Recursive Expectation Maximization Algorithm Research Articles

Online state and unknown inputs estimation for nonlinear systems with particle filter based recursive expectation‐maximization algorithm

AbstractThe article presents an innovative approach to simultaneously estimate states and unknown inputs (UIs) in nonlinear systems using a particle filter (PF) based recursive expectation‐maximization (EM) algorithm. This method is distinct from traditional iterative EM algorithms. During the E‐step, it calculates the Q‐function recursively within the maximum likelihood framework, while the PF estimates the system states. The M‐step involves local maximization of the recursive Q‐function to online estimate the UIs. The effectiveness of the PF‐based recursive EM algorithm is demonstrated with a numerical example, and comparisons with the augmented state PF are made to highlight its advantages. Finally, the proposed algorithm is implemented in a real application for the estimation of the continuous fermentation process.

Intelligent Modeling for Batch Polymerization Reactors with Unknown Inputs

While system identification methods have developed rapidly, modeling the process of batch polymerization reactors still poses challenges. Therefore, designing an intelligent modeling approach for these reactors is important. This paper focuses on identifying actual models for batch polymerization reactors, proposing a novel recursive approach based on the expectation-maximization algorithm. The proposed method pays special attention to unknown inputs (UIs), which may represent modeling errors or process faults. To estimate the UIs of the model, the recursive expectation-maximization (EM) technique is used. The proposed algorithm consists of two steps: the E-step and the M-step. In the E-step, a Q-function is recursively computed based on the maximum likelihood framework, using the UI estimates from the previous time step. The Kalman filter is utilized to calculate the estimates of the states using the measurements from sensor data. In the M-step, analytical solutions for the UIs are found through local optimization of the recursive Q-function. To demonstrate the effectiveness of the proposed algorithm, a practical application of modeling batch polymerization reactors is presented. The performance of the proposed recursive EM algorithm is compared to that of the augmented state Kalman filter (ASKF) using root mean squared errors (RMSEs). The RMSEs obtained from the proposed method are at least 6.52% lower than those from the ASKF method, indicating superior performance.

Identification of time-delay Markov jumps autoregressive system with recursive expectation maximum and convex optimization algorithm

A novel proposed approach of the recursive expectation–maximization (REM)–convex optimization algorithm is developed to solve the parameter identification problem of the Markov jump autoregressive exogenous (MJARX) systems with unknown time delays. First, the REM algorithm is employed for identifying the unknown time delay of the system, based on which the time-delay sequence is obtained. Subsequently, the model parameter vector is determined by the convex optimization algorithm. Finally, the transition probability matrix and variance are calculated based on the minimized mean square error. A numerical example and simulated continuous fermentation reactor process are implemented to demonstrate the effectiveness of the proposed algorithm. The identification results are better than the expectation–maximization algorithm and REM algorithm results.

Offline and Online Parameter Learning for Switching Multirate Processes With Varying Delays and Integrated Measurements

It is difficult to measure properties of certain key/quality variables at a fast rate due to technical constraints or economical considerations. Such variables are measured infrequently through laboratory analysis with a considerable delay. Also, sample collection for laboratory analysis may be extended over a significant time interval. In this article, the objective is to solve the parameter estimation problem along with real-time output prediction for switching multirate sampled processes with unknown varying delays and unknown varying sampling intervals. First, under the framework of the expectation–maximization (EM) algorithm, offline parameter estimation problem of dual-rate switching augmented regression models is handled. The delays, sampling intervals, and operating modes are considered as the hidden variables modeled by a nonparametric-distribution-based approach. In addition, as the fast-rate prediction of the slow-rate sampled variables is often used for process control applications, a recursive EM algorithm is used to predict the fast-rate outputs in real time. The efficacy of the proposed algorithms is shown through an experimental study on a laboratory hybrid tank system. The results show a satisfactory fast-rate prediction of the slow-rate sampled variables, and the significance of considering different sampling intervals is highlighted. Also, estimates of the occurrence probabilities of each possible delay, sampling interval, and mode are obtained without being limited by the assumption of a prior distribution.

A recursive expectation-maximization algorithm for speaker tracking and separation

The problem of blind and online speaker localization and separation using multiple microphones is addressed based on the recursive expectation-maximization (REM) procedure. A two-stage REM-based algorithm is proposed: (1) multi-speaker direction of arrival (DOA) estimation and (2) multi-speaker relative transfer function (RTF) estimation. The DOA estimation task uses only the time frequency (TF) bins dominated by a single speaker while the entire frequency range is not required to accomplish this task. In contrast, the RTF estimation task requires the entire frequency range in order to estimate the RTF for each frequency bin. Accordingly, a different statistical model is used for the two tasks. The first REM model is applied under the assumption that the speech signal is sparse in the TF domain, and utilizes a mixture of Gaussians (MoG) model to identify the TF bins associated with a single dominant speaker. The corresponding DOAs are estimated using these bins. The second REM model is applied under the assumption that the speakers are concurrently active in all TF bins and consequently applies a multichannel Wiener filter (MCWF) to separate the speakers. As a result of the assumption of the concurrent speakers, a more precise TF map of the speakers’ activity is obtained. The RTFs are estimated using the outputs of the MCWF-beamformer (BF), which are constructed using the DOAs obtained in the previous stage. Next, using the linearly constrained minimum variance (LCMV)-BF that utilizes the estimated RTFs, the speech signals are separated. The algorithm is evaluated using real-life scenarios of two speakers. Evaluation of the mean absolute error (MAE) of the estimated DOAs and the separation capabilities, demonstrates significant improvement w.r.t. a baseline DOA estimation and speaker separation algorithm.

Online state and inputs identification for stochastic systems using recursive expectation-maximization algorithm

In this paper, an online joint state estimation and unknown inputs (UIs) identification approach for industrial processes represented by the state-space model is proposed. The UIs identification is achieved by applying the recursive expectation-maximization (REM) technique. In E-step, a recursively calculated Q-function is derived based on the maximum likelihood framework, and the Kalman filter (KF) is adopted to estimate the states. In M-step, analytical solutions for UIs are obtained via locally maximizing the recursive Q-function. A numerical example of a quadrupled water tank process and practice application to system modeling of a distillation tower are employed to illustrate the proposed REM-KF algorithm's effectiveness. It is also demonstrated that the REM-KF algorithm is more accurate than existing online solutions.

Robust identification of linear ARX models with recursive EM algorithm based on Student’s t-distribution

This paper considers the robust identification issue of linear systems represented by autoregressive exogenous models using the recursive expectation-maximization (EM) algorithm. In this paper, a recursive Q-function is formulated based on the maximum likelihood principle. Meanwhile, the outliers that frequently appear in practical processes are accommodated with the Student’s t-distribution. The parameter vector, variance of noise, and the degree of freedom are recursively estimated. Finally, a numerical example, as well as a simulated continuous stirred tank reactor (CSTR) system, is performed to verify the effectiveness of the proposed recursive EM algorithm.

Nonnegative HMM for Babble Noise Derived From Speech HMM: Application to Speech Enhancement

Deriving a good model for multitalker babble noise can facilitate different speech processing algorithms, e.g. noise reduction, to reduce the so-called cocktail party difficulty. In the available systems, the fact that the babble waveform is generated as a sum of N different speech waveforms is not exploited explicitly. In this paper, first we develop a gamma hidden Markov model for power spectra of the speech signal, and then formulate it as a sparse nonnegative matrix factorization (NMF). Second, the sparse NMF is extended by relaxing the sparsity constraint, and a novel model for babble noise (gamma nonnegative HMM) is proposed in which the babble basis matrix is the same as the speech basis matrix, and only the activation factors (weights) of the basis vectors are different for the two signals over time. Finally, a noise reduction algorithm is proposed using the derived speech and babble models. All of the stationary model parameters are estimated using the expectation-maximization (EM) algorithm, whereas the time-varying parameters, i.e. the gain parameters of speech and babble signals, are estimated using a recursive EM algorithm. The objective and subjective listening evaluations show that the proposed babble model and the final noise reduction algorithm significantly outperform the conventional methods.

Recursive online EM estimation of mixture autoregressions

This paper develops a recursive expectation–maximization (REM) algorithm for estimating a mixture autoregression (MAR) with an independent and identically distributed regime transition process. The proposed method, which is useful for long time series as well as for data available in real time, follows a recursive predictor error-type scheme. Based on a slightly modified system to the expectation–maximization (EM) equations for an MAR model, the REM algorithm consists of two steps at each iteration: the expectation step, in which the current unobserved regime transition is estimated from new data using previous recursive estimates, and the minimization step, in which the MAR parameter estimates are recursively updated following a minimization direction. Details of implementation of the REM algorithm are given and its finite-sample performance is shown via simulation experiments. In particular, the EM and REM provide roughly similar estimates, especially for moderate and long time series.

Online Updating With a Probability-Based Prediction Model Using Expectation Maximization Algorithm for Reliability Forecasting

Recently, a novel prediction model based on the evidential reasoning (ER) approach is developed to forecast reliability in engineering systems. In order to determine the parameters of the ER-based prediction model, some optimization models have been proposed to train the ER-based prediction model. However, these models are implemented in an offline fashion and thus it is very expensive to train and retrain them when new information is available. This correspondence paper is concerned with developing the recursive algorithms for updating the ER-based prediction model from the probability-based point of view. Using the recursive expectation maximization algorithm, two recursive algorithms are proposed for updating the parameters of the ER-based prediction model under judgmental and numerical outputs, respectively. As such, the proposed algorithms can be used to fine tune the ER-based prediction model online once new information becomes available. We verify the proposed method via a realistic example with missile reliability data.

Maximum Likelihood State Estimation of Semi-Markovian Switching System in Non-Gaussian Measurement Noise

In the work presented here, we consider state and parameter estimation of a semi-nonlinear Markov jump system in a non-Gaussian noise environment. The non-Gaussian measurement noise is approximated by a finite Gaussian mixture model (GMM). We propose a maximum likelihood (ML) solution to this state estimation problem which leads to two expectation-maximization (EM) algorithms. The first is a batch EM method which takes all the available data in the conditional expectation of the state in the E-step. An interacting multiple model (IMM) smoother is employed to evaluate the conditional expectation of the state by which a suboptimal estimate of system state is directly obtained. The Gaussian mixture parameters are then updated in the M-step. The second is a recursive EM algorithm which results from a stochastic approximation procedure and uses a standard IMM filter. For performance evaluation, posterior Cramer-Rao bound (PCRB) on the state estimation is adopted. Simulation results verify the effectiveness of the proposed algorithms.

HMM-Based Gain Modeling for Enhancement of Speech in Noise

Accurate modeling and estimation of speech and noise gains facilitate good performance of speech enhancement methods using data-driven prior models. In this paper, we propose a hidden Markov model (HMM)-based speech enhancement method using explicit gain modeling. Through the introduction of stochastic gain variables, energy variation in both speech and noise is explicitly modeled in a unified framework. The speech gain models the energy variations of the speech phones, typically due to differences in pronunciation and/or different vocalizations of individual speakers. The noise gain helps to improve the tracking of the time-varying energy of nonstationary noise. The expectation-maximization (EM) algorithm is used to perform offline estimation of the time-invariant model parameters. The time-varying model parameters are estimated online using the recursive EM algorithm. The proposed gain modeling techniques are applied to a novel Bayesian speech estimator, and the performance of the proposed enhancement method is evaluated through objective and subjective tests. The experimental results confirm the advantage of explicit gain modeling, particularly for nonstationary noise sources

Adaptive nonlinear filters for narrow-band interference suppression in spread-spectrum CDMA systems

This paper presents a novel nonlinear filter and parameter estimator for narrow band interference suppression in code division multiple access spread-spectrum systems. As in the article by Rusch and Poor (1994), the received sampled signal is modeled as the sum of the spread-spectrum signal (modeled as a finite state independently identically distributed (i.i.d.) process-here we generalize to a finite state Markov chain), narrow-band interference (modeled as a Gaussian autoregressive process), and observation noise (modeled as a zero-mean white Gaussian process). The proposed algorithm combines a recursive hidden Markov model (HMM) estimator, Kalman filter (KF), and the recursive expectation maximization algorithm. The nonlinear filtering techniques for narrow-band interference suppression presented in Rusch and Poor and our proposed HMM-KF algorithm have the same computational cost. Detailed simulation studies show that the HMM-KF algorithm outperforms the filtering techniques in Rusch and Poor. In particular, significant improvements in the bit error rate and signal-to-noise ratio (SNR) enhancement are obtained in low to medium SNR. Furthermore, in simulation studies we investigate the effect on the performance of the HMM-KF and the approximate conditional mean (ACM) filter in the paper by Rusch and Poor, when the observation noise variance is increased. As expected, the performance of the HMM-KF and ACM algorithms worsen with increasing observation noise and number of users. However, HMM-KF significantly outperforms ACM in medium to high observation noise.

EM-based recursive estimation of channel parameters

Recursive (online) expectation-maximization (EM) algorithm along with stochastic approximation is employed in this paper to estimate unknown time-invariant/variant parameters. The impulse response of a linear system (channel) is modeled as an unknown deterministic vector/process and as a Gaussian vector/process with unknown stochastic characteristics. Using these models which are embedded in white or colored Gaussian noise, different types of recursive least squares (RLS), Kalman filtering and smoothing and combined RLS and Kalman-type algorithms are derived directly from the recursive EM algorithm. The estimation of unknown parameters also generates new recursive algorithms for situations, such as additive colored noise modeled by an autoregressive process. The recursive EM algorithm is shown as a powerful tool which unifies the derivations of many adaptive estimation methods.

Estimation of Markov-modulated time-series via EM algorithm

We consider the estimation of various Markov-modulated time series. We obtain maximum likelihood estimates of the time-series parameters including the Markov chain transition probabilities and the time-series coefficients using the expectation maximization (EM) algorithm. In addition, the recursive EM algorithm is used to obtain on-line parameter estimates. Simulation studies show that both algorithms yield satisfactory results. >