Support Vector Echo-State Machine for Chaotic Time-Series Prediction

Variability of echo state network prediction horizon for partially observed dynamical systems

Study of dynamical systems using partial state observation is an important problem due to its applicability to many real-world systems. We address the problem by studying an echo state network (ESN) framework with partial state input with partial or full state output. Application to the Lorenz system and Chua's oscillator (both numerically simulated and experimental systems) demonstrate the effectiveness of our method. We show that the ESN, as an autonomous dynamical system, is capable of making short-term predictions up to a few Lyapunov times. However, the prediction horizon has high variability depending on the initial condition-an aspect that we explore in detail using the distribution of the prediction horizon. Further, using a variety of statistical metrics to compare the long-term dynamics of the ESN predictions with numerically simulated or experimental dynamics and observed similar results, we show that the ESN can effectively learn the system's dynamics even when trained with noisy numerical or experimental data sets. Thus, we demonstrate the potential of ESNs to serve as cheap surrogate models for simulating the dynamics of systems where complete observations are unavailable.

A double-cycle echo state network topology for time series prediction

Echo state network (ESN) has gained wide acceptance in the field of time series prediction, relying on sufficiently complex reservoir connections to remember the historical features of the data and using these features to obtain the outputs by a simple linear readout. However, the randomness of its input and reservoir connections pose negative impacts on the prediction performance and performance stability of the models, the complexity of reservoir connections brings high time consumption during network computing, and the presence of randomness and complexity makes the hardware implementation of the ESN difficult. In response, we propose a double-cycle ESN (DCESN) based on the Li-ESN model, which has fixed weights to improve prediction performance and performance stability and simpler reservoir connections compared to the classical ESN to reduce the time consumption. The existence of both greatly reduces the difficulty of hardware implementation of the ESN and provides many conveniences for the future application of the ESN. Experimental results on many widely used time series datasets show that the DCESN has comparable or even better prediction performance than the ESN and good robustness against noise and parameter fluctuations.

Computational Modeling of Structural Synaptic Plasticity in Echo State Networks

Most existing studies on computational modeling of neural plasticity have focused on synaptic plasticity. However, regulation of the internal weights in the reservoir based on synaptic plasticity often results in unstable learning dynamics. In this article, a structural synaptic plasticity learning rule is proposed to train the weights and add or remove neurons within the reservoir, which is shown to be able to alleviate the instability of the synaptic plasticity, and to contribute to increase the memory capacity of the network as well. Our experimental results also reveal that a few stronger connections may last for a longer period of time in a constantly changing network structure, and are relatively resistant to decay or disruptions in the learning process. These results are consistent with the evidence observed in biological systems. Finally, we show that an echo state network (ESN) using the proposed structural plasticity rule outperforms an ESN using synaptic plasticity and three state-of-the-art ESNs on four benchmark tasks.

Human recognition with the optoelectronic reservoir-computing-based micro-Doppler radar signal processing

Current perception and monitoring systems, such as human recognition, are affected by several environmental factors, such as limited light intensity, weather changes, occlusion of targets, and public privacy. Human recognition using radar signals is a promising direction to overcome these defects; however, the low signal-to-noise ratio of radar signals still makes this task challenging. Therefore, it is necessary to use suitable tools that can efficiently deal with radar signals to identify targets. Reservoir computing (RC) is an efficient machine learning scheme that is easy to train and demonstrates excellent performance in processing complex time-series signals. The RC hardware implementation structure based on nonlinear nodes and delay feedback loops endows it with the potential for real-time fast signal processing. In this paper, we numerically study the performance of the optoelectronic RC composed of optical and electrical components in the task of human recognition with noisy micro-Doppler radar signals. A single-loop optoelectronic RC is employed to verify the application of RC in this field, and a parallel dual-loop optoelectronic RC scheme with a dual-polarization Mach-Zehnder modulator (DPol-MZM) is also used for performance comparison. The result is verified to be comparable with other machine learning tools, which demonstrates the ability of the optoelectronic RC in capturing gait information and dealing with noisy radar signals; it also indicates that optoelectronic RC is a powerful tool in the field of human target recognition based on micro-Doppler radar signals.

Comparing deep and shallow neural networks in forecasting call center arrivals

Forecasting volumes of incoming calls is the first step of the workforce planning process in call centers and represents a prominent issue from both research and industry perspectives. We investigate the application of Neural Networks to predict incoming calls 24 hours ahead. In particular, a Machine Learning deep architecture known as Echo State Network, is compared with a completely different rolling horizon shallow Neural Network strategy, in which the lack of recurrent connections is compensated by a careful input selection. The comparison, carried out on three different real world datasets, reveals better predictive performance for the shallow approach. The latter appears also more robust and less demanding, reducing the inference time by a factor of 2.5 to 4.5 compared to Echo State Networks.

Learn to synchronize, synchronize to learn

In recent years, the artificial intelligence community has seen a continuous interest in research aimed at investigating dynamical aspects of both training procedures and machine learning models. Of particular interest among recurrent neural networks, we have the Reservoir Computing (RC) paradigm characterized by conceptual simplicity and a fast training scheme. Yet, the guiding principles under which RC operates are only partially understood. In this work, we analyze the role played by Generalized Synchronization (GS) when training a RC to solve a generic task. In particular, we show how GS allows the reservoir to correctly encode the system generating the input signal into its dynamics. We also discuss necessary and sufficient conditions for the learning to be feasible in this approach. Moreover, we explore the role that ergodicity plays in this process, showing how its presence allows the learning outcome to apply to multiple input trajectories. Finally, we show that satisfaction of the GS can be measured by means of the mutual false nearest neighbors index, which makes effective to practitioners theoretical derivations.

Using machine learning to predict statistical properties of non-stationary dynamical processes: System climate,regime transitions, and the effect of stochasticity

We develop and test machine learning techniques for successfully using past state time series data and knowledge of a time-dependent system parameter to predict the evolution of the "climate" associated with the long-term behavior of a non-stationary dynamical system, where the non-stationary dynamical system is itself unknown. By the term climate, we mean the statistical properties of orbits rather than their precise trajectories in time. By the term non-stationary, we refer to systems that are, themselves, varying with time. We show that our methods perform well on test systems predicting both continuous gradual climate evolution as well as relatively sudden climate changes (which we refer to as "regime transitions"). We consider not only noiseless (i.e., deterministic) non-stationary dynamical systems, but also climate prediction for non-stationary dynamical systems subject to stochastic forcing (i.e., dynamical noise), and we develop a method for handling this latter case. The main conclusion of this paper is that machine learning has great promise as a new and highly effective approach to accomplishing data driven prediction of non-stationary systems.

Echo Memory-Augmented Network for time series classification

Echo State Networks (ESNs) are efficient recurrent neural networks (RNNs) which have been successfully applied to time series modeling tasks. However, ESNs are unable to capture the history information far from the current time step, since the echo state at the present step of ESNs mostly impacted by the previous one. Thus, ESN may have difficulty in capturing the long-term dependencies of temporal data. In this paper, we propose an end-to-end model named Echo Memory-Augmented Network (EMAN) for time series classification. An EMAN consists of an echo memory-augmented encoder and a multi-scale convolutional learner. First, the time series is fed into the reservoir of an ESN to produce the echo states, which are all collected into an echo memory matrix along with the time steps. After that, we design an echo memory-augmented mechanism employing the sparse learnable attention to the echo memory matrix to obtain the Echo Memory-Augmented Representations (EMARs). In this way, the input time series is encoded into the EMARs with enhancing the temporal memory of the ESN. We then use multi-scale convolutions with the max-over-time pooling to extract the most discriminative features from the EMARs. Finally, a fully-connected layer and a softmax layer calculate the probability distribution on categories. Experiments conducted on extensive time series datasets show that EMAN is state-of-the-art compared to existing time series classification methods. The visualization analysis also demonstrates the effectiveness of enhancing the temporal memory of the ESN.

Chaotic Time Series Prediction Using a Novel Echo State Network Model with Input Reconstruction, Bayesian Ridge Regression and Independent Component Analysis

This paper presents a novel Echo State Network (ESN) model for chaotic time series prediction, which consists of three steps including input reconstruction, dimensionality reduction and regression. First, phase-space reconstruction is used to reconstruct the original ‘attractor’ of the input time series. Then, Independent Component Analysis (ICA) is used to identify independent components, reduce dimensionality and overcome multicollinearity problem of the reconstructed input matrix. Finally, Bayesian Ridge Regression provides accurate predictions thanks to its regularization effect to avoid over-fitting and its robustness to noise owing to its probabilistic strategy. Our experimental results show that our model significantly outperforms other ESN models in predicting both artificial and real-world chaotic time series.

Interval Type-2 Fuzzy Neural Networks for Chaotic Time Series Prediction: A Concise Overview

Chaotic time series widely exists in nature and society (e.g., meteorology, physics, economics, etc.), which usually exhibits seemingly unpredictable features due to its inherent nonstationary and high complexity. Thankfully, multifarious advanced approaches have been developed to tackle the prediction issues, such as statistical methods, artificial neural networks (ANNs), and support vector machines. Among them, the interval type-2 fuzzy neural network (IT2FNN), which is a synergistic integration of fuzzy logic systems and ANNs, has received wide attention in the field of chaotic time series prediction. This paper begins with the structural features and superiorities of IT2FNN. Moreover, chaotic characters identification and phase-space reconstruction matters for prediction are presented. In addition, we also offer a comprehensive review of state-of-the-art applications of IT2FNN, with an emphasis on chaotic time series prediction and summarize their main contributions as well as some hardware implementations for computation speedup. Finally, this paper trends and extensions of this field, along with an outlook of future challenges are revealed. The primary objective of this paper is to serve as a tutorial or referee for interested researchers to have an overall picture on the current developments and identify their potential research direction to further investigation.

Time series classification with Echo Memory Networks

Echo state networks (ESNs) are randomly connected recurrent neural networks (RNNs) that can be used as a temporal kernel for modeling time series data, and have been successfully applied on time series prediction tasks. Recently, ESNs have been applied to time series classification (TSC) tasks. However, previous ESN-based classifiers involve either training the model by predicting the next item of a sequence, or predicting the class label at each time step. The former is essentially a predictive model adapted from time series prediction work, rather than a model designed specifically for the classification task. The latter approach only considers local patterns at each time step and then averages over the classifications. Hence, rather than selecting the most discriminating sections of the time series, this approach will incorporate non-discriminative information into the classification, reducing accuracy. In this paper, we propose a novel end-to-end framework called the Echo Memory Network (EMN) in which the time series dynamics and multi-scale discriminative features are efficiently learned from an unrolled echo memory using multi-scale convolution and max-over-time pooling. First, the time series data are projected into the high dimensional nonlinear space of the reservoir and the echo states are collected into the echo memory matrix, followed by a single multi-scale convolutional layer to extract multi-scale features from the echo memory matrix. Max-over-time pooling is used to maintain temporal invariance and select the most important local patterns. Finally, a fully-connected hidden layer feeds into a softmax layer for classification. This architecture is applied to both time series classification and human action recognition datasets. For the human action recognition datasets, we divide the action data into five different components of the human body, and propose two spatial information fusion strategies to integrate the spatial information over them. With one training-free recurrent layer and only one layer of convolution, the EMN is a very efficient end-to-end model, and ranks first in overall classification ability on 55 TSC benchmark datasets and four 3D skeleton-based human action recognition tasks.

Interactive Genetic Algorithm Oriented toward the Novel Design of Traditional Patterns

To create alternative complex patterns, a novel design method is introduced in this study based on the error back propagation (BP) neural network user cognitive surrogate model of an interactive genetic algorithm with individual fuzzy interval fitness (IGA-BPFIF). First, the quantitative rules of aesthetic evaluation and the user’s hesitation are used to construct the Gaussian blur tool to form the individual’s fuzzy interval fitness. Then, the user’s cognitive surrogate model based on the BP neural network is constructed, and a new fitness estimation strategy is presented. By measuring the mean squared error, the surrogate model is well managed during the evolution of the population. According to the users’ demands and preferences, the features are extracted for the interactive evolutionary computation. The experiments show that IGA-BPFIF can effectively design innovative patterns matching users’ preferences and can contribute to the heritage of traditional national patterns.

An Ant-Lion Optimizer-Trained Artificial Neural Network System for Chaotic Electroencephalogram (EEG) Prediction

The prediction of future events based on available time series measurements is a relevant research area specifically for healthcare, such as prognostics and assessments of intervention applications. A measure of brain dynamics, electroencephalogram time series, are routinely analyzed to obtain information about current, as well as future, mental states, and to detect and diagnose diseases or environmental factors. Due to their chaotic nature, electroencephalogram time series require specialized techniques for effective prediction. The objective of this study was to introduce a hybrid system developed by artificial intelligence techniques to deal with electroencephalogram time series. Both artificial neural networks and the ant-lion optimizer, which is a recent intelligent optimization technique, were employed to comprehend the related system and perform some prediction applications over electroencephalogram time series. According to the obtained findings, the system can successfully predict the future states of target time series and it even outperforms some other hybrid artificial neural network-based systems and alternative time series prediction approaches from the literature.

Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronization and cryptography

Using the machine learning approach known as reservoir computing, it is possible to train one dynamical system to emulate another. We show that such trained reservoir computers reproduce the properties of the attractor of the chaotic system sufficiently well to exhibit chaos synchronization. That is, the trained reservoir computer, weakly driven by the chaotic system, will synchronize with the chaotic system. Conversely, the chaotic system, weakly driven by a trained reservoir computer, will synchronize with the reservoir computer. We illustrate this behavior on the Mackey-Glass and Lorenz systems. We then show that trained reservoir computers can be used to crack chaos based cryptography and illustrate this on a chaos cryptosystem based on the Mackey-Glass system. We conclude by discussing why reservoir computers are so good at emulating chaotic systems.

Using echo state networks for classification: A case study in Parkinson's disease diagnosis

Despite having notable advantages over established machine learning methods for time series analysis, reservoir computing methods, such as echo state networks (ESNs), have yet to be widely used for practical data mining applications. In this paper, we address this deficit with a case study that demonstrates how ESNs can be trained to predict disease labels when stimulated with movement data. Since there has been relatively little prior research into using ESNs for classification, we also consider a number of different approaches for realising input-output mappings. Our results show that ESNs can carry out effective classification and are competitive with existing approaches that have significantly longer training times, in addition to performing similarly with models employing conventional feature extraction strategies that require expert domain knowledge. This suggests that ESNs may prove beneficial in situations where predictive models must be trained rapidly and without the benefit of domain knowledge, for example on high-dimensional data produced by wearable medical technologies. This application area is emphasized with a case study of Parkinson's disease patients who have been recorded by wearable sensors while performing basic movement tasks.

Investigating Echo-State Networks Dynamics by Means of Recurrence Analysis

In this paper, we elaborate over the well-known interpretability issue in echo-state networks (ESNs). The idea is to investigate the dynamics of reservoir neurons with time-series analysis techniques developed in complex systems research. Notably, we analyze time series of neuron activations with recurrence plots (RPs) and recurrence quantification analysis (RQA), which permit to visualize and characterize high-dimensional dynamical systems. We show that this approach is useful in a number of ways. First, the 2-D representation offered by RPs provides a visualization of the high-dimensional reservoir dynamics. Our results suggest that, if the network is stable, reservoir and input generate similar line patterns in the respective RPs. Conversely, as the ESN becomes unstable, the patterns in the RP of the reservoir change. As a second result, we show that an RQA measure, called , is highly correlated with the well-established maximal local Lyapunov exponent. This suggests that complexity measures based on RP diagonal lines distribution can quantify network stability. Finally, our analysis shows that all RQA measures fluctuate on the proximity of the so-called edge of stability, where an ESN typically achieves maximum computational capability. We leverage on this property to determine the edge of stability and show that our criterion is more accurate than two well-known counterparts, both based on the Jacobian matrix of the reservoir. Therefore, we claim that RPs and RQA-based analyses are valuable tools to design an ESN, given a specific problem.

Laplacian Echo State Network for Multivariate Time Series Prediction

Echo state network is a novel kind of recurrent neural networks, with a trainable linear readout layer and a large fixed recurrent connected hidden layer, which can be used to map the rich dynamics of complex real-world data sets. It has been extensively studied in time series prediction. However, there may be an ill-posed problem caused by the number of real-world training samples less than the size of the hidden layer. In this brief, a Laplacian echo state network (LAESN), is proposed to overcome the ill-posed problem and obtain low-dimensional output weights. First, an echo state network is used to map the multivariate time series into a large reservoir. Then, assuming that an unknown underlying manifold is inside the reservoir, we employ the Laplacian eigenmaps to estimate the manifold by constructing an adjacency graph associated with the reservoir states. Finally, the output weights are calculated by the low-dimensional manifold. In addition, some criteria of transient stability, local controllability, and local observability are given. Experimental results based on two real-world data sets substantiate the effectiveness and characteristics of the proposed LAESN model.

Multilayered Echo State Machine: A Novel Architecture and Algorithm

In this paper, we present a novel architecture and learning algorithm for a multilayered echo state machine (ML-ESM). Traditional echo state networks (ESNs) refer to a particular type of reservoir computing (RC) architecture. They constitute an effective approach to recurrent neural network (RNN) training, with the (RNN-based) reservoir generated randomly, and only the readout trained using a simple computationally efficient algorithm. ESNs have greatly facilitated the real-time application of RNN, and have been shown to outperform classical approaches in a number of benchmark tasks. In this paper, we introduce a novel criteria for integrating multiple layers of reservoirs within the ML-ESM. The addition of multiple layers of reservoirs are shown to provide a more robust alternative to conventional RC networks. We demonstrate the comparative merits of this approach in a number of applications, considering both benchmark datasets and real world applications.

Growing Echo-State Network With Multiple Subreservoirs

An echo-state network (ESN) is an effective alternative to gradient methods for training recurrent neural network. However, it is difficult to determine the structure (mainly the reservoir) of the ESN to match with the given application. In this paper, a growing ESN (GESN) is proposed to design the size and topology of the reservoir automatically. First, the GESN makes use of the block matrix theory to add hidden units to the existing reservoir group by group, which leads to a GESN with multiple subreservoirs. Second, every subreservoir weight matrix in the GESN is created with a predefined singular value spectrum, which ensures the echo-sate property of the ESN without posterior scaling of the weights. Third, during the growth of the network, the output weights of the GESN are updated in an incremental way. Moreover, the convergence of the GESN is proved. Finally, the GESN is tested on some artificial and real-world time-series benchmarks. Simulation results show that the proposed GESN has better prediction performance and faster leaning speed than some ESNs with fixed sizes and topologies.

Echo State Networks With Orthogonal Pigeon-Inspired Optimization for Image Restoration

In this paper, a neurodynamic approach for image restoration is proposed. Image restoration is a process of estimating original images from blurred and/or noisy images. It can be considered as a mapping problem that can be solved by neural networks. Echo state network (ESN) is a recurrent neural network with a simplified training process, which is adopted to estimate the original images in this paper. The parameter selection is important to the performance of the ESN. Thus, the pigeon-inspired optimization (PIO) approach is employed in the training process of the ESN to obtain desired parameters. Moreover, the orthogonal design strategy is utilized in the initialization of PIO to improve the diversity of individuals. The proposed method is tested on several deteriorated images with different sorts and levels of blur and/or noise. Results obtained by the improved ESN are compared with those obtained by several state-of-the-art methods. It is verified experimentally that better image restorations can be obtained for different blurred and/or noisy instances with the proposed neurodynamic method. In addition, the performance of the orthogonal PIO algorithm is compared with that of several existing bioinspired optimization algorithms to confirm its superiority.

Adaptive Elastic Echo State Network for Multivariate Time Series Prediction

Echo state network (ESN) is a new kind of recurrent neural network with a randomly generated reservoir structure and an adaptable linear readout layer. It has been widely employed in the field of time series prediction. However, when high-dimensional reservoirs are utilized to predict multivariate time series, there may be a collinearity problem. In this paper, to overcome the collinearity problem and obtain a sparse solution, we propose a new model-adaptive elastic ESN, in which adaptive elastic net algorithm is used to calculate the unknown weights. It combines the strengths of the quadratic regularization and the adaptively weighted lasso shrinkage. Hence, the proposed model can deal with the collinearity problem and enjoy the oracle property with an unbiased estimation. We exhibit the merits of our model on two benchmark multivariate chaotic datasets and two real-world applications. Experimental results substantiate the effectiveness and characteristics of the proposed model.

Chaotic time series prediction for prenatal exposure to polychlorinated biphenyls in umbilical cord blood using the least squares SEATR model

Chaotic time series prediction based on nonlinear systems showed a superior performance in prediction field. We studied prenatal exposure to polychlorinated biphenyls (PCBs) by chaotic time series prediction using the least squares self-exciting threshold autoregressive (SEATR) model in umbilical cord blood in an electronic waste (e-waste) contaminated area. The specific prediction steps basing on the proposal methods for prenatal PCB exposure were put forward, and the proposed scheme's validity was further verified by numerical simulation experiments. Experiment results show: 1) seven kinds of PCB congeners negatively correlate with five different indices for birth status: newborn weight, height, gestational age, Apgar score and anogenital distance; 2) prenatal PCB exposed group at greater risks compared to the reference group; 3) PCBs increasingly accumulated with time in newborns; and 4) the possibility of newborns suffering from related diseases in the future was greater. The desirable numerical simulation experiments results demonstrated the feasibility of applying mathematical model in the environmental toxicology field.