Impact of noise on a dynamical system: prediction and uncertainties from a swarm-optimized neural network

In this study, an artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey--Glass chaotic time series in the short-term $x(t+6)$. The performance prediction was evaluated and compared with another studies available in the literature. Also, we presented properties of the dynamical system via the study of chaotic behaviour obtained from the predicted time series. Next, the hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called {\it stochastic} hybrid ANN+PSO) in order to obtain a new estimator of the predictions, which also allowed us to compute uncertainties of predictions for noisy Mackey--Glass chaotic time series. Thus, we studied the impact of noise for several cases with a white noise level ($\sigma_{N}$) from 0.01 to 0.1.Comment: 11 pages, 8 figure

Recognizing recurrent neural networks (rRNN): Bayesian inference for recurrent neural networks

Recurrent neural networks (RNNs) are widely used in computational neuroscience and machine learning applications. In an RNN, each neuron computes its output as a nonlinear function of its integrated input. While the importance of RNNs, especially as models of brain processing, is undisputed, it is also widely acknowledged that the computations in standard RNN models may be an over-simplification of what real neuronal networks compute. Here, we suggest that the RNN approach may be made both neurobiologically more plausible and computationally more powerful by its fusion with Bayesian inference techniques for nonlinear dynamical systems. In this scheme, we use an RNN as a generative model of dynamic input caused by the environment, e.g. of speech or kinematics. Given this generative RNN model, we derive Bayesian update equations that can decode its output. Critically, these updates define a 'recognizing RNN' (rRNN), in which neurons compute and exchange prediction and prediction error messages. The rRNN has several desirable features that a conventional RNN does not have, for example, fast decoding of dynamic stimuli and robustness to initial conditions and noise. Furthermore, it implements a predictive coding scheme for dynamic inputs. We suggest that the Bayesian inversion of recurrent neural networks may be useful both as a model of brain function and as a machine learning tool. We illustrate the use of the rRNN by an application to the online decoding (i.e. recognition) of human kinematics

Online Natural Gradient as a Kalman Filter

We cast Amari's natural gradient in statistical learning as a specific case of Kalman filtering. Namely, applying an extended Kalman filter to estimate a fixed unknown parameter of a probabilistic model from a series of observations, is rigorously equivalent to estimating this parameter via an online stochastic natural gradient descent on the log-likelihood of the observations. In the i.i.d. case, this relation is a consequence of the "information filter" phrasing of the extended Kalman filter. In the recurrent (state space, non-i.i.d.) case, we prove that the joint Kalman filter over states and parameters is a natural gradient on top of real-time recurrent learning (RTRL), a classical algorithm to train recurrent models. This exact algebraic correspondence provides relevant interpretations for natural gradient hyperparameters such as learning rates or initialization and regularization of the Fisher information matrix.Comment: 3rd version: expanded intr

Inferring unknown unknowns: Regularized bias-aware ensemble Kalman filter

Because of physical assumptions and numerical approximations, low-order models are affected by uncertainties in the state and parameters, and by model biases. Model biases, also known as model errors or systematic errors, are difficult to infer because they are `unknown unknowns', i.e., we do not necessarily know their functional form a priori. With biased models, data assimilation methods may be ill-posed because either (i) they are 'bias-unaware' because the estimators are assumed unbiased, (ii) they rely on an a priori parametric model for the bias, or (iii) they can infer model biases that are not unique for the same model and data. First, we design a data assimilation framework to perform combined state, parameter, and bias estimation. Second, we propose a mathematical solution with a sequential method, i.e., the regularized bias-aware ensemble Kalman Filter (r-EnKF), which requires a model of the bias and its gradient (i.e., the Jacobian). Third, we propose an echo state network as the model bias estimator. We derive the Jacobian of the network, and design a robust training strategy with data augmentation to accurately infer the bias in different scenarios. Fourth, we apply the r-EnKF to nonlinearly coupled oscillators (with and without time-delay) affected by different forms of bias. The r-EnKF infers in real-time parameters and states, and a unique bias. The applications that we showcase are relevant to acoustics, thermoacoustics, and vibrations; however, the r-EnKF opens new opportunities for combined state, parameter and bias estimation for real-time and on-the-fly prediction in nonlinear systems.Comment: 22 Figure

Wind Power Forecasting Based on Echo State Networks and Long Short-Term Memory

Wind power generation has presented an important development around the world. However, its integration into electrical systems presents numerous challenges due to the variable nature of the wind. Therefore, to maintain an economical and reliable electricity supply, it is necessary to accurately predict wind generation. The Wind Power Prediction Tool (WPPT) has been proposed to solve this task using the power curve associated with a wind farm. Recurrent Neural Networks (RNNs) model complex non-linear relationships without requiring explicit mathematical expressions that relate the variables involved. In particular, two types of RNN, Long Short-Term Memory (LSTM) and Echo State Network (ESN), have shown good results in time series forecasting. In this work, we present an LSTM+ESN architecture that combines the characteristics of both networks. An architecture similar to an ESN is proposed, but using LSTM blocks as units in the hidden layer. The training process of this network has two key stages: (i) the hidden layer is trained with a descending gradient method online using one epoch; (ii) the output layer is adjusted with a regularized regression. In particular, the case is proposed where Step (i) is used as a target for the input signal, in order to extract characteristics automatically as the autoencoder approach; and in the second stage (ii), a quantile regression is used in order to obtain a robust estimate of the expected target. The experimental results show that LSTM+ESN using the autoencoder and quantile regression outperforms the WPPT model in all global metrics used

Deep Learning for Environmentally Robust Speech Recognition: An Overview of Recent Developments

Eliminating the negative effect of non-stationary environmental noise is a long-standing research topic for automatic speech recognition that stills remains an important challenge. Data-driven supervised approaches, including ones based on deep neural networks, have recently emerged as potential alternatives to traditional unsupervised approaches and with sufficient training, can alleviate the shortcomings of the unsupervised methods in various real-life acoustic environments. In this light, we review recently developed, representative deep learning approaches for tackling non-stationary additive and convolutional degradation of speech with the aim of providing guidelines for those involved in the development of environmentally robust speech recognition systems. We separately discuss single- and multi-channel techniques developed for the front-end and back-end of speech recognition systems, as well as joint front-end and back-end training frameworks