Expectation-maximization Procedure Research Articles

A Flexible EM-Like Clustering Algorithm for Noisy Data.

Though very popular, it is well known that the Expectation-Maximisation (EM) algorithm for the Gaussian mixture model performs poorly for non-Gaussian distributions or in the presence of outliers or noise. In this paper, we propose a Flexible EM-like Clustering Algorithm (FEMCA): a new clustering algorithm following an EM procedure is designed. It is based on both estimations of cluster centers and covariances. In addition, using a semi-parametric paradigm, the method estimates an unknown scale parameter per data point. This allows the algorithm to accommodate heavier tail distributions, noise, and outliers without significantly losing efficiency in various classical scenarios. We first present the general underlying model for independent, but not necessarily identically distributed, samples of elliptical distributions. We then derive and analyze the proposed algorithm in this context, showing in particular important distribution-free properties of the underlying data distributions. The algorithm convergence and accuracy properties are analyzed by considering the first synthetic data. Finally, we show that FEMCA outperforms other classical unsupervised methods of the literature, such as k-means, EM for Gaussian mixture models, and its recent modifications or spectral clustering when applied to real data sets as MNIST, NORB, and 20newsgroups.

A comparison of three algorithms in the filtering of a Markov-modulated non-homogeneous Poisson process

ABSTRACT A Markov-modulated non-homogeneous Poisson process (MMNPP), whose intensity process is designed to capture both the cyclical and nonrecurring trends, is considered for modelling the total count of cyber incidents. Extending the Expectation-Maximisation (EM) algorithm for the current MMPP literature, we derive the filters and smoothers to support the MMNPP online parameter estimation. A scaling transformation is introduced to address the numerical issue for large data sizes whilst maintaining accuracy. The filter- and smoother-based EM algorithms are then benchmarked to the maximum likelihood-based EM algorithm at the theoretical level. The differences emerge in the E-step of the EM procedure. Both the filtering and smoothing schemes, in conjunction with the change-of-measure technique, avoid the computing complication caused by the hidden regimes. In contrast to the usual EM algorithm, the said two algorithms could be implemented given only the incident counts data without the specific times of jumps. Within the data compiled by the U.S. Department of Health and Human Services, the filter-based algorithm performs better than the algorithm involving smoothers. The benchmarked algorithm may do well in calibration under the presence of extreme incident counts with an extremely low frequency; however, overfitting may occur. For most practical applications involving 2 or 3 regimes, both algorithms are superior when it comes to efficiency, real-time update, and low computational cost. The benchmarked algorithm is better when there are more regimes under relatively closer intensities. Overall, the filter-based algorithm gives better estimation, especially if there is a low-frequency regime and the flexible binning of the data set is an important consideration.

A Dirichlet Process Mixture of Robust Task Models for Scalable Lifelong Reinforcement Learning.

While reinforcement learning (RL) algorithms are achieving state-of-the-art performance in various challenging tasks, they can easily encounter catastrophic forgetting or interference when faced with lifelong streaming information. In this article, we propose a scalable lifelong RL method that dynamically expands the network capacity to accommodate new knowledge while preventing past memories from being perturbed. We use a Dirichlet process mixture to model the nonstationary task distribution, which captures task relatedness by estimating the likelihood of task-to-cluster assignments and clusters the task models in a latent space. We formulate the prior distribution of the mixture as a Chinese restaurant process (CRP) that instantiates new mixture components as needed. The update and expansion of the mixture are governed by the Bayesian nonparametric framework with an expectation maximization (EM) procedure, which dynamically adapts the model complexity without explicit task boundaries or heuristics. Moreover, we use the domain randomization technique to train robust prior parameters for the initialization of each task model in the mixture; thus, the resulting model can better generalize and adapt to unseen tasks. With extensive experiments conducted on robot navigation and locomotion domains, we show that our method successfully facilitates scalable lifelong RL and outperforms relevant existing methods.

Unsupervised Classification under Uncertainty: The Distance-Based Algorithm

This paper presents a method for unsupervised classification of entities by a group of agents with unknown domains and levels of expertise. In contrast to the existing methods based on majority voting (“wisdom of the crowd”) and their extensions by expectation-maximization procedures, the suggested method first determines the levels of the agents’ expertise and then weights their opinions by their expertise level. In particular, we assume that agents will have relatively closer classifications in their field of expertise. Therefore, the expert agents are recognized by using a weighted Hamming distance between their classifications, and then the final classification of the group is determined from the agents’ classifications by expectation-maximization techniques, with preference to the recognized experts. The algorithm was verified and tested on simulated and real-world datasets and benchmarked against known existing algorithms. We show that such a method reduces incorrect classifications and effectively solves the problem of unsupervised collaborative classification under uncertainty, while outperforming other known methods.

HMM-Based Joint Modeling of Condition Monitoring Signals and Failure Event Data for Prognosis

Accurate estimation of remaining useful life (RUL) of a unit is critical to fulfill reliability commitments. In the presence of hard failures (i.e., absence of a predefined failure threshold), accurate prognosis of RUL using condition monitoring (CM) signals becomes challenging. To tackle this problem, we present a prognostic framework by jointly modeling CM signals and failure event data. Development of the presented method depends on the idea that while the unit operates, it continually degrades through a series of hidden states and the CM signals are functionally related to this hidden failure process. The unit fails once the hidden failure process reaches a dead state. Through this modeling, requirement of a failure threshold on CM signals is eliminated. We provide a modified expectation-maximization procedure to estimate parameters, and through a comprehensive set of numerical as well as real-world experiments, we demonstrate superior prognosis performance against some benchmark methods.

Parameter Estimation of KST-IRT Model under Local Dependence

A mantra often repeated in the introductory material to psychometrics and Item Response Theory (IRT) is that a Rasch model is a probabilistic version of a Guttman scale. The idea comes from the observation that a sigmoidal item response function provides a probabilistic version of the characteristic function that models an item response in the Guttman scale. It appears, however, more difficult to reconcile the assumption of local independence, which traditionally accompanies the Rasch model, with the item dependence existing in a Guttman scale. In recent work, an alternative probabilistic version of a Guttman scale was proposed, combining Knowledge Space Theory (KST) with IRT modeling, here referred to as KST-IRT. The present work has, therefore, a two-fold aim. Firstly, the estimation of the parameters involved in KST-IRT models is discussed. More in detail, two estimation methods based on the Expectation Maximization (EM) procedure are suggested, i.e., Marginal Maximum Likelihood (MML) and Gibbs sampling, and are compared on the basis of simulation studies. Secondly, for a Guttman scale, the estimates of the KST-IRT models are compared with those of the traditional combination of the Rasch model plus local independence under the interchange of the data generation processes. Results show that the KST-IRT approach might be more effective in capturing local dependence as it appears to be more robust under misspecification of the data generation process, but it comes with the price of an increased number of parameters.

Neural-Network-Based Pose Estimation During Noncooperative Spacecraft Rendezvous Using Point Cloud

The article presents Deep Coherent Point Drift (DeepCPD), a neural-network approach for estimating the six-degrees-of-freedom pose of noncooperative spacecraft during autonomous rendezvous and docking through point cloud data. The method registers unorganized scan point clouds with their reference model point clouds. DeepCPD replaces the Expectation-Maximization procedure in the Gaussian Mixture Model registration algorithm with a neural network that learns point-to-component correspondence, achieving better estimation performance and acceleration of the registration process. The proposed method is also robust to perturbation, corruption, occlusion, and distance, as validated by simulated experimental results. Our code will be available at https://github.com/Zhang-CV/DeepCPD.

Modeling Emotion Dynamics in Song Lyrics with State Space Models

Abstract Most previous work in music emotion recognition assumes a single or a few song-level labels for the whole song. While it is known that different emotions can vary in intensity within a song, annotated data for this setup is scarce and difficult to obtain. In this work, we propose a method to predict emotion dynamics in song lyrics without song-level supervision. We frame each song as a time series and employ a State Space Model (SSM), combining a sentence-level emotion predictor with an Expectation-Maximization (EM) procedure to generate the full emotion dynamics. Our experiments show that applying our method consistently improves the performance of sentence-level baselines without requiring any annotated songs, making it ideal for limited training data scenarios. Further analysis through case studies shows the benefits of our method while also indicating the limitations and pointing to future directions.

Multimodal probabilistic modeling of melt pool geometry variations in additive manufacturing

Laser Powder Bed Fusion (LPBF) is commonly used to fabricate metal additive manufacturing (AM) parts. Advanced sensing empowers the collection of in-situ signals (e.g., melt pools) for real-time monitoring of AM processes. The variations of melt pool geometry are closely pertinent to mechanical and functional properties of final AM builds. Most of previous works are concerned about the extraction of melt pool features (e.g., geometric statistics, size, shape descriptors) for AM process monitoring. However, very little has been done to investigate the emission physics of photons that leads to the generation of melt pool images. To fill the gap, this work presents a novel approach to simulate the emission of photons for statistical estimation and modeling of the multimodal probability distribution function (PDF) of a melt pool. Specifically, the complex geometry of a melt pool is represented as a multimodal PDF. Photons emitted to generate melt pool images are treated as independent and identically distributed (iid) random samples drawn from the multimodal PDF. Further, we propose a new Gaussian mixture model to represent and estimate the multimodal PDF via the expectation–maximization procedures. In addition, we investigate how process conditions influence the geometric variations of melt pools. Experimental results show that the proposed methodology effectively builds the multimodal distribution model of melt pool geometric variations. Most importantly, the proposed methodology is flexible enough to take raw images as the input and shows great potentials to be generally applicable for different types of melt pool images and AM processes.

Regularization of Mixture Models for Robust Principal Graph Learning.

A regularized version of Mixture Models is proposed to learn a principal graph from a distribution of D-dimensional datapoints. In the particular case of manifold learning for ridge detection, we assume that the underlying structure can be modeled as a graph acting like a topological prior for the Gaussian clusters turning the problem into a maximum a posteriori estimation. Parameters of the model are iteratively estimated through an Expectation-Maximization procedure making the learning of the structure computationally efficient with guaranteed convergence for any graph prior in a polynomial time. We also embed in the formalism a natural way to make the algorithm robust to outliers of the pattern and heteroscedasticity of the manifold sampling coherently with the graph structure. The method uses a graph prior given by the minimum spanning tree that we extend using random sub-samplings of the dataset to take into account cycles that can be observed in the spatial distribution.

Decision Feedback Modulation Recognition with Channel Estimation for Amplify and Forward Two-Path Consecutive Relaying Systems.

Automatic modulation recognition (AMR) is an essential component in the design of smart radios that can intelligently communicate with their surroundings in order to make the most efficient use of available resources. Throughout the last few decades, this issue has been subjected to in-depth examination in the published research literature. To the best of the authors’ knowledge, there have only been a few studies that have been specifically devoted to the task of performing AMR across cooperative wireless transmissions. In this contribution, we examine the AMR problem in the context of amplify-and-forward (AAF) two-path consecutive relaying systems (TCRS) for the first time in the literature. We leverage the property of data redundancy associated with AAF-TCRS signals to design a decision feedback iterative modulation recognizer via an expectation-maximization procedure. The proposed recognizer incorporates the soft information produced by the data detection process as a priori knowledge to generate the a posteriori expectations of the information symbols, which are employed as training symbols. The proposed algorithm additionally involves the development of an estimate of the channel coefficients as a secondary activity. The simulation outcomes have validated the feasibility of the proposed design by demonstrating its capacity to achieve an excellent recognition performance under a wide range of running conditions. According to the findings, the suggested technique converges within six rounds, achieving perfect recognition performance at a signal-to-noise ratio of 14 dB. Furthermore, the minimal pilot-to-frame-size ratio necessary to successfully execute the iterative procedure is 0.07. In addition, the proposed method is essentially immune to time offset and performs well throughout a broad range of frequency offset. Lastly, the proposed strategy beats the existing techniques in recognition accuracy while requiring a low level of processing complexity.

Clustering acoustic emission data streams with sequentially appearing clusters using mixture models

The interpretation of unlabeled acoustic emission (AE) data classically relies on general-purpose clustering methods. While several criteria have been used in the past to select the hyperparameters of those algorithms, few studies have paid attention to the development of dedicated objective functions in clustering methods able to cope with the specificities of AE data. We investigate how to explicitly represent clusters onsets in mixture models in general, and in Gaussian Mixture Models (GMM) in particular. We propose the first clustering method able to provide, through parameters estimated by an expectation–maximization procedure, information about when clusters occur (onsets), how they grow (kinetics) and their level of activation through time. This new objective function accommodates continuous timestamps of AE signals and, thus, their order of occurrence. The method, called GMMSEQ, is experimentally validated to characterize the loosening phenomenon in bolted structure under vibrations. A comparison with four standard clustering methods on raw streaming data from five experimental campaigns shows that GMMSEQ not only provides useful qualitative information about the timeline of clusters, but also shows better performance in terms of cluster characterization.

Planar Array Diagnosis Based on Bayesian Learning With a Bernoulli–Gaussian Prior Model

A new approach is proposed for the planar array failure diagnosis from a set of far-field measurements. Under the framework of Bayesian learning, a Bernoulli–Gaussian (BG) model is constructed to exploit the binary prior information of the unknown failure elements. Subsequently, the variational expectation maximization (EM) strategy is used to perform the Bayesian inference. Moreover, the damped generalized approximate message passing (GAMP) is further adopted to circumvent the matrix inversion in the variational EM procedure, thus making the overall process computationally very efficient. A set of representative numerical experiments are provided to validate the accuracy, efficiency, and flexibility of the proposed method.

Power of Error Correcting Codes for SFBC-OFDM Classification Over Unknown Channels

The development of intelligent radios in wireless applications is mainly driven by the growing need for higher data rates, along with constrained spectrum resources. An intelligent radio is one that can autonomously assess the communication environment and automatically update the communication parameters to achieve optimal performance. The problem of determining the type of space-frequency block coding (SFBC) for orthogonal frequency division multiplexing (OFDM) transmissions is one of the main tasks of an intelligent receiver. Previous approaches to this problem are restricted to uncoded communications; nevertheless, existing systems typically utilize error-correcting codes. This study develops a maximum-likelihood (ML) classifier that discriminates among SFBC-OFDM signals using the soft outputs of a channel decoder. The mathematical analysis shows that the maximization of the likelihood function can be carried out by employing an iterative expectation-maximization (EM) procedure. A channel estimator is also included in the proposed classifier as a vital step. The findings show that the classification performance of the proposed algorithm is considerably better than the classical classifiers reported in the literature, at the cost of an acceptable increase in computing complexity.

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.

Improving Gradient Flow with Unrolled Highway Expectation Maximization

Integrating model-based machine learning methods into deep neural architectures allows one to leverage both the expressive power of deep neural nets and the ability of model-based methods to incorporate domain-specific knowledge. In particular, many works have employed the expectation maximization (EM) algorithm in the form of an unrolled layer-wise structure that is jointly trained with a backbone neural network. However, it is difficult to discriminatively train the backbone network by backpropagating through the EM iterations as they are prone to the vanishing gradient problem. To address this issue, we propose Highway Expectation Maximization Networks (HEMNet), which is comprised of unrolled iterations of the generalized EM (GEM) algorithm based on the Newton-Rahpson method. HEMNet features scaled skip connections, or highways, along the depths of the unrolled architecture, resulting in improved gradient flow during backpropagation while incurring negligible additional computation and memory costs compared to standard unrolled EM. Furthermore, HEMNet preserves the underlying EM procedure, thereby fully retaining the convergence properties of the original EM algorithm. We achieve significant improvement in performance on several semantic segmentation benchmarks and empirically show that HEMNet effectively alleviates gradient decay.

Sequential Sampling and Estimation of Approximately Bandlimited Graph Signals.

Graph signal sampling has been widely studied in recent years, but the accurate signal models required by most of the existing sampling methods are usually unavailable prior to any observations made in a practical environment. In this paper, a sequential sampling and estimation algorithm is proposed for approximately bandlimited graph signals, in the absence of prior knowledge concerning signal properties. We approach the problem from a Bayesian perspective in which we formulate the signal prior by a multivariate Gaussian distribution with unknown hyperparameters. To overcome the interconnected problems associated with the parameter estimation, in the proposed algorithm, hyperparameter estimation and sample selection are performed in an alternating way. At each step, the unknown hyperparameters are updated by an expectation maximization procedure based on historical observations, and then the next node in the sampling operation is chosen by uncertainty sampling with the latest hyperparameters. We prove that under some specific conditions, signal estimation in the proposed algorithm is consistent. Subsequent validation of the approach through simulations shows that the proposed procedure yields performances which are significantly better than existing state-of-the-art approaches notwithstanding the additional attribute of robustness in the presence of a broad range of signal attributes.

Turbo Modulation Identification Algorithm for OFDM Software-Defined Radios

The problem of automatic modulation identification (AMI) is one of the main challenges that faces orthogonal frequency division multiplexing (OFDM) software-defined radio (SDR) systems. The previously reported studies of this aspect have limited to uncoded transmission; however, practical systems typically employ error-correcting codes. In this work, we exploit the properties of bit-interleaved coded modulation iterative decoding (BICM-ID) schemes, which are widely used in wireless communications systems, to develop a novel AMI algorithm for OFDM-SDR systems. Since the exact maximum-likelihood (ML) solution is intractable in practice, we resort to an expectation-maximization (EM) procedure as a statistical analysis tool. The proposed identification algorithm works in an iterative fashion, in which the soft information provided by the channel decoder of BICM-ID scheme is utilized as a priori information to the proposed identifier. The results indicate that the identification performance of the proposed algorithm achieves a significant improvement with iterations and outperforms that of the conventional uncoded algorithms. These favorable results are achieved with reduced processing time and acceptable sensitivity to transmission impairments, such as clock timing offset, frequency offset, Doppler shift, and phase noise.

MmWave Channel Estimation via Compressive Covariance Estimation: Role of Sparsity and Intra-Vector Correlation

In this work, we address the problem of multiple-input multiple-output mmWave channel estimation in a hybrid analog-digital architecture, by exploiting both the underlying spatial sparsity as well as the spatial correlation in the channel. We accomplish this via compressive covariance estimation, where we estimate the channel covariance matrix from noisy low dimensional projections of the channel obtained in the pilot transmission phase. We use the estimated covariance matrix as a plug-in to the linear minimum mean square estimator to obtain the channel estimate. We present a new Gaussian prior model, inspired by sparse Bayesian learning (SBL), which incorporates parameters to capture the channel correlation in addition to sparsity. Based on this prior, we develop the Corr-SBL algorithm, which uses an expectation maximization procedure to learn the parameters of the prior and update the posterior channel estimates. A closed form solution is obtained for the maximization step based on fixed-point iterations. To facilitate practical implementation, an online version of the algorithm is developed which significantly reduces the latency at a marginal loss in performance. The efficacy of the prior model is studied by analyzing the normalized mean squared error in the channel estimate. Our results show that, when compared to a genie-aided estimator and other existing sparse recovery algorithms, exploiting both sparsity and correlation results in significant performance gains, even under imperfect covariance estimates obtained using a limited number of samples.

Multifractal point processes and the spatial distribution of wildfires in French Mediterranean regions

We introduce a simple and wide class of multifractal spatial point patterns as Cox processes which intensity is multifractal, i.e., the class of Poisson processes with a stochastic intensity corresponding to a random multifractal measure. We then propose a maximum likelihood approach by means of a standard Expectation–Maximization procedure in order to estimate the distribution of these intensities at all scales. This provides, as validated on various numerical examples, a simple framework to estimate the scaling laws and therefore the multifractal properties for this class of spatial point processes. The wildfire distribution gathered in the Prométhée French Mediterranean wildfire database is investigated within this approach that notably allows us to compute the statistical moments associated with the spatial distribution of annual likelihood of fire event occurrence. We show that for each order q, these moments display a well defined scaling behavior with a non-linear spectrum of scaling exponents ζq. From our study, it thus appears that the spatial distribution of the wildfire ignition annual risk can be described by a non-trivial, multifractal singularity spectrum and that this risk cannot be reduced to providing a number of events per km2. Our analysis is confirmed by a direct spatial correlation estimation of the intensity logarithms whose the peculiar slowly decreasing shape corresponds to the hallmark of multifractal cascades. The multifractal features appear to be constant over time and similar over the three regions that are studied.