1 Introduction

Artificial neural network (ANN) imitates the behavioral and logic properties of biological neural networks (NNs) in information treating [12]. The information treating is fulfilled by regulating the interrelations of nodes in the NN with the capability to self-adapt and self-learn [47]. The ANN is stimulated via the organism NNs that comprise biological brains (Amer and Maul [3]. It is joined via numerous flexible linkage weights of hidden-neurons and retains the features of decentralized message preservation, vast-scope parallel handling, self-studying, and self-organize capability [43]. In addition, some scholars have utilized ANN and its mergence to generate substitute system which may be realized on good placement [37].

Furthermore, on the basis of the pure topological construction and the potential to reveal how knowing holds on a definite treat, the RBFnet has been diffusely utilized as the global region function approximator to settle nonlinear issues (Lin and Wu [44]. In addition, according to the regenerating kernel approach, an inspiring numerical method has been mentioned by Arqub et al. [5] to decide groups of fuzzy fractional integrodifferential equations with Atangana–Baleanu–Caputo (ABC) fractional dispersed order derivatives [5]. Also, the tasks of regenerating kernel for numerical explanation of fuzzy fractional primary value is exploited in Arqub et al. [6]. Besides, several of the superiorities of the RBF kernel [64] are as subsequent: (1) different from the linear kernel, the RBF kernel may process the instance when there is a nonlinear correlation between category tags and properties; (2) with the impact of the count of over-dynamic parametric on the model extract complication, the count of RBF kernel parameters is lower than the count of polynomial kernel. As a result, this study focused on the coaching and modification on the related parameters of RBFnet, and aimed to propose a new HIAO algorithm for RBFnet to gain a distinguished imitating expression.

Next, evolutionary algorithms (EAs) as a type of optimization algorithm have indicated the usefulness in ANNs [71]. On the other hand, computational intelligence (CI) had a speedy growth and turned into a diversity subdomain of artificial intelligence (AI) that offered a variation of methods (i.e., EAs, ANNs, and fuzzy logic (FL) inference systems etc.), permanently enhanced and the new ones, stimulated from natural ecosystems or biological [58]. Consequently, numerous investigations emphasized the appearing challenges and tendencies of adopting CI optimization algorithms for resolving prediction issues in particular domains with superior accurateness [58].

Within last decade, numerous scholars have been keeping attention to the natural-selection based issue resolving methods [23]. Further, EA may seek numerous optimum solutions in one solitary imitation execution owing to its seeking method foundation on population [13]. Besides, the immune system (IS) is constituted of physical shelter alike the respiratory system and epidermis. Two nature organic protection procedures which affect mutually: (1) B-cells to terminate the damaging attacker; (2) the natural resistance retaining when born, and the particular adaptability resistance using a group of T-cells to identify the bacterium [36]. Alike to evolution methods, artificial immune system (AIS) adopts the agency of vertebrate IS to establish new intelligence optimization algorithms, which offers several new ways to resolve optimization issues [62]. Moreover, immune calculation is a comparatively new discipline in CI when compared to the evolutionary computation (EC), ANNs and FL inference systems. Stimulated via the biological IS, AIS has been investigated for decades and has fascinated raising interest among academics [49].

On the other hand, swarm intelligence (SI) technique illustrates major potentiality in resolving numerous optimization issues and is an evaluative program that is affected via the innate ecology [29]. Also, a trusty and effectual method for resolving optimization issues which have multiple objects attained through adopting EAs. These EAs acquire diverse explanations concurrently during the evolution. Thus, for conquering intricately optimization issues, ACO approach exists to be one of the successful answers [29].

The reason that prediction is so significant is that forecasting of future incidents is a crucial input into numerous classes of decision and planning making. Previously, traditional statistically techniques were adopted to predict time series material. But, the time series materials are often full of non-uniformity and are nonlinear [38]. Thus, enhancing prediction particularly for time series prediction in accurateness is a significant yet often challenging problem for researchers [40]. Recently, the NN technique has also been widely adopted to predict the crude oil prices. It should be noted that the NN technique also has its native drawback, such as worse generalization ability, local optimum solution, and over-fitting. Consequently, in the subsequent research, the incorporated prediction models combining the NN technique tend to be more preferred [84]. In addition, to settle some parameters in RBFnet, it is required to have certain degrees of training and adjustment to achieve superior performance, so it can be applied on more theoretical and practical problem solving.

Therefore, through adopting the advantages of CI and SI learning methods, this paper incorporates the AIS and ACO approaches to propose the HIAO algorithm for regulating corresponding parameters (i.e., centers and widths within hidden layer; weights in output layer) of RBFnet. In detail, AIS approach executes exploitation in local zone of resolving space to avoid insufficient convergence (i.e., merely to regain the secondary best solution) and to make resolving space more intensified. On the other hand, ACO approach executes exploitation in global zone of resolving space to refrain the phenomenon of immature convergence, which would make the resolving space more diversified. This research was then followed by applying the crude oil spot price prediction as a practical instance with the purpose to justify the dominance of the proposed HIAO algorithm. Ultimately, the HIAO algorithm received the best estimate outcome in accurateness than the other algorithms in literature.

2 Literature Review

The biological immune system (BIS) is an inherently dispersed model with decentralized instruction mechanism (dispersion property). It can take actions such as observation, recognition, response, and assessment (combination property) to protect the body from a large diversity of hazards foundation on a restricted count of principles and ideas (universal property). It can remember antigens (studying property), and to resort this information in the process of hereafter when it encounters with similar materials (adjustment property) [53]. Also, the clonal selection (CS) is excited by the CS explanation for gaining immunity introduced by Burnet [16]. The concept behind the principle is to only outspread cells that identify the antigen. Once antibodies (e.g., B cells) bind with antigen, antibody clones are generated and engage with purely somatic mutation. Thus, memory cells with strong variation are generated to keep the message for one particular antigen in the instance of catching owner yet another time [25].

Since 1980s, a developing interest has been facilitated to exploit computational modes excited by less immunological procedures. The particular AIS technique caught high attention in literature is CS [24]. Additionally, AIS-based approaches originate from the principium of the BIS, which attempts to resolve the complex issue of protecting a creature against natural threats by dispersing information to a swarm encompassing an enormous numeral of proxies [70]. And then, the CS explanation is utilized to grabble the optimum attribute dataset, which may gain a dominant manifestation than both present local and complete attribute extraction algorithms [78]. For example, Diao and Passino [26] recommended the CS algorithm (CSA)-based on AIS for parametric modulating on RBFnet construction. But, for further evolution, the CSA would be got caught in the local optimum district and loses the global investigation potential [76].

On the other hand, moving average (MA)-genetic algorithm (GA) and auto-regressive (AR)-GA modelling methods are presented in Abo-Hammour et al. [1] as innovative system recognition and modelling approach of linearly dynamic systems. Suppose the input–output (I/O) data sequence of the model in the absence of any information about the order, the correct order of the model as well as the exact parameters is evaluated synchronously by adopting GA [1]. Further, Arqub and Abo-Hammour [4] have proposed continuous GA for numerical explanation of systems of second-order boundary value tasks. The GA and a few of its evolutions were utilized in mathematics and machine learning (ML) method for optimization tasks [4].

Recently, SI-based algorithms have turn into one of the most advisable resolving methods for challenge engineering or realistic issues [59]. SI is a further type of metaheuristic (MH), which simulates the societal activity of animals in swarms. The major property of this type is sharing of group message of all units during the optimization operation [31]. Lately, several optimization algorithms built on the agility, cooperation, and adaptability features of the species. The members of a group display a specific pattern to be accompanied via every member of the population without any centralization for dominating the transmission between the units appear in the group [75]. For instance, the algorithm foundation on group is merged with artificial seek proxies each with individual acknowledge understanding [60].

SI includes some intimate algorithms such as ACO which is refined since the daily practice of ants [29]. The activity of ants to seek shortest journey between their location and food is the major property of ACO. The ACO algorithm was applied to seek an explanation to the traveling salesman problem [61]. Moreover, Singh and Singh [68] proposed a novel characteristic pick procedure foundation on ACO for the categorization of power standard turbulences. Next, Peng et al. [63] presented a ACO-based property choose technique and called feature ACO (FACO). This technique through interpreting an adaptation function and two-phase pheromone updating criterion to avoid the ACO from sinking into a local optimum solution. Besides, Mousa and Hussein [54] have proposed an unmanned aerial vehicles (UAVs)-based offloading scheme. Then, a discrete differential evolution (DDE) algorithm with novel crossover and mutation manners is proposed for the prescribed optimization task. Further, the ACO algorithm is utilized to recognize the shortest road over the nest heads for the UAV to travel around [54]. Additionally, Mousa and Hussein [55] have proposed a graphics processing unit (GPU)-based parallelization of particle swarm optimization (PSO) to reach equilibrium of the group sizes and recognize the shortest road along these groups while reducing the UAV navigating time and energy spent [55]. Also, Mousa and Hussein [56] present a methodology to establish a surface approximation for an orientation set of examples that may be fully utilized on GPUs (Mousa and Hussein [56].

And then, normally combination is exercised owing to the inadequate complementarity sole mode in catching diverse modes in sampling [35]. Latter, on the basis of Atsalakis et al. [8], hybrid studying algorithm is a very efficient calculation tool to deal with nonlinearity and inexactness. For instance, Ojha et al. [57] have offered a broad range of survey on optimization-based hybrid algorithms exploited on ANN models. Besides, in some investigations both pre-extracting methods and learning methods have been utilized simultaneously to enhance the prediction capability of mixed algorithms [35]. Subsequently, a further method exploited by Sun et al. [72] adopts a combination of two MH learning algorithms, one concentrating on prospecting and the other on regional purification of the solving method [77]. For instance, AIS has been utilized as a structural fragment to hybridize with other optimization algorithms to enhance the performance [83]. In addition, the EA and AIS are merged to refine the association rules within item series [22].

Time series prediction is a wide and vigorous study field which has caught brand attention from variety of areas such as statistics, engineering, finance etc. With its outstanding accuracy, employing mixed algorithms for time series prediction and modeling has become more common in latter discussions [35]. Furthermore, NNs as one of the major prevalent prediction techniques, has lately fascinated scholars to predict the oil price tendency [10]. For example, Mirmirani and Li [52] predicted oil prices in the American market adopting genetic algorithms (GAs) and NNs. Thereafter, Haidar et al. [34] utilized a three-tiered feed-forward ANN (FFNN) for short-term prediction of crude oil prices. They purposely took a list of properties such as S&P 500 index, gold spot price, crude oil futures prices, United States (US) dollar index, and heating oil spot price as indicators. Additionally, Kristjanpoller and Minutolo [42] favorably applied a hybrid ANN-generalized autoregressive conditional heteroscedasticity (GARCH) mode to predict the uncertainty of oil price. Besides, Ma [50] exploited an immune-based clustering neural method for prediction of oil price time series in which the position and count of hidden layers were determined via symbiosis progressive and immune algorithm. Also, Azadeh et al. [9] imitated long-time oil prices with an intelligent adaptability method.

Concurrently, incorporating distinct modes is one of the major general treatments appeared in the literature. It purposing to take the superior ones from separate modes in identification utilized and methods modeling in numerous of time series prediction dissertations [35]. Xie et al. [80] adopted linear auto-regressive integrated moving average (ARIMA) mode to predict West Texas Intermediate (WTI) prices, and debated that oil prices reveal nonlinear feature which is unable to be tackled by linear methods. Further, several studies have compared the ability of AI methods with traditional methods such as regression and ARIMA models in the field of prediction and they have discovered that AI-based models have more accurate consequences than ordinary techniques like Regression and ARIMA models [33]. Also, the forecasting of crude oil price is a vigorous field of research in the literature, and efforts have been made to exploit a trusty model that may forecast its feature. Thereby, it is critical to offer decision makers with forecasts of coming incidents of its models so that they may be adopted for domestic and global expansion programs and decrease the tribulation classically due to the excursion of the crude oil price [17].

3 Methodology

The proposed HIAO algorithm regarding how to cultivate RBFnet by AIS and ACO optimization approaches will be elaborated in the following part. Next, an exhaustive declaration on how the evolution program for the HIAO algorithm was expounded as follows. Subsequently, the data flow diagram (DFD) for the HIAO algorithm is exhibited in Fig. 1. Then, the pseudo-code for the HIAO algorithm is annotated in Fig. 2, and the evolutionary sequences for the HIAO algorithm were implemented and explained as follows.

Fig. 1
figure 1

The DFD for the proposed HIAO algorithm

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Fig. 2
figure 2

The pseudo-code for the proposed HIAO algorithm

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3.1 The Exhaustive Explanation of the Proposed HIAO Algorithm

In order to resolve some challenging nonlinear optimization problems, this paper proposed the HIAO algorithm which combines the advantages of evolutionary algorithm (EA) and swarm intelligence (SI). The proposed HIAO algorithm is able to implement parallel searching with exploitation and exploration in the resolving space, and therefore to avoid the solution from falling into the local optimum dilemma. Further, the HIAO algorithm also meets the requirements of intensification and diversification inside population when resolving. With these reasons, it allows the solution to converge gradually and to obtain the global optimum solutions. Lastly, this proposed HIAO algorithm may be utilized on function approximation as well as to apply on the problem for the crude oil spot price prediction. Furthermore, a detailed explanation on how the evolution sequence for the HIAO algorithm was fulfilled is stated and illustrated as follows.

3.1.1 Initialize Immune Cell Population

(a) Initialize: generate a population which comprises a random numeral of immunoglobulin (Ig). The Igs generated in the population may be considered as the neurons generated in the hidden layer of RBFnet. With several distinct standard testing functions, different input domains will generate distinct training instances [i.e., pathogen (Pg)] as well. These different training instances will be produced inside mapping input domain on N dimension space for several standard testing functions.

(b) To avoid the autoimmunity phenomenon, it is necessary to keep a proper arithmetic mean distance among each generated Ig in the population. The value of width inside individual Ig can be adjusted through Eq. (3).

$$\varphi_{j} = \varphi_{{\text{D}}} \left[ {1 + (1 + \exp^{ - t} )^{ - 1} \cdot \left( {\varphi_{0}^{ - 1} \cdot n^{ - 1} \sum\limits_{i = 1}^{n} {\left\| {d_{i} } \right\|} - 1} \right)} \right].$$
(1)

Here adopted Sigmoid function is a bounded differentiable real function, which is defined as: \(S(t) = (1 + \exp^{ - t} )^{ - 1}\) [32] and its scope is between 0 and 1. This Sigmoid function is blunt and simply to realize [7]. Further, \(\varphi_{0}\) is the initial Euclidean distance for width, and it can be considered as an inhibition threshold. The arithmetic mean Euclidean distance is equivalent to \(n^{ - 1} \sum\nolimits_{i = 1}^{n} {\left\| {d_{i} } \right\|}\) between any Ig with others. \(\varphi_{{\text{D}}}\) is the default width constant within the initialization term for the specific resolving standard testing function. Lastly, \(\varphi_{j}\) is the width value of jth Ig within RBFnet. It ensures the scope value of \(\sigma_{j} = \left[ {\sigma_{D} ,3\sigma_{D} } \right]\) is caused by the restricted output value of \(S(t) = [0,1]\). It thereby prevents the autoimmunity phenomenon among Igs, so the homogeneous immune cells would not fall into being destroyed.

(c) The orthogonal least squares (OLS) algorithm [18] is adopted to resolve the value of weight between hidden and output layers in RBFnet.

3.1.2 Fitness Assessment and Affinity Maturation

The affinity maturation between Ig and Pg will be assessed when resolving several standard testing functions. And then some candidate solutions with superior suit values will be derived. Accordingly, the affinity maturation between an Ig confronted with foreign Pg can be calculated through Eq. (2):

$${\text{Fitness(Ig)}} = \left\| {{\text{fitness}}(c_{j} ) - \overline{{{\text{affinity}}_{{{\text{TrainingSet}}}} }} } \right\|,$$
(2)

where \(\overline{{{\text{affinity}}_{{{\text{TrainingSet}}}} }}\) is the averaged affinity surface assignation with Euclidean distance between Ig and Pg from initial generated training subsets on N dimension space for distinct standard testing functions. In addition, the \({\text{fitness}}(c_{j} )\) is the suit value of center point \(c_{j}\) within generated Ig. The \({\text{Fitness(Ig)}}\) is the Euclidean distance between the \({\text{fitness}}(c_{j} )\) within Ig and \(\overline{{{\text{affinity}}_{{{\text{TrainingSet}}}} }}\), which is a standardized vector value between 0 and 1. Once Ig is more able to recognize Pg, it has higher suit value for Ig which indicates proportional affinity maturation and will induce immune response.

3.1.3 Mutation Asexually

Once the affinity maturation between Ig and Pg is higher, the Ig inside population will have lower mutation ratio. This sequence will prompt Ig with higher affinity to survive inside population in next generation, and further to trigger evolution and resolving.

3.1.4 Clonal Proliferation

A fixed quantity of Igs with higher affinity from parent population can then be duplicated and inherited to next generation through Eq. (3), which further promotes subsequent evolution and resolving:

$$\psi = \left\lfloor {c_{j} } \right\rfloor \cdot \vartheta ,$$
(3)

where \(\vartheta\) is the clonal ratio, \(c_{j}\) is the center point of jth neuron inside hidden layer on RBFnet, \(\psi\) is clonal numeral and its purpose is to proliferate a fixed proportion quantity of Igs with higher affinity from current population to offspring population. Accordingly, such clonal proliferation process allows more elite Igs to duplicate in the subsequent population, and facilitates the resolving to be able to converge toward the global optimum solution.

3.1.5 ACO Approach

(a) In the beginning of optimization process, let \(L\) ants proceed from their nest to seek for food. It will initialize retaining equivalent amount of pheromone at each edge of any ant’s passing paths. Ants proceed from any node of nest, and terminate at the destination node throughout each generation, where any ant may select the next node to visit by Eq. (4):

$$R_{\ell } (f,g) = \left\{ \begin{gathered} \frac{{\lambda (f,g)^{\chi } \times \partial (f,g)^{\gamma } }}{{\sum\nolimits_{{y \in U_{\ell } (f)}} {\lambda (f,y)^{\chi } \times \partial (f,y)^{\gamma } } }}\mathop {}\limits_{{}} ,{\text{if}}\mathop {}\nolimits_{{}} g \in U_{\ell } (f) \hfill \\ \mathop {}\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop 0\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop {}\nolimits_{{}} \mathop {}\limits_{{}} \mathop ,\limits_{{}} {\text{Otherwise}} \hfill \\ \end{gathered} \right.,$$
(4)

where \(U_{\ell } (f)\) is the memory of all unvisited nodes once ant \(\ell\) proceeds from the beginning node on edge \(f\) of passing paths in traveling. \(R_{\ell } (f,g)\) is the random probability value of uniform distribution for ant \(\ell\) proceeding from the beginning node on edge \(f\) to next random visited node on edge \(g\). Next, \(\chi\) and \(\gamma\) are the empirical parameters to control the sensitivity for pheromone value (\(\chi \ge 0\)) and heuristic value (\(\gamma \le 1\)) respectively. \(\lambda (f,y)\) is the pheromone test value from any node on edge \(f\) to any node on edge \(y\), while \(\partial (f,y)\) is the heuristic value from any node on edge \(f\) to any node on edge \(y\).

(b) For the advanced solution, it is certain that it will execute with the local and global updates on pheromone trial. For any ant \(\ell\) proceeding from the nest has visited any path, it will deposit fixed pheromone on the path through the local updating rule as shown in Eq. (5):

$$\lambda (f,g) = \lambda (f,g) + \Delta \lambda^{\ell } (f,g).$$
(5)

Subsequently, once all ants in the nest have completed path seeking on the travel, the pheromone will be revised through the global updating rule as shown in Eq. (6):

$$\lambda (f,g) = (1 - \sigma ) \cdot \lambda (f,g) + \sum\limits_{\ell = 1}^{L} {\Delta \lambda^{\ell } (f,g)} ,$$
(6)

where \(\sigma\) is pheromone’s decaying ratio in pheromone trial. \(\Delta \lambda^{\ell } (f,g) = (\lambda_{\ell } )^{ - 1}\) is the numeral of times on pheromone trials for ant \(\ell\) to travel from any node on edge \(f\) to any node on edge \(g\), where the numeral of times on pheromone trials is resolved through the reciprocal of legacy pheromone \(\lambda_{\ell }\) that ant \(\ell\) has attempted path seeking on the travel.

3.1.6 Inter-Ig Inhibition

During the evolutionary resolving process, if the affinity between any two Igs is lower than the inhibition threshold, any Ig will be inhibited. The Ig who retains higher affinity with Pg will be more likely to recognize the features of data. Once it retains proliferated Igs with higher affinity maturation inside population, more memory cells will be enlisted and will allow secondary immune response to recognize more Pgs.

3.1.7 Enlistment

To retain the balance between diversity and stability inside population during the resolving process, Eq. (7) can be used to enlist a fixed percentage of Igs in the evolutionary population:

$$N_{{{\text{En}}}} = \mu \cdot \tau \cdot {\text{Max}}[2^{ - 1} \cdot (F_{1} ,Z_{1} ),50],$$
(7)

where \((F_{1} ,Z_{1} )\) is the numeral of data points inside training set in the experimentation, \({\text{Max}}[2^{ - 1} \cdot (F_{1} ,Z_{1} ),50]\) is the maximum (max.) numeral of Igs inside population. \(\tau\) is the decaying factor, and it may promote the stability during the resolving period. Next, \(\mu\) is the enlisting ratio, and \(N_{{{\text{En}}}}\) is the enlistment numeral of Igs inside population.

The numeral of new enlisted Igs inside the population is controlled through the enlistment sequence. In this way, it may prevent enlisted Igs to be excessive (too few) and cause solution’s premature divergence (convergence).

3.1.8 Update the Global Optimal Solution

While the randomly generated population has been examined through the proposed HIAO algorithm and material feature has been conformed to the adaptation function, the global optimum solution will be resolved gradually.

3.1.9 Ending Basis

The proposed HIAO algorithm will continue to execute and return to Step (2), until the adaptation function is satisfied or a specific numeral of generations is reached.

The part solutions received from original population through ACO approach are added into the other part of solutions received from AIS approach. In this way, it will decelerate the convergence speed of solving and expand the solution space. Furthermore, applying ACO approach alone to handle and feedback information would drag the overall computation sequences of solving. Followed by AIS approach to handle and feedback information will complement ACO approach and strengthen the diversity of entire solution sets. With this, it would achieve complementary integration for the solving process of whole population. Moreover, it’ll allow the solution space to have the exploitation in local search and exploration in global search. Simultaneously, it allows the solution sets to be generated under intensification and diversification, and increases the opportunity to receive the optimum solution. Next, how the proposed HIAO algorithm cultivate RBFnet through AIS and ACO approaches will be elaborated exhaustively in the following section.

4 Experimental Consequences

This chapter focuses on modifying and adapting the advisable parameters in RBFnet for estimation of function approximation issue. The objective is to verify the optimum applicable values which refers to the parametric resolved by the RBFnet. The intent is then to judge the advisable values of the parametric set from the probing district in the experimentation. The proposed HIAO algorithm may be modified and accordingly meet the explanations of parametric values set for RBFnet.

4.1 Standard Nonlinear Problems Experimentation

Experimental testing function incites significant resemblance to recompense RBFnet for the consequence of continuous mapping conjunction. This research introduces theoretically nonlinear problems that are constantly utilized in the literature to be the rival standard of tested algorithms.

The uni-modal functions are examined for measuring the developing of algorithms as them have merely one entire optimum. Consequently, the multi-modality or composite functions, they possess numerous regional optimum which in backtrack are advisable for measuring the manifestation of algorithms and can avoid regional optimum and make investigation judgment [65]. Then, the experimentation comprises the subsequent five standard testing functions, including Mackey–Glass time series [46], B2 [67], Rosenbrock, Sphere, and Griewank [14] nonlinear problems.

The first experimentation, the Mackey–Glass time series [46], is introduced as below:

$$\frac{{{\text{d}}x(t)}}{{{\text{d}}(t)}} = 0.1x(t) + \frac{0.2x(t - 17)}{{1 + x(t - 17)^{10} }},$$
(8)

where x(t) is the value of time series at time pace t. This study for the retrieval t scopes be amount [118, 1118] with the Mackey–Glass time series, from which one thousand samples were stochastically emerged [79]. The data set is created with second-order Runge–Kutta formula and with pace size of 0.1 [69].

In the second experimentation, B2 [67] function is introduced as below:

$$B2(x_{j} ,x_{j + 1} ) = x_{j}^{2} + 2x_{j + 1}^{2} - 0.3\cos (3\pi x_{j} ) - 0.4\cos (4\pi x_{j + 1} ) + 0.7.$$
(9)
  1. (a)

    Probing district: \(- 100 \le x_{j} \le 100\), j = 1;

  2. (b)

    One universal minimum (min.): (\(x_{1} ,x_{2}\)) = (0, 0); \({\text{B2}}(x_{1} ,x_{2} )\) = 0.

In the third experimentation, Rosenbrock [14] function is introduced as below:

$${\text{RS}}(x_{j} ,x_{j + 1} ) = \sum\limits_{j = 1}^{n - 1} {[100(x_{j}^{2} - x_{j + 1} )^{2} + (x_{j} - 1)^{2} ]} .$$
(10)
  1. (a)

    Probing district: \(- 30 \le x_{j} \le 30\), j = 1;

  2. (b)

    One universal min.: (\(x_{1} ,x_{2}\)) = (1, 1); \({\text{RS}}(x_{1} ,x_{2} )\) = 0.

In the fourth experimentation, Sphere [14] function is introduced as below:

$${\text{SP}}(x) = \sum\limits_{i = 1}^{n} {x_{i}^{2} } .$$
(11)
  1. (a)

    Probing district: \(- 100 \le x_{i} \le 100\), \(i = 1\).

  2. (b)

    One universal min.: (\(x_{1} ,x_{2}\)) = (0, 0); \({\text{SP(}}x_{1} ,x_{2} {)}\) = 0.

In the fifth experimentation, Griewank [14] function is introduced as below:

$${\text{GR}}(x_{j} ,x_{j + 1} ) = \sum\limits_{j = 1}^{n} {\frac{{x_{j}^{2} }}{4000} - \prod\limits_{j = 1}^{n} {\cos \left( {\frac{{x_{j + 1} }}{{\sqrt {j + 1} }}} \right)} + 1} .$$
(12)
  1. (a)

    Probing district: \(- 100 \le x_{j} \le 100\), j = 1;

  2. (b)

    One universal min.: (\(x_{1} ,x_{2}\)) = (0, 0); \({\text{GR}}(x_{1} ,x_{2} )\) = 0.

4.2 Parametric Determination

There is a numeral of correlative parametric values on RBFnet that are required to define advance to carry out cultivating and adaptation for function approximation. Consequently, the HIAO algorithm possesses the superior technique to cultivate RBFnet than other corresponding methods in existing literature, since it retains a pre-setting extent for all standard testing functions correlative to its own probing districts. These estimated algorithms are initiated with the estimate of the parameter sets for five nonlinear problems enumerated in Table 1.

Table 1 The parameter sets for five nonlinear problems
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In the HIAO algorithm, four parameters (i.e., the clonal ratio, the enlisting ratio, the pheromone test, and pheromone’s decaying ratio), which have significant impact on estimation results are examined. Concurrently, this experimentation consulted to relevant literature for the spacing of the parametric values. Furthermore, the formation of the parametric values for the HIAO algorithm is regulated by referring to the Taguchi methodology [73] layout where examination mode was used for experimental purpose [82]. The Taguchi methodology [73] is introduced where orthogonal matrixes (arrays) are utilized to remarkably lower the counts of experiments [74]. Accordingly, the Taguchi experimental parsing were deployed in a L9 (34) (i.e., 9 trials and 3 levels with 4 factors) orthogonal matrix for the HIAO algorithm where the experiments were successfully executed for sixty times. Subsequently, the MINITAB 18 was adopted to parse parametric values for the HIAO algorithm, where the stability of system status in the experiment is evaluated by the signal-to-noise (S/N) ratio [45]. Later, the max. numeral of generations is set at one thousand to regulate as ending point for the experiment. In result, the estimation values of the parameters for the HIAO algorithm are listed in Table 2.

Table 2 Parameter values for the HIAO algorithm
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4.3 Performance Parsing and Comparison

The regulating of all utilized algorithms on parametric solution sets for RBFnet emerged from the population during the adaptation of the evolution program in the experiment are discoursed in this part. One thousand stochastically emerged datasets are divided into three parts (i.e., training set (65%), testing set (25%), and validation set (10%)) [48] to cultivate RBFnet. With this, it may inspect the imitating situation and regulate the parametric configuration. This paper then introduces those algorithms mentioned to parse the optimum sets of parametric solution for RBFnet. Afterward, it randomly emerges dissimilar training set (65%) from one thousand emerged samples and takes the dataset into RBFnet for cultivating. Using the identical process, it randomly emerges dissimilar testing set (25%) to inspect individual’s parametric solution inside population and evaluates the adaptation function. As yet, RBFnet has adopted 90% dataset in imitating period. Succeeding one thousand generations in the evolution manipulation, the optimum solution sets of parameters for RBFnet are gained. Lastly, it randomly emerges dissimilar validation set (10%) to certify how the parametric solution of individual resembles the five nonlinear problems and reserves the root mean square error (RMSE) values to interpret the imitating period of RBFnet. Once the data extraction phase mentioned above has performed, all algorithms utilized are carried out. The imitating and examination periods involved above were conducted sixty times before the arithmetic mean of RMSE (i.e., \(\overline{{{\text{RMSE}}}}\)) values were evaluated. The contents of the \(\overline{{{\text{RMSE}}}}\) and standard deviation (SD) for all algorithms used and evaluated from the experiment are disclosed in Table 3.

Table 3 Result comparison among corresponding algorithms utilized in this experiment
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As illustrated in Table 3, since AIS and ACO approaches have conducted exploitation and exploration. respectively in the solution region, therefore the solution set has the advantages of intensification and diversification and allows the proposed HIAO to show its complimentary capability. Accordingly, the result shows that HIAO algorithm gains the exact adequate values with certain signification during the imitating period of the experiment. Moreover, RBFnet can comply the single set of parametric solution from the promotion of proceeding population inside, which has achieved the setting to conduct function approximation. When the cultivating of RBFnet through the HIAO algorithm is accomplished, the individual with the optimum execution of parametric solution set (i.e., the centers on RBFnet hidden layers, widths, and weights) in imitating period will be the determined RBFnet circumstance.

5 Realistic Instance for the Crude Oil Spot Price Prediction

While oil market participator forecasts are straightly influenced by the variations of the crude oil prices, it has become critical to exploit prediction patterns for these spectrums. Therefore, the crude oil prediction has drawn both academic and industry’s attentions on probing new methods that may be beneficial to comprehend the intrinsic trends of oil prices [39]. Accordingly, the prediction of crude oil spot price can be considered as a significant issue.

It has been evidenced that RBFnet is able to attain exact resemblance on theoretically standard nonlinear problems via the proposed HIAO algorithm. The consequences are contrasted with corresponding algorithms in academic literatures and the accurateness of the HIAO algorithm is shown.

Furthermore, the WTI crude oil spot price is diffusely utilized as the base of numerous crude oil forms, and the WTI is the celebrated basis price for majority [81]. Thus, this valuation strives to discuss the accurateness of prediction on crude oil spot price tendency data of the WTI (https://www.fxempire.com/commodities/wti-crude-oil) from 01/02/2008 to 02/02/2009 (258 tuples in approximately one year) which is applied as tuples in this paper. The detailed material assignation of this instance is acquired from the annual database of the U.S. Department of Energy and is presented in Table 4. Further, the examination has assumed that the influence of foreign investigational factors did not exist and the tendency of the crude oil spot price was not interfered by any exceptional incidents.

Table 4 The tuples assignation of the crude oil spot price tendency data
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5.1 Establish ARIMA Models

Time series materials are regularly estimated with the assumption of inventing a historic mode that may be utilized in the prediction. In Box and Jenkins [15], they explored and used the ARIMA approach to predict time series incidents. Additionally, ARIMA is a statistical method that forecasts value of material through refining linear correlation and extracting annually [35]. Then, with the target to assure the forecasts of ARIMA models can be accomplished, whether or not the time series material is stock-still (stationary) was taken into account, and the crude oil spot price prediction modes were applied to examine the material. Moreover, to ensure the forecasts of ARIMA models can be performed, the instance study for crude oil spot price prediction was applied to confirm the models. SPSS 16.0 and EViews 11.0 (statistical software) were utilized for the parsing of ARIMA models to evaluate the material consequences.

Next, this research realized the crude oil spot price prediction foundation on ARIMA models. The ARIMA (p, d, q) modelling sequence retains three stages: (1) recognizing p, d, and q orders of the model; (2) evaluating the modulus (coefficients) of the model; and (3) prediction the material [11]. If the material pattern is stock-still, model assessment should be realized straightly, otherwise, differencing should be carried out to allow it to be stationary. Thus, this paper conducted the material certification of ARIMA models through the augmented examination [27]. Therefore, it is proper to introduce ARIMA (p, d, q) models to carry out estimation and prediction of the crude oil spot price material. On the other hand, the basis of Akaike [2]’s information criterion (IC) (i.e., Akaike IC = 5.226) was utilized to refine the optimum pattern model [30]. Based on the results, it has shown that the value of Akaike IC of ARIMA (2, 1, 2) model is the least (i.e., regulated R-square = 0.0059) among all possible ARIMA models, illustrating that it is the optimum model and therefore the most advisable one as the crude oil spot price tendency data. The result of model identification demonstrates that the Kmenta [41]’s Ljung-Box statistic (i.e., Q-statistic) values are 0.05 superior than the outcome of ARIMA models, in which are not serial associated and had been sufficiently suited. This paper applied the most appropriate ARIMA (2, 1, 2) model, which has been assessed and recognized to carry out the crude oil spot price prediction.

5.2 Parametric Determination for the Crude Oil Spot Price Prediction Instance

There are a few parameter values for RBFnet that should be established in advance of executing cultivation for the practical instance of prediction evaluation. Thus, certain parameters’ conditions for the HIAO algorithm is obtained on the basis of Taguchi methodology [73] and corresponding literatures. And then, the MINITAB 18 (statistical software) was utilized in the inspection of parametric configuration. The Taguchi checks (trials) were arranged in an L9 (34) orthogonal matrix (array) for the HIAO algorithm and the experiment was followed by sixty times of execution. Lastly, the HIAO algorithm was executed with parametric set enumerated in Table 5.

Table 5 Parametric set for the HIAO algorithm in the crude oil spot price prediction instance
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5.3 Variance Estimate for the Crude Oil Spot Price Prediction Performance

In Looney [48], it proposes to acquire 65% subset of material patterns for training, 25% subset for testing, and 10% subset for validation. Besides, majority of researches in literatures utilize appropriate proportion to divide in- and out-of-patterns including 70–30%, 80–20%, or 90–10% [85]. Therefore, this research utilizes the proportion of 90% (233 tuples for imitation)–10% (25 tuples for prediction) to divide material. The period of crude oil spot price tendency data acquired is from 01/02/2008 to 02/02/2009. The practical instance with the crude oil spot price prediction is based on time series material assignment and is applied for prediction inspection.

The imitating period of RBFnet will be based on daily crude oil spot price data; it includes training (65% dataset) and testing (25% dataset) subsets. The cultivating was initiated by entering four tuples in turn, which were retrieved from training subset for RBFnet. In the operation, the solution of individual parameter inside the population was examined solely with the entire development sequence, then the values of adaptation function of all individuals inside the population should be assessed with the testing subset. Accordingly, 90% of the dataset of the crude oil spot price data was utilized to the imitating period of RBFnet, which emerged a solution of individual parameter with the most exact prediction. In the meantime, it was required for the representation of the RBFnet prediction to be evaluated with the validation (10% dataset) subset. Then, the expected values passed were emerged in succession from the moving shutter program. The initial 90% dataset was utilized for model assessment while the rest 10% dataset was utilized for validation and by degrees to move against prediction. In summary, this part declares how material is feed in to RBFnet for prediction via a few corresponding algorithms, and how the achievement is compared with ARIMA (2, 1, 2) model.

Furthermore, the RMSE indicated the SD of material pattern inside the disparities between expected (i.e., \(\hat{q}_{i}\)) and observed (i.e., \(q_{i}\)) values. As one of the regularly adopted variance indices in statistics, the RMSE is described as [19]:

$${\text{RMSE = }}\sqrt {\frac{{1}}{N}\sum\limits_{i = 1}^{N} {(q_{i} - \hat{q}_{i} )^{2} } } .$$
(13)

Moreover, the RMSE, mean absolute error (MAE), and mean absolute percentage error (MAPE) are the major variance assessment indicators which are frequently utilized in commerce, and thence were adopted to identify the prediction models [21]. As such, the MAE was the arithmetic mean of the absolute variance between expected and observed values. It is described as below [19]:

$${\text{MAE = }}\frac{{1}}{N}\sum\limits_{i = 1}^{N} {\left| {q_{i} - \hat{q}_{i} } \right|} .$$
(14)

Lastly, the MAPE is a statistical assessment index for the accuracy of forecasting of a prediction process. It generally indicates the percentage ratio of the emerging variance and is described as below [19]:

$${\text{MAPE = }}\frac{{1}}{N}\sum\limits_{i = 1}^{N} {\left| {\frac{{q_{i} - \hat{q}_{i} }}{{q_{i} }}} \right|} \, \times {\text{ 100\% }}{.}$$
(15)

This paper also utilizes the HIAO algorithm for the instance of prediction on the tendency of crude oil spot price. Among corresponding algorithms, the results acquired from three variances of the HIAO algorithm were the optimum ones, which are illustrated in Table 6. Additionally, comparisons with relevant algorithms and the ARIMA models were also considered, the result shows the instance by the HIAO algorithm has the highest accuracy. With this, the results obtained shows that compared to the ARIMA models, the HIAO algorithm has significantly improved the accuracy of the crude oil spot price prediction.

Table 6 The comparison of prediction variances for all relevant algorithms utilized in the crude oil spot price instance
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As for the validation of statistical significance, it obtained the significant result when carrying out the matched coupled material pattern tests of T test with the absolute variance from the evaluated dataset of the tendency data in corresponding algorithms. Accordingly, the T test result (significance level = 5%) and the verification of each algorithm’s prediction among all corresponding algorithms are demonstrated in Table 7, which shows that ARIMA (2, 1, 2) model and the HIAO algorithm are not statistical significant (i.e., it does not exist significant bias between the real and forecasted values; p value bigger than 0.05) and thus offer better prediction than the rest. Further, the statistical significance shows that the HIAO algorithm has the best result with the best prediction among all relevant algorithms.

Table 7 The statistical result for T-test among all relevant algorithms
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Accordingly, the proposed HIAO algorithm offers notably the best result of the crude oil spot price prediction while the compared results for the instance is converged in Fig. 3. Note that the prediction and verification of the crude oil spot price is denoted in U.S. dollar (US$).

Fig. 3
figure 3

The prediction results compared to the proposed HIAO algorithm and ARIMA (2, 1, 2) model for the crude oil spot price prediction instance

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6 Conclusions

This research proposed the HIAO algorithm through incorporating the abilities of exploitation and exploration with the AIS and ACO approaches based algorithm, which provides the settings of parametric on RBFnet. The complementation of some evolutionary operations that heightens the diversity of populations also raises the accurateness of the consequences. The estimated experimental consequences have been compared with those gained through the HIAO algorithm cultivated by the comparative algorithms. The HIAO algorithm retains superior parametric setting of network and consequently enables RBFnet to implement excellent imitating and resemblance in theoretically standard nonlinear problems experimentation. Besides, the crude oil spot price practical instance indicates that the HIAO algorithm exceeded corresponding algorithms in prediction accurateness.