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Review

Chaotic Time Series Forecasting Approaches Using Machine Learning Techniques: A Review

School of Electrical Engineering, Vellore Institute of Technology, Vellore 632014, India
*
Author to whom correspondence should be addressed.
Symmetry 2022, 14(5), 955; https://doi.org/10.3390/sym14050955
Submission received: 4 April 2022 / Revised: 25 April 2022 / Accepted: 2 May 2022 / Published: 7 May 2022
(This article belongs to the Special Issue Nonlinear Symmetric Systems and Chaotic Systems in Engineering)

Abstract

:
Traditional statistical, physical, and correlation models for chaotic time series prediction have problems, such as low forecasting accuracy, computational time, and difficulty determining the neural network’s topologies. Over a decade, various researchers have been working with these issues; however, it remains a challenge. Therefore, this review paper presents a comprehensive review of significant research conducted on various approaches for chaotic time series forecasting, using machine learning techniques such as convolutional neural network (CNN), wavelet neural network (WNN), fuzzy neural network (FNN), and long short-term memory (LSTM) in the nonlinear systems aforementioned above. The paper also aims to provide issues of individual forecasting approaches for better understanding and up-to-date knowledge for chaotic time series forecasting. The comprehensive review table summarizes the works closely associated with the mentioned issues. It includes published year, research country, forecasting approach, application, forecasting parameters, performance measures, and collected data area in this sector. Future improvements and current studies in this field are broadly examined. In addition, possible future scopes and limitations are closely discussed.

1. Introduction

The first section of this paper provides a brief description of the chaos and the properties of chaotic systems. In addition, the importance of chaotic time series forecasting in significant areas is addressed. Finally, this section covers the previous and current literature surveys on chaotic time series forecasting.

1.1. Chaotic Systems

The behavior of a nonlinear dynamical system that may be extremely sensitive to small changes in initial conditions is known as chaos. This sensitivity to initial conditions means that a slight change in the starting point can lead to different outcomes. For example, the butterfly effect shows how a small change in one state of a deterministic nonlinear system may result in enormous deviations in a subsequent state [1]. The other characteristic of a chaotic system is no periodic behavior. The symmetric property of these nonlinear dynamic systems can play a vital role in producing the systems’ chaotic behavior. Due to this fact, various researchers have recently shown much interest in the symmetric properties of chaotic systems. In [2], the authors have proposed a chaotic oscillator with both odd and even symmetries. Similarly, some of the other applications of symmetric properties of chaotic systems lie in image processing, security, and communications [3]. The symmetric and asymmetric behavior has been observed in many natural phenomena. Due to these characteristics, the chaotic motion is difficult to forecast. For instance, predicting the butterfly effect for the long term is impossible [1]. This is because these systems are deterministic, i.e., the future behavior of these systems is entirely defined by their initial conditions. Hence, these systems are wholly deterministic and unpredictable.
On the other hand, a chaotic time series is generated when the variable changes with time in a chaotic system. This chaotic time series provides extensive information about the nonlinear system and helps evaluate and analyze the chaotic system’s behavior. The phase space reconstruction technique reveals this dynamic information hidden in the chaotic time series and transforms the existing data into a more describable framework [4]. As a result, it is essential to have approaches that can forecast chaotic time series and differentiate chaotic data from stochastic data [5,6,7]. The traditional prediction methods for this purpose have failed to produce satisfactory performance. Thus, many advanced techniques using machine learning-based approaches have been proposed recently. Therefore, this paper presents a comprehensive review of the performance of traditional and machine learning-based methods for chaotic time series forecasting and their implementation on nonlinear dynamical systems, such as photovoltaic systems, wind farms, communication signals and systems, oil and gas, hydrological systems, weather, and other systems.

1.2. Importance of Chaotic Time Series Forecasting

Forecasting is an approach for creating predictions to determine the direction of future trends using historical data and current trend analysis as inputs [8]. Forecasting is the most significant optimization concept related to energy savings, material savings, increasing efficiency, making appropriate and suitable accurate decisions [8,9]. On the other hand, chaos theory is an essential part of nonlinear science, developed in the 1970s [10]. Chaos is a long-term non-periodic behavior in a predictable system with a high sensitivity to initial conditions. It shows the order and regularity hidden behind disorganized and complex occurrences. This tendency permeates and promotes many subjects. As a result, chaos research has access to a solution. In the meantime, chaos theory applications are becoming increasingly popular. They are significantly used in diverse scientific applications such as wind farms [11], PV systems, oil and gas [12,13], hydrological systems [14], etc. A brief description of the need for chaotic time series forecasting in each of these applications is explained below.

1.2.1. Chaotic Time Series Forecasting in Power and Energy

Electricity demand and market price predictions have played a significant role in the electric power industry for over a century [15]. Moreover, due to the worldwide energy crisis and alarmingly rising air, water, and soil pollution levels, renewable energy has become increasingly popular for power generation in recent years. This popularity is because renewable energy is a pure and limitless energy source [11]. As a result, a rising number of nations are becoming involved, and investors are committing to developing renewable energy plants. However, the lack of consistent energy sources due to intermittent nature represents renewable energies’ main problem. Thus, forecasting renewable generation is the key to integrating these intermittent energies into the electricity grid for several reasons [9,16]. The main advantage of predicting the intermittent nature of renewable energy resources is that the number of backup systems can be reduced, thus, reducing the investments and need for electricity to meet the demand. Many forecasting approaches have been proposed using ANN, fuzzy, etc. These approaches are based entirely on time series analysis in which the chaotic time series data of renewable energy are one of the most challenging dynamics to be forecast.

1.2.2. Chaotic Time Series Forecasting in Oil and Gas

It is well known that the intake flow of a gasoline engine directly impacts the accuracy of the air–fuel ratio management under transient situations [12]. As a result, precise control becomes extremely difficult because the air ratio is far from stoichiometry for various reasons. Thus, forecasting the engine’s intake flow with greater accuracy in less time can improve the convergence rate. Additionally, it will be able to overcome the shortcomings of the airflow sensor’s lag. This is because it allows an accurate forecast of the future airflow. Similarly, it is also well known that there are abnormal fluctuations in the ventilation air in the nonlinear coal mines’ ventilation systems [17]. These fluctuations in the air are due to the mining depth and intensity gradually increasing and equipment aging. The abnormalities, as mentioned earlier, can affect the entire system, resulting in various underground accidents and lost coal mines’ ventilation system stability. Therefore, timely air quality prediction in coal mines’ ventilation systems can help adequately manage systems, which directly influences the safety and output of the coal mine.

1.2.3. Chaotic Time Series Forecasting in Hydrological Systems

Hydrological forecasting plays a critical role in reducing future flood impacts, also helps produce more benefits for hydropower production, and enhances water resource management [18]. It is worth noting that predicting the destiny of a river inflow is an essential concern for water quality management [19].

1.2.4. Chaotic Time Series Forecasting in Other Systems

The distributed control system and information technology, which comprises supervisory information technology and management information systems, are commonly used technologies in thermal power plants [20]. The real-time data collected from power plant equipment and personnel controls using these technologies are a chaotic time series. Further, the instantaneous generator output power is critical to indicate the adjusting and controlling equipment’s status. As a result, predicting the immediate generator power time series could provide decision-making, maintenance, and incident-handling information. Further, it positively impacts plant production, optimal operation, and problem detection and maintenance technology.
Natural hazards, such as earthquakes, severe floods, fires, and volcanic eruptions, and the destruction they create are worldwide issues that impose a high cost in terms of human lives and financial damages [21]. The wireless sensor networks monitor the urban river levels and other natural environmental conditions for predicting the floods before they occur so that the people at risk evacuate in time.
Similarly, the nonlinear spacecraft system contains various fields with advanced technology, and it has a significant impact on national economies, research, and technology. Faults in the spacecraft system are challenging to detect and rectify. As a result, studying the trend of spacecraft telemetry metrics and the variation law is essential for the early prediction of spacecraft problems.

1.3. Previous and Current Literature Survey

Few reviews have focused on applying chaos theory in multiple applications. For instance, in [14,22,23], a study on the application of the chaos concept in hydrology was reported. The study also reveals some critical issues raised while applying the chaos concept in hydrology. Similarly, a review of the application of chaos theory in traffic flow patterns was reported in [24]. In both works, some of the reviewed methods reported for the short-term forecasting are correlation dimension, Lyapunov exponent, Kolmogorov entropy, SVM, ANN, nonlinear prediction, and dynamic neural network. These reviews overlapped with elements of this field, though none have brought together all material related to chaotic time series forecasting approaches using machine learning techniques for various applications.
Considering the above research scope, the authors in this paper reviewed chaotic time series forecasting approaches using machine learning techniques in various applications. At first, the importance of chaotic time series forecasting is identified in multiple applications, including all the recently published methods, and addresses issues of individual techniques. The review of these chaotic time series forecasting approaches in the past three decades is summarized in Section 2. Section 3 gives a comprehensive review of machine-learning-based chaotic time series forecasting approaches developed using ANN, FNN, WNN, and optimization algorithms. The study on forecasting various chaotic parameters in multiple applications is detailed in Section 4. Section 5 discusses the various performance measures used for chaotic time series forecasting approaches. Finally, Section 6 concludes the current works, highlighting the shortcomings and suggesting possible future research perspectives.

2. Review on Chaotic Time Series Forecasting

In the past three decades, many researchers have rigorously researched forecasting of chaos in various areas, such as wind farms, photovoltaic systems, hydrological systems, communication systems, and oil and gas fields, using ANN. Thus, there is a scope for a critical review of chaotic time series forecasting in various areas using machine learning techniques. This manuscript critically reviews various works published from 1992 to 2021. The decade-wise research contributions to chaotic time series forecasting during this period are shown in Figure 1.
According to the literature review collected from Table 1, 43% of works have employed the ANN-based approaches in the literature for chaotic time series forecasting. The additional techniques are based on the following: FNN 24%, optimization algorithms 15%, WNN 6%, and other approaches 4%. In the 15% of optimization-algorithm-based techniques, the various algorithms used are GA, PSO, SSA, SA, SOM, CGO, GWO, CBAS, etc. The objective of these techniques is to improve accuracy, computational efficiency, and concerns due to the presence of uncertainties in various applications. Some novel techniques reported in Table 1 focused on efficiently tackling multiple objectives. Table 1 also shows that these articles have dealt with several forecasting parameters, such as load, power, speed, traffic flow, signals, etc. In some of these works, real-time data were also collected from various countries, including Australia, Belgium, Canada, China, Iran, Laos, Morocco, Thailand, and the USA, as shown in Figure 2. In the first decade, research on chaotic time series forecasting relied on statistical data to forecast the system’s future behavior. The rest of the decades used artificial intelligence and other novel models for chaotic time series forecasting in various applications. The detailed analysis of various forecasting approaches in different applications is explained in the following sections.

3. Neural Network-Based Forecasting Approaches

As mentioned in Section 2, various researchers have developed ANN, FNN, WNN, and optimization-based approaches for chaotic time series forecasting. The multiple techniques developed using these approaches are shown in Figure 3. A detailed explanation of these techniques, including the objectives and performance analysis, is presented underneath. The future scope of the method is also highlighted.

3.1. ANN-Based Forecasting Approaches

ANN has multiple perceptrons’ or nodes at each layer. For example, the network with two input nodes, two hidden layers with four nodes in each, and one output node is shown in Figure 4. This network can be called FFNN when its inputs are processed forward (refer to the red dotted line in Figure 4). The FFNN is one of the most straightforward neural networks, and it passes information in one direction through various input nodes until the output node [178]. This type of neural network may or may not have hidden layers, making its functioning more understandable. Some advantages of FFNN include storing information on the entire network, working with incomplete knowledge, offering tolerance, and having distributed memory. However, the disadvantages of FFNN include having hardware dependency and unexplained behavior that can leave us tormented with results. No particular rule for deciding the network’s structure and the appropriate network structure is achieved through experience and trial and error.
BPNN is an essential mathematical tool for improving the accuracy of predictions in data mining and machine learning. In FFNN, the network propagates forward to obtain the output and compares it with real value to obtain the error. However, to minimize the error, the BPNN will propagate backward by finding the error derivative for each weight and then subtracting this value from the weight value. The architecture of a BPNN is also shown in Figure 4, and the direction of propagation is shown in the green dotted line. On the other hand, RNN is more complex than FFNN and BPNN. Here, the RNN’s every node acts as a memory cell and continues the operations computation [4]. The RNN saves the output of processing nodes and feeds them back into the network, and hence, they do not pass the information in one direction only (refer to the blue dotted line in Figure 4). If the network’s prediction is incorrect, the system self-learns and continually works toward correcting the forecast during backpropagation.
Researchers have utilized ANNs in numerous applications to predict or forecast various chaotic systems’ behavior. For instance, in [25], the researchers developed the complex weighted neural network method for high-resolution adaptive bearing prediction. It is observed that this concept is especially effective in circumstances where the hermit matrix progressively changes over time due to adaptive tracking. Jae-Gyan Choi et al. proposed the application of ANN in power systems for predicting the one-day-ahead daily peak load based on chaotic time series data using absolute error as a performance measure [26]. It is to be noted that the proposed technique can also be used for other forecasting applications, such as predicting the special days, hourly load, temperature, etc. In [27], the researchers have presented the RNN model for Mackey–Glass chaotic time series. The proposed model’s experimental results are more practicable and effective in making short-term predictions for chaotic time series than the multi-dimension embedding phase space method.
Guichao Yang et al. developed a multilayer neural network adaptive control algorithm for disturbance compensation in nonlinear systems. The work remarks that this developed algorithm can also be used simultaneously for nonlinear systems with mismatched uncertainties. Additionally, an extended state observer was employed to estimate the exogenous disturbance and predict the system’s state [179]. In extension, the authors presented the integration of a full-state feedback control algorithm, adaptive neural network, and extended state observer to handle the unknown nonlinear dynamics and external disturbances. In addition, the output feedback control algorithm was combined with an adaptive neural network, extended state observer, and nonlinear disturbance observer to estimate the unknown nonlinear dynamics, unmeasured states, and external disturbances [180]. In both works, a double-rod hydraulic servo system was chosen to validate the two control schemes’ high-performance control effect. The authors also introduced a neuroadaptive learning method for disturbance rejection in constrained nonlinear systems. Moreover, the neural network adaptive control and the extended state observer to estimate endogenous uncertainties and external disturbances in real time and correct them feed-forwardly were presented in [181]. Further, the filtering problems and nonlinearity of the input were accounted for by adding an auxiliary system. Finally, the overall closed-loop stability was precisely ensured, and the accomplished control performance was validated by real-time nonlinear systems application results.
Another forecasting method known as delay-based ANN for predicting the turbulent flow temporal signals was proposed in [32]. These signals are obtained from a hot wire anemometer at a single point inside the cylinder to detect coherent structures. In [34], the authors presented the RBFNN model for forecasting the time series of the logistic map, Henon map, Mackey–Glass, and Duffing’s systems. In [36], the researchers developed the recurrent predictor neural network model for predicting the annual and monthly sunspot time series. The experimental results of the proposed model are better than the Kalman filter and universal learning network in terms of accuracy and RMSE. The authors of [39] developed the KIII-chaotic neural network for forecasting the multistep time series data on a benchmark system. In [40], the researchers presented the RNN for predicting the electricity price of the power system. The work highlights that this approach is equally relevant to Chinese electrical market data. In [43], the authors presented the RBFNN model for forecasting the Henon map, Lorenz map, four real-time series discharge data, and sea-surface temperature anomaly data collected from various rivers. The work remarks that this presented model can also be used for geological time series. In [48], the authors raised the time delay neural network method for predicting the future behavior of the solar activity. In [51], the researchers demonstrated the BPNN to forecast the multistep nonlinear time series of the diode resonator circuit. From the presented work, it is to be noted that the approach can also be used in other chaotic time series.
Qian-Li Ma et al. presented the evolving RNN model for the Lorenz series, logistic, Mackey–Glass, and real-world sunspots series [53]. The experimental results of the proposed model showed to be better than the boosted RNN. Bao Rong Chang and Hsiu Fen Tsai proposed an optimal BPNN model for time series of signal deviation in the stock market [55]. The proposed method is based on SVM and AR models. The experimental results of the proposed model showed better performance than the ARMA, RBFNN, and other models in terms of MAD. In [57], the authors developed the NARX neural network model for empirically predicting chaotic laser, variable bit rate, and video traffic time series of real-world datasets. The simulation results of the developed model reliably performed better than the Elman architectures. Further, the work highlights that this model can also be used for electric load forecasting, financial time series, and signal processing tasks.
Yagang Zhang et al. developed an ANN model for predicting the stochastic generating sequences in a chaotic unimodal dynamical system [58]. It is observed that the presented strategy can also be further applicable for applications such as DNA-based groupings, protein structure arrangement, and financial market time series. The authors of [61] proposed an ensemble ANN model for forecasting the turning points in the Mackey–Glass system. An expectation–maximization parameter learning algorithm for the developed model was used for probability threshold prediction during the out-of-sample validation. The experimental result from the system proves the viability of the proposed technique and shows better results than the ANN model alone. In [68], the work presented the RBFNN model to predict the Shanghai Composite index that is chaotic according to the phase diagram analysis. The proposed technique’s experimental results are better than the BPNN. In [78], the hybrid Elman–NARX neural network model is presented to chaotic systems, such as Mackey–Glass, Lorenz equations, and the real-life sunspot time series, for predicting the chaotic time series. The proposed method has performed more effectively and accurately than the AR model, GA, and fuzzy methods. Gao Shuang et al. presented the rough set neural network model for long-term wind power prediction [80]. The experimental results show that the rough set method has the least NMAE compared to the other three methods, the chaos neural network model, persistence model, and rough set neural network model. Another forecasting method for gas emission rate prediction, known as the global method based on the BPNN, was proposed in [83]. The proposed model showed good accuracy and stability predictions than the first-order weighted local prediction method. In [84], the researchers developed the chaotic RBFNN method for predicting the power systems’ short-term load. The results of the proposed method showed promising results better than conventional RBFNN. In [12], a chaos RBFNN method was presented for forecasting the gasoline engine intake flow’s transient condition. The simulation results showed more accuracy compared to conventional RBFNN.
In [101], the application of ANN in a chaotic dynamical system for forecasting embedded dimension and robust location was presented. In [103], the selecting and combining models with the SOM neural network model for long-term chaotic time series prediction from the Mackey–Glass equation, NN5 tournament, AR model, and sine function were presented. It is to be noted that this model can be used to assess the selected outcomes of the modeling techniques by considering the best-predicted SMAPE. In [21], the MLP model for enhancing the accuracy of a flood prediction through machine learning and chaos theory was presented. The experimental results of the proposed method performed better than the Elman-RNN method. It is to be noted that this concept is also applicable to sensors, allowing for more individual action in severe conditions. Further, the idea can also lower the system’s total operational costs and ensure next-generation power grids’ effective and reliable functioning. In [182], the authors developed a hybrid machine learning technique for forecasting the time series of NN5 using the nearest trajectory model, one-year-cycle model, and neural network. In [128], the self-adaptive chaotic BPNN algorithm was proposed based on Chebyshev’s chaotic map for predicting the electrical power system’s load. The presented algorithm results showed better global optimization performance than conventional BPNN, RBFNN, and Elman networks. The work highlights that the chaotic neural network regression using the probability density forecast method can predict the electricity demand. In [131], the deep CNN model was proposed for forecasting Lyapunov exponents from observed time series in discrete dynamical systems. In [157], the authors presented the RNN-based LSTM model to predict the mutation rate in a human body affected by COVID-19. The proposed approach can be extended further by inserting and deleting mutation rates in the model.
The authors of [162] presented the Deep CNN model using meteorological data to forecast flight delays. The results of Deep CNN showed to be better than the CNN, which is proven in terms of weight gradient error and hidden layer error. The authors of [168] presented the LSTM neural network model to forecast the delay time in a chaotic optical system and compared the model with the delayed mutual information method and autocorrelation function method. It is worth noting that the proposed model can also enhance the security and maturity of optical chaos secure communications. In [183], the authors presented a LSTM-based forecasting model by integrating ensemble and reinforcement learning techniques. Further, an adaptive gradient algorithm was used to train the network and validated on the Lorenz, Duffing, and Rössler systems. The authors of [184] developed a FFNN-based prediction model to estimate the change in future state values of a Rössler system. In [169], the authors presented a gate recurrent unit-based Deep RNN model to forecast time series of three chaotic systems, (i) Lorenz, (ii) Rabinovich–Fabrikant, and (iii) Rössler, which showed better performance than the LSTM-based Deep RNN model. This model can also be used for real-time applications to predict the hyper-turbulent frameworks to control the turbulence or synchronize the framework model.

3.2. Fuzzy with ANN-Based Forecasting Approaches

FNN is a hybrid network developed using ANN’s learning ability and fuzzy logic’s noise handling capability. The architecture of the FNN is also shown in Figure 4. The figure shows that the network has four layers: the input layer, fuzzification layer, inference layer, and defuzzification layer (refer to the yellow dotted lines in Figure 4). FNN uses two approaches, namely (i) Mamdani and (ii) Takagi and Sugeno. Fuzzy logic is represented using the neural network’s structure and trained using either a BP or an optimization algorithm. The FNN is implemented in the following three ways:
  • Real inputs with fuzzy weights;
  • Fuzzy inputs with real weights;
  • Fuzzy inputs and fuzzy weights.
In [50], the authors developed the self-organizing Takagi and Sugeno-type FNN model for predicting the short-term traffic flow. The experimental results of the developed model showed to be feasible and more effective than RBFNN. In [54], the researchers developed the distributed chaotic fuzzy RBFNN method applied to fault section estimation in the distribution network. The simulation results of the developed strategy achieved better efficiency, learning ability, fault-tolerance, and low convergence rates than the BPNN model. On the other hand, in [66], the work presented a subtractive clustering-based FNN for forecasting the traffic flow and used the GA for deciding the clustering radius. Ding Guan-bin and Ding Jia-Feng introduced an adaptive neural network-based fuzzy inference system for predicting the monthly average flow in a hydrological station, which showed better results than the AR model [67]. The authors of [69] developed the fuzzy descriptor model integrated with singular spectrum analysis for predicting the various time series, including Mackey–Glass, Lorenz, Darwin sea level pressure, and the disturbance storm time index. The presented model results showed to be better than the MLP and RBFNN models. Another forecasting method known as the FNN model based on chaos theory for predicting the hydraulic pumps’ vibration signal was proposed in [75]. It is to be noted that this model can also be used to improve prediction accuracy by readjusting the minimal embedding dimension optimally. The dynamic recurrent FNN model used to predict the power systems’ short-term load was developed in [76]. It was proved that the developed model’s convergence rate and forecasting accuracy are enhanced compared to the conventional FNN model. In [97], the researchers presented the interval type-2 fuzzy cerebellar model articulation controller for forecasting the Henon system of chaotic time series and the chaos synchronization of the Duffing–Holmes system. The proposed model of simulation results showed to be better than the FNN and interval type-2 FNN.
In [104], the researchers proposed the saliency back-emf-based wavelet FNN model for a torque observer, using a new maximum torque per ampere control for forecasting the speed of a sensorless interior permanent magnet synchronous motor. In [110], the authors presented the embedding theorem-repetitive fuzzy method for predicting the time series data of Mackey–Glass, Lorenz, and sunspot numbers. The proposed model’s experimental results provided better forecasting than the simple fuzzy, adaptive neuro-fuzzy inference and other models in terms of error indices. Qinghai Li and Rui-Chang Lin presented the self-constructing recurrent FNN model for forecasting the logistic and Henon time series [112]. The proposed model had a worthier performance in convergence rate and forecasting accuracy than the self-constructing FNN. The authors of [117] presented the interactively recurrent fuzzy functions model for predicting the time series data of Lorenz, Mackey–Glass, and real-time lung sound signal modeling. The benchmark and real-time models’ results showed to be better than the recurrent networks, such as fuzzy WNN, self-evolving FNN, ESN, and LS. Luo Chao and Wang Haiyue presented the application of
  • Generalized zonary time-variant fuzzy information granule;
  • LSTM mechanism with FNN model.
For Zurich monthly sunspot numbers, Mackey–Glass time series, and daily maximum temperatures in Melbourne were used for predicting the granules [141]. The results of the proposed methods showed better performance than the AR and nonlinear AR neural network models. In [176], the researchers presented the adaptive RBFNN model for forecasting the online vehicle velocity, showing better prediction accuracy and computational efficiency than the LSTM, NARX, and deep neural network models.

3.3. Optimization Algorithms with ANN-Based Forecasting Approaches

The authors of [29] developed the temporal difference GA-based reinforcement learning neural network model to predict and control two chaotic systems, i.e., the Henon map and the logistic map. The advantage of the proposed concept is that it can apply directly to control chaotic physical systems in real-world models. Mohammad Farzad et al. proposed the GA for forecasting the Mackey–Glass chaotic time series, and the model showed better performance than the ANN and polynomials methods [47]. The proposed model may also be used to forecast any other chaotic systems. In [71], a modified bee evolution using a PSO-based chaotic neural network model was presented to predict the load in the power system. The proposed model’s simulation results showed better outcomes than the PSO algorithm used to develop the power system’s proper planning and has good prospects. In [96], a hybrid approach using the chaotic self-adaptive PSO algorithm and BPNN was presented to forecast the polymers’ gas solubility. The proposed model is reliable, accurate, and practicable for analyzing and designing polymer processing technology, compared to PSO-tuned BPNN models. It is to be noted that the proposed approach can also be further extended to tackle actual difficulties. The chaotic PSO tuned ANN model was presented in [102] to forecast air quality by predicting the particulate concentration. It is to be noted that this model can also be used to prove the meteorological condition of wind speed, which has a significant effect at urban intersections for specific matter concentrations. In [107], the researchers developed the improved GA for forecasting the synchronous parameters of chaotic time series to achieve higher accuracy and efficiency than GA alone.
The authors of [115] proposed the modified BPNN based on chaotically optimized GA and simulated annealing algorithms to forecast electrical energy demand in a smart grid. It is to be noted that this concept can also be relevant to lowering the system’s total operational costs and ensuring the effective and reliable functioning of next-generation power grids. Akhmad Faqih et al. developed the extreme learning mechanism using RBFNN and SOM models to predict the multistep ahead time series of Lorenz’s chaotic system [133]. It is to be noted that this proposed model can also combine with several behaviors to provide the best behavior. In [135], the researcher presented the GA and LS-based SVM method to control fractional-order systems, which achieved better effectiveness and feasibility than the conventional LS-based SVM. The authors of [139] proposed the principal component analysis using the chaotic immune PSO tuned GRNN for forecasting the corrosion of circulating cooling water in a petrochemical enterprise. The approach achieved better forecasting accuracy and convergence speed than the traditional PSO-tuned GRNN model. The advantage of the proposed model is that it can also be employed to forecast other nonlinear systems. In [147], the authors proposed the chaotic PSO algorithm for predicting the mobile location and achieved better location accuracy and faster convergence rate than such algorithms as those of Chan, Taylor, and PSO. Ji Jin et al. developed the fractal dimension-based EMD method and GA tuned BPNN model for predicting the wind speed in wind farms by considering the atmospheric motions’ fractal feature [152]. The proposed models showed better performance than LSTM, GA tuned BPNN, and ensemble EMD-GA-BPNN. It is to be observed that this model can also require further study to optimize the computational time. It is also necessary to analyze the model on various time scales to decide the proposed models’ suitability to wind speed series on any timescale. Happy Aprillia et al. proposed the SSA tuned CNN for predicting the short-term power of PV systems [158]. The presented algorithm’s results showed better accuracy than the SSA tuned SVM and LSTM methods. Further, the work highlights that this proposed model can also address uncertainty, particularly for wet weather, heavy overcast weather, peak time, and forecasting on typhoon days. Shuzhi Gao et al. developed the soft sensor model using the CBAS algorithm and Elman neural networks to forecast the conversion rate of vinyl chloride monomer [171]. The developed model’s performance can be extended by utilizing the deep neural network approaches.

3.4. Wavelet NN-Based Forecasting Approaches

The merits of wavelet and neural networks are hybridized to form a new WNN to achieve better forecasting ability. WNNs have been used with great success in a wide range of applications. In some applications, it was proven that if the combination of a neural network and wavelet is used, the proposed model’s efficiency is increased. The WNN architecture also follows the same fashion as the network shown in Figure 4. However, in the hidden layer, wavelet basis functions are used as activation functions instead of the conventional function of the FFNN.
Antonis K. Alexandridis et al. proposed machine learning algorithms, namely wavelet network and genetic programming, for forecasting the average temperature precisely when it comes to weather derivative pricing, compared to SVM and RBF [185]. Wei Wu et al. developed a WNN model for electricity-based chaotic time series data to predict the spot market prices [38]. In [45], the researchers proposed the WNN model and compared it with the BPNN model for single-step forecasting of Lorenz and Mackey–Glass chaotic time series. It is to be noted that this approach can be extended further to be used for real-world chaotic data. In [74,81], the authors proposed the forecasting models for wind farms. The wavelet decomposition method and ITSM in [74] showed an improved accuracy compared to ANN in predicting wind speed and power. Similarly, the developed hybrid algorithm using wavelet transform, chaotic theory, and grey model in [81] showed better prediction than the direct prediction method. The models in [74,81] can be further optimized and applied in various countries’ wind farms, such as the Dongtai wind farm in China. Bo Zhou and Aiguo Shi presented the phase space reconstruction-based WNN method to predict Henon and Lorenz’s chaotic time series [95]. The significant benefit of this proposed method over a WNN is the improvement in SMAPE. It is to be noted that this concept can also help optimize the process parameters and the execution time during the simulation. Tian Zhongda et al. presented the wavelet transform and multiple model fusion for forecasting the Lorenz and Mackey–Glass time series and achieved more effective performance in terms of SMAPE [120]. The models can be applied to real-world chaotic systems, such as geomagnetic series, network traffic series, etc. In [130], the ANN-discrete wavelet transform method was presented for forecasting the photovoltaic system’s power based on chaos theory. The significant benefit of this method over the ANN and ANN-phase space reconstruction is the improvement in the Theil index.

3.5. Other Approaches

The authors of [56] developed the generalized EKF for forecasting the Lorenz time series with various Bernoulli distribution probabilities, which achieved an acceptable prediction precision and good robustness. Xue-dong Wu et al. proposed the GPF and compared it with UKF and EKF to forecast the Mackey–Glass time series [73]. In [106], the researchers proposed the EKF-based MPSV method to estimate the transmitted signal in power line communications and confirmed the better efficiency than the inverse filter-based MPSV method. However, the real-time validation of the proposed approach is the research gap. In [129], the authors developed the equivalent model using EKF to predict the state of charge in power Li-ion batteries. Yijun Xu et al. proposed the polynomial chaos-based Kalman filter to predict the nonlinear system dynamics [146]. In [160], the authors presented the UKF for forecasting the parameters of the gray-box model for dynamic EEG system modeling and achieved the lowest RMSE compared to the particle filter and EKF.

4. Forecasting of Chaotic Time Series in Various Applications

As mentioned in Table 1, various parameters have been forecast in multiple applications using the machine learning-based approaches detailed in Section 3. The list of these forecasting parameters categorized into the different applications is given in Figure 5. The detailed description of these forecasting parameters using various approaches in various fields is explained underneath.

4.1. Power and Energy

This section describes power and energy forecasting techniques, using various chaotic time series approaches applied in wind farms, solar, photovoltaic systems, etc.

4.1.1. Wind Farms

Many applications of wind power and speed forecasting approaches have been developed based on chaotic characteristics or chaotic time series and applied on various wind farms. For instance, the statistical type of forecasting approaches for predicting wind power, speed, and load for short-term and long-term wind power, speed, and load prediction in Beijing in China are as follows:
  • ITSM with wavelet decomposition method [74];
  • SVM [16];
  • Rough set neural network [80];
  • BFA tuned double-reservoir ESN [166].
The approaches, as mentioned earlier, showed good short-term and long-term performance, but the computational complexity is high to complete the task. Similarly, many other works have been attempted to predict weather conditions for wind farms whose operations are more complex.
Various researchers have proposed hybrid prediction methods to enhance accuracy. These approaches were made by integrating the following:
  • Wavelet transforms with chaotic time series and grey model [81];
  • Hilbert–Huang transforms with Hurst analysis [108];
  • Hybrid neuro evolutionary [175].
The hybrid approaches mentioned above with multistep chaotic characteristics were validated for short-term forecasting of wind power at the Dongtai wind farm and Hebei province in the east of China. The work highlights that EMD-based combined forecasting methods can improve short-term forecasting accuracy based on their characteristics. The surrogate data technique and spectral analysis methods are applied to forecast wind wave height, period, and direction for three-hourly chaotic time series from three stations in the Caspian’s southern, central, and northern parts of the sea [109]. The hybrid approach developed using ensemble EMD-sample entropy and the full parameters continued fraction model were developed for predicting the wind power of farm location at Xinjiang, China [119]. Moreover, the Markov chain switching regime model developed in [144] used hourly, short-term, and long-term chaotic time series data for predicting wind speed and direction of the farm located at Bonneville Power Administration control area in the Northwest USA. It is to be noted that these proposed approaches can also be used for the proper planning and scheduling of wind power. The self-adaptive and artificial intelligence type forecasting techniques for predicting the wind speed are as follows:
  • Fractal dimension-Lorenz stenflo-Ensemble EMD;
  • GA tuned BPNN model;
  • Empirical dynamical model [152,165].
For short-term prediction of wind speed considers the atmospheric motion and fractal feature at Abbotsford in Canada, and Kansas and Missouri in the USA. It is to be noted that better results can be generated using exogenous variables in the ANN approach.

4.1.2. Solar and Photovoltaic Systems

The recurrent predictor neural network model presented in [36] is based on an extended algorithm of self-adaptive BP through a time learning algorithm for predicting the annual sunspot time series in Skylab. Similarly, the time-delay neural network model [48] and the multi-layered neural network-based co-evolutionary algorithm [52] are used for predicting the annual sunspot time series of the space laboratory launched by the USA in 1973 and the sunspot index data center in Belgium, respectively. The ANN-based discrete transform using chaos theory [130], ensemble EMD based on optimized chaotic phase space reconstruction [155], SSA tuned CNN [158], and CGO [172] are used for predicting the power, voltage, and current of PV system in Beni Mellal, Morocco, St Lucia campus PV station, Australia. The k-fold cross-validation with GRNN reported in [134] is used for predicting the accuracy of sunspot under different embedding dimensions for phase space reconstruction of chaotic time series according to the Takens theorem in the Solar Influences Data Analysis Center, Belgium.

4.1.3. Other Power Systems

Under certain circumstances, chaos in an electrical power system can show abnormal oscillations, threatening the electrical grid’s reliability and stability. Because of the nonlinearities of electricity networks, chaos theory is a high priority. Hence, the application of chaos theory and several forecasting approaches to improve the accuracy and reliability of load forecasting. The proposed forecasting approaches for predicting the electrical daily peak load of the power systems, such as South Korea Electric Power Corporation, Daqing oilfield company in China, New South Wales in Australia, and North China city, are as follows:
  • ANN [26];
  • Bee evolution modifying PSO tuned chaotic neural network [71];
  • Adding-weighted LLE [72];
  • Dynamic recurrent FNN [76];
  • Chaotic RBFNN [84];
  • Chaotic local weighted linear forecast algorithm based on angle cosine [88].
The self-adaptive chaotic BPNN and parallel chaos algorithm reported in [128], and [118], respectively, are used for forecasting the short-term electrical power load in the China network. The limitations of the proposed approaches are eliminated with the application of hybridized chaotic RBFNN-quantile regression model for forecasting the weather, seasons, wind power, and electricity price. The hybrid forecasting approaches developed for predicting the dynamic characteristics of electricity are the wavelet decomposition methodology [120], variational mode decomposition-maximum relevance minimum redundancy based BPNN-LS-SVM [156], and short- and medium-term load in the Xi’an power grid corporation, China.
Short-term electricity price forecasting has become crucial in the power markets, as it allows for the foundation for market participants’ profit maximization. The proposed methods for forecasting the short-term electricity spot market prices and the marginal price at the New England and California electricity markets in the USA are as follows:
  • Nonlinear auto-correlated chaotic model-based WNN [38];
  • RNN [40];
  • LS-SVM algorithm [60];
  • Add-weighted one-rank multi-steps prediction model [63].
The generation companies can decide on scheduling generators and provide high-quality power services to customers. Thus, the validation algorithm presented in [93] is based on the voltage sensor applied to a DC zonal shipboard electric power system, using decentralized polynomial chaos theory for the sensor validation decentralized state prediction. In addition, it is to be reported that the presented conventional algorithms were improved using artificial intelligence techniques. The independent component analysis method reported in [94] for predicting the amplitude and frequency of highly chaotic distorted power system signals is presented based on duffing oscillator solutions. The proposed approach can be used for the real-time control and measurement of the fundamental frequency of a power system while focusing mainly on chaotic disturbances. The maximum velocity criterion method, sinusoidal wave frequency modulation, and chaotic control algorithm are for forecasting the chaos and suppressing the predicted chaos to increase the security for cyber–physical power systems [112]. The modified BPNN, chaos-search GA, and SA algorithms are applied to predict a smart grid’s short-term electrical energy demand in New South Wales, Australian grid [115]. The proposed approach can also lower the system’s total operational costs and ensure the next-generation power grids’ effective and reliable functioning. The polynomial chaos expansion-based Langevin Markov chain Monte Carlo and multi fidelity-surrogate-based Bayesian inference via adaptive importance sampling predict decentralized dynamic parameters, such as inertia, exciter gains, damping ratio, and the droop of the synchronous generator in New England, USA [138,151].

4.2. Hydrological Systems

The RBFNN model is developed to estimate the Mekong River’s nonlinear hydrological time series in Thailand and Laos, the Chao Phraya River in Thailand, and sea-surface temperature anomaly data [43]. In addition, the presented approach can also be applied to other geological time series. In [67], an adaptive fuzzy inference-based neural network model is developed to predict the medium- and long-term hydrological residual time series. The data are collected from the Guantai hydrological station, Zhang River, China. An empirical, statistical, and chaotic nonlinear dynamic model in [19] was applied to forecast the stream water temperature from the available solar radiation and air temperature in the Lake Tahoe basin, California, Nevada, USA. The chaotic FNN for predicting the hydraulic pump’s vibration signal was presented in [75]. It is to be reported that the proposed approach can be extended further to improve prediction accuracy by readjusting the minimal embedding dimension optimally. The coupled quantity–pattern similarity model reported in [18] predicts the monthly precipitation of hydrological systems in the Danjiangkou reservoir basin, China. The proposed approach can also be applied to time series with various lead time scales.

4.3. Communication Signals and Systems

The complex weighted neural network algorithm in [25] solves the principal component analysis problem and high-resolution adaptive bearing prediction. The proposed approach is especially effective in circumstances where the hermit matrix progressively changes over time due to adaptive tracking. The BPNN tuned SVM grey model in [55] is used for forecasting the signal deviation time series. The anchor selection method is based on polynomial chaos expansions [86] for angle-of-arrival prediction-based positioning systems. The chaos algorithm in [92] is proposed for forecasting the radio wave propagation in the ionosphere. The proposed algorithm can also forecast a set of radio transmission signals at a fading amplitude time series location. The phase space reconstruction-LS-SVM in [98] is developed to predict FM radio’s band occupancy rate in German Rohde, Schwarz company, and fixed radio monitoring station of Xihua University, USA. The proposed approach can be extended further to improve multistep time series prediction. The minimum phase-space volume-EKF equalization method presented in [106] is for forecasting the chaos in power line communications. The LLE, Higuchi’s fractal dimension, and sample entropy techniques are used for predicting the fractals, chaos, and parametric entropy features of surface electromyography signals during dynamic contraction of biceps muscles under a varying load [127]. The proposed process can also be helpful in physiotherapy and athletic biomechanics for testing muscular fitness. A deterministic chaotic sequences method is developed to forecast quadrature baseband signals and orthogonal frequency division multiplexing-based cognitive radio channel [137]. The proposed approach can also be applicable to bit error rate performance, which is projected to improve if an appropriate power management method is used.

4.4. Oil and Gas

The global prediction method uses a BPNN model for forecasting the gas emission rate in the Hegang Nanshan mine located in China [83]. The proposed model showed better step, accuracy, and stability predictions. The improved Duffing oscillator chaotic traffic prediction model in [85] was developed for coal gas’ traffic flow prediction for a coal mine. The proposed approach can also increase signal detection accuracy. The chaos RBFNN method in [12] predicts the intake airflow of the gasoline engine. The coal mine ventilation systems’ management technology reported in [17] can predict the gas concentration in Jining, Shandong, China. As a result, the system can provide reliable assurance for mine safety production.

4.5. Other Systems

The multistep time series prediction in diode resonator circuits is made by integrating the nonlinear signal prediction method with a BPNN [51]. The proposed approach can be extended further to be used in other chaotic time series. The integration of nonlinear time series analysis and backpropagation MLP for multistep nonlinear time series forecasting of chaotic diode resonator circuits was reported in [59]. The distributed chaotic fuzzy RBFNN is exploited for distributed network fault section prediction [54]. The global prediction of chaos method forecasts the chaotic instantaneous generator output power in Liaoning province in China [20]. The chaotic adding-weight dynamic local predict model predicts the pseudo-random number generator of the initial sequence number in the transmission control protocol stack [62]. The chaos-based Rivest Shamir Adleman algorithm and chaos-based random number generator forecast the security vulnerabilities of the cryptosystem [142].

5. Performance Measures

This section discusses the various performance measures used for chaotic time series forecasting approaches. According to the literature review summary in Table 1, it can be concluded that there are many approaches for chaotic time series forecasting. However, it is challenging to choose one proposed method that performs better based on the performance measures. Table 1 also shows that the researchers have evaluated the performance of the forecasting approach using various statistical errors. The different classifications of statistical performance measures used for chaotic time series forecasting are mean, relative, percentage, prediction, and coefficients. The classification and its subcategories are shown in Figure 6. The formula for computing these performance measures is demonstrated in Figure 7. In Figure 7, n denotes the number of samples, and Y a , i and Y p , i are the actual and predicted outputs by the chaotic time series forecasting model. Further, Y ¯ a , and Y ¯ p are the averages of Y a , i and Y p , i .
As shown in Figure 6 and Figure 7, most of the performance measures used for chaotic time series forecasting are mean errors. Further, the review summary in Table 1 shows that MSE and its variants are the most widely used performance measures for chaotic time series forecasting. The MSE and its variants measure the error between Y a and Y p , and the closest value to zero indicates a better estimation of the forecasting approach [186,187]. After mean errors, the percentage errors are the second most used performance measure for chaotic time series forecasting. The percentage errors measure the percentage error between Y a and Y p . The closer values of percentage error to zero also indicate a better estimation of the forecasting approach. On the other hand, the coefficient of determination R 2 is most widely used to indicate the forecasting approach’s predictive ability in fitting the actual data Y a [188,189]. Thus, the values of R 2 range from zero to one, and the value equal to 1.0 indicates a perfect fit. The summary in Table 1 also shows that most of the researchers used a combination of different performance measures for evaluating the forecasting approach. The combinations are MAE, MAPE, and RMSE; MSE, MAPE, and RMSE; MAE and RMSE; MSE and RMSE; R 2 and MSE; R and MSE, etc.

6. Conclusions

This article reviewed various approaches for chaotic time series forecasting based on machine learning in multiple areas, such as wind farms, PV systems, hydrological systems, communication signals and systems, oil and gas, and other systems. At the beginning of this paper, the chaotic system/time series and the importance of chaos forecasting were introduced. Next, the various machine learning-based chaotic time series forecasting approaches were presented. These approaches use WNN, FNN, CNN, LSTM, and Markov chain models. Then, a review of the prediction of various parameters in multiple applications using machine learning-based techniques is presented. This review concludes that traditional prediction methods can hardly obtain satisfactory results. Hence, many chaotic time series prediction methods were developed using machine learning-based approaches, which enhanced their efficiency and accuracy.

6.1. Findings

This review summarizes the findings of various approaches developed for multiple applications as follows:
  • The wavelet decomposition method predicted wind speed and power accurately and effectively using improved time series, chaotic time series, and grey models [74,81]. The false nearest neighbor analysis method forecast the chaotic behavior of the wind–wave characteristics, including wave period and height [109].
  • Hilbert–Huang transform and Hurst analysis is a proper choice to forecast the multi-scale chaotic characteristics of wind power [108]. In contrast, ensemble EMD and full parameters continued fraction is appropriate for predicting wind power’s nonlinear chaotic time series [119].
  • The empirical dynamic model presented in [165] forecast the wind speed for various height levels. At the same time, the fractal dimensional-based self-adaptive model for wind speed predicted atmospheric motion and fractal features [152].
  • The approaches such as the ordinary least square method [28], recurrent predictor neural network [36], hybrid Elman–NARX neural network [78], and embedding theorem-repetitive fuzzy [21] forecast the sunspot number (chaotic time series) effectively. In all these cases, the sunspot data were collected from the world data center for Belgium’s sunspot index.
  • The combination of chaos theory and techniques, such as ensemble EMD and CNN-SSA, effectively forecast the PV system’s output power under certain conditions, such as rainy, heavy cloudy, lightly cloudy, and sunny conditions [155,158]. The data were collected from the St Lucia campus PV station, Australia, in all these cases.
  • The integration of the BPNN with GA, SA algorithms [115], parallel chaos [118], wavelet decomposition-based methods [120,157] was successfully used to forecast the deregulated power system’s short-term electrical energy demand. These methods help in proper economic power dispatching with an enhanced demand response that assists in efficient spot price-fixing in the deregulated power market.
  • The regression analysis models using ANN and chaotic nonlinear dynamic [73] and coupled quantity-pattern similarity [18] were validated to predict the stream water temperature and monthly precipitation.
  • The minimum phase space-based EKF method was used to forecast the blind equalization in power line communication systems to overcome channel noise [106].
  • The response surface-based Bayesian inference [149] and PCE-based hybrid MCMC [163] approaches were used to predict the generator’s dynamic parameters, such as inertia, exciter gains, damping ratio, and droop.
  • The independent component analysis method in [94] adequately estimated the amplitude and frequency of power systems’ highly distorted signals to avoid the ferroresonance effect.
  • The Markov chain switching regime model enhanced the precision accuracy and is helpful for wind power forecasting during scheduling and planning [144].

6.2. Future Directions

This comprehensive review helped open up new scopes in the field of chaotic time series forecasting approaches in various applications and is highlighted underneath.
  • Chaotic time series analysis and SVM can estimate short-term wind speeds while considering weather conditions and more complex scenarios of wind farm operations [16].
  • To the dispersed power resource system, the wind power generation unit can be connected to the grid of this system through high-quality forecasting of the parameters using the Jacobian matrix estimate method and weather data optimal points using deterministic chaos [104].
  • EMD-based forecasting approaches can increase short-term wind power prediction accuracy based on their behavior characteristics. Furthermore, the relationship between different scale subsequences and numerical weather forecasting can improve the accuracy of this short-term wind power forecasting [108].
  • The hybrid neuro evolutionary approach, i.e., adaptive variational mode decomposition-AOA-LSTM proposed for wind farms, has employed multiple outlier identification methods with optimization and decomposition procedures to improve forecasting outcomes [175]. This method can also be adaptable to other geographies.
  • The independent component analysis method can be extended for real-time monitoring and controlling the power system’s fundamental frequency with an appropriate time delay between observed data frames [94].
  • The precision accuracy of the response surface-based Bayesian inference method proposed for the power systems to predict the dynamic parameters has to be improved when there is a substantial outrageous deviation in the boundaries [149].
  • The coupled quantity pattern similarity model proposed for the prediction of monthly precipitation can also be applied to the time series with different lead time scales [18].
  • The hybrid algorithms proposed using CNN and wavelet transforms for predicting the chaotic time series of Chen, Lorenz, Mackey–Glass, and sunspot numbers can also be used for real-time series, such as geomagnetic, network traffic, and weather systems [13,170].
  • The forecasting accuracy of an online vehicle velocity prediction approach proposed using adaptive RBFNN can be enhanced using additional data, such as driving time, climate, gas, and brake pedals [176].

Author Contributions

Conceptualization, B.R. and K.B.; formal analysis, B.R.; investigation, B.R.; resources, K.B.; data curation, K.B.; writing—original draft preparation, B.R.; writing—review and editing, K.B.; visualization, K.B.; supervision, K.B. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:
ACFAuto correlation function
ANNArtificial neural networks
AOAArithmetic optimization algorithm
APEAbsolute percentage error
APSKAmplitude phase shift keying
ARAutoregressive
AREAverage relative error
ARIMAAutoregressive integrated moving average
ARMAAutoregressive moving average
ARMSEAverage root mean square error
BFABacterial foraging algorithm
BPBackpropagation
BPNNBackpropagation neural network
CBAMConvolutional block attention module
CBASChaos beetle antennae search algorithm
CCOCluster chaotic optimization
CGOChaos game optimization
CMSECumulative mean square error
CVRMSECoefficient of variance of the root mean square error
DCSKDifferential chaos shift keying
DMIDelayed mutual information
EKFExtended Kalman filter
EMDEmpirical mode decomposition
ESNEcho state network
FFNNFeed-forward neural network
GAGenetic algorithm
GPFGaussian particle filtering
GRNNGeneralized regression neural network
GWOGrey wolf optimization
HBOHoney bee optimization
HEAHybrid evolutionary adaptive
HFDHiguchi’s fractal dimension
IGWOImproved grey wolf optimizer
ITSMImproved time series method
LLELargest Lyapunov exponent
LLNFLocally linear neuro-fuzzy
LSLeast square
MADMean absolute deviation
MAEMean absolute error
MAPEMean absolute percentage error
MAREMean absolute relative error
MCMCMonte Carlo Markov chain
MLEMachine learning ensembles
MLPMultilayer perceptron
MMSEMinimum mean square error
MPSVMinimum phase space volume
MREMean relative error
MRFOManta ray foraging optimization
MRPEMaximal relative percentage error
MSDMean squared deviation
MSEMean squared error
MSLEMean squared logarithmic error
MSP d EMean squared prediction error
NARXNonlinear autoregressive exogenous model
NMAENormalized mean absolute error
NMAPENormalized mean absolute percentage error
NMSENormalized mean square error
NRMSENormalized root mean square error
NWPNumerical weather prediction
PCRPrincipal component regression
PCSPolynomial chaos surrogates
P d EPrediction error
PEPercentage error
PIDProportional–integral–derivative
PLSPartial least square
PREPercentage relative error
PSOParticle swarm optimization
PVPhotovoltaic
QAMQuadrature amplitude modulation
RCoefficient of correlation
R 2 Coefficient of determination
RBFRadial basis function
RBFNNRadial basis function neural network
RERelative error
RMSERoot mean squared error
RNNRecurrent neural network
RRRidge regression
RRMSERelative root mean squared error
SASimulated annealing
SMAPESymmetric mean absolute percentage error
SOMSelf-organizing map
SSASalp swarm algorithm
SVMSupport vector machine
TCNTemporal convolutional network
TLBOTeaching–learning-based optimization
TTLSTruncated total least squares
UKFUnscented Kalman filter
ULNUniversal learning network
YCOYield-constrained optimization

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Figure 1. Decade-wise research contributions to chaotic time series forecasting from 1992 to 2021.
Figure 1. Decade-wise research contributions to chaotic time series forecasting from 1992 to 2021.
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Figure 2. Locations of real-time data collected from various parts of the world.
Figure 2. Locations of real-time data collected from various parts of the world.
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Figure 3. Summary of multiple techniques developed using ANN, FNN, WNN, and optimization-based approaches for chaotic time series forecasting.
Figure 3. Summary of multiple techniques developed using ANN, FNN, WNN, and optimization-based approaches for chaotic time series forecasting.
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Figure 4. Architecture of various neural networks.
Figure 4. Architecture of various neural networks.
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Figure 5. List of forecasting parameters categorized into the different applications.
Figure 5. List of forecasting parameters categorized into the different applications.
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Figure 6. Classification of various performance measures used for chaotic time series forecasting.
Figure 6. Classification of various performance measures used for chaotic time series forecasting.
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Figure 7. Formula for computing the various performance measures.
Figure 7. Formula for computing the various performance measures.
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Table 1. Summary of works focused on chaos forecasting using machine-learning-based approaches.
Table 1. Summary of works focused on chaos forecasting using machine-learning-based approaches.
Ref., YearCountryJournal/ ConferenceForecasting ApproachApplicationForecasting Parameter(s)Comparison TechniquesPerformance MeasuresData
[25], 1992ChinaIEEE ConferenceComplex weighted neural networkMusic formulaArrival direction
[26], 1996South KoreaIEEE ConferenceANNPower systemDaily peak loadMAPESouth Korea electric power corporation
[27], 1998ChinaIEEE ConferenceEmbedding phase space using RNNMackey-Glass modelTime seriesMSE
[28], 1998NorwayPhysica D: Nonlinear PhenomenaOrdinary least square methodSunspot, R-R intervals of human ECG signalsTime seriesPCR, PLS, TTLS, RRNRMSE
[29], 1999ChinaIEEE Transactions on Neural NetworksTemporal difference GA based reinforcement learning neural networkHenon map, Logistic mapExternal reinforcement signalPrediction error
[30], 2000ChinaIEEE ConferenceNovel noise reductionChaotic interferenceFrequencyResidual error
[31], 2001AustraliaIEEE ConferenceStandard Gaussian approximationAsynchronous DS-CDMA systemsAccuracyImproved GA
[32], 2001SpainIEEE ConferenceANNHot wire anemometerTurbulent flow temporal signalsMSE
[33], 2002UKIEEE ConferenceGaussian processesHenon mapTime seriesSVMNMSEFar infrared-laser
[34], 2004IranChaos, Solitons and FractalsRBFNNLogistic map, Henon map, Mackey-Glass modelTime series-MSE, NMSE
[35], 2004CanadaIEEE Transactions on Biomedical EngineeringANNSilico modelOnset of state transitions
[36], 2004ChinaIEEE Transactions on Signal ProcessingRecurrent predictor neural networkSunspot numberTime seriesKalman filter, ULNRMSE, PE
[37], 2004ChinaChemical Engineering ScienceChaotic forecastingEvaporator with two-phase flowHeat-transfer coefficientARE
[38], 2004ChinaIEEE ConferenceWNNElectricitySpot market pricesMSE, APESouth china
[39], 2004ChinaIEEE ConferenceKIII-chaotic neural networkIJCNN CATS benchmark test dataTime seriesN-based methodMSEIJCNN’O4 CATS benchmark set
[40], 2005ChinaIEEE ConferenceRNNPower systemPriceMean and maximum percentage errorsNew England electricity market, USA 1
[41], 2005ChinaIEEE ConferenceSVMMarket priceExchange rateANNMSE
[42], 2005JapanIEEE Transactions on Circuits and SystemsMaster–slave synchronization schemeFitzHugh–Nagumo model, Chua’s oscillatorChaotic behaviorPrediction error
[43], 2006ItalyHydrological SciencesRBFsHenon map, Lorenz map, Sea-surface temperatureTime seriesCMSEMekong river in Thailand and Laos, Chao phraya river in Thailand
[44], 2006ChinaIEEE ConferenceSigmoid and wavelet hybrid transfer functionESNMemory capacityESN predictorNRMSE
[45], 2006MexicoIEEE ConferenceWNNLorenz system, Mackey–Glass modelTime seriesBPNNMSE
[46], 2006SpainPhysica D: Nonlinear PhenomenaDiscrete-time recursive updateLorenz systemOn-line parameterMaybhate’s technique, d’Anjou’s techniqueNMAE
[47], 2006IranIEEE ConferenceGAMackey-Glass modelTime seriesANNNMSE
[48], 2006CanadaIEEE ConferenceTime delay neural networkSolar systemNumber of dark spotsWeight elimination FFNN, Dynamical RNN, Hybrid clusteringNMSESkylab, Solar influences data analysis center, Belgium 2
[49], 2007South KoreaIEEE ConferenceTerminal sliding mode controllerDuffing, Lorenz systemsTracking errorClassical sliding mode controlMSE
[50], 2007ChinaIEEE ConferenceSelf-organizing Takagi and Sugeno-type FNNTraffic systemTraffic flowRBFNNRMSEZizhu Bridge in Beijing
[51], 2007GreeceIEEE ConferenceBPNNDiode resonator circuitsTime seriesRMSE
[52], 2007IranIEEE ConferenceCo-evolutionarySolar systemSunspot number time seriesAR, Threshold AR modelNMSESolar influences data analysis center, Belgium 1
[53], 2007ChinaIEEE ConferenceEvolving RNNLorenz, Logistic, Mackey–Glass, Real-world sun spots seriesTime seriesLLNF, Bidirectional RNNNMSE, RMSESolar influences data analysis center, Belgium 1
[54], 2008ChinaIEEE ConferenceDistributed chaotic fuzzy RBFNNDistribution networkFault sectionBPNN
[55], 2008ChinaExpert Systems with ApplicationsOptimal BPNNSignal deviationTime seriesGrey model, ARMA, RBFNNMAD, MAPE, MSE
[56], 2008ChinaIEEE ConferenceGeneralized EKFLorenz systemTime seriesMLP networkMSE
[57], 2008BrazilNeurocomputingNARX neural networkChaotic laser, Real-world video trafficTime seriesTime delay neural network, Elman RNNNMSEChaotic laser, Variable bit rate video traffic time series
[58], 2008ChinaIEEE ConferenceANNUnimodal surjective map systemGenerating sequencesPRE
[59], 2008GreeceEngineering Applications of Artificial IntelligenceNonlinear time series analysis, BP-MLPChaotic diode resonator circuitsTime seriesNMSE
[60], 2008ChinaIEEE ConferenceLS-SVMPower systemMarginal priceBPNNAPE, MAPECalifornia electricity market, USA
[61], 2008ChinaIEEE ConferenceEnsemble ANNMackey–Glass modelTurning pointsSingle ANN
[62], 2008ChinaIEEE ConferenceChaotic adding-weight dynamic local predict modelPseudo random number generatorISN valueScope error, Margin error
[63], 2008ChinaIEEE ConferenceAdd-weighted one-rank multi-steps predictionElectricityPriceMutual information, False neighbors methodsMaximum percentage error, Average error
[64], 2008ChinaIEEE ConferenceHybrid accelerating GARiver flow modelRoughness parameterStandard binary-encoded and real-valued accelerating GAAREYangtse river upstream flow, China
[65], 2008GreeceChaos, Solitons and FractalsNearest neighborSingle transistor chaotic circuitTime series cross
[66], 2008ChinaIEEE ConferenceSubtractive clustering based FNN modelingTraffic systemTraffic flowBPNN, FNNMAE, MAPE, MSE, MSP d E
[67], 2009ChinaIEEE ConferenceAdaptive neural network fuzzy inference systemHydrological stationsAverage monthly flowAR methodPREGuantai hydrological station of zhang river, China
[68], 2009ChinaIEEE ConferenceRBFNNShanghai composite indexEconomic time seriesBPNNMAPEShanghai composite index, China
[69], 2009IranNeural Computing and ApplicationsFuzzy descriptor singular spectral analysisMackey–Glass, Lorenz, Darwin sea level pressure, Disturbance storm modelsTime seriesMLP, LLNF, RBFNNNMSEDarwin sea level pressure in Australia, Solar influences data analysis center, Belgium, US national oceanic and atmospheric administration 1
[70], 2009IranChaos, Solitons and FractalsLevenberg–Marquardt learningMackey–Glass modelTime seriesMSE, NMSE
[71], 2009ChinaIEEE ConferenceBee evolution modifying PSO chaotic networkPower systemLoadPSORMSEDaqing oil field company, China
[72], 2009ChinaIEEE ConferenceAdding-weighted LLEGridLoadAdding-weighted one-rank localMaximum and minimal relative errors, AREGrid of New South Wales, Australia
[19], 2009USAJournal of HydrologyRegression analysis, ANN, Chaotic nonlinear dynamic modelsHydrological systemsTemperatureR 2 , RMSE, MSELake Tahoe basin, California and Nevada, USA
[73], 2010ChinaIEEE ConferenceGaussian particle filteringMackey–Glass modelTime seriesEKF, UKFPrediction error
[74], 2010ChinaRenewable EnergyWavelet decomposition method, ITSMWind farmPower, SpeedBPNNMAE, MSE, MAPE
[75], 2010ChinaIEEE ConferenceChaos theory, FNNHydraulic pumpVibration signalAPE, MSE
[76], 2010ChinaIEEE ConferenceDynamic recurrent FNNPower systemLoadFNNMSENorth china city
[77], 2010ChinaIEEE ConferenceParallel RBFNNLorenz system, Hydraulic pumpTime seriesRBFNNAPE
[78], 2010ChinaNeurocomputingHybrid Elman–NARX neural networkMackey–Glass, Lorenz, Real life sunspot modelsTime seriesAR model, GA, FuzzyMSE, RMSE, NMSESolar influences data analysis center, Belgium 1
[79], 2010ChinaIEEE ConferenceNonlinear ARChaotic systemExchange rateBPNN, SVM modelAPEFX data of USD
[16], 2010ChinaIEEE ConferenceSVMWind farmSpeedANNRRMSE
[80], 2011ChinaIEEE ConferenceRough set neural networkWind farmPowerChaos neural network, Persistence modelsNMAEWind farm in Beijing area, China
[81], 2011ChinaExpert Systems with ApplicationsChaotic wavelet decomposition–Grey modelWind farmPowerDirect prediction methodMAPE, NMAE, NRMSEDongtai wind farm, East China
[82], 2011USAIEEE ConferenceProbabilistic collocationPower systemSparse grid pointsMonte Carlo methodMeasurement errorNASA
[83], 2011ChinaProcedia EngineeringGlobal prediction method based on BPNNGasEmission rateFirst-order weighted local prediction methodMSE, RMSEHegang nanshan mine, China
[84], 2011ChinaIEEE ConferenceChaotic RBFNNPower systemLoadRBFNNAbsolute error
[85], 2011ChinaIEEE ConferenceImproved duffing oscillator-chaotic traffic prediction modelCoal gasTraffic flowPeak-to-peak errorCoal mine northwest edge router room, China
[86], 2012FranceIEEE ConferenceAnchor selection based on polynomial chaos expansionsAnchorAngle-of-arrivalRMSE, Median error
[87], 2012ChinaPhysics ProcediaMutative scale chaos optimizationSVM parametersChaotic time seriesChaos optimization algorithmRMSE
[88], 2012ChinaSystems Engineering ProcediaChaotic local weighted linear forecast algorithmElectricityDaily loadWeighted first order local methodARESouth china city
[89], 2012ChinaIEEE ConferenceHierarchic ESNLorenz, Sunspot, Yellow river annual runoff modelsTime seriesESNRMSE
[90], 2012South KoreaIEEE ConferenceMLPDC electric arc furnaceVoltage, Current signals, Arc resistanceRBFNNAutocorrelationDC electric arc furnace
[91], 2012ChinaIEEE Transactions on Systems, Man, CyberneticsH-infinity state estimationDiscrete time chaotic systemsH-infinity stateEKFEstimation error
[92], 2012ChinaIEEE ConferenceChaos algorithmRadio waveAmplitudeTraditional chaotic time series prediction methodRMSE
[93], 2012ItalyIEEE ConferenceDecentralized polynomial chaos theoryPower systemVoltage sensor validationDecentralized polynomial chaos theoryLocal covariance error
[94], 2013TurkeyElectric Power Systems ResearchIndependent component analysisPower systemAmplitude, Frequency signalsZero-crossing, Discrete Fourier transform, Orthogonal filters, Kalman filterMSE
[95], 2013ChinaIEEE ConferenceWNN with phase space reconstructionLorenz, Henon modelsTime seriesWNN without phase space reconstructionSMAPE
[20], 2013ChinaIEEE ConferenceGlobal prediction of chaosGeneratorOutput powerPREThermal power plant in Liaoning province, China
[12], 2013ChinaIEEE ConferenceChaotic RBFNNGasolineIntake flowRBFNNMSD, MAE, ARE
[96], 2013ChinaFluid Phase EquilibriaSelf-adaptive PSO based BPNNPolymersGas solubilityBPNN, PSO-BPNNMSE
[97], 2014TaiwanIEEE Transactions on CyberneticInterval type-2 fuzzy cerebellar model articulation controllerHenon systemTime seriesFNN, Interval type-2 FNNMSE
[98], 2014ChinaThe Scientific World JournalPhase space reconstruction-LS-SVMFM radioBand occupancy rateGA-LS-SVM, Monte Carlo-LS-SVMNMSE, RMSE, MAPEFixed radio monitoring station of Xihua university, China
[99], 2014ChinaIEEE ConferenceChaos elitism estimation of distributionChaotic systemElitism strategyEstimation of distribution algorithm for large scale global optimizationStandard deviation
[100], 2014EgyptJournal of the Egyptian Mathematical SocietyAdaptive chaos synchronization techniqueHyperchaotic systemSystem parametersError dynamics
[101], 2014GreeceSimulation Modeling Practice and TheoryANNChaotic dynamical systemEmbedding dimensionRMSE
[102], 2014Hong KongBuilding and EnvironmentANN-chaotic PSOAir qualityParticulate matter concentrationMulleven Levenberg–MarquardtR, MSE
[103], 2014MexicoIEEE ConferenceSOM tuned neural networkMackey–Glass, NN5Time seriesRMSE, MAE, SMAPE
[104], 2014JapanIEEE ConferenceJacobian matrix estimationWind farmSpeed, PowerANN, GARMSEJapan meteorological agency, Aomori area, North of Honshu, Japan
[105], 2014ChinaMathematical Problems in EngineeringGeneralized Liu systemChaotic secure communication, implementation of electronic circuits, numerical simulationsGlobal exponential stabilityWeighted first order local methodRMSE
[106], 2014CanadaIEEE Transactions on Power DeliveryMinimum phase space volume-EKF equalizationPower line communicationsBlind equalizationInverse filter-based MPSV methodMSE
[107], 2015ChinaJournal of Engineering Science and Technology ReviewImproved GALorenz modelTime seriesGAPercentage coordinate error
[108], 2015ChinaApplied EnergyHilbert–Huang transform and Hurst analysisWind farmPowerEMD model, LS-SVMNMAE, NRMSEWind farm of Hebei province, China
[109], 2015IranOcean EngineeringFalse nearest neighborWind farmWave characteristicsPort and maritime organization, Iran
[110], 2015IranJournal of Intelligent & Fuzzy SystemsEmbedding theorem-repetitive fuzzyMackey–Glass, Lorenz, Sunspot number modelsTime seriesMLP gradient, Adaptive neuro fuzzy inference, AR, FuzzyMSE, RMSE, NMSESolar influences data analysis center, Belgium 1
[21], 2015BrazilNeural Computing & ApplicationsMLPFloodRiver levelElman-RNNMAE, RMSE, R 2 Urban rivers by means of wireless sensor networks
[111], 2016ChinaJournal of Parallel and Distributed ComputingMaximum velocity criterion, Sinusoidal wave frequency modulation, Chaotic control using fuzzySmart gridChaosRaw smart grid
[112], 2016ChinaMathematical Problems in EngineeringSelf-constructing recurrent FNNLogistic, Henon mapsTime seriesSelf-constructing FNNRMSE
[113], 2016ChinaIEEE ConferenceChaos RBFNN predictionBlast furnaceCarbon-monoxide utilization ratioRMSE
[114], 2016ChinaMathematical Problems in EngineeringChattering-free sliding mode controlPower systemDisturbancesNonlinear disturbance observer based sliding mode controlSteady state error
[115], 2016MalaysiaNeural Computing & ApplicationsBPNN, Chaos search GA, Simulated annealingSmart gridElectrical energy demandANNMAE, RMSE, MSE, MAPEGrid of New South Wales, Australian
[116], 2016RussiaIEEE ConferenceGuaranteedOne-dimensional chaotic systemGuaranteed state, ParameterLS methodMeasurement errors
[117], 2016IranJournal of Intelligent & Fuzzy SystemsInteractively recurrent fuzzy functionsLorenz, Noisy Mackey–Glass, Real lung sound signalsTime seriesFNN, WNN, ESN, LSRMSE, PREDepartment of pneumology in Shariati hospital collected by Amirkabir University’s researchers
[118], 2016ItalyChemical Engineering TransactionsParallel chaosPower systemLoadANNEast China power grid enterprise
[119], 2017ChinaEnergyEnsemble EMD, Full-parameters continued fractionWind farmPowerHEA, MLE, RBFNRMSE, NMAEFarm in Xinjiang, China
[13], 2017ChinaChaos, Solitons and FractalsWavelet transform, Multiple model fusionLorenz, Mackey–Glass modelsTime seriesImproved free search-LS-SVM, Direct superposition without Gauss–Markov fusionRMSE, MAE, SMAPE
[120], 2017ChinaRenewable and Sustainable Energy ReviewsWavelet decomposition, EMDElectricityElectricity demandANN, SVM
[121], 2017ChinaIEEE ConferenceRBFNN, Volterra filterSpacecraft systemSpacecraft telemetry parameterAbsolute error, RE
[122], 2017ChinaChaos, Solitons and FractalsRecursive Levenberg–MarquardtNeural networksChaotic time seriesOn-line Levenberg–Marquardt algorithmMSE
[123], 2017South KoreaSustainabilityInverse model, Chaos time series inverseBuilding energy management systemElectric energy consumptionSVMMAE, CVRMSE
[124], 2017ChinaComputer Methods in Applied Mechanics and EngineeringFast initial solution predictionSheet metal stampingInverse isogeometric analysisOne-step inverse finite element method
[17], 2017ChinaInternational Journal of Mining Science and TechnologyCoal mine ventilation systems management technologyCoal mineGas concentrationMSECoal mine in Jining, Shandong, China
[125], 2017IranIEEE ConferenceTakens embedding theoryChaotic Henon mapTime seriesPyragas methodEstimation error
[126], 2017New ZealandWireless Communications and Mobile ComputingAdaptive multiuser transceiver schemeDS-CDMA SystemBit error rateLeast mean squareMMSE
[127], 2017IndiaIEEE ConferenceLLE, HFD, SampEnElectromyography signalsChaos, Fractal dimension, EntropyGrassberger–Procaccia algorithm, Approximate entropy
[128], 2018ChinaNeural Computing & ApplicationsChaotic BPNNPower systemLoadBPNN, RBFNN, Elman, PSO-BPNN, RBFNN-Quantile regressionMRPE, MAPEElectrical load data of a city in china network
[129], 2018ChinaIEEE ConferenceEquivalent circuit model, EKFLi-ion batteriesState of chargeEstimation error
[130], 2018MoroccoIEEE ConferenceANN–Discrete wavelet transformPV systemPowerANN, ANN–Phase space reconstructionMSE, MAPE, RMSEPhotovoltaic park, faculty of science and technology, Beni Mellal, Morocco
[131], 2018RussiaIEEE ConferenceDeep CNNDiscrete dynamic systemsLyapunov exponentMAPE, MPERussian central bank 1
[132], 2018ChinaSensorsSATime series interferometric synthetic aperture radarDeformationBeijing area, china
[133], 2018IndonesiaIEEE ConferenceSOM extreme learning mechanism-RBFNNLorenz systemMulti-step ahead time seriesAR, ARIMA modelsMultiple correlation coefficient
[134], 2018ChinaIEEE ConferenceGeneralized regression neural network of k-fold cross validationSunspotTime seriesRBFNNLeast generalization error, Normalized errorSolar influences data analysis center, Belgium 2
[135], 2018ChinaIEEE ConferenceGA-LS-SVMFractional order systemsNonlinear functionLS-SVMMSE
[136], 2019IndonesiaIEEE ConferenceRoberts edge detectionWeatherTornadoes
[18], 2019ChinaJournal of HydrologyCoupled quantity–pattern similarityHydrological applicationMonthly precipitationLocal approximation prediction, Autoregressive modelsR, RMSE, MARE, MSEDanjiangkou reservoir basin, China
[137], 2019MexicoIEEE ConferenceSuperimposed chaos sequenceQuadratic base band, Orthogonal frequency division multiplexing-based cognitive radio ChannelFrequencyPilot design method, Wavelet pilot design
[138], 2019USAIEEE ConferencePolynomial chaos expansion–Langevin MCMCPower systemInertia, Exciter gains, Damping ratio, DroopMetropolis–Hastings algorithm
[139], 2019ChinaIEEE ConferencePrincipal component analysis–chaotic immune PSO-GRNNCooling waterCorrosionPSO-GRNN algorithmAREPetrochemical enterprises
[140], 2019UKElectric Power Systems ResearchHarmonic robust grid synchronizationGridVoltage signalSecond-order generalized integrator-frequency locked loop techniquePhase estimation error
[141], 2019ChinaApplied Soft ComputingFuzzy information granules, LSTM-FNNZurich monthly sunspot numbers, Mackey–Glass model, Daily maximum temperatures in MelbourneTime series, GranulesAR, Nonlinear AR neural networkRMSE, MAPE, MAE
[142], 2019SwitzerlandIEEE ConferenceChaos–Rivest shamir adleman, Chaos–Random number generatorCrypto systemSecurity vulnerabilities
[143], 2019ChinaIEEE ConferenceCorrelation matrix augmentationBistatic co-prime MIMO arrayDirections of departure and arrivalESPRIT-Root MUSIC and RD-Root MUSICRMSE
[144], 2019ChinaRenewable EnergyMarkov chain switching regimeWind farmSpeed, directionNeural network, SVMMAE, RMSE, MAPEBonneville power administration, Washington, USA
[145], 2019USAIEEE ConferenceTrue random number generatorChaotic jerk systemSampling periodPseudo random number generator
[146], 2019USAIEEE Signal Processing LettersKalman filterSynchronous generatorComputing timeEKFRMSE
[147], 2019ChinaIEEE AccessChaotic optimized-PSOMobileLocationChan, Taylor, PSORMSE, MSE
[148], 2019ChinaJournal of Power SourcesFractional-orderLi-ion battery and ultra-capacitor hybrid power source systemLoad current, powerMAE, RMSE, MRE
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Ramadevi, B.; Bingi, K. Chaotic Time Series Forecasting Approaches Using Machine Learning Techniques: A Review. Symmetry 2022, 14, 955. https://doi.org/10.3390/sym14050955

AMA Style

Ramadevi B, Bingi K. Chaotic Time Series Forecasting Approaches Using Machine Learning Techniques: A Review. Symmetry. 2022; 14(5):955. https://doi.org/10.3390/sym14050955

Chicago/Turabian Style

Ramadevi, Bhukya, and Kishore Bingi. 2022. "Chaotic Time Series Forecasting Approaches Using Machine Learning Techniques: A Review" Symmetry 14, no. 5: 955. https://doi.org/10.3390/sym14050955

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