Chaotic Time Series Prediction Research Articles

Quantized kernel least inverse hyperbolic sine adaptive filtering algorithm

In the last few decades, the kernel method has been successfully used in the field of adaptive filtering to solve nonlinear problems. Mercer kernel is used to map data from input space to reproducing kernel Hilbert space (RKHS) by kernel adaptive filter (KAF). In regenerated kernel Hilbert spaces, the inner product can be easily calculated by computing the so-called kernel trick. The Kernel adaptive filtering algorithm is superior to common adaptive filtering algorithm in solving nonlinear problems and nonlinear channel equalization. For nonlinear problems, a robust kernel least inverse hyperbolic sine (KLIHS) algorithm is proposed by combining the kernel method with the inverse of hyperbolic sine function.The main disadvantage of KAF is that the radial-basis function (RBF) network grows with every new data sample, which increases the computational-complexity and requires more momories. The vector quantization (VQ) has been proposed to address this problem and has been successfully applied to the current kernel adaptive filtering algorithm. The main idea of the VQ method is to compress the input space through quantization to curb the network-size growth. In this paper, vector quantization is used to quantify the input spatial data, and a quantized kernel least inverse hyperbolic sine (QKLIHS) algorithm is constructed to restrain the growth of network scale. The energy conservation relation and convergence condition of quantized kernel least inverse hyperbolic sine algorithm are given. The simulation results of Mackey-Glass short-time chaotic time series prediction and nonlinear channel equalization environment show that the proposed kernel least inverse hyperbolic sine algorithm and quantized kernel least inverse hyperbolic sine algorithm have advantages in convergence speed, robustness and computational complexity.

The Generalized HR $q$-Derivative and Its Application to Quaternion Least Mean Square Algorithm

Quaternion adaptive filters have been widely used in processing three-dimensional and four-dimensional signals. To improve the performance of quaternion adaptive filtering algorithms, this letter first proposes a novel derivation rule named generalized HR (GHR) <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative based on the concept of <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative and quaternion GHR derivative. Then, the product rules of GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative are deduced, and the results of GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative regarding some common univariate and multivariable functions are given. In addition, based on the proposed GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative, the cost function of the quaternion least mean square (QLMS) algorithm is minimized to generate the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-QLMS algorithm. Finally, an example of Lorentz chaotic time-series prediction verifies the effectiveness of the proposed <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-QLMS algorithm.

Microring resonators with external optical feedback for time delay reservoir computing.

Microring resonators (MRRs) are a key photonic component in integrated devices, due to their small size, low insertion losses, and passive operation. While the MRRs have been established for optical filtering in wavelength-multiplexed systems, the nonlinear properties that they can exhibit give rise to new perspectives on their use. For instance, they have been recently considered for introducing optical nonlinearity in photonic reservoir computing systems. In this work, we present a detailed numerical investigation of a silicon MRR operation, in the presence of external optical feedback, in a time delay reservoir computing scheme. We demonstrate the versatility of this compact, passive device, by exploiting different operating regimes and solving computing tasks with diverse memory requirements. We show that when large memory is required, as it occurs in the Narma 10 task, the MRR nonlinearity does not play a significant role when the photodetection nonlinearity is involved, while the contribution of the external feedback is significant. On the contrary, for computing tasks such as the Mackey-Glass and the Santa Fe chaotic timeseries prediction, the MRR and the photodetection nonlinearities contribute both to efficient computation. The presence of optical feedback improves the prediction of the Mackey-Glass timeseries while it plays a minor role in the Santa Fe timeseries case.

Photonic neural field on a silicon chip: large-scale, high-speed neuro-inspired computing and sensing

Photonic neural networks have significant potential for high-speed neural processing with low latency and ultralow energy consumption. However, the on-chip implementation of a large-scale neural network is still challenging owing to its low scalability. Herein, we propose the concept of a photonic neural field and implement it experimentally on a silicon chip to realize highly scalable neuro-inspired computing. In contrast to existing photonic neural networks, the photonic neural field is a spatially continuous field that nonlinearly responds to optical inputs, and its high spatial degrees of freedom allow for large-scale and high-density neural processing on a millimeter-scale chip. In this study, we use the on-chip photonic neural field as a reservoir of information and demonstrate a high-speed chaotic time-series prediction with low errors using a training approach similar to reservoir computing. We show that the photonic neural field is potentially capable of executing more than one peta multiply–accumulate operations per second for a single input wavelength on a footprint as small as a few square millimeters. The operation of the neural field is energy efficient due to a passive scattering process, for which the required power comes only from the optical input. We also show that in addition to processing, the photonic neural field can be used for rapidly sensing the temporal variation of an optical phase, facilitated by its high sensitivity to optical inputs. The merging of optical processing with optical sensing paves the way for an end-to-end data-driven optical sensing scheme.

Model-free inference of unseen attractors: Reconstructing phase space features from a single noisy trajectory using reservoir computing.

Reservoir computers are powerful tools for chaotic time series prediction. They can be trained to approximate phase space flows and can thus both predict future values to a high accuracy and reconstruct the general properties of a chaotic attractor without requiring a model. In this work, we show that the ability to learn the dynamics of a complex system can be extended to systems with multiple co-existing attractors, here a four-dimensional extension of the well-known Lorenz chaotic system. We demonstrate that a reservoir computer can infer entirely unexplored parts of the phase space; a properly trained reservoir computer can predict the existence of attractors that were never approached during training and, therefore, are labeled as unseen. We provide examples where attractor inference is achieved after training solely on a single noisy trajectory.

Prediction of Chaotic Time Series Based on BEN-AGA Model

Aiming at the prediction problem of chaotic time series, this paper proposes a brain emotional network combined with an adaptive genetic algorithm (BEN-AGA) model to predict chaotic time series. First, we improve the emotional brain learning (BEL) model using the activation function to change the two linear structures the amygdala and the orbitofrontal cortex into the nonlinear structure, and then we establish the brain emotional network (BEN) model. The brain emotional network model has stronger nonlinear calculation ability and generalization ability. Next, we use the adaptive genetic algorithm to optimize the parameters of the brain emotional network model. The weights to be optimized in the model are coded as chromosomes. We design the dynamic crossover probability and mutation probability to control the crossover process and the mutation process, and the optimal parameters are selected through the fitness function to evaluate the chromosome. In this way, we increase the approximation capability of the model and increase the calculation speed of the model. Finally, we reconstruct the phase space of the observation sequence based on the short-term predictability of the chaotic time series; then we establish a brain emotional network model and optimize its parameters with an adaptive genetic algorithm and perform a single-step prediction on the optimized model to obtain the prediction error. The model proposed in this paper is applied to the prediction of Rossler chaotic time series and sunspot chaotic time series. The experimental results verify the effectiveness of the BEN-AGA model and show that this model has higher prediction accuracy and more stability than other methods.

High-efficiency chaotic time series prediction based on time convolution neural network

The prediction of chaotic time series is important for both science and technology. In recent years, this type of prediction has improved significantly with the development of deep learning. Here, we propose a temporal convolutional network (TCN) model for the prediction of chaotic time series. Our TCN model offers highly stable training, high parallelism, and flexible perception field. Comparative experiments with the classic long short-term memory (LSTM) network and hybrid (CNN-LSTM) neural network show that the TCN model can reduce the training time by a factor of more than two. Furthermore, the network can focus on more important information because of the attention mechanism. By embedding the convolutional block attention module (CBAM), which combines the spatial and channel attention mechanisms, we obtain a new model, TCN-CBAM. This model is comprehensively better than the LSTM, CNN-LSTM, and TCN models in the prediction of classical systems (Chen system, Lorenz system, and sunspots). In terms of prediction accuracy, the TCN-CBAM model obtains better results for the four main evaluation indicators: root mean square error, mean absolute error, coefficient of determination, and Spearman's correlation coefficient, with a maximum increase of 41.4%. The TCN-CBAM has also the shortest training times among the previous classic four models.

Stochastic approach for assessing the predictability of chaotic time series using reservoir computing.

The applicability of machine learning for predicting chaotic dynamics relies heavily upon the data used in the training stage. Chaotic time series obtained by numerically solving ordinary differential equations embed a complicated noise of the applied numerical scheme. Such a dependence of the solution on the numeric scheme leads to an inadequate representation of the real chaotic system. A stochastic approach for generating training time series and characterizing their predictability is suggested to address this problem. The approach is applied for analyzing two chaotic systems with known properties, the Lorenz system and the Anishchenko-Astakhov generator. Additionally, the approach is extended to critically assess a reservoir computing model used for chaotic time series prediction. Limitations of reservoir computing for surrogate modeling of chaotic systems are highlighted.

Quantized kernel Lleast lncosh algorithm

This paper introduces the kernel least lncosh (KLL) algorithm, in which the lncosh (logarithm of hyperbolic cosine) cost function is successfully applied in the reproducing-kernel-Hilbert space. The online vector quantization (VQ) is then used to quantize the input space to construct the algorithm which can curb the growth of network size. As a result, the quantized kernel least lncosh (QKLL) algorithm is developed, which is robust in non-Gaussian environments. The sufficient condition for mean-square convergence of the QKLL algorithm has been conducted. The performance of the KLL and QKLL algorithms is demonstrated by the short-term chaotic time-series prediction and non-linear channel-equalization (NCE).

Voltage-controlled superparamagnetic ensembles for low-power reservoir computing

We propose thermally driven, voltage-controlled superparamagnetic ensembles as low-energy platforms for hardware-based reservoir computing. In the proposed devices, thermal noise is used to drive the ensembles' magnetization dynamics, while control of their net magnetization states is provided by strain-mediated voltage inputs. Using an ensemble of CoFeB nanodots as an example, we use analytical models and micromagnetic simulations to demonstrate how such a device can function as a reservoir and perform two benchmark machine learning tasks (spoken digit recognition and chaotic time series prediction) with competitive performance. Our results indicate robust performance on timescales from microseconds to milliseconds, potentially allowing such a reservoir to be tuned to perform a wide range of real-time tasks, from decision making in driverless cars (fast) to speech recognition (slow). The low energy consumption expected for such a device makes it an ideal candidate for use in edge computing applications that require low latency and power.

Chaotic time series prediction using echo state network based on selective opposition grey wolf optimizer

Chaos prediction of nonlinear system is of great significance for proposing control strategies early. On the other hand, echo state network (ESN) as an artificial neural recursive network is widely used in time series prediction, while it has significant disadvantages due to the random input weight matrix. This work proposes a chaos prediction model based on ESN optimized by the selective opposition grey wolf optimizer (SOGWO). Firstly, the input weight matrix of ESN is restructured to express the position of gray wolf optimizer (GWO). Secondly, the selective opposition strategy is introduced into the GWO to enhance the exploration and exploitation capability. Thirdly, the optimal network is achieved by iteratively updating the position of SOGWO where the input weight is replaced by optimal search agent. Finally, taking two typical chaotic time series of Mackey–Glass and Lorenz as the objects, the effectiveness of the proposed SOGWO-ESN is verified by numerical simulation. The experimental results of SOGWO-ESN are compared with those of the traditional echo state network, the network model optimized by the particle swarm optimization (PSO) and the traditional gray wolf optimizer. The results imply that the proposed SOGWO-ESN has better prediction performance and is able to accurately predict a much longer period of time.

Predictive Analysis of Economic Chaotic Time Series Based on Chaotic Genetics Combined with Fuzzy Decision Algorithm

The irreversibility in time, the multicausality on lines, and the uncertainty of feedbacks make economic systems and the predictions of economic chaotic time series possess the characteristics of high dimensionalities, multiconstraints, and complex nonlinearities. Based on genetic algorithm and fuzzy rules, the chaotic genetics combined with fuzzy decision-making can use simple, fast, and flexible means to complete the goals of automation and intelligence that are difficult to traditional predicting algorithms. Moreover, the new combined method’s ergodicity can perform nonrepetitive searches in a global scope, hence improving the algorithm’s accuracy and efficiency. On the basis of summarizing and analyzing previous research works, this paper expounded the research status and significance of the prediction of economic chaotic time series, elaborated the development background, current status, and future challenges of the combined algorithm of chaotic genetics with fuzzy decision, introduced the basic principles of chaotic genetic algorithm and fuzzy decision algorithm, constructed a prediction model for economic chaotic time series, performed parameter synchronization optimization and moderate function construction, analyzed the prediction processes of economic chaotic time series, conducted phase space reconstruction and correlation dimension calculation, and finally carried out a simulation experiment with its result analysis. The study results show that the algorithm of chaotic genetics combined with fuzzy decision-making can dynamically adjust chaotic mutation operators and summarize fussy expert experiences. The phase space of its reconstructed chaotic attractor has high-precision predictability and can find orderly processes from changeable economic results, which in turn can be used to analyze and predict the complex economic chaotic time series. The study results of this paper provide a reference for further research on predictive analysis of economic chaotic time series based on chaotic genetics combined with fuzzy decision algorithm.

Hierarchical delay-memory echo state network: A model designed for multi-step chaotic time series prediction

Predicting for long-term dynamics of complex systems from observations is a challenging topic in the field of time series modeling and analysis, and is continually under research. Noteworthily, multi-step prediction requires accurate learning of dynamics and correlations between historical data for predicting future behavior. In this paper, we proposed a modified recurrent neural network named hierarchical delay-memory echo state network (HDESN) for solving the task of multi-step chaotic time series prediction. The HDESN uses multiple reservoirs with delay-memory capabilities, which can simultaneously discover and explore the information of short-term and long-term memory hidden in the historical sequence, and extract the valuable evolution patterns through deep topology and hierarchical processing. Moreover, to ensure high-quality prediction results and reduce the computational burden as much as possible, we further design a phase-space representation strategy which can calculate a compact topology and delay-memory coefficient according to the chaotic characteristics of the data. Compared with other improved ESN-based models, the proposed HDESN does not have a larger memory capacity to capture potential evolution law hidden in the complex system layer by layer, but can also adaptively determine a suitable network architecture to reflect the mapping relations in chaotic phase space. The experimental results on two benchmark chaotic systems and a real-world meteorological dataset demonstrate that the proposed HDESN model obtains satisfactory performance in multi-step chaotic time series prediction.

A deep learned type-2 fuzzy neural network: Singular value decomposition approach

The main objective of this study is to present a novel dynamic fractional-order deep learned type-2 fuzzy logic system (FDT2-FLS) with improved estimation capability. The proposed FDT2-FLS is constructed based on the criteria of singular value decomposition and uncertainty bounds type-reduction. The upper and the lower singular values of the set of inputs are estimated by a simple filter and the output is obtained by fractional-order integral of the uncertainty bounds type-reduction. Using stability criteria of fractional-order systems, the adaptation rules of the consequent parameters are extracted such that the globally Mittag-Leffler stability is achieved. The proposed FDT2-FLS is employed for online dynamic identification of a hyperchaotic system, online prediction of chaotic time series and online prediction of glucose level in type-1 diabetes patients and its performance is compared with other well-known methods. It is shown that the proposed mechanism results in significantly better prediction and estimation performance with less tunable parameters in just one learning epoch.

Prediction of Chaotic Time Series Based on SALR Model with Its Application on Heating Load Prediction

With the development of nonlinear science, improving the prediction performance of chaotic time series is of great significance in industrial production and daily life. Now, researchers have to develop effective models to achieve accurate prediction performance. The echo state network (ESN) has been proven to be an excellent prediction tool. However, the ESN has been criticized for not being principled enough. Thus, a novel ESN model namely self-join adjacent-feedback loop reservoir (SALR) is proposed. This model achieves the simplest topology structure on the premise of ensuring that all the connection modes of the classic ESN are available. In addition, in order to ensure the prediction performance of the network, the whale optimization algorithm was used to solve the parameter selection problems in the traditional cycle reservoir (SCR) model, the adjacent-feedback loop reservoir (ALR) model, and the SALR model. Finally, we use the proposed SALR model to solve classic benchmark chaotic time series as well as practical heating load prediction problems, and compare the SALR with the ESN, SCR, and ALR, respectively. Experimental results show that the proposed model can obtain higher accuracy with relatively low complexity than the ESN, SCR, and ALR.

Selection of Embedding Dimension and Delay Time in Phase Space Reconstruction via Symbolic Dynamics.

The modeling and prediction of chaotic time series require proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Thus, one must choose appropriate time delay and embedding dimension p for phase space reconstruction. The value of can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay , which is the optimal time delay for independence of the time series. The C-C method, based on Correlation Integral, is a method simpler than Mutual Information and has been proposed to select optimally and . In this paper, we suggest a simple method for estimating and based on symbolic analysis and symbolic entropy. As in the C-C method, is estimated as the first local optimal time delay and as the time delay for independence of the time series. The method is applied to several chaotic time series that are the base of comparison for several techniques. The numerical simulations for these systems verify that the proposed symbolic-based method is useful for practitioners and, according to the studied models, has a better performance than the C-C method for the choice of the time delay and embedding dimension. In addition, the method is applied to EEG data in order to study and compare some dynamic characteristics of brain activity under epileptic episodes

Application of particle swarm optimization with ANFIS model for double scroll chaotic system

The predictions for the original chaos patterns can be used to correct the distorted chaos pattern which has changed due to any changes whether from undesired disturbance or additional information which can hide under chaos pattern. This information can be recovered when the original chaos pattern is predicted. But unpredictability is most features of chaos, and time series prediction can be used based on the collection of past observations of a variable and analysis it to obtain the underlying relationships and then extrapolate future time series. The additional information often prunes away by several techniques. This paper shows how the chaotic time series prediction is difficult and distort even if Neuro-Fuzzy such as Adaptive Neural Fuzzy Inference System (ANFIS) is used under any disturbance. The paper combined particle swarm (PSO) and (ANFIS) to exam the prediction model and predict the original chaos patterns which comes from the double scroll circuit. Changes in the bias of the nonlinear resistor were used as a disturbance. The predicted chaotic data is compared with data from the chaotic circuit.

Reservoir Computing Based on Two Parallel Reservoirs Under Identical Electrical Message Injection

In this work, we propose and numerically investigate a scheme for reservoir computing (RC) based on two parallel reservoirs under identical electrical message injection, in which two semiconductor lasers (SLs) under optical feedback are utilized as two parallel reservoirs. For simplifying the system, only one mask signal is employed in this scheme. After multiplying with input data, the masked information is injected into two parallel reservoir lasers (SL1 and SL2). The virtual node states can be obtained from the temporal outputs of two SLs. RC can be accomplished by three ways, namely RC1/RC2 (the virtual node states originating from SL1/SL2 are used for training and testing) and RCM (the merged virtual node states originating from two SLs are used for training and testing). Via chaotic time series prediction task and memory capacity (MC) test, the performance of the RC system is simulated and assessed. The results show that, for a given data processing rate, better prediction performance and higher MC can be realized by RCM through setting suitable mismatched parameters between the two SLs. Under satisfying the requirement for achieving a good performance, the highest data processing rate can be doubled for RCM with respect to that for RC1/RC2.

Prediction of Sea Clutter Based on Recurrent Neural Network

As radar sea clutter has chaotic characteristics, a prediction model can be established to predict sea clutter according to short-term predictability of chaotic time series. Therefore, this paper proposes a sea clutter amplitude prediction model based on LSTM, Stacked LSTM and Nested LSTM. This paper first obtains the best state space for sea clutter prediction according to the phase space reconstruction theory and establishes the prediction equation, then uses three networks to predict the sea clutter amplitude, and finally compares the effects of the three neural networks and analyzes the factors that affect the prediction precision. The experimental results on the IPIX radar data and P-band radar data show that, the model used in this paper has a smaller prediction error than RBF, and the prediction performance is better. The model can achieve high-precision and high-efficiency prediction of sea clutter and has verified the effectiveness of the method.

Study on fault identification of mechanical dynamic nonlinear transmission system

Abstract To solve the problems of large mechanical powertrain such as complex structure, serious accident, strong nonlinear characteristics of running state, bad operating environment, non-Gaussian noise, and various uncertain factors, it is difficult to make an accurate fault diagnosis. This paper proposes a method for dealing with nonlinear characteristics using nuclear waves, as well as a system, deeply conducted nuclear base fault feature extraction, classification, and decision making, such as nuclear base state trend prediction technology research, focusing on exploring and improving the accuracy of fault diagnosis under nonlinear conditions, technical method, and way to state prediction accuracy. It offers effective technical assistance for the advancement and use of mechanical power train monitoring and diagnosis technology. A fault detection method based on kernel method is proposed. Based on the characteristics of this method in dealing with nonlinear problems, the research on kernel feature extraction, kernel fault classification and decision making, and kernel state trend prediction are carried out systematically. The experimental results show that the simulation analysis of typical chaotic time series prediction and the application of the operation state prediction of a certain ship main steam turbine unit have achieved good results, among which the average relative error of the single-step prediction of the unit state is 1.7881%, and the average relative error of the 30-step prediction is 3.3983%. Proved that the nuclear methods systematically applied to mechanical power transmission system fault diagnosis and state prediction, effectively enhancing some traditional methods and techniques dealing with nonlinear feature extraction, the nonlinear prediction capability for fault identification, and nonlinear state, to deal with nonlinear fault diagnosis problems of engineering practice, a large number of explored effective solution.