameter estimation of EMS systems for all three noise variations, however, the BSA-3 outperforms the rest of the schemes. Further analysis of accuracy level is carried out for 100 independent runs of different variants of BSA through statistical operators based on the best, mean, and worst magnitudes of fitness forσ 2 = 0.0012 , 0.012 , and 0.12 in EMSMs. Statistical results for all six variants of BSA are provided in Tables 3, 4 and 5 for Examples 4.1, 4.2 and 4.3, respectively, along with the true

parameters. With the rise in noise level σ 2 = 0.0012 –0.12 dropoff in the performance is observed for all six variants of BSA in each scenario, however, all six variants are still applicable for optimizing EMS parameters with reasonable accuracy. Analysis on the performance proceeds further by determining the global performance indices as defined in Eqs. (30), (31), (32), and (33), respectively. Results on 100 runs of all six variants of BSA are listed in Table 6 for all three EMSMs in case of all three noise scenarios. Generally, the accuracy levels for global indices in terms of mean fitness lie around 10−02 to 10−09 in case of

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

15

Table 4 Performance comparison through results of statistics for EMS system in Example 4.3. Example

Variant

BSA-1

BSA-2

BSA-3

Noise

BSA-5

BSA-6

True values

Approximate parameter vector i=1

i=2

i=3

i=4

i=5

i=6

i=7

i=8

i=9

0.0012

Best Mean Worst

−1.094 −1.094 −1.096

0.109 0.109 0.107

1.000 1.000 0.998

−0.249 −0.249 −0.251

−0.028 −0.028 −0.030

1.97E−03 1.91E−03 2.01E−03

−9.32E−06 −1.82E−06 −1.00E−05

−1.00E−05 −1.63E−06 4.69E−06

2.36E−08 2.36E−08 2.38E−08

0.0102

Best Mean Worst

−1.094 −1.094 −1.093

0.109 0.109 0.108

1.000 1.000 1.000

−0.249 −0.249 −0.240

−0.028 −0.028 −0.018

1.89E−03 1.94E−03 1.86E−03

3.59E−06 −1.30E−06 3.82E−06

3.17E−06 −1.36E−06 6.50E−06

2.37E−08 2.36E−08 2.06E−08

0.1002

Best Mean Worst

−1.092 −1.091 −1.088

0.112 0.112 0.112

1.000 1.000 1.000

−0.249 −0.249 −0.247

−0.028 −0.028 −0.032

2.73E−03 2.83E−03 3.16E−03

1.80E−06 3.85E−06 1.00E−05

1.00E−05 9.00E−06 7.27E−06

2.43E−08 2.46E−08 2.63E−08

0.0012

Best Mean Worst

−1.094 −1.094 −1.094

0.109 0.109 0.109

1.000 1.000 1.000

−0.249 −0.249 −0.249

−0.028 −0.028 −0.029

1.90E−03 1.90E−03 1.95E−03

−7.04E−06 −3.05E−06 3.00E−06

−5.63E−07 −1.87E−06 −1.36E−06

2.36E−08 2.36E−08 2.38E−08

0.0102

Best Mean Worst

−1.094 −1.094 −1.093

0.109 0.109 0.109

1.000 1.000 1.000

−0.249 −0.249 −0.249

−0.028 −0.028 −0.029

1.90E−03 1.90E−03 1.78E−03

1.57E−06 −5.48E−06 1.00E−05

6.24E−07 4.85E−06 −1.00E−05

2.36E−08 2.36E−08 2.40E−08

0.1002

Best Mean Worst

−1.091 −1.091 −1.091

0.112 0.112 0.112

1.000 1.000 1.000

−0.249 −0.249 −0.249

−0.028 −0.028 −0.028

2.79E−03 2.79E−03 2.88E−03

9.83E−06 9.75E−06 8.89E−06

1.00E−05 9.99E−06 1.00E−05

2.45E−08 2.45E−08 2.47E−08

0.0012

Best Mean Worst

−1.094 −1.094 −1.094

0.109 0.109 0.109

1.000 1.000 1.000

−0.249 −0.249 −0.249

−0.028 −0.028 −0.028

1.90E−03 1.90E−03 1.90E−03

−7.98E−06 −7.09E−06 −6.44E−07

1.37E−07 −4.32E−07 −2.59E−06

2.36E−08 2.36E−08 2.36E−08

0.0102

Best Mean Worst

−1.094 −1.094 −1.094

0.109 0.109 0.109

1.000 1.000 1.000

−0.249 −0.249 −0.249

−0.028 −0.028 −0.028

1.89E−03 1.89E−03 1.89E−03

−1.00E−05 −9.96E−06 −1.00E−05

1.00E−05 9.94E−06 9.01E−06

2.36E−08 2.36E−08 2.36E−08

0.1002

Best Mean Worst

−1.091 −1.091 −1.091

0.112 0.112 0.112

1.000 1.000 1.000

−0.249 −0.249 −0.249

−0.028 −0.028 −0.028

2.79E−03 2.79E−03 2.79E−03

1.00E−05 1.00E−05 1.00E−05

1.00E−05 1.00E−05 1.00E−05

2.45E−08 2.45E−08 2.45E−08

0.0012

Best Mean Worst

−1.076 −1.090 −1.194

0.131 0.103 0.089

1.000 0.954 0.524

−0.261 −0.279 −0.468

−0.029 −0.015 −0.111

1.80E−03 3.26E−03 7.16E−03

−1.90E−06 −3.61E−07 5.50E−06

−8.82E−06 −1.11E−06 −6.86E−06

3.18E−08 1.87E−08 −9.23E−08

0.0102

Best Mean Worst

−1.105 −1.102 −1.134

0.110 0.104 0.143

0.993 0.956 0.694

−0.248 −0.297 −0.646

−0.041 −0.024 −0.031

3.25E−03 3.15E−03 1.70E−03

1.00E−05 −1.09E−06 −3.96E−07

9.81E−06 −1.90E−08 1.00E−05

2.17E−08 2.52E−08 2.79E−07

0.1002

Best Mean Worst

−1.095 −1.096 −1.149

0.136 0.131 0.241

1.000 0.956 0.637

−0.262 −0.289 −0.577

−0.028 −0.031 −0.033

8.72E−03 3.66E−03 1.00E−02

−7.70E−06 −1.80E−06 9.18E−06

−5.25E−06 2.75E−07 1.00E−05

3.42E−08 4.20E−08 5.13E−07

0.0012

Best Mean Worst

−1.097 −1.095 −1.143

0.129 0.113 0.137

1.000 0.988 1.000

−0.268 −0.256 −0.450

−0.030 −0.028 −0.071

1.25E−03 2.75E−03 2.85E−03

5.94E−06 −8.49E−07 1.00E−05

−4.07E−06 −1.16E−06 1.00E−05

2.68E−08 2.45E−08 2.92E−08

0.0102

Best Mean Worst

−1.093 −1.097 −0.982

0.101 0.107 0.189

1.000 0.988 0.995

−0.255 −0.260 −0.399

−0.033 −0.028 −0.077

2.76E−03 2.92E−03 5.21E−03

4.00E−06 −1.38E−07 −1.00E−05

3.18E−06 −2.95E−07 1.00E−05

2.35E−08 2.40E−08 8.01E−08

0.1002

Best Mean Worst

−1.082 −1.095 −1.163

0.104 0.114 0.134

1.000 0.988 1.000

−0.247 −0.255 −0.340

−0.014 −0.029 0.042

8.35E−04 3.50E−03 0.00E+00

−2.42E−06 −7.41E−09 −5.11E−06

3.77E−06 1.67E−06 −3.80E−06

1.92E−08 2.46E−08 −1.22E−08

0.0012

Best Mean Worst

−1.083 −1.101 −1.064

0.128 0.115 0.088

1.000 0.990 0.990

−0.249 −0.254 −0.356

−0.029 −0.025 −0.160

7.84E−04 2.85E−03 7.44E−03

−9.36E−06 −1.14E−06 9.88E−06

−1.01E−06 −1.11E−06 1.00E−05

3.58E−08 2.31E−08 7.50E−08

0.0102

Best Mean Worst

−1.095 −1.096 −1.035

0.113 0.110 0.088

0.996 0.992 0.984

−0.241 −0.253 −0.390

−0.028 −0.034

6.79E−03 2.84E−03 8.46E−04

8.31E−06 −3.13E−07 −9.57E−06

−6.99E−06 −2.18E−07 −1.00E−05

2.57E−08 2.53E−08 2.35E−08

0.1002

Best Mean Worst

−1.095 −1.096 −1.140

0.111 0.117 0.153

1.000 0.993 0.948

−0.254 −0.253 −0.280

0.052

3.21E−03 3.56E−03 0.00E+00

1.00E−05 −7.39E−07 −3.23E−06

6.78E−06 2.16E−06 6.64E−06

2.03E−08 2.48E−08 −7.12E−09

−1.094

0.109

1.000

−0.249

−0.028

1.90E−03

−7.83E−06

1.78E−08

2.36E−08

4.3

BSA-4

Model

Example 4.1, 10−04 to 10−13 in case of Example 4.2, and 10−04 to 10−11 in case of Example 4.3, for all six variants of BSA, while the trend of performance indices of GMAEθ , GRMSEθ , and GTICθ also lie in the same range. The acquired values of the global metrics are found close to their optimal magnitudes that establish the consistency of the BSAs for parameter estimation of EMS systems. Computational complexity analyses for all six variants of BSA is performed in terms of average time consumed, cycles executed and merits functions evaluated by the algorithms to obtain the optimized parameters of all three EMS case studies. All three

0.017

−0.020 −0.028

complexity measures are computed for 100 independent runs of each variant of BSAs and results are tabulated in Table 7 in terms of mean and standard deviation gauges. The range of respective average execution time, generations and functions evaluated are around 3.65 ± 0.07, 3000 and 180060 for EMS system in Example 4.1, 0.89 ± 0.03, 264 ± 50.43 and 41189.17 ± 15950.43 for Example 4.2 and 5.34 ± 0.08, 1000 and 280280 for Example 4.3. It is noticed that the magnitudes of complexity metrics based on fitness functions evaluated for EMS in Example 4.2 is greater than rest of the two models due to higher number of generations

16

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

Table 5 Performance of BSA variants based on the best run through accuracy and complexity measures for all three case studies of EMS systems. Example

Variant

Noise σ 2

Accuracy operators

Complexity operators

ε

δ

MAE

RMSE

TIC

Time

Gens

Functions

2

BSA-1

0.001 0.0102 0.1002

4.73E−06 8.98E−06 3.13E−02

2.23E−06 2.49E−06 5.18E−06

1.95E−03 2.02E−03 3.55E−03

2.17E−03 2.43E−03 5.05E−03

1.11E−06 1.24E−06 2.59E−06

0.363 0.354 0.356

200 200 200

8040 8040 8040

BSA-2

0.0012 0.0102 0.1002

3.62E−11 3.13E−06 3.13E−02

5.90E−09 1.19E−08 1.28E−06

4.59E−06 8.33E−06 5.66E−04

5.75E−06 1.16E−05 1.25E−03

2.95E−09 5.96E−09 6.40E−07

0.696 0.712 0.700

400 400 400

16040 16040 16040

BSA-3

0.0012 0.0102 0.1002

3.13E−12 3.13E−06 3.13E−02

3.15E−10 1.31E−08 1.33E−06

1.71E−08 6.22E−06 5.87E−04

2.08E−08 1.14E−05 1.30E−03

1.06E−11 5.83E−09 6.66E−07

1.044 1.212 1.038

600 600 600

24040 24040 24040

BSA-4

0.0012 0.0102 0.1002

1.54E−05 1.41E−05 3.13E−02

3.30E−05 1.25E−05 5.91E−05

3.29E−03 1.77E−03 2.99E−03

3.92E−03 3.31E−03 3.45E−03

2.01E−06 1.70E−06 1.77E−06

0.191 0.185 0.217

200 200 200

4020 4020 4020

BSA-5

0.0012 0.0102 0.1002

3.34E−06 5.21E−06 3.13E−02

7.36E−06 8.21E−06 2.40E−05

1.28E−03 1.31E−03 3.24E−03

1.83E−03 1.45E−03 4.17E−03

9.37E−07 7.44E−07 2.14E−06

0.359 0.358 0.375

200 200 200

8040 8040 8040

BSA-6

0.0012 0.0102 0.1002

5.42E−06 8.44E−06 3.13E−02

1.62E−05 1.54E−05 1.19E−04

1.74E−03 1.62E−03 3.72E−03

2.33E−03 2.30E−03 4.29E−03

1.19E−06 1.18E−06 2.20E−06

0.528 0.533 0.565

200 200 200

12060 12060 12060

BSA-1

0.0012 0.0102 0.1002

7.61E−08 9.65E−08 1.80E−04

4.35E−05 1.62E−05 9.04E−04

8.82E−05 2.78E−05 1.57E−03

1.34E−04 5.00E−05 2.78E−03

2.18E−05 8.11E−06 4.52E−04

2.31 2.31 2.33

1000 1000 1000

40040 40040 40040

BSA-2

0.0012 0.0102 0.1002

1.73E−12 4.30E−08 1.67E−04

1.19E−07 6.28E−06 1.56E−03

2.59E−07 1.26E−05 2.67E−03

3.67E−07 1.93E−05 4.79E−03

5.96E−08 3.14E−06 7.78E−04

4.58 4.68 4.59

2000 2000 2000

80040 80040 80040

BSA-3

0.0012 0.0102 0.1002

2.93E−14 1.67E−08 1.67E−04

4.99E−09 1.55E−05 1.56E−03

1.04E−08 2.66E−05 2.67E−03

1.54E−08 4.79E−05 4.80E−03

2.50E−09 7.77E−06 7.79E−04

6.93 6.88 7.02

3000 3000 3000

120040 120040 120040

BSA-4

0.0012 0.0102 0.1002

4.74E−09 7.48E−08 2.30E−04

3.32E−06 3.71E−05 4.20E−04

6.90E−06 7.24E−05 9.23E−04

1.02E−05 1.14E−04 1.29E−03

1.66E−06 1.86E−05 2.10E−04

0.59 0.61 0.61

1000 1000 1000

10010 10010 10010

BSA-5

0.0012 0.0102 0.1002

1.30E−06 6.48E−08 1.77E−04

2.32E−05 2.84E−05 1.06E−03

5.33E−05 4.89E−05 1.88E−03

7.13E−05 8.74E−05 3.25E−03

1.16E−05 1.42E−05 5.28E−04

2.31 2.32 2.31

1000 1000 1000

40040 40040 40040

BSA-6

0.0012 0.0102 0.1002

4.36E−07 3.05E−07 1.83E−04

5.49E−05 4.96E−05 9.91E−04

1.03E−04 9.36E−05 1.81E−03

1.69E−04 1.53E−04 3.05E−03

2.74E−05 2.48E−05 4.96E−04

3.46 3.47 3.51

1000 1000 1000

60060 60060 60060

BSA-1

0.0012 0.0102 0.1002

2.90E−07 1.80E−07 8.18E−05

1.08E−04 1.58E−04 2.09E−03

3.13E−04 2.41E−04 1.22E−03

5.44E−05 7.93E−05 1.05E−03

5.41E−05 7.90E−05 1.05E−03

1.44 1.46 1.44

500 500 500

20040 20040 20040

BSA-2

0.0012 0.0102 0.1002

1.28E−09 1.08E−08 8.10E−05

3.86E−06 1.77E−05 2.66E−03

1.55E−05 2.57E−05 7.47E−04

1.94E−06 8.87E−06 1.34E−03

1.93E−06 8.83E−06 1.33E−03

2.14 2.14 2.15

750 750 750

30040 30040 30040

BSA-3

0.0012 0.0102 0.1002

2.86E−12 7.75E−09 8.10E−05

4.68E−07 2.20E−05 2.79E−03

8.92E−07 7.91E−06 7.53E−04

2.35E−07 1.10E−05 1.40E−03

2.34E−07 1.10E−05 1.40E−03

2.85 2.86 2.87

1000 1000 1000

40040 40040 40040

BSA-4

0.0012 0.0102 0.1002

1.55E−03 5.22E−03 6.70E−03

2.03E−02 1.25E−02 1.59E−02

1.20E−02 3.61E−02 5.35E−02

1.02E−02 6.29E−03 8.00E−03

1.02E−02 6.25E−03 7.96E−03

0.16 0.16 0.16

200 200 200

2010 2010 2010

BSA-5

0.0012 0.0102 0.1002

2.04E−03 1.97E−04 4.59E−04

1.64E−02 7.84E−03 1.29E−02

2.85E−02 7.06E−03 1.02E−02

8.24E−03 3.94E−03 6.48E−03

8.22E−03 3.92E−03 6.47E−03

0.59 0.59 0.59

200 200 200

8040 8040 8040

BSA-6

0.0012 0.0102 0.1002

1.42E−03 7.37E−04 5.25E−04

1.48E−02 7.33E−03 6.60E−03

1.59E−02 1.21E−02 1.39E−02

7.44E−03 3.68E−03 3.31E−03

7.42E−03 3.67E−03 3.30E−03

0.87 0.87 0.87

200 200 200

12060 12060 12060

4.1

4.2

4.3

executed for each variant. Moreover, it is observed that with the increase in the cycle consumption, average execution time increases which is quite evident due to increase in function counts. Furthermore, increase in population size also directly affects the magnitude of complexity operators.

4.3. Analysis through single objective for EMS The performance of the proposed BSA heuristic algorithm for optimization of EMS models is also studied using single objective based fitness function given in (36), (39) and (42) for Examples 4.1, 4.2 and 4.3 respectively. The results of BSA-1 to 6 in terms of learning curves are shown graphically in Figs. 7, 8 and

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

17

Table 6 Performance comparison through global operators for all three case studies of EMS systems. Example

Variant

Noise σ 2

Global Operators Fitness ε

MWD δ

RMSE

TIC

Mean

STD

Mean

STD

Mean

STD

Mean

STD

2

BSA-1

0.001 0.0102 0.1002

3.68E−04 3.91E−04 3.14E−02

6.56E−04 7.03E−04 7.62E−04

1.14E−02 1.22E−02 4.16E−02

9.17E−03 8.80E−03 3.54E−02

1.50E−02 1.59E−02 5.56E−02

1.21E−02 1.17E−02 4.78E−02

7.67E−06 8.15E−06 2.85E−05

6.19E−06 5.98E−06 2.45E−05

BSA-2

0.0012 0.0102 0.1002

2.20E−08 3.14E−06 2.99E−02

7.39E−08 4.45E−08 5.37E−04

7.44E−05 7.56E−05 7.21E−02

8.36E−05 6.31E−05 2.53E−02

9.64E−05 9.61E−05 1.03E−01

1.13E−04 9.21E−05 3.60E−02

4.94E−08 4.92E−08 5.28E−05

5.81E−08 4.72E−08 1.84E−05

BSA-3

0.0012 0.0102 0.1002

3.75E−09 3.13E−06 2.97E−02

3.75E−08 5.30E−08 4.11E−04

2.88E−06 1.17E−05 7.64E−02

2.50E−05 3.51E−05 1.75E−02

6.60E−06 2.49E−05 1.10E−01

6.12E−05 8.44E−05 2.50E−02

3.38E−09 1.28E−08 5.62E−05

3.14E−08 4.32E−08 1.28E−05

BSA-4

0.0012 0.0102 0.1002

7.54E−04 1.02E−03 3.20E−02

8.91E−04 1.95E−03 1.31E−03

1.72E−02 1.90E−02 3.76E−02

9.99E−03 1.48E−02 3.34E−02

2.37E−02 2.50E−02 4.91E−02

1.39E−02 2.00E−02 4.33E−02

1.22E−05 1.28E−05 2.52E−05

7.14E−06 1.02E−05 2.22E−05

BSA-5

0.0012 0.0102 0.1002

3.62E−04 3.10E−04 3.13E−02

5.06E−04 4.39E−04 7.71E−04

1.19E−02 1.11E−02 4.53E−02

8.13E−03 7.06E−03 3.78E−02

1.56E−02 1.46E−02 6.03E−02

1.10E−02 9.69E−03 5.08E−02

7.98E−06 7.50E−06 3.09E−05

5.61E−06 4.97E−06 2.60E−05

BSA-6

0.0012 0.0102 0.1002

2.02E−04 2.36E−04 3.11E−02

2.07E−04 3.29E−04 6.74E−04

9.92E−03 9.62E−03 4.69E−02

4.90E−03 6.18E−03 3.74E−02

1.28E−02 1.27E−02 6.36E−02

6.29E−03 8.48E−03 5.08E−02

6.54E−06 6.52E−06 3.26E−05

3.22E−06 4.35E−06 2.60E−05

BSA-1

0.0012 0.0102 0.1002

1.73E−05 2.00E−05 1.78E−04

2.59E−05 4.24E−05 1.46E−05

1.26E−03 1.42E−03 3.05E−03

1.04E−03 1.62E−03 6.96E−04

1.92E−03 2.11E−03 5.23E−03

1.55E−03 2.30E−03 1.04E−03

3.11E−04 3.42E−04 8.48E−04

2.51E−04 3.73E−04 1.69E−04

BSA-2

0.0012 0.0102 0.1002

4.03E−09 2.10E−08 1.67E−04

9.96E−09 1.24E−08 3.17E−11

1.97E−05 3.45E−05 2.67E−03

2.51E−05 2.13E−05 3.39E−07

2.94E−05 5.85E−05 4.80E−03

3.55E−05 3.31E−05 6.93E−07

4.77E−06 9.51E−06 7.79E−04

5.76E−06 5.38E−06 1.12E−07

BSA-3

0.0012 0.0102 0.1002

5.06E−13 1.67E−08 1.67E−04

1.37E−12 1.92E−14 1.32E−17

2.02E−07 2.67E−05 2.67E−03

2.74E−07 9.21E−09 6.53E−10

3.07E−07 4.80E−05 4.80E−03

3.89E−07 1.65E−08 1.16E−09

4.98E−08 7.79E−06 7.79E−04

6.32E−08 2.68E−09 1.88E−10

BSA-4

0.0012 0.0102 0.1002

1.80E−03 1.57E−01 1.02E−03

5.64E−03 1.10E+00 3.67E−03

9.82E−03 1.50E−02 7.38E−03

2.05E−02 6.07E−02 1.42E−02

1.52E−02 2.47E−02 1.20E−02

3.18E−02 1.03E−01 2.22E−02

2.44E−03 4.25E−03 1.93E−03

5.09E−03 1.84E−02 3.55E−03

BSA-5

0.0012 0.0102 0.1002

1.76E−05 1.88E−05 1.81E−04

3.47E−05 3.15E−05 2.74E−05

1.28E−03 1.33E−03 3.09E−03

1.44E−03 1.36E−03 8.02E−04

1.93E−03 2.01E−03 5.28E−03

2.10E−03 2.01E−03 1.23E−03

3.12E−04 3.25E−04 8.57E−04

3.40E−04 3.25E−04 1.99E−04

BSA-6

0.0012 0.0102 0.1002

1.29E−05 9.76E−06 1.75E−04

1.82E−05 9.82E−06 9.04E−06

1.19E−03 1.03E−03 2.90E−03

1.09E−03 7.04E−04 4.42E−04

1.79E−03 1.59E−03 4.97E−03

1.57E−03 1.08E−03 6.83E−04

2.91E−04 2.58E−04 8.06E−04

2.55E−04 1.75E−04 1.11E−04

BSA-1

0.0012 0.0102 0.1002

6.44E−07 9.66E−07 8.21E−05

9.52E−07 2.35E−06 1.65E−06

3.39E−04 3.67E−04 9.99E−04

2.50E−04 3.28E−04 2.72E−04

5.56E−04 6.16E−04 1.61E−03

4.06E−04 5.84E−04 3.95E−04

5.54E−04 6.13E−04 1.61E−03

4.04E−04 5.82E−04 3.94E−04

BSA-2

0.0012 0.0102 0.1002

2.75E−09 1.38E−08 8.10E−05

1.32E−08 2.40E−08 3.99E−09

1.77E−05 2.74E−05 7.59E−04

2.24E−05 3.09E−05 1.15E−05

2.72E−05 4.27E−05 1.40E−03

3.83E−05 5.41E−05 1.54E−05

2.71E−05 4.25E−05 1.40E−03

3.82E−05 5.39E−05 1.53E−05

BSA-3

0.0012 0.0102 0.1002

1.27E−11 7.76E−09 8.10E−05

2.74E−11 5.42E−11 2.91E−11

1.40E−06 8.67E−06 7.53E−04

1.22E−06 1.46E−06 7.23E−07

1.94E−06 1.29E−05 1.40E−03

1.76E−06 1.84E−06 8.49E−07

1.93E−06 1.29E−05 1.40E−03

1.75E−06 1.83E−06 8.46E−07

BSA-4

0.0012 0.0102 0.1002

4.82E−03 3.99E−03 5.16E−03

6.03E−03 4.33E−03 5.46E−03

3.08E−02 3.00E−02 3.27E−02

1.87E−02 1.60E−02 1.95E−02

5.29E−02 5.06E−02 5.49E−02

3.21E−02 2.78E−02 3.29E−02

5.33E−02 5.06E−02 5.50E−02

3.31E−02 2.82E−02 3.35E−02

BSA-5

0.0012 0.0102 0.1002

1.09E−03 1.08E−03 1.04E−03

1.12E−03 1.02E−03 7.52E−04

1.56E−02 1.54E−02 1.46E−02

7.26E−03 7.77E−03 6.41E−03

2.63E−02 2.59E−02 2.47E−02

1.22E−02 1.29E−02 1.07E−02

2.62E−02 2.58E−02 2.46E−02

1.21E−02 1.29E−02 1.06E−02

BSA-6

0.0012 0.0102 0.1002

7.49E−04 7.56E−04 8.29E−04

6.66E−04 6.97E−04 5.33E−04

1.32E−02 1.25E−02 1.33E−02

6.44E−03 5.84E−03 5.61E−03

2.23E−02 2.11E−02 2.22E−02

1.03E−02 1.02E−02 9.18E−03

2.22E−02 2.10E−02 2.21E−02

1.02E−02 1.01E−02 9.16E−03

4.1

4.2

4.3

9 for Examples 4.1, 4.2 and 4.3 respectively using all three noise variations. In all these figures, it is found that for all three levels of noise variances, all the variants of BSA are convergent but the accuracy and convergence of the BSA-3 is much superior than all other schemes for each EMS models. Further analysis through single objective for EMS parameter estimation is carried out by 100 independent runs of different variants of BSA using statistical operators based on the best,

mean, and worst magnitudes of fitness for σ 2 = 0.0012 , 0.012 , and 0.12 . Statistical results for all six variants of BSA are provided in Tables 8, 9 and 10 for Examples 4.1, 4.2 and 4.3, respectively, along with the true parameters. With an increase in the noise level, σ 2 = 0.0012 –0.12 , drop-off in the performance is observed for all six variants of BSA in each scenario. Moreover, the results obtained through single objective for EMS given in Tables 3 to 5 are compared with supporting objective based results presented

18

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

Table 7 Comparison through complexity operators for all three case studies of EMS systems. Example

Variant

Noise σ 2

Complexity operators Time

Generations

Function Counts

Mean

STD

Mean

STD

Mean

STD

2

BSA-1

0.001 0.0102 0.1002

0.358 0.362 0.358

0.003 0.004 0.002

200 200 200

0 0 0

8040 8040 8040

0 0 0

BSA-2

0.0012 0.0102 0.1002

0.699 0.711 0.700

0.004 0.006 0.003

400 400 400

0 0 0

16040 16040 16040

0 0 0

BSA-3

0.0012 0.0102 0.1002

1.038 1.127 1.044

0.004 0.096 0.012

600 600 600

0 0 0

24040 24040 24040

0 0 0

BSA-4

0.0012 0.0102 0.1002

0.189 0.186 0.203

0.003 0.001 0.022

200 200 200

0 0 0

4020 4020 4020

0 0 0

BSA-5

0.0012 0.0102 0.1002

0.358 0.359 0.379

0.003 0.006 0.008

200 200 200

0 0 0

8040 8040 8040

0 0 0

BSA-6

0.0012 0.0102 0.1002

0.528 0.529 0.551

0.002 0.002 0.026

200 200 200

0 0 0

12060 12060 12060

0 0 0

BSA-1

0.0012 0.0102 0.1002

2.421 2.311 2.315

1.067 0.016 0.016

1000 1000 1000

0 0 0

40040 40040 40040

0 0 0

BSA-2

0.0012 0.0102 0.1002

4.639 4.596 4.602

0.056 0.020 0.028

2000 2000 2000

0 0 0

80040 80040 80040

0 0 0

BSA-3

0.0012 0.0102 0.1002

6.942 6.878 6.916

0.082 0.021 0.064

3000 3000 3000

0 0 0

120040 120040 120040

0 0 0

BSA-4

0.0012 0.0102 0.1002

0.594 0.596 0.593

0.004 0.007 0.005

1000 1000 1000

0 0 0

10010 10010 10010

0 0 0

BSA-5

0.0012 0.0102 0.1002

2.322 2.318 2.313

0.017 0.021 0.016

1000 1000 1000

0 0 0

40040 40040 40040

0 0 0

BSA-6

0.0012 0.0102 0.1002

3.486 3.472 3.470

0.030 0.025 0.024

1000 1000 1000

0 0 0

60060 60060 60060

0 0 0

BSA-1

0.0012 0.0102 0.1002

1.442 1.448 1.440

0.008 0.014 0.003

500 500 500

0 0 0

20040 20040 20040

0 0 0

BSA-2

0.0012 0.0102 0.1002

2.148 2.164 2.151

0.007 0.022 0.012

750 750 750

0 0 0

30040 30040 30040

0 0 0

BSA-3

0.0012 0.0102 0.1002

2.856 2.858 2.864

0.007 0.007 0.010

1000 1000 1000

0 0 0

40040 40040 40040

0 0 0

BSA-4

0.0012 0.0102 0.1002

0.160 0.160 0.160

0.001 0.001 0.001

200 200 200

0 0 0

2010 2010 2010

0 0 0

BSA-5

0.0012 0.0102 0.1002

0.587 0.589 0.587

0.002 0.004 0.002

200 200 200

0 0 0

8040 8040 8040

0 0 0

BSA-6

0.0012 0.0102 0.1002

0.872 0.873 0.873

0.003 0.002 0.003

200 200 200

0 0 0

12060 12060 12060

0 0 0

4.1

4.2

4.3

in Tables 8 to 10. It is observed that using supporting objective based fitness function for EMS provides better parameter estimates than single objective based fitness function and this accuracy difference is quite evident in case of log sigmoid nonlinearity, taken in Example 4.3 of EMS model. However, all six variants are still applicable for optimizing EMS parameters with reasonable accuracy.

5. Conclusions Novel application of Backtracking search optimization heuristic, i.e., BSA is presented for accurate, reliable and robust parameter estimation problem of EMS systems required for rehabilitation of paralyzed muscles. The EMS systems are modeled with Hammerstein control autoregressive structure in which linear dynamics of muscle are given by transfer function and nonlinear

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

19

Table 8 Performance comparison through results of statistics using single objective based fitness function for EMS system in Example 4.1. Example

Variant

Noise

0.0012

BSA-1

0.0102

0.1002

0.0012

BSA-2

0.0102

0.1002

0.0012

BSA-3

0.0102

0.1002 4.1 0.0012

BSA-4

0.0102

0.1002

0.0012

BSA-5

0.0102

0.1002

0.0012

BSA-6

0.0102

0.1002 True values

Model

Approximate parameter vector i=1

i=2

i=3

i=4

i=5

i=6

Best Mean Worst Best Mean Worst Best Mean Worst

−2.00 −1.19 −1.26 −0.71 −1.01 −0.63 −1.20 −0.95 −1.03

1.00 0.17 0.22 0.34 0.06 0.42 0.22 0.14 0.08

0.02 0.05 0.02 1.98 0.28 0.03 6.96 1.04 0.20

0.83 0.14 0.08 0.13 0.78 1.66 2.00 1.51 4.39

0.02 0.23 0.35 1.59 0.54 2.00 2.00 0.89 1.17

2196.09 2337.04 1181.09 2420.33 2134.39 1691.20 2524.85 1203.06 1249.11

Best Mean Worst Best Mean Worst Best Mean Worst

−1.97 −1.24 −1.74 −0.54 −1.08 −0.67 −1.04 −1.16 −0.97

0.96 0.24 0.83 0.59 0.01 0.46 0.08 0.09 0.12

0.05 0.08 0.00 0.26 0.34 1.58 1.20 0.38 1.92

0.09 0.07 1.08 0.21 0.22 0.46 5.44 3.06 0.61

0.14 0.22 1.37 1.56 0.80 0.40 1.17 0.74 1.38

2502.72 1531.01 1377.10 2133.01 2723.11 1377.45 2003.57 1290.76 1417.13

Best Mean Worst Best Mean Worst Best Mean Worst

−1.92 −1.62 −1.69 −1.38 −1.17 −1.36 −1.61 −1.33 −1.42

0.92 0.65 0.75 0.31 0.09 0.27 0.57 0.27 0.37

2.00 0.07 0.00 0.16 1.12 3.71 3.19 0.80 1.15

2.00 0.24 0.01 0.51 0.01 1.17 1.98 2.16 4.35

0.00 0.97 2.00 1.10 0.98 0.06 1.15 1.12 0.99

2288.71 2349.85 1614.40 2388.33 2066.76 1223.66 2468.62 1864.05 1960.33

Best Mean Worst Best Mean Worst Best Mean Worst

−0.82 −0.78 −0.41 −0.40 −0.75 −0.88 −1.40 −0.66 −0.38

0.01 0.23 0.40 0.84 0.40 0.43 0.34 0.41 0.55

0.11 0.07 0.01 1.17 0.99 0.22 4.30 1.15 0.07

0.01 0.14 0.09 5.51 0.85 3.18 2.00 0.48 0.56

1.31 0.39 1.30 0.09 0.38 0.41 1.07 1.38 7.00

2569.25 2619.78 1353.52 2237.44 2225.24 1103.36 2255.14 1286.77 1002.00

Best Mean Worst Best Mean Worst Best Mean Worst

−1.46 −1.29 −0.78 −1.85 −1.46 −1.66 −1.61 −0.86 −0.35

0.40 0.29 0.30 0.83 0.40 0.60 0.56 0.22 0.68

0.21 0.18 2.00 0.10 1.06 2.00 0.42 2.04 −0.01

0.44 0.11 0.00 0.82 0.01 0.06 1.97 1.74 3.33

0.01 0.34 0.32 0.69 0.24 0.23 1.85 1.10 6.72

2206.11 1995.29 778.01 2333.43 2002.83 395.84 2458.23 1429.30 −2.00

Best Mean Worst Best Mean Worst Best Mean Worst

−0.51 −0.91 −1.17 −1.05 −1.13 −0.91 −1.69 −1.23 −1.64

0.51 0.08 0.27 0.09 0.04 0.26 0.65 0.16 0.62

0.00 0.68 0.44 0.37 0.82 5.18 1.49 3.05 5.44

0.95 0.03 0.00 5.01 0.14 0.36 7.00 0.14 1.77

0.82 0.50 2.00 0.22 0.02 0.14 0.83 1.12 0.46

2291.04 2230.56 1582.23 2669.13 2358.64 1219.58 2390.94 1809.02 1392.68

−1.9985

0.9985

0.0022

6.8994

0.0410

2389.70

characteristic are based on polynomial, sigmoid and cubic spline kernels. The six variants of BSA are designed by exploiting the memory size and population dynamics, and effectively applied to three cases studies of EMS models for noise variances σ 2 = 0.0012 , 0.012 , and 0.12 . Comparative study from true parameter of systems based on performance indices of MAEθ , RMSEθ , MWDθ and TICθ established the worth of each algorithm however, the performance of each variant degraded with increase in the level of noise. Generally, the BSA-3 provides better results in terms of performance measures from the rest. Verification and validations for the variants of BSA is ascertained through statistical interpretations of the results based on large number of trials

for optimization of parameters of EMS systems of all three case studies, and BSA-3 is found to be the most accurate, reliable and robust performer. The variants of BSA algorithm based on more number of iterations as well as operated with bigger population of individual perform grander estimation of EMS parameters and vice versa. Additionally, the complexity of the BSAs for EMS system in Example 4.2 is higher from other two case studies because the design algorithms for the said system based on larger memory, i.e., number of iterations. The proposed BSA is accurate and reliable for both single objective and supporting objective based fitness functions, but the later provides the more accurate parameter estimates of EMS system, especially in case

20

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

Table 9 Performance comparison through results of statistics using single objective based fitness function for EMS system in Example 4.2. Example

Variant

BSA-1

BSA-2

BSA-3

Noise

BSA-5

BSA-6

True values

Approximate parameter vector i=1

i=2

i=3

i=4

i=5

i=6

i=7

0.0012

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.990 0.976

2.798 2.824 2.872

−4.800 −4.848 −4.921

5.702 5.756 5.843

0.600 0.594 0.585

0.0102

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

0.999 0.987 0.971

2.805 2.838 2.881

−4.806 −4.864 −4.943

5.706 5.778 5.870

0.599 0.592 0.583

0.1002

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.991 0.965

2.808 2.827 2.907

−4.806 −4.847 −4.972

5.697 5.754 5.905

0.600 0.594 0.579

0.0012

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

0.998 0.986 0.965

2.806 2.840 2.901

−4.811 −4.869 −4.974

5.713 5.781 5.906

0.599 0.592 0.579

0.0102

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.991 0.979

2.800 2.826 2.860

−4.801 −4.845 −4.902

5.701 5.754 5.821

0.600 0.594 0.588

0.1002

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

0.999 0.985 0.967

2.802 2.844 2.896

−4.803 −4.875 −4.965

5.704 5.790 5.896

0.600 0.591 0.580

0.0012

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

0.998 0.991 0.982

2.805 2.825 2.851

−4.808 −4.843 −4.888

5.710 5.751 5.804

0.599 0.595 0.589

0.0102

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.988 0.975

2.801 2.834 2.871

−4.802 −4.858 −4.921

5.702 5.769 5.844

0.600 0.593 0.585

0.1002

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

0.998 0.993 0.988

2.806 2.821 2.833

−4.810 −4.836 −4.856

5.712 5.742 5.767

0.599 0.596 0.593

0.0012

Best Mean Worst

−1.000 −1.000 −0.999

0.800 0.800 0.800

0.999 0.990 0.960

2.809 2.830 3.000

−4.808 −4.846 −5.000

5.704 5.751 5.886

0.599 0.596 0.582

0.0102

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.995 0.967

2.800 2.812 2.899

−4.800 −4.822 −4.964

5.699 5.725 5.892

0.600 0.598 0.581

0.1002

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.986 0.959

2.798 2.839 2.906

−4.797 −4.871 −5.000

5.699 5.778 5.944

0.600 0.593 0.577

0.0012

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.989 0.961

2.799 2.830 2.922

−4.799 −4.854 −5.000

5.700 5.765 5.926

0.600 0.594 0.577

0.0102

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.990 0.971

2.801 2.828 2.875

−4.800 −4.850 −4.944

5.700 5.758 5.873

0.601 0.594 0.583

0.1002

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

0.997 0.988 0.971

2.806 2.836 2.888

−4.813 −4.861 −4.940

5.716 5.772 5.871

0.598 0.593 0.583

0.0012

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

0.995 0.985 0.975

2.814 2.841 2.871

−4.824 −4.873 −4.923

5.729 5.787 5.845

0.597 0.591 0.585

0.0102

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.986 0.965

2.802 2.838 2.905

−4.800 −4.867 −4.976

5.698 5.779 5.906

0.600 0.592 0.579

0.1002

Best Mean Worst

−1.000 −1.000 −1.000

0.800 0.800 0.800

1.000 0.989 0.972

2.803 2.828 2.878

−4.801 −4.854 −4.938

5.699 5.767 5.867

0.600 0.593 0.583

−1.00

0.80

1.00

2.80

−4.80

5.70

0.60

4.2

BSA-4

Model

of sigmoid type nonlinearities. The proposed study shows that nature/bio inspired heuristic is an alternate, accurate and reliable computing paradigm that effectively estimates the parameters of (EMS) model. Therefore, the proposed study motivates the researchers to investigate in applying other metaheuristics based on fractional order Darwinian particle swarm optimization, fractional order firefly algorithm, fireworks and new cuckoo search algorithms for optimization of EMS model. Moreover, one may

explore in developing modified forms of BSA for more accurate and robust estimation of EMS parameters in case of real and online scenarios. Declaration of competing interest No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

21

Table 10 Performance comparison through results of statistics using single objective based fitness function for EMS system in Example 4.3. Example

Variant

BSA-1

BSA-2

BSA-3

Noise

BSA-5

BSA-6

True values

Approximate parameter vector i=1

i=2

i=3

i=4

i=5

i=6

i=7

i=8

i=9

0.0012

Best Mean Worst

−1.1902 −1.1873 −1.5009

0.1908 0.2010 0.5128

1.0000 0.4764 0.0628

−0.4373 −0.4546 −0.1441

0.1100 −0.3203 −0.3591

0.00E+00 1.05E−03 0.00E+00

−1.00E−05 1.06E−06 1.00E−05

−1.00E−05 −4.44E−06 6.33E−06

−1.65E−08 −2.05E−09 1.97E−08

0.0102

Best Mean Worst

−1.3403 −1.2852 −1.5371

0.3542 0.2982 0.5480

0.6948 0.3117 0.0145

−0.6534 −0.2908 −0.0035

0.4584 0.1908 0.8498

5.62E−03 4.69E−03 1.00E−02

1.00E−05 1.30E−07 −1.00E−05

−1.00E−05 1.55E−07 1.00E−05

6.61E−08 8.67E−08 3.35E−07

Best Mean Worst

−1.2183 −1.2886 −1.2362

0.2367 0.3015 0.2428

0.9988 0.4689 0.0231

−0.6815 −0.4448

0.0918

0.1002

0.00E+00 3.23E−03 0.00E+00

1.00E−05 3.59E−06 −1.00E−05

9.91E−06 −1.05E−06 1.00E−05

6.58E−09 1.05E−07 4.11E−07

0.0012

Best Mean Worst

−1.1205 −1.1438 −1.2677

0.1394 0.1598 0.2854

0.6968 0.2511 0.0749

0.0084

5.76E−03 5.70E−03 1.00E−02

−1.00E−05 −1.28E−07 −1.00E−05

1.00E−05 0.00E+00 1.00E−05

5.99E−08 5.63E−08 3.71E−07

0.0102

Best Mean Worst

−1.1469 −1.1314 −1.2310

0.1641 0.1469 0.2513

0.9498 0.4109 0.6383

−0.6968 −0.4731 −0.6062

7.91E−03 4.59E−03 3.35E−03

1.00E−05 5.76E−06 1.00E−05

1.00E−05 −1.53E−07 −1.41E−06

1.16E−07 4.48E−08 4.25E−08

0.1002

Best Mean Worst

−1.1905 −1.2717 −1.2458

0.2036 0.2856 0.2564

0.8172 0.4216 0.0000

−0.7380 −0.5268 0.0332

−0.2669 −1.0000

9.40E−03 3.78E−03 0.00E+00

1.00E−05 7.96E−06 −1.00E−05

−1.00E−05 −6.27E−06 1.00E−05

1.22E−07 9.04E−08 5.88E−07

0.0012

Best Mean Worst

−1.1809 −1.1681 −1.1903

0.1982 0.1842 0.2066

0.4532 0.5336 0.0209

−0.1768 −0.5176 −0.0708

−0.0077 −0.2133 −0.9500

3.77E−03 5.37E−03 6.10E−03

−1.00E−05 −2.49E−06 1.00E−05

−1.00E−05 4.60E−06 1.00E−05

4.16E−08 7.42E−08 6.93E−08

0.0102

Best Mean Worst

−1.1296 −1.1405 −1.1857

0.1455 0.1562 0.1994

0.5396 0.3530 0.3371

−0.2975 −0.3256 −0.3722

−0.1476 −0.3333 −0.9719

6.37E−03 5.34E−03 0.00E+00

−1.00E−05 2.84E−06 1.00E−05

1.00E−05 4.78E−06 7.24E−06

9.01E−08 7.47E−08 −7.45E−09

Best Mean Worst

−1.2809 −1.2789 −1.3046

0.2945 0.2926 0.3200

0.9052 0.4886 0.6942

−0.4044 −0.4541 −0.6547

−0.2302

0.1002

6.53E−03 5.18E−03 2.57E−03

−1.00E−05 6.00E−06 1.00E−05

1.00E−05 −3.75E−06 −7.52E−06

8.72E−08 4.70E−08 2.41E−08

0.0012

Best Mean Worst

−1.0966 −1.4770 −2.0000

0.0958 0.4904 1.0000

0.6066 0.3168 0.0000

−0.2870 −0.3063 −0.0050

−0.3443 −0.0008 1.0000

1.00E−02 4.86E−03 4.39E−05

9.26E−06 3.05E−06 1.00E−05

−6.76E−06 3.94E−07 −8.73E−06

1.34E−07 8.62E−08 −4.00E−07

0.0102

Best Mean Worst

−1.3329 −1.4559 −2.0000

0.3229 0.4686 1.0000

1.0000 0.5024 0.0403

−0.9449 −0.4403 −0.0549

0.0443 0.0480 1.0000

9.51E−03 3.70E−03 0.00E+00

−1.00E−05 3.88E−06 8.29E−06

3.83E−06 −1.00E−06 −1.00E−05

1.13E−07 2.12E−08 −3.81E−07

0.1002

Best Mean Worst

−1.1508 −1.5108 −1.9635

0.1665 0.5229 1.0000

0.0637 0.3590 0.0998

−0.1156 −0.2205 −0.0889

−0.1073 −0.1487 −1.0000

1.12E−03 5.21E−03 1.00E−02

−5.66E−06 −3.35E−06 −1.00E−05

−1.00E−05 0.00E+00 1.00E−05

−1.76E−07 −9.05E−08 5.95E−07

Best Mean Worst

−1.2754 −1.3155 −1.7303

0.2967 0.3310 0.7453

0.9966 0.2478 0.0065

−0.5683 −0.0752 −0.0105

−0.3646

0.0012

0.0404 −1.0000

1.00E−02 2.92E−03 0.00E+00

−1.00E−05 −2.21E−06 −1.00E−05

−1.00E−05 −1.20E−06 1.00E−05

1.32E−07 −7.23E−08 −6.48E−07

0.0102

Best Mean Worst

−1.1190 −1.3371 −1.5592

0.1303 0.3480 0.5726

0.8795 0.4978 0.5941

−0.2422 −0.3355 −0.5801

0.1128 0.2532 1.0000

8.74E−04 4.63E−03 1.00E−02

4.28E−06 −1.29E−06 −2.08E−06

−2.59E−06 7.41E−07 −1.00E−05

−1.71E−08 9.29E−08 1.14E−07

Best Mean Worst

−1.4021 −1.4460 −1.4932

0.4043 0.4547 0.4992

0.0262 0.3558 0.0421

−0.1929 −0.1937

−0.0862

0.1002

0.0146

7.37E−03 7.21E−03 1.00E−02

9.58E−06 −1.37E−07 −1.00E−05

−1.00E−05 2.05E−06 1.00E−05

−1.21E−08 1.04E−07 4.22E−07

0.0012

Best Mean Worst

−1.1017 −1.1840 −1.4535

0.1006 0.1985 0.4740

0.5662 0.3520 0.2311

−0.1792 −0.4151 −1.0000

0.1140 −0.4250

1.00E−02 4.61E−03 8.02E−03

−7.37E−07 −4.16E−06 −1.00E−05

1.00E−05 4.39E−06 1.00E−05

2.36E−07 −2.45E−08 1.16E−07

0.0102

Best Mean Worst

−1.3893 −1.2219 −1.4959

0.3949 0.2353 0.5064

0.9314 0.3591 0.0000

0.0890 −0.0124 −0.2307

0.0058 −0.0848 −0.8628

0.00E+00 4.31E−03 1.00E−02

−8.75E−06 −2.85E−06 −1.00E−05

−2.22E−06 3.48E−06 1.00E−05

5.78E−09 −2.65E−08 2.28E−07

0.1002

Best Mean Worst

−0.9698 −1.2162 −1.3446

0.0000 0.2340 0.3424

1.0000 0.5295 0.0681

−0.1319 −0.3286 0.5045

−0.0078 −0.3501 −0.7156

2.51E−03 3.42E−03 1.00E−02

1.00E−05 2.67E−06 −1.00E−05

1.00E−05 2.56E−07 1.00E−05

1.90E−08 7.61E−08 2.25E−07

−1.094

0.109

1.000

−0.249

−0.028

1.90E−03

−7.83E−06

1.78E−08

2.36E−08

4.3

BSA-4

Model

0.0547

−0.5723 −0.1378

−0.0457 −1.0000 0.0160

−0.0929 −0.8440 −0.2430 −0.0831 0.9658 0.0380

0.1387 0.7292

0.0805 −1.0000

−0.7043

impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105705.

of Hammerstein structure obtained through real time experimentation conducted at rehabilitation center of the Southampton University subject to proper citing the relevant literature [12–14].

Acknowledgments

Appendix A. Supplementary data

Authors would like to thank Prof. Ivan Markovsky and his research team for giving the consent to use the desired parameters

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.asoc.2019.105705.

22

A. Mehmood, A. Zameer, N.I. Chaudhary et al. / Applied Soft Computing Journal 84 (2019) 105705

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