Improved whale trainer for sonar datasets classification using neural network

Applied Acoustics 154 (2019) 176–192

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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust

Improved whale trainer for sonar datasets classification using neural network M. Khishe a, M.R. Mosavi b,⇑ a b

Department of Electrical Engineering, Imam Khomeini Marine Science University, Nowshahr, Iran Department of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran 13114-16846, Iran

a r t i c l e

i n f o

Article history: Received 11 November 2018 Received in revised form 5 April 2019 Accepted 2 May 2019 Available online 10 May 2019 Keywords: Classifier MLP NN Improved whale trainer Sonar

a b s t r a c t To classify various sonar dataset, this paper proposes the use of the newly developed Whale Optimization Algorithm (WOA) algorithm for training Multi-Layer Perceptrons Neural Network (MLPs NN). Similar to other evolutionary classifiers, trapping in local minima, slow convergence rate, and non-real-time classification are three shortcomings it confronts in solving high-dimensional problems. Due to the novelty of WOA trainer, there is little in the literature regarding decreasing aforementioned deficiencies. In this paper, we also utilize seven spiral shapes to improve the performance of the WOA trainer. To assess the performance of the proposed classifiers, these networks will be evaluated using the three realworld practical sonar dataset. For endorsement, the results are compared to five popular metaheuristics trainers include Particle Swarm Optimization (PSO), Gravitational Search Algorithm (GSA), Ant Colony Optimization (ACO), Gray Wolf Optimization (GWO), and WOA. The results show that new classifiers indicate better performance than the other benchmark algorithms, in terms of avoidance from getting stuck at local minima, classification accuracy, and convergence speed. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction The classification of underwater targets has received much attention in recent years due to its high complexities which offer designing powerful classifiers. This classification includes discrimination between the echoes of targets and non-targets as well as the background clutter. The classification of different underwater targets and marine vessels, is a challenging problem due to dependence on the mode of operation and engine regimes. These characteristics, as well as time-varying propagation channels, time dispersions, and ambient noises, cause to change in the spectrum of the received acoustic signals [1]. In recent years, two main classification models have been proposed by researchers. Firstly, deterministic methods which are based on oceanography, sonar modeling, and statistical processing [2]. Secondly, stochastic methods, including the artificial intelligence approaches, that they have various applications such as the prediction and approximation of oceanic phenomena, feature extraction methods, and development of new classifiers [3].

⇑ Corresponding author. E-mail addresses: [email protected] (M. Khishe), [email protected] (M.R. Mosavi). https://doi.org/10.1016/j.apacoust.2019.05.006 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.

In the first category, researchers try to consider environmental conditions, multi-path effects, sound propagation models, topographic effects, and non-stationary clutter resources to calculate the approximated statistical distribution. In these methods, the classification was performed by calculating the distribution parameters. These deterministic methods have two major disadvantages. First, these models need the costly investment in equipment and human resources. Secondly, these they cannot guarantee the generality and robustness for other experimental areas [4]. The second category proposes stochastic methods to reduce complexities and costs. They have significant feature such as high accuracy, versatility, and inherently parallel structure, result in being applicable in the sonar target classification problems [5]. One of the best-known classifiers in this field of study is MultiLayer Perceptron Neural Networks (MLP NNs). There has been extensive research regarding the use of MLP NNs in the sonar targets’ classification [6–11] on account of the fact that they have outstanding features such as high accuracy [12,13], compliance with the fast changing of the environmental conditions [14], inherently parallel structure, which is very useful in hardware implementation (specifically FPGA) and then real-time processing [15,16]. Despite their applications, the distinct ability of MLP NNs is learning [17]. Learning means that these MLP NNs can learn like the human’s brain from experience or experiments. Learning can

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be divided into two main categories: supervised [18] and unsupervised [19]. Generally speaking, a trainer is an algorithm that provides learning for MLP NNs [11]. Trainers are responsible for training an MLP NN; so that it takes the highest performance for a new input dataset. For many years, the back-propagation technique and other its variants [20] play a major role between researchers in training MLP NNs. However, there are serious drawbacks in using the gradient-based methods such as slow convergence rate, trapping in local minima, and high adherence to the initial parameters [21]. Therefore, using meta-heuristic algorithm is one of the most applicable trainers for training the MLP NNs in the highdimensional problems (sonar targets’ classification) because of its stochastic nature and so higher efficiency in avoiding local optima [22]. As another point of view, single-solution and multi-solution algorithms are the two main categories of the stochastic methods. The literature indicates that multiple solutions algorithms prevent the trainer from becoming stuck at local optima higher than single solution stochastic algorithms in high dimensional problems [23]. In the literature, considerable number of well-known multisolution meta-heuristic algorithms have been utilized to train MLP NNs such as PSO [24], Genetic Algorithm (GA) [25], Artificial Bee Colony (ABC) [26], ACO [27,28], and Differential Evolution (DE) [29]. The recently proposed meta-heuristic trainer algorithms are: Social Spider Optimization algorithm (SSO) [30], Jaya Algorithm (JA) [31], Charged System Search (CSS) [32], Chemical Reaction Optimization (CRO) [33], Invasive Weed Optimization (IWO) [34], Teaching-Learning Based Optimization (TLBO) trainer [35], Biogeography-Based Optimizer (BBO) [36], GWO [37,38], Lightning Search Algorithm (LSA) [39], and WOA [40]. In spite of differences between various meta-heuristic algorithms, they divide the search space into two separate phases named ‘‘exploration” and ‘‘exploitation” [41–43]. The exploration phase refer to the primary phase in which the searching agents explore the whole search space as thoroughly as possible to obtain the promising area of the search space, whiles the exploitation phase refers to the second phase in which the searching agents try to converge toward the best solution taken in the exploration phase, quickly. It is worth noting that, No Free Lunch (NFL) theorem [44] causes this field of study become extremely active. This theorem has mathematically indicated that there is no specific meta-heuristic algorithm well fitted to solve all optimization problems. NFL theorem motivated many researchers to propose novel meta-heuristic algorithms, use existing algorithms for solving various problems, and improve the performance of the present algorithm. This also motivates us to investigate the efficiencies of the newly proposed WOA trainer in training MLP NN and then improve its performance for the sake of sonar dataset classification (high dimensional problem). WOA is inspired by the bubble-net hunting strategy of humpback whales [45]. Despite the excellent performance in the exploration phase, WOA is not efficient in the exploitation phase especially in the high dimensional problem such as sonar dataset classification [40,45]. Hence, this paper tries to improve the performance of WOA by proposing the novel spiral shapes of walls’ bubble net as its real natural shape. In the other words, improved version of WOA named Improved Wall Trainer (IWT) converges faster than conventional one by modifying the agent of the exploitation phase, i.e. the shape of bubble net. The paper is organized as follows: MLP NNs are introduced briefly in Section 2. Section 3 describes MLP NN training algorithms (i.e. WOA and IWT). In Section 4, an MLP NN is trained by trainers. Section 5 is devoted to sonar datasets. The simulation results are described in Section 6. Finally, conclusions are presented in Section 7.

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2. Multilayer perceptron neural networks MLP NNs are the most practical and common type of feedforward neural networks. In an MLP NN, neurons are interconnected in a one-directional mode. Fig. 1 shows an MLP NN with two layers. R is the number of input nodes, S1 is the number of hidden neurons and S2 is the number of output neurons. The outputs of an MLP NN are calculated as Eq. (1):

n1 ¼ IW  P þ b1

ð1Þ

where IW is the connection weight matrix from the input nodes to the neurons of the hidden layer. b1 is the bias matrix of neurons in the hidden layer and P is the input matrix. The output of hidden layer’s neurons is calculated using a sigmoid function as given in Eq. (2):

a1 ¼ Sigmoidðn1 Þ ¼

1 ð1 þ expðn1 ÞÞ

ð2Þ

Final outputs after calculating the output of the hidden node can be defined as:

n1 ¼ LW  P þ b2 y ¼ Sigmoidðn2 Þ ¼

ð3Þ 1 ð1 þ expðn2 ÞÞ

ð4Þ

where LW is the matrix of interconnection weight between the hidden layer and the output layer and b2 is the matrix of the neuron’s bias in the output layer. The most important parts of the MLP NNs are the connection weights and neuron’s biases. As previously stated, the final output of an MLP NN is defined by connection weights and neuron’s biases. The training process is set out to find the best values for connection weights and neuron’s biases; so that the desired outputs are obtained from specific inputs. 3. Trainer algorithms This section is intended to provide necessary information to WOA and its improved trainer, i.e. IWT which is used for designing the classifier. 3.1. Whale optimization algorithm As mentioned before, WOA is a novel population-based metaheuristic optimization algorithm in which their search agents are utilized to obtain the global optimum [45]. This algorithm initiates the search process by generating a set of random candidate solutions and then updating and improving the set over and over until the end criterion is satisfied. The updating rule which is used for improving the candidate solutions is the main difference between WOA and other population-based meta-heuristic optimization algorithms. Generally speaking, WOA mathematically models the hunting behaviour of humpback whales. This model includes finding and attacking preys called ‘‘bubble-net feeding” [45]. Fig. 2 shows the bubble-net hunting behaviour. This figure indicates that this intelligent foraging is done by producing a trap with moving in a spiral way around preys and generating bubbles along the ‘‘9shape” path. The encircling mechanism is another hunting behaviour of humpback whales. Humpback whales start preys’ hunting by encircling around them using the bubble-net mechanism. The mathematical model of this algorithm is described as follows [45]:




X ðt þ 1Þ ¼

r < 0:5 X  ðtÞ  AD D0 Mcosð2ptÞ þ X  ðtÞ r > 0:5

D ¼ jCX  ðtÞ  XðtÞj

ð5Þ ð6Þ

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Input

Layer 1 IW

a1

S1×R

n1 S1×1

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b1 S1×1

f1 (Sigmoid)

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S1×1

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y S2×1

S2

R

Fig. 1. An MLP NN with one hidden layer.

Fig. 2. Bubble-net hunting behaviour.

D0 ¼ jX  ðtÞ  XðtÞj

ð7Þ

A ¼ 2QR  Q

ð8Þ

C ¼ 2R

ð9Þ

M ¼ ekl

ð10Þ

where X is the changing positions of whale during iterations and X* is the best position obtained so far (the position of the prey), r is a random number in the range of [0, 1], D0 shows the distance of the ith whale and the best solution obtained so far (the prey), k is a constant which determines the spiral shape of the bubble-net, l is a random number in the range of [1, 1], the current iteration is shown by t, Q linearly decreases from 2 to 0 during iterations and R is a random vector in the range of [0, 1]. When r < 0.5, Eq. (5) was intended to simulate the encircling mechanism, otherwise (r > 0.5) the equation mimics the bubble-net spiral movement. The random number r switches the model between these two main phases, equally. The feasible positions of a search agent using these two main phases are shown in Fig. 3. Generally speaking, in the population-based optimization algorithms, the global optimum finding process can be divided into two main phases: the exploration and exploitation. In the first phase (exploration), searching agents aim to scatter thoroughly across

the search space in order to discover the whole search space which WOA uses shrinking encircling mechanism for this purpose. In other words, they need to explore the search space instead of clustering around local minima. In the next phase, data is gathered for general minimum convergence. WOA set out to do this phase (exploitation) by using spiral updating position. However, although WOA acts fairly in two aforementioned phases for the common optimization problem, little attention has been paid to the selection of an appropriate spiral shape in the exploitation phase. Considering the importance of exploitation phase especially for a high-dimensional problem like sonar dataset classification, following section aims to investigate the determination of the best spiral shape for WOA; so that it has the better performance than standard WOA in addition to other benchmark optimization algorithms. 3.2. Improved whale trainer As stated in the previous section, there is almost no study for choosing an appropriate spiral shape to improve the WOA’s exploitation phase. Therefore, in this section, some well-known spiral shapes were described and their mathematical models were investigated. Generally speaking, a spiral is a curve on a plane winding about a fixed point. There are many various types of spirals. In the follow-

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a) Shrinking encircling mechanism

b) Spiral updating position

Fig. 3. Bubble-net search mechanism.

ing, seven of the most interesting types of spirals were presented. In the standard WOA, a special spiral form is used for modelling bubble-net hunting behaviour so that this spiral model is formulated as Eq. (10) [45]. In order to scrutinize the exact behaviour of this spiral model, its two and three-dimensional plots were illustrated in Fig. 4. As shown in this figure, the radius of the spiral is constant in all ascending circle and there is little convergence behaviour in this shape. On the other hand, all of the optimization algorithms need a powerful convergence operator, especially in the high dimension problems. To tackle this shortcoming, in this study, some converging spiral models were substituted for the original spiral model such a way that they have various convergence behaviour. 3.2.1. Other spirals 3.2.1.1. Archimedean spiral. An Archimedean spiral is a curve therein each turn has the same distance from the previous one. This spiral is traced using polar equations of the form r = a h. Where, r is spiral’s radius, a can be any constant and h represent polar angle [46]. 3.2.1.2. Logarithmic spiral. The Logarithmic spiral is a type of spiral therein the angle between the line to the origin and the tangent

line is always identical. Logarithmic spiral can be traced using polar equations of the form log(r) = a h [47].

3.2.1.3. Fermat spiral. This type of spiral was described by Fermat in 1636 [48]. Fermat spiral can be traced using polar equations of the form r2 = a2h.

3.2.1.4. Lituus spiral. The Lituus spiral was described by Cotes in 1722 [48]. This spiral is the inverse of Fermat type. It can be traced using polar equations of the form r2 = a2/h.

3.2.1.5. Equiangular spiral. This type of spiral is same as Logarithmic spiral except that log is replaced by Ln. Therefore, Equiangular spiral can be traced using polar equations of the form Ln(r) = ah[47].

3.2.1.6. Random spiral. This type of spiral is almost same as the Archimedean spiral, considering the fact that, the constant a is determined by random numbers. Hence, the Random spiral can be traced using polar equations of the form r = rand()h. To sum up and to compare the various spiral shapes, their spiral curves are indicated in Fig. 5.

Original WOA: Logaritmic Spiral

Original WOA: Logaritmic Spiral 2.5 2 1.5

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Fig. 4. Spiral model used in the conventional WOA.

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4. Training MLP NNs using the trainers There are two important phases in training an MLP NN using the meta-heuristic trainer: (1) the representation of the problem’s parameters by the search agents of meta-heuristic trainers and (2) choosing the fitness function. The representation of problems is the first and most important phase in training an MLP NN using meta-heuristic trainers. In other words, the MLP NN’s training problems must be formulated obviously in such a way that essential parameters of the problem are

clearly determined. It is known from the literature that weights and biases are the two most important parameters in the training an MLP NN [49]. A trainer should be used in such a way that it find a set of values for the weights and biases of an MLP NN that provides the highest approximation or classification accuracy. To sum up, the unknown parameters here are the biases and weights. Generally, there are three methods to represent biases and weights of an MLP NN as searching agents of meta-heuristic trainers: (1) vector, (2) matrix and (3) binary state [12]. Since the WOA trainer accepts the parameters in a vector form, in this study the

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parameters of an MLP NN shown in Fig. 6 were represented as Eq. (11):

 ! Searching Agent ¼ SA ¼ ½W 11 ; W 12 ; :::; W nh ; b1 ; :::; bh ; M11 ; :::; Mhm  ð11Þ where n is the number of the input nodes, Wij indicates the connection weight between the ith input node and the jth hidden neuron, bj is the bias of the jth hidden neuron and Mjo shows the connection weight from the jth hidden neuron to the oth output neuron.

After defining the problem’s parameters, a fitness function for the meta-heuristic trainer must be defined. As mentioned previously, the final goal in training an MLP NN is to obtain the highest testing accuracy. Mean Square Error (MSE) is one of the most common metrics for the estimation of an MLP NN. MSE is formulated as Eq. (12), so that a given vector of training samples is exerted to the MLP NN and then the difference between the desired and calculated output is obtained from the following equation:

MSE ¼

m X i¼1

k 2

ðoki  di Þ

ð12Þ

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W11

W12

. . .

Wnh

b1

...

bh

M11

M12

...

Mhm

A typical vector of the problem’s parameters (i.e. biases and weights of the MLP NN ) which was provided for a searching agent of meta-heuristic trainer Fig. 6. Assigning the problem’s parameters to meta-heuristic’s searching agent.

In the next section, the merits of the IWTs was practically investigated in training an MLP NN. 5. Sonar dataset

Fig. 7. IWT provides optimal weights and biases for MLP NN and receives MSE for all training samples.

k

where m shows the number of outputs, oki and di indicate respectively the actual and desired output of the ith input unit when the kth training sample is exerted. Finally, the problem of training an MLP NN was formulated with aforementioned parameters and fitness function (MSE) as Eq. (13):

Minimize F ðSAÞ ¼ minðMSEÞ

ð13Þ

The whole process of training an MLP NN using IWT is shown in Fig. 7. As can be seen from this figure, the IWT provides optimal weights and biases for MLP NN and receives MSE for all training samples. In other words, the IWT optimally updates weights and biases of the MLP NN in each iteration and for all training samples to minimize the MSE. Consequently, the weights and biases of the MLP NN are updated over the course of iterations and tended to move toward the best MSE by IWT algorithm. In other words, the overall MSE of the MLP NN is iteratively decreased. Therefore, in the long run, IWT converges to the best solution (i.e. weights and biases) that is better than the other best solutions obtained so far.

To test the designed classifier, it is benchmarked on three sonar datasets as follow:  Sejnowski & Gorman’s Dataset [50]: this dataset is chosen for the sake of having a reference dataset to compare the proposed classifier with other benchmark algorithms and classifiers.  Passive Sonar Dataset: To have a comprehensive investigation in the passive sonar scope, a new passive sonar dataset will be proposed in this section. The source audio of this dataset is available in reference [51].  Active Sonar Dataset: To have a comprehensive investigation in the active sonar area, authors designed a sonobuoy to collect a general active sonar dataset for the sake of sonar target classification. The complementary information about this dataset and test scenario planning is available in [52–54]. 5.1. The Sejnowski & Gorman’s dataset This dataset is derived from Sejnowski & Gorman’s experiment [50]. In this test, there are two kinds of echoes; the first is related to a metal cylinder (real target) and the second is a rock with the same size as the cylinder (false target). In this experiment, a metal cylinder with the length of 5 feet and a rock with the same size is placed on a sandy seabed and a wideband linear modulated chirp ping is sent towards the aforementioned targets. The number of

Fig. 8. An example of the echoes returned from the rock and the metal cylinder.

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a) Sampling apertures

b) Applying the set of sampling apertures on the two-dimensional spectrogram Fig. 9. Preprocessing used to obtain the spectral envelope.

208 echoes of 1200 collected echoes was chosen based on the Signal to Noise Ratio (SNR) of the received echo so that the chosen echoes have SNR in the range of 4–15 dB. 111 of the 208 echoes belong to the metal cylinder and 97 echoes belong to the rock. Fig. 8 shows the typical received echoes from the rock and the metal cylinder.

To obtain the spectral envelope, special preprocessing is used as shown in Fig. 8. Fig. 9a indicates a set of sampling apertures. In Fig. 9b, a set of sampling apertures are located on the twodimensional spectrogram of Fourier transform of received echoes. The spectral envelope is calculated by gathering the effects of each apertures’ valve. In this experiment, the spectral envelope consists

Fig. 10. The experimental setup and test scenario design for gathering passive propeller noise.

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4

x 10

Small oil tanker Fig. 11 (continued)

of 60 samples which have been normalized in the range of 0 to 1. Each of these numbers represents the total energy contained in its associated sampling aperture. 5.2. Passive dataset

a) Designed sonobuoy

b) Oceanographic sonobuoy

Fig. 12. The sonobuoys used for sonar dataset acquisition.

To carry out this experiment and obtain valid dataset, seven types of propellers, which are the laboratory sample of original one, were tested in the cavitation tunnel as shown in Fig. 10. In the experiment, which was performed in the hydrodynamic cavitation tunnel, for simulating the various working conditions of different vessels, seven propellers with different Revolutions Per Minute (RPM), different water velocity, water density were examined. This dataset was collected by the authors using the cavitation tunnel model B&K 8103. The complementary information and whole dataset are available in [51].

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a. target 1

b. target 2

d. target 4

c. target 3

e. non-target 1

f. non-target 2

Fig. 13. Examples of echoes returning from target and non-target objects.

Table 1 Sonar datasets. Datasets

# Attributes

#Training Samples

#Test Samples

#Classes

Sejnowski & Gorman Active Passive

60 23 140

150 200 400

58 150 250

2 6 7

The typical hydrophone noise level, its Fourier transform, and Power Spectral Density (PSD) estimation of recorded propeller noise were shown in Fig. 11 from left to right, respectively. 5.3 Active Dataset This dataset was collected by the authors using their designed sonobuoy and the port and maritime organization’s sonobuoy as shown in Fig. 12 [52–54]. In this experiment, 6 objects including 4 targets and 2 nontargets have been placed on the sandy seabed. In this section, the transmitted signal is a wideband linear frequency modulated ping

which covers the frequency range of 5–110 Hz. An electric motor rotates the objects (at the bottom of the sea) through 180 degrees, with a precision of one degree. Due to the high volume of raw data obtained in the previous step, there is a high complexity in calculations. In order to reduce the complexity of the classifier, a detection process is needed. The detection process has been declared in author’s previous work [49]. Fig. 13 indicates the received echoes from different objectives, which are the functions of their frequency and direction. After target detection, feature extraction stage must be utilized.

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Preprocessing

Detection Ping

Raw Data Echo

Detection

Scaling

Inverse Filtering

Normalization

Windowing 512 Points

S(k) S (k )

Matched-Filter

Down sampling 4

Feature Extraction

Mel-Scale Filters Energy Spectrum

FFT

Classifier

MFCC E(l)

2

e(l)

Eq.14

Log

Eq.15

ANN

C(n)

DCT

Eq.17

Eq.13

IWT

Fig. 14. The entire classification process: preprocessing, detection, feature extraction and classifier.

Convergence Curves

Convergence Curves

0.2

WOA GSA ACO GWO PSO

0.16

0.09

Best score obtained so far

Best score obtained so far

0.18

0.14 0.12 0.1 0.08 0.06

0.08 0.07 0.06 0.05 0.04 0.03

0.04

0.02

0.02

0.01

10

20

30

40

50

60

70

80

90

IWT-Archimedean IWT-Fermat IWT-Lituus IWT-Random IWT-Logarithmic IWT-Equiangular

0.1

100

10

20

30

Iteration

40

50

60

70

80

90

100

Iteration

Fig. 15. Comparison of convergence curve for Sejnowski & Gorman dataset.

5.3. Feature extraction

EðlÞ ¼

After preprocessing, the detected sounds are transferred to the frequency domain (named S(k)) and then they are delivered to the feature extraction stage. In this step, first, the spectral density function of the detected signal is calculated by Eq. (14): 2

jSðkÞj ¼ S2r ðkÞ þ S2i ðkÞ S2r (k)

S2i (k)

ð14Þ

where and are the real and the imaginary parts of the detected signal’s Fourier transform, respectively. In the next step, the spectral energy is filtered by a Mel-scaled triangular filter. Therefore, the output energy of the lth filter is calculated by Eq. (15):

N1 X

2

jSðkÞj Hl ðkÞ

ð15Þ

k¼0

where N is the number of discrete frequencies used for FFT in the preprocessing stage and Hl ðkÞ is the transfer function of the given filter, where l = 0, 1, . . ., Ml. The dynamic range of Mel-scale filtered energy spectrum is compacted by the logarithmic function as Eq. (16):

EðlÞ ¼ log ðEðlÞÞ

ð16Þ

Mel-scaled Frequency Cepstral Coefficients (MFCC) are converted back to the time domain using Discrete Cosine Transform (DCT) as Eq. (17):

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Table 2 The primary parameters of the benchmark algorithms. Algorithms

Parameters

Value

GWO

a Maximum number of iterations Population size Layout Cognitive constant (C1) Social constant (C2) Local constant (W) Population size Number of masses a Gravitational constant Maximum number of iterations Primary pheromone (s0 ) Pheromone updating constant (Q) Pheromone constant (q0) Decreasing rate of the overall pheromone (Pg) Decreasing rate of local pheromone (Pt) Pheromone sensitivity (a) Observable sensitivity (b) Population size a Maximum number of iterations Population size

linearly decreased from 2 to 0 250 200 Full connection 1 1 0.3 200 70 20 1 250 0.000001 20 1 0.9 0.5 1 5 200 linearly decreased from 2 to 0 250 200

PSO

GSA

ACO

WOA

10

MSE

5 0 -5

-10

WOA

GSA

ACO

GWO

PSO

Algorithm 10

MSE

5 0 -5 -10

IWT-Archimedean

IWT-Fermat

IWT-Lituus

IWT-Random

IWT-Logarithmic

IWT-Equiangular

Algorithm Fig. 16. Boxplot of MSE of various trainer for Sejnowski & Gorman dataset.

CðnÞ ¼

M X l¼1

    1 p eðlÞcos n l  2 M

ð17Þ

Finally, the feature vector will be in the form of Eq. (18):

X m ¼ ½cð0Þ:cð1Þ:    :cðP  1ÞT

ð18Þ

The entire classification process includes preprocessing, detection, feature extraction, and classifier designing is shown in Fig. 14. Consequently, obtained benchmark sonar datasets are shown in Table 1.

6. Experimental setup and simulation result As shown in Table 1, three practical benchmark sonar datasets are utilized to testify the performance of IWTs. The objective is to find the best spiral shape for improving the performance of

the designed classifier. These datasets are divided into three groups: Sejnowski & Gorman, active, and passive dataset. In this table, attribute indicates the number of the features determining the number of MLP NNs’ input nodes, and also the number of training and test samples are given. Note that Sejnowski & Gorman dataset is chosen in order to have a benchmark sonar dataset for comparing with other researchers’ works. In addition, passive and active sonar datasets have also been selected to provide a comprehensive real world practical experiment. A detailed description of the passive and active datasets and their audio source is available in the references [51] and [52–54], respectively. To validate the results of IWTs, they are compared with the WOA, PSO, ACO, GWO, and GSA trainers. Required parameters and initial values are presented in Table 2. Each designed classifier is executed 20 times and the typical convergence curves and boxplot of MSE for various sonar dataset are shown in Figs. 15–20. Also, the statistical result is illustrated in Tables 3–5.

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Convergence Curves

Convergence Curves

0.266

WOA GSA ACO GWO PSO

0.262

IWT-Archimedean IWT-Fermat IWT-Lituus IWT-Random IWT-Logarithmic IWT-Equiangular

0.05

Best score obtained so far

0.264

Best score obtained so far

0.06

0.26 0.258 0.256 0.254 0.252

0.04

0.03

0.02

0.25 0.01

0.248 10

20

30

40

50

60

70

80

90

100

10

20

30

Iteration

40

50

60

70

80

90

100

Iteration

Fig. 17. Comparison of convergence curve for passive dataset.

10

MSE

5 0 -5 -10

WOA

GSA

ACO

GWO

PSO

Algorithm 10

MSE

5 0 -5 -10

IWT-Archimedean

IWT-Fermat

IWT-Lituus

IWT-Random

IWT-Logarithmic

IWT-Equiangular

Algorithm Fig. 18. Boxplot of MSE of various trainer for passive dataset.

Having a fair comparison, all algorithms stop after reaching a maximum of 250 iterations. Since there are no standards regarding the hidden nodes’ number, the Eq. (19) is utilized to select the number of hidden node in the designed classifier as follow:

H ¼ 2N þ 1

ð19Þ

where N is the number of inputs and H represents the number of hidden nodes. It should be noted that the simulations were performed in MATLAB using a PC with a 2.3 GHz processor and 4 GB RAM. The Average (AVE) and Standard Deviation (STD) of MSE in the table of results get calculated. To reach a greater capability of the algorithm in order to avoid local minima, the value of AVE ± STD must be lower. Note that the best results are highlighted in bold type in Tables 3–5. According to Derrac et al. [55], statistical tests are essential to have a fair evaluation of meta-heuristic algorithms’ performance as well as AVE ± STD. In order to see whether the

obtained results of IWTs differ from other benchmark trainers in a statistically significant way, a non-parametric statistical test, Wilcoxon’s rank-sum test [56], was accomplished at 5% significance level. Therefore, the obtained P-values are shown in the table of results as well. In the tables, N/A indicates ‘‘Not Applicable’’ which means that the specified algorithm cannot be compared with itself in the Wilcoxon’s rank-sum test. Logically, P-values less than 0.05 are considered as powerful evidence against the null hypothesis. Therefore, P-values greater than 0.05 are underlined in the tables. Another comparative measure shown in the results are classification rates. Tables 3, 4 and Figs. 15, 16 show the results of the five wellknown benchmark trainers and six improved version of WOA (i.e. IWT) for Sejnowski & Gorman dataset. As may be seen from this tables and figures, WOA has the best results in this dataset among the classical benchmark trainers. In general, the results of IWT-Lituus and IWT-Fermat are much better than the other

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Convergence Curves

Convergence Curves WOA GSA ACO GWO PSO

IWT-Archimedean IWT-Fermat IWT-Lituus IWT-Random IWT-Logarithmic IWT-Equiangular

0.06

Best score obtained so far

Best score obtained so far

0.15

0.1

0.05

0.05

0.04

0.03

0.02

0.01

0

10

20

30

40

50

60

70

80

90

100

10

20

30

Iteration

40

50

60

70

80

90

100

Iteration

Fig. 19. Comparison of convergence curve for active dataset.

10

MSE

5 0 -5 -10

WOA

GSA

ACO

GWO

PSO

Algorithm 10

MSE

5 0 -5 -10

IWT-Archimedean

IWT-Fermat

IWT-Lituus IWT-Random Algorithm

IWT-Logarithmic

IWT-Equiangular

Fig. 20. Boxplot of MSE of various trainer for active dataset.

improved algorithms. These results show that Lituus and Fermat spiral shape could improve the performance of WOA algorithm for training the designed classifiers much better than other spiral shapes. As can be also seen from Tables 5, 6 and Figs. 17, 18, these outstanding results are same for passive dataset so that IWT_Lituus and IWT-Fermat have the best result among other trainers. Considering Fig. 5 and special spiral shape of Lituus and Fermat map, these improvements are reasonable. Because these two spiral shapes have a unique ability in the exploitation phase; so that these special spiral curves could converge faster than conventional one. Figs. 16, 17 and 19 show the convergence curves of the trainers. As can be seen from these curves, IWTLituus has the best convergence rates for all of the benchmark datasets, followed by IWT-Fermat has the best results in two out of three datasets. It is worth noting that the boxplot is a standardized way of indicating the distribution of the dataset. Therefore, the boxplots of the trainers’ MSE indicate that how a result

diverges from the global minimum. Consequently, the more compact boxplot, the more powerful for avoiding local minima. Considering Figs. 16, 18 and 20, the superior performance of Lituus rather than Fermat is justifiable, because its boxplot is more compact and without outlier data rather than Fermat boxplot. The reason for the superior results is that the special spiral shapes are able to exploit knowledge of the location of near optimal solutions effectively and then avoid getting stuck in local minima (Tables 7 and 8). To sum up, The IWT_Lituus algorithm has the best convergence rates of the IWTs and other conventional benchmark trainers. The reason for the improved capability in avoiding local minima is that the Lituus spiral shape gives WOA more accelerated search in comparison with the popular benchmark trainer and recent modifications of the WOA in exploitation phase. Hence, the searching agents are not easily trapped in local minima and reach global minima faster than normal searching agent strategies.

M. Khishe, M.R. Mosavi / Applied Acoustics 154 (2019) 176–192 Table 3 Experimental results of benchmark trainers for Sejnowski & Gorman dataset. Algorithm

MSE (AVE ± STD)

P-values

Classification rate %

WOA GSA ACO GWO PSO

0.0212 ± 0.0111 0.1049 ± 0.0965 0.2233 ± 0.1172 0.0794 ± 0.0112 0.1311 ± 0.1076

N/A 9.2798e20 0.0079 0.0079 7.2239e04

96.4311 92.7500 90.6666 95.7692 93.6741

Table 4 Experimental results of newly proposed trainers for Sejnowski & Gorman dataset. Algorithm

MSE (AVE ± STD)

P-values

Classification rate %

IWT-Archimedean IWT-Fermat IWT-Lituus IWT-Random

0.1425 ± 0.1113 0.1112 ± 0.3366 0.0011 ± 0.0571 0.1425 ± 0.1323

0.0471 0.0479 N/A

95.9333 97.2211 97.9833 94.1915

IWT-Logarithmic IWT-Equiangular

0.0799 ± 0.1106 0.1425 ± 0.1323

0.4429 0.0471 0.0471

93.3223 95.1155

Table 5 Experimental results of benchmark trainers for passive dataset. Algorithm

MSE(AVE ± STD)

P-values

Classification rate %

WOA GSA ACO GWO PSO

0.0002 ± 0.0001 0.1149 ± 0.0005 0.2111 ± 0.1452 0.0094 ± 0.0012 0.0311 ± 0.0076

N/A 9.2798e20 9.2798e20 0.049 7.2239e04

97.1111 92.7124 92.6446 96.9922 95.8712

Table 6 Experimental results of newly proposed trainers for passive dataset. Algorithm

MSE (AVE ± STD)

P-values

Classification rate %

IWT-Archimedean IWT-Fermat IWT-Lituus IWT-Random IWT-Logarithmic IWT-Equiangular

0.0425 ± 0.0113 0.0002 ± 0.4256 0.0001 ± 0.0001 0.0025 ± 0.1323 0.1119 ± 0.1306 0.0025 ± 0.0023

0.0471 0.0479 N/A 0.0429 0.0471 0.0471

92.2415 97.4351 98.1133 95.8815 93.1123 96.4185

Table 7 Experimental results of benchmark trainers for active dataset. Algorithm

MSE(AVE ± STD)

P-values

Classification rate %

WOA GSA ACO GWO PSO

0.1112 ± 0.0111 0.1449 ± 0.1165 0.1233 ± 0.1172 0.1194 ± 0.1112 0.1311 ± 0.1176

N/A 8.41e12 7.22e04 0.048 7.22e04

96.8811 93.9840 92.7854 95.9992 94.7741

Table 8 Experimental results of newly proposed trainers for active dataset. Algorithm

MSE (AVE ± STD)

P-values

Classification rate %

IWT-Archimedean IWT-Fermat IWT-Lituus IWT-Random IWT-Logarithmic IWT-Equiangular

0.0025 ± 0.0013 0.0111 ± 0.4266 0.0001 ± 0.0011 0.1225 ± 0.1323 0.0799 ± 0.1106 0.0045 ± 0.0023

0.0471 0.0479 N/A 0.0471 0.0471 0.0471

97.3333 95.3334 97.9993 93.1115 94.5623 96.1111

7. Conclusions In this paper, new modifications of WOA entitled IWTs is suggested using the concept of spiral curves inspired by a humpback whale produces a trap with moving in a spiral path around preys

191

and producing bubbles along the path. Seven versions of IWT with various spiral trapping shape were proposed. To evaluate their performance, three benchmark real-word sonar datasets were utilized, and the results compared with the conventional, and some recently proposed meta-heuristic trainers. The results show that IWT-Lituus has merit compared to other conventional and IWT trainers in terms of classification rate, local minima avoidance and convergence speed, especially for high dimensional realworld problems. For future studies, it would be interesting to employ IWTs in solving real-world optimization problems. Moreover, utilizing different types of spiral shapes with the greater variety of curvatures, slopes, and interception points is suggested for future studies.

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.apacoust.2019.05.006.

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