Chaotic Flower Pollination and Grey Wolf Algorithms for parameter extraction of bio-impedance models

Highlights

• Chaotic FPA and Chaotic GWO are investigated.

• Performance is mathematically tested over CEC 2017 benchmark function.

• The parameters of simplified Hayden and double shell bio-impedance models are extracted.

• Set of experimental data from carrot, cucumber and eggplant are employed.

• The results of the integer and fractional order models are compared.

• Non-Linear Square and Particle Swarm Optimization are also used for comparison.

• Recommend which model provides a better fitting for the experimental data.

• Recommend which algorithm is more accurate and faster in extracting the model parameters.

Abstract

Precise parameter extraction of the bio-impedance models from the measured data is an important factor to evaluate the physiological changes of plant tissues. Traditional techniques employed in the literature for this problem are not robust which reflects on their accuracy. In this paper, the Flower Pollination Algorithm (FPA), the Grey Wolf Optimizer (GWO) and ten of their chaotic variants are employed to extract the parameters of bio-impedance models. These models are the simplified Hayden and Double-Shell models in their integer and fractional order forms. Experimental data sets of three different vegetables are employed and the performance of the chaotic variants of FPA and GWO are tested mathematically over CEC 2017 benchmark functions. The optimization results are compared based on fitting error and execution time to visualize the trade-off between accuracy, consistency, and speed of convergence. The fitting results are compared with the traditional Non-Linear Least Square (NLLS) fitting and Particle Swarm Optimization (PSO) algorithms. The target is to select the most adequate algorithm for extracting the parameters for each presented model. For the simplified Hayden model, both CGWO and CFPA achieved good results. However, there is a trade off between time and consistency of the results. If the time factor (consistency of the results) is the dominant factor, the CGWO (CFPA) algorithm will perform better. For the double shell model, the GWO and CGWO achieve more accurate and more consistent results and less execution time than the FPA and CFPA algorithms. The fractional order version of the studied models provid a better fitting to the measured data compared with their integer order counterparts.

Introduction

Electrical impedance measurements have been widely used to detect the chemical and the physiological changes in plant tissues [1], [2]. It has been found in the literature that impedance parameters of fruits and vegetables are affected by freeze, chill or bruise damages [3], [4], [5]. In [3], the amount of bruising sustained by a Granny Smith apple was assessed by bio-impedance measurements. A strong correlation between the electrical impedance and the bruise damage on apples was reported in [4]. It was found in [5], that after freezing–thawing treatment of egg-plant, the impedance was different from the original sample because the membranes of the cells were severely damaged during the freezing process. Impedance measurements can also be used to estimate plant health, maturity, damages, structural variation during ripening, the presence of viruses and measurement of tree root growth [6], [7], [8]. These analyses can provide valuable insights in the field of quality assessments of fruits and vegetables for better characterization of these agricultural products [9]. A non-destructive investigation to study the electrical impedance variations in banana ripening was presented in [6]. The electrical impedance of kiwifruit was studied during ripening in [8]. The relationship between ethylene biosynthesis induced by direct current in cucumber fruit and bio-impedance was investigated in [10].

The data obtained from bio-impedance measuring is commonly analyzed by fitting it to an equivalent circuit model. There are several electrical models which were proposed in literature to model bio-impedance [11]. The selection of each model to be applied to a given plant depends on how accurately this model fits the measured data. The Hayden model presented in [12] was one of the firstly introduced model but was shown to be unable to adequately fit some measured impedance data [13]. Another model was presented to overcome the flaws in the Hayden model which is the Double shell model introduced in [1], [14]. These models provide useful physiological information in simple tissues [7]. However, for highly heterogeneous tissues, these models fail to provide a good fit for the measured data [7]. Therefore, fractional order models are preferred in order to give better description of tissues.

Fractional calculus provides a powerful tool to model complex biological systems with non-linear behavior and long-term memory. It provides a more adequate description of many actual dynamical processes modeled by fractional order differential equations [15]. It is now employed in many fields in the area of science and engineering such as: chaotic systems [16], encryption [17] , control [18], super-capacitor modeling [19], analog circuit theory [20], and bioengineering [21], [22]. The Cole-impedance model is a fractional-order model for bio-impedance which is considered among the best models to fit the measured impedance data [23]. It provides a better fit in both simple and complex tissues [11]. In [22], a generalization of the simplified Hayden model and the Double Shell model into the fractional order domain was presented. The models were fitted over a set of $6$ types of fruits and vegetables (banana, cucumber, orange, mandarin, lemon, and tomato). The fractional models proved to provide better fitting than their integer order counterparts [22].

There are many deterministic algorithms used to extract the parameters of the integer order Hayden and Double Shell models such as Complex Nonlinear-Least Square (CNLS) [12], [13], [24]. Unfortunately, the fitting performance of these methods is not robust to the noise present in experimentally measured data which reflects on the accuracy of the estimated parameters. These conventional methods only catch the near optimal solution without trying to exhaust the whole search space. Thus searching for more robust and reliable algorithms is a vitally important issue.

Consequently, another category of algorithms namedbiologically-inspired optimization techniques have appeared [25], [26]. These algorithms are mostly inspired by nature to emulate the philosophy of intelligent swarming behavior of animals, ants, whales, and wolves or to mimic the biological processes in nature [27]. They became attractive techniques for researchers to solve problems in several fields such as feature selection [28], photo-voltaic systems [29], neural network optimization [30], and email spam filtering [31] because of the global search capability of these algorithms. Examples of these techniques include but are not limited to Particle Swarm Optimizer [32], Grasshopper Optimizer [33], Salp Swarm Algorithm [34] and Simulated Annealing Algorithm [35].

In the context of biologically-inspired algorithms, the Flower Pollination Algorithm (FPA) has been proposed to mimic the pollination process in plants [25]. This algorithm was employed in various applications because of its simplicity and efficiency [21], [36]. Likewise, the Grey Wolf Optimizer (GWO) [26] was developed to simulate the hunting process in wolves. It became one of the most popular algorithms in literature [37], [38]. However, as mentioned in the no-free-lunch theorem that there is no specific algorithm for solving all optimization problems [39]. Thus, meta-heuristic algorithms achieve unpredictable performance in some applications while unfortunately, they fail in others due to their dependency on some specific control parameters. These parameters have their impact on the exploration and exploitation phases of the algorithms. Therefore, by changing these parameters, the performance of the algorithm is varied. That is why algorithm developers attempt via several approaches to balance between the diversification and intensification phases of the algorithms during the search process such as in [27], [40], [41], [42], [43], [44] to introduce adequate algorithms for their applications.

Lately, a new approach has been introduced to help in balancing between the exploitation and exploration phases of the stochastic algorithms. It is based on a combination between the chaotic maps and the basic algorithm version to adaptively tune some of their random parameters for improving their consistency and accuracy [45], [46]. Researchers suggest to replace the uniform or Gaussian distributions in the standard algorithm by chaotic maps to benefit from their better statistical and dynamical properties. This strategy has proved its efficiency in improving the accuracy, consistency and convergence speed of the original versions of several optimization algorithms such as Chaotic Gravitational Search Optimizer [45], Chaotic Whale Optimization Algorithm [47], Chaotic Grasshopper Optimizer [48], [49] and Chaotic Salp Optimizer [50]. The basic versions of FPA and GWO have undergone the same approach of modifications to tune some of their parameters as in [51]. The authors employed four chaotic maps in the global and local pollination process to enrich the performance of FPA. Ten chaotic maps have been combined with FPA in [52], [53] to generate the initial population. The authors in [54] proposed three chaotic variants for FPA. The chaotic maps were used in the first variant for the initialization while in the second variant the probability switch between the global and local search was adapted chaotically and the third variant is a combination of both. In [55], chaotic maps were used with FPA to adjust its switching probability between the global and local search for maximizing area coverage in wireless sensor networks. Regarding the merge between the chaotic maps and GWO, the CGWO was reported for the reproduction of desired robot motion trajectories in [56]. Moreover, the CGWO was mixed with two chaotic maps for position control of a robotic manipulator in [57]. The CGWO was used to solve five different constrained optimization engineering design problems in [58]. However, there are still other possible combinations of control parameters that may lead to other chaotic variants of FPA and GWO algorithms.

In the current work, the authors are motivated to introduce other chaotic variants of FPA and GWO to improve their efficiency in estimating the parameters of bio-impedance models. To test, validate and demonstrate the performance of CFPA and CGWO mathematically, they are applied to the CEC2017 benchmark functions. Moreover, non-parametric statistical analysis has been performed to validate the superiority of the introduced techniques. Furthermore, in this work, the FPA, CFPA, GWO, and CGWO have been employed to identify the parameters of the integer order and the fractional order simplified Hayden and double shell models based on sets of experimental data for three different vegetables (carrot, cucumber, and eggplant). To the authors’ best knowledge, these algorithms have not been applied to this application yet. For further verification, Nonlinear Least Square, and Particle Swarm Optimizer (PSO) are applied for the same problem for comparison. The results in case of the integer order models are compared with the fractional order ones. Moreover, the comparison between the results of the utilized techniques is carried out to recommend which model provides a better fitting for the experimental data and which algorithm is more adequate to extract the model parameters accurately and with shorter execution time. Fig. 1 shows a block diagram for the complete process.

The paper is organized as follows; Section 2 reviews the plant cell anatomy and related physical properties. It also presents the mathematical and physical aspects of the electrical bio-impedance models investigated in this paper. Section 3 introduces the formulation of the optimization problem solved using the bio-inspired optimization algorithms reviewed in Section 4. Section 5 presents the mathematical CEC2017 function results of the introducedchaotic variants. Section 6 contains parameter extraction results of the optimization algorithms when applied to experimental data-sets and compared to the traditional non-linear least square algorithm. The concluding remarks are summarized in Section 7.