Comparative Analysis of Deep Learning, and Quantum Models for Time Series Forecasting

Echo state networks (ESN), a type of reservoir computing (RC) architecture, are efficient and accurate artificial neural systems for time series processing and learning. An ESN consists of a core of recurrent neural networks, called a reservoir, with a small number of tunable parameters to generate a high-dimensional representation of an input, and a readout layer which is easily trained using regression to produce a desired output from the reservoir states. Certain computational tasks involve real-time calculation of high-order time correlations, which requires nonlinear transformation either in the reservoir or the readout layer. Traditional ESN employs a reservoir with sigmoid or tanh function neurons. In contrast, some types of biological neurons obey response curves that can be described as a product unit rather than a sum and threshold. Inspired by this class of neurons, we introduce a RC architecture with a reservoir of product nodes for time series computation. We find that the product RC shows many properties of standard ESN such as short-term memory and nonlinear capacity. On standard benchmarks for chaotic prediction tasks, the product RC maintains the performance of a standard nonlinear ESN while being more amenable to mathematical analysis. Our study provides evidence that such networks are powerful in highly nonlinear tasks owing to high-order statistics generated by the recurrent product node reservoir.

Reservoir computing (RC) is considered as a suitable alternative for descending gradient methods in recursive neural networks (RNNs) training. The echo state network (ESN) is a platform for RC and nonlinear system simulation in the cloud environment with many external users. In the past researches, the highest eigenvalue of reservoir connection weight (spectral radius) was used to predict reservoir dynamics. Some researchers have illustrated; the characteristics of scale-free and small-world can improve the approximation capability in echo state networks; however, recent studies have shown importance of the infrastructures such as clusters and the stability criteria of these reservoirs as altered. In this research, we suggest a high clustered ESN called HCESN that its internal neurons are interconnected in form of clusters. Each of the clusters contains one backbone and a number of local nodes. We implemented a classical clustering algorithm, called K-means, and three optimization algorithms including genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO) to improve the clustering efficiency of the new reservoir and compared them with each other. For investigating the spectral radius and predictive power of the resulting reservoirs, we also applied them to the laser time series and the Mackey-Glass dynamical system. It is demonstrated that new clustered reservoirs have some specifications of biologic neural systems and complex networks like average short path length, high clustering coefficient, and power-law distribution. The empirical results illustrated that the ESN based on PSO could strikingly enhance echo state property (ESP) and obtains less chaotic time series prediction error compared with other works and the original version of ESN. Therefore, it can approximate nonlinear dynamical systems and predict the chaotic time series.

Nonlinear model predictive control (NMPC) of industrial processes is changeling in part because the model of the plant may not be completely known but also for being computationally demanding. This work proposes an extremely efficient reservoir computing (RC)-based control framework that speeds up the NMPC of processes. In this framework, while an echo state network (ESN) serves as the dynamic RC-based system model of a process, the practical nonlinear model predictive controller (PNMPC) simplifies NMPC by splitting the forced and the free responses of the trained ESN, yielding the so-called ESN-PNMPC architecture. While the free response is generated by the forward simulation of the ESN model, the forced response is obtained by a fast and recursive calculation of the input-output sensitivities from the ESN. The efficiency not only results from the fast training inherited by RC but also from a computationally cheap control action given by the aforementioned novel recursive formulation and the computation in the reduced dimension space of input and output signals. The resulting architecture, equipped with a correction filter, is robust to unforeseen disturbances. The potential of the ESN-PNMPC is shown by application to the control of the four-tank system and an oil production platform, outperforming the predictive approach with a long-short term memory (LSTM) model, two standard linear control algorithms, and approximate predictive control.

Reservoir Computing (RC) is a paradigm for efficient training of Recurrent Neural Networks (RNNs). The Echo State Network (ESN), a type of RC paradigm, has been widely used for time series forecasting. Whereas, few works exist on classification with ESN. In this paper, we shed light on the use of ESN for pattern recognition problem, i.e. emotion recognition from Electroencephalogram (EEG). We show that the reservoir with its recurrence is able to perform the feature extraction step directly from the EEG raw. Such kind of recurrence rich of nonlinearities allows the projection of the input data into a high dimensional state space. It is well known that the ESN fails due to the poor choices of its initialization. Nevertheless, we show that pretraining the ESN with the Intrinsic Plasticity (IP) rule remedies the shortcoming of randomly initialization. To validate our approach, we tested our system on the benchmark DEAP containing EEG signals of 32 subjects and the results were promising.

Echo state networks (ESNs), as efficient and powerful computational models for approximating nonlinear dynamical systems, have been successfully applied in financial time series forecasting. Reservoir constructions in standard ESNs rely on trials and errors in real applications due to a series of randomized model building stages. A novel form of ESN with deterministically constructed reservoir is competitive with standard ESN by minimal complexity and possibility of optimizations for ESN specifications. In this paper, forecasting performances of deterministic ESNs are investigated in stock price prediction applications. The experiment results on two benchmark datasets (Shanghai Composite Index and S&P500) demonstrate that deterministic ESNs outperform standard ESN in both accuracy and efficiency, which indicate the prospect of deterministic ESNs for financial prediction.

In this paper, we derive a theoretical upper bound on the generalization error of reservoir computing (RC), a special category of recurrent neural networks (RNNs). The specific RC implementation considered in this paper is the echo state network (ESN), and an upper bound on its generalization error is derived via the empirical Rademacher complexity (ERC) approach. While recent work in deriving risk bounds for RC frameworks makes use of a non-standard ERC measure and a direct application of its definition, our work uses the standard ERC measure and tools allowing fair comparison with conventional RNNs. The derived result shows that the generalization error bound obtained for ESNs is tighter than the existing bound for vanilla RNNs, suggesting easier generalization for ESNs. With the ESN applied to symbol detection in MIMO-OFDM (Multiple Input Multiple Output-Orthogonal Frequency Division Multiplexing) systems, we show how the derived generalization error bound can guide underlying system design. Specifically, the derived bound together with the empirically characterized training loss is utilized to identify the optimum reservoir size in neurons for the ESN-based symbol detector. Finally, we corroborate our theoretical findings with results from simulations that employ 3GPP standards-compliant wireless channels, signifying the practical relevance of our work.

Reservoir Computing is a computing model ideal for performing computation on varied physical substrates. However, these physical reservoirs can be difficult to scale up. We propose joining various reservoirs together as an approach to solving this problem, simulating physical reservoirs with Echo State Networks (ESNs). We investigate various methods of combining ESNs to form larger reservoirs, including a method that we dub Restricted ESNs. We provide a notation for describing Restricted ESNs, and use it to benchmark a standard ESN against restricted ones. We investigate two methods to keep the weight matrix density consistent when comparing a Restricted ESN to a standard one, which we call overall consistency and patch consistency. We benchmark restricted ESNs on NARMA10 and the sunspot prediction benchmark, and find that restricted ESNs perform similarly to standard ones. We present some application scenarios in which restricted ESNs may offer advantages over standard ESNs. We then test restricted ESNs on a version of the multi-timescale Multiple Superimposed Sines tasks, in order to establish a baseline performance that can be improved upon in further work. We conclude that we can scale up reservoir performance by linking small homogeneous subreservoirs together without significant loss in performance over a single large reservoir, justifying future work on using heterogeneous subreservoirs for greater flexibility.

Image segmentation refers to the process of dividing an image into multiple regions which represent meaningful areas. Image segmentation is an essential step for most image analysis tasks such as object recognition and tracking, pattern recognition, content-based image retrieval, etc. In recent years, a large number of image segmentation algorithms have been developed, but achieving accurate segmentation still remains a challenging task. Recently, reservoir computing (RC) has drawn much attention in machine learning as a new model of recurrent neural networks (RNN). Echo State Network (ESN) represents one efficient realization of RC, which is initially designed to facilitate learning in Recurrent Neural Networks. In this paper we investigate the viability of ESN as feature extractor for pixel classification based colour image segmentation. Extensive experiments are conducted on real world colour image datasets and the global ESN reservoir parameters are varied to identify their operating ranges that allow the use of the reservoir nodes internal activations as new pixel features for the colour image segmentation task. A simple feed forward neural network is used to realize the ESN readout function and classify these new features. The experimental results show that the proposed method achieves high performance image segmentation comparing with state-of-the-art techniques. In addition, a set of empirically derived guidelines for setting the reservoir global parameters are proposed.

Deterministic echo state network (ESN) models integrated with particle swarm optimization (PSO) are proposed to improve the accuracy and efficiency of stock price forecasting. ESNs have been successfully applied to financial time series forecasting because of their efficient and powerful computational ability in approximating nonlinear dynamical systems. However, reservoir construction in standard ESNs is primarily driven by a series of randomized model-building stages, because of which both researchers and practitioners have to rely on a series of trials and errors to determine parameters. An ESN with a deterministically constructed reservoir is comparable in performance to a standard ESN and has minimal complexity as well as potential for optimizations with regard to ESN parameters. In this paper, forecasting performances of the proposed PSO-DESN models are compared with those of standard ESNs for stock price prediction on the benchmark dataset of S&P 500. The comparison results demonstrate that the proposed PSO-DESNs exhibit better performance in stock price forecasting in terms of both accuracy and efficiency, thereby verifying the potential of PSO-DESNs for financial predictions.

In this paper, we derive an upper bound on the generalization error of reservoir computing (RC), a special category of recurrent neural networks (RNNs). The particular RC implementation considered in this paper is the echo state network (ESN), and the generalization bound is derived via the empirical Rademacher complexity (ERC) approach. While recent work in deriving risk bounds for RC frameworks makes use of a non-standard empirical Rademacher Complexity (ERC) measure and a direct application of its definition, our work uses the standard ERC measure and tools which allow easier extension to deep RC structures and a fair comparison with other RNN structures including the vanilla RNNs. Finally, we train an ESN for the task of symbol detection, a key component in the receive processing of 5G systems and show with an example how the derived generalization bound can guide the underlying system design. To be specific, we utilize the derived generalization error together with the characterized training loss to analytically identify the optimum number of neurons in the reservoir for the symbol detection task. Simulation results using 3GPP specified channels corroborate our theoretical findings, illustrating the significance and practical relevance of our work.

Photonic reservoir computing has been intensively investigated to solve machine learning tasks effectively. A simple learning procedure of output weights is used for reservoir computing. However, the lack of training of input-node and inter-node connection weights limits the performance of reservoir computing. The use of multiple reservoirs can be a solution to overcome this limitation of reservoir computing. In this study, we investigate parallel and deep configurations of delay-based all-optical reservoir computing using semiconductor lasers with optical feedback by combining multiple reservoirs to improve the performance of reservoir computing. Furthermore, we propose a hybrid configuration to maximize the benefits of parallel and deep reservoirs. We perform the chaotic time-series prediction task, nonlinear channel equalization task, and memory capacity measurement. Then, we compare the performance of single, parallel, deep, and hybrid reservoir configurations. We find that deep reservoirs are suitable for a chaotic time-series prediction task, whereas parallel reservoirs are suitable for a nonlinear channel equalization task. Hybrid reservoirs outperform other configurations for all three tasks. We further optimize the number of reservoirs for each reservoir configuration. Multiple reservoirs show great potential for the improvement of reservoir computing, which in turn can be applied for high-performance edge computing.

Recurring neural networks (RNNs) are a class of neural networks that are naturally suitable for processing time series and other sequential data. RNNs and specifically Reservoir Computing (RC) has attracted much attention over the past decade, particularly in the context of time series prediction. However, most time series have a multi-scale structure. The Echo State Networks (ESN) are one of two pioneering methods of reservoir computing. ESN requires some experiences and insights to achieve the acclaimed accuracy in many tasks. In this paper, we present a practical application of case prediction using several time series data, multiple parameters of our ESN system are scaled to compare the accuracy of the proposed model.

In this paper we present a unified framework for extreme learning machines and reservoir computing (echo state networks), which can be physically implemented using a single nonlinear neuron subject to delayed feedback. The reservoir is built within the delay-line, employing a number of “virtual” neurons. These virtual neurons receive random projections from the input layer containing the information to be processed. One key advantage of this approach is that it can be implemented efficiently in hardware. We show that the reservoir computing implementation, in this case optoelectronic, is also capable to realize extreme learning machines, demonstrating the unified framework for both schemes in software as well as in hardware.

Cyberattack Detection in Smart Grids based on Reservoir Computing

Real-time processing of space-and-time-variant signals is imperative for perception and real-world problem-solving. In the brain, spatio-temporal stimuli are converted into spike trains by sensory neurons and projected to the neurons in subcortical and cortical layers for further processing. Reservoir Computing (RC) is a neural computation paradigm that is inspired by cortical Neural Networks (NN). It is promising for real-time, on-line computation of spatio-temporal signals. An RC system incorporates a Recurrent Neural Network (RNN) called reservoir, the state of which is changed by a trajectory of perturbations caused by a spatio-temporal input sequence. A trained, non- recurrent, linear readout-layer interprets the dynamics of the reservoir over time. Echo-State Network (ESN) [1] and Liquid-State Machine (LSM) [2] are two popular and canonical types of RC system. The former uses non-spiking analog sigmoidal neurons – and, more recently, Leaky Integrator (LI) neurons – and a normalized random connectivity matrix in the reservoir. Whereas, the reservoir in the latter is composed of Leaky Integrate-and-Fire (LIF) neurons, distributed in a 3-D space, which are connected with dynamic synapses through a probability function. The major difference between analog neurons and spiking neurons is in their neuron model dynamics and their inter-neuron communication mechanism. However, RC systems share a mysterious common property: they exhibit the best performance when reservoir dynamics undergo a criticality [1–6] – governed by the reservoirs’ connectivity parameters, |λmax| ≈ 1 in ESN, λ ≈ 2 and w in LSM – which is referred to as the edge of chaos in [3–5]. In this study, we are interested in exploring the possible reasons for this commonality, despite the differences imposed by different neuron types in the reservoir dynamics. We address this concern from the perspective of the information representation in both spiking and non-spiking reservoirs. We measure the Mutual Information (MI) between the state of the reservoir and a spatio-temporal spike-trains input, as well as that, between the reservoir and a linearly inseparable function of the input, temporal parity. In addition, we derive Mean Cumulative Mutual Information (MCMI) quantity from MI to measure the amount of stable memory in the reservoir and its correlation with the temporal parity task performance. We complement our investigation by conducting isolated spoken-digit recognition and spoken-digit sequence-recognition tasks. We hypothesize that a performance analysis of these two tasks will agree with our MI and MCMI results with regard to the impact of stable memory in task performance. It turns