An experimental evaluation of echo state network for colour image segmentation

In this chapter we review two additional types of Recurrent Neural Network, which present important differences with respect to the architectures described so far. More specifically, we introduce the nonlinear auto-regressive with eXogenous inputs (NARX) neural network and the Echo State Network. Both these networks have been largely employed in Short Term Load Forecast applications and they have been shown to be more effective than other methods based on statistical models. The main differences of NARX networks and Echo State Networks with respect to the other previously described models, are both in terms of their architecture and, in particular, in their training procedure. Indeed, both these architectures are designed in such a way that Back Propagation Through Time is not necessary. Specifically, in NARX the network output is replaced by the expected ground truth and this allows to train the network like a feedforward architecture. On the other hand, in a Echo State Network only the outermost linear layer is trained, usually by means of ridge regression. Due to these fundamental differences, some of the properties and training approaches discussed in the previous sections do not hold for the NARX and Echo State Network models and we reserved a separate chapter to review these models.

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.

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.

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.

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.

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.

This thesis deals with recurrent neural networks, a particular class of artificial neural networks which can learn a generative model of input sequences. The input is mapped, through a feedback loop and a non-linear activation function, into a hidden state, which is then projected into the output space, obtaining either a probability distribution or the new input for the next time-step. This work consists mainly of two parts: a theoretical study for helping the understanding of recurrent neural networks framework, which is not yet deeply investigated, and their application to non-linear prediction problems, since recurrent neural networks are really powerful models suitable for solving several practical tasks in different fields. For what concerns the theoretical part, we analyse the weaknesses of state-of-the-art models and tackle them in order to improve the performance of a recurrent neural network. Firstly, we contribute in the understanding of the dynamical properties of a recurrent neural network, highlighting the close relation between the definition of stable limit cycles and the echo state property of an echo state network. We provide sufficient conditions for the convergence of the hidden state to a trajectory, which is uniquely determined by the input signal, independently of the initial states. This may help extend the memory of the network and increase the design options for the network. Moreover, we develop a novel approach to address the main problem in training recurrent neural networks, the so-called vanishing gradient problem. Our new method allows us to train a very simple recurrent neural network, making the gradient not to vanish even after many time-steps. Exploiting the singular value decomposition of the vanishing factors in the gradient and random matrices theory, we find that the singular values have to be confined in a narrow interval and derive conditions about their root mean square value. Then, we also improve the efficiency of the training of a recurrent neural network, defining a new method for speeding up this process. Thanks to a least square regularization, we can initialize the parameters of the network, in order to set them closer to the minimum and running fewer epochs of classical training algorithms. Moreover, it is also possible to completely train the network with our initialization method, running more iterations of it without losing in performance with respect to classical training algorithms. Finally, it is also possible to use it as a real-time learning algorithm, adjusting the parameters to the new data through one iteration of our initialization. In the last part of this thesis, we apply recurrent neural networks to non-linear prediction problems. We consider prediction of numerical sequences, estimating the following input choosing it from a probability distribution. We study an automatic text generation problem, where we need to predict the following character in order to compose words and sentences, and a path prediction of walking mobile users in the central area of a city, as a sequence of crossroads. Then, we analyse the prediction of video frames, discovering a wide range of applications related to the prediction of movements. We study the collision problem of bouncing balls, taking into account only the sequence of video frames without any knowledge about the physical characteristics of the problem, and the distribution over days of mobile user in a city and in a whole region. Finally, we address the state-of-the-art problem of missing data imputation, analysing the incomplete spectrogram of audio signals. We restore audio signals with missing time-frequency data, demonstrating via numerical experiments that a performance improvement can be achieved involving recurrent neural networks.

As efficient recurrent neural network (RNN) models, echo state networks (ESNs) have attracted widespread attention and been applied in many application domains in the last decade. Although they have achieved great success in modeling time series, a single ESN may have difficulty in capturing the multitimescale structures that naturally exist in temporal data. In this paper, we propose the convolutional multitimescale ESN (ConvMESN), which is a novel training-efficient model for capturing multitimescale structures and multiscale temporal dependencies of temporal data. In particular, a multitimescale memory encoder is constructed with a multireservoir structure, in which different reservoirs have recurrent connections with different skip lengths (or time spans). By collecting all past echo states in each reservoir, this multireservoir structure encodes the history of a time series as nonlinear multitimescale echo state representations (MESRs). Our visualization analysis verifies that the MESRs provide better discriminative features for time series. Finally, multiscale temporal dependencies of MESRs are learned by a convolutional layer. By leveraging the multitimescale reservoirs followed by a convolutional learner, the ConvMESN has not only efficient memory encoding ability for temporal data with multitimescale structures but also strong learning ability for complex temporal dependencies. Furthermore, the training-free reservoirs and the single convolutional layer provide high-computational efficiency for the ConvMESN to model complex temporal data. Extensive experiments on 18 multivariate time series (MTS) benchmark datasets and 3 skeleton-based action recognition datasets demonstrate that the ConvMESN captures multitimescale dynamics and outperforms existing methods.

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.

As a novel recurrent neural network (RNN), an echo state network (ESN) that utilizes a reservoir with many randomly connected internal units and only trains the readout, avoids increased complexity of training procedures faced by traditional RNN. The ESN can cope with complex nonlinear systems because of its dynamical properties and has been applied in hydrological forecasting and load forecasting. Due to the linear regression algorithm usually adopted by generic ESN to train the output weights, an ill-conditioned solution might occur, degrading the generalization ability of the ESN. In this study, the ESN with Bayesian regularization (BESN) is proposed for short-term power production forecasting of small hydropower (SHP) plants. According to the Bayesian theory, the weights distribution in space is considered and the optimal output weights are obtained by maximizing the posterior probabilistic distribution. The evidence procedure is employed to gain optimal hyperparameters for the BESN model. The recorded data obtained from the SHP plants in two different counties, located in Yunnan Province, China, are utilized to validate the proposed model. For comparison, the feed-forward neural networks with Levenberg-Marquardt algorithm (LM-FNN) and the generic ESN are also employed. The results indicate that BESN outperforms both LM-FNN and ESN.

Echo State Networks (ESNs) are recurrent neural networks that map an input signal to a high-dimensional dynamical system, called reservoir, and possess adaptive output weights. The output weights are trained such that the ESN’s output signal fits the desired target signal. Classical reservoirs are sparse and randomly connected networks. In this article, we investigate the effect of different network topologies on the performance of ESNs. Specifically, we use two types of networks to construct clustered reservoirs of ESN: the clustered Erdös–Rényi and the clustered Barabási-Albert network model. Moreover, we compare the performance of these clustered ESNs (CESNs) and classical ESNs with the random reservoir by employing them to two different tasks: frequency filtering and the reconstruction of chaotic signals. By using a clustered topology, one can achieve a significant increase in the ESN’s performance.

A recurrent neural network (RNN) model is developed for simulating and predicting shear behavior of both a fine-grained residual soil and a dune sand. The RNN model with one hidden layer of 20 nodes appears very effective in modeling complex soil behavior, due to its feedback connections from a hidden layer to an input layer. A dynamic gradient descent learning algorithm is used to train the network. By training part of the experimental data, which include strain-controlled undrained tests and stress-controlled drained tests performed on a residual Hawaiian volcanic soil, the network is able to capture significant variability of shear behavior existing in the residual soil. The unusual characteristics that the denser soil samples dilate under a higher stress level and the looser soil samples contract under a lower stress level are well represented by the RNN model. The RNN model also shows encouraging results in simulation and prediction of behavior of a dune sand which experienced loading-unloading-reloading conditions. Excellent agreements between the measured data and the modeling results are observed in both stress-strain behavior and volumetric-change characteristics. As compared with a traditional model, the RNN model shows more effectiveness and less effort.Key words: neural network, modeling, soil behavior, shear tests, simulation, prediction.

Accurate production forecasting is crucial for effective reservoir management and field development planning. Traditional methods; such as numerical simulation, material balance equations coupled with well models, and advanced decline curve analysis (DCA) require extensive data and involve tradeoffs between accuracy and complexity. Recently, deep learning techniques, specifically artificial neural network (ANN) and recurrent neural network (RNN) models have demonstrated promising performance in this domain. This study applies ANN and RNN for production forecasting and compares their effectiveness with traditional DCA models.An actual production dataset for a gas field containing 13 wells is used to compare ANN and RNN models to traditional decline curve analysis models. After thorough data cleaning and preparation, the dataset was divided into training (historical) and test data sets. The training data was used to build ANN and RNN models in addition to estimating the model parameters for traditional DCA models. Traditional DCA models that were investigated include Arps (exponential, hyperbolic, and harmonic), Duong, Power Law Exponential Decline (PLE), and Stretched Exponential Decline (SEPD). Finally, the prediction of different models was evaluated and compared using the testing dataset. The comparison also included long, medium, and short-term prediction.Based on the comparison of daily gas production rate forecast, the ANN model demonstrated notable improvements over Arps models for the majority of wells, with a mean absolute percent error (MAPE) of 3.9% compared to 9.1%. However, across all wells, the average MAPE was slightly lower at 6.3% versus 8.1%. Conversely, the RNN model exhibited varied performance compared to Arps models. It showed poorer long-term performance (13% versus 4.5%), similar performance in the medium term (4.4% versus 4.2%), and superior performance in the short term (2.7% versus 4.2%). Furthermore, among DCA models specifically developed for unconventional wells, SEPD exhibited nearly identical performance to Arps (8.3% compared to 8.1%). In contrast, Duong model demonstrated the poorest performance with MAPE of 21.9%.This study provides insights into model analysis and introduces guidelines in using artificial and recurrent neural networks for predicting production rates. The ANN and RNN models were developed through careful testing with different designs and techniques. All the models in this research were created using Python scripts and powerful open-source libraries like SciPy and TensorFlow.

The echo state network (ESN) employs a huge reservoir with sparsely and randomly connected internal nodes and only trains the output weights, which avoids the suboptimal problem, exploding and vanishing gradients, high complexity and other disadvantages faced by traditional recurrent neural network (RNN) training. In light of the outstanding adaption to nonlinear dynamical systems, ESN has been applied into a wide range of applications. However, in the era of Big Data, with an enormous amount of data being generated continuously every day, the data are often distributed and stored in real applications, and thus the centralized ESN training process is prone to being technologically unsuitable. In order to achieve the requirement of Big Data applications in the real world, in this study we propose an algorithm and its implementation for distributed ESN training. The mentioned algorithm is based on the parallel particle swarm optimization (P-PSO) technique and the implementation uses Spark, a famous large-scale data processing framework. Four extremely large-scale datasets, including artificial benchmarks, real-world data and image data, are adopted to verify our framework on a stretchable platform. Experimental results indicate that the proposed work is accurate in the era of Big Data, regarding speed, accuracy and generalization capabilities.

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