Statistical signal processing applications usually require the estimation of some parameters of interest given a set of observed data. These estimates are typically obtained either by solving a ...multi-variate optimization problem, as in the maximum likelihood (ML) or maximum a posteriori (MAP) estimators, or by performing a multi-dimensional integration, as in the minimum mean squared error (MMSE) estimators. Unfortunately, analytical expressions for these estimators cannot be found in most real-world applications, and the Monte Carlo (MC) methodology is one feasible approach. MC methods proceed by drawing random samples, either from the desired distribution or from a simpler one, and using them to compute consistent estimators. The most important families of MC algorithms are the Markov chain MC (MCMC) and importance sampling (IS). On the one hand, MCMC methods draw samples from a proposal density, building then an ergodic Markov chain whose stationary distribution is the desired distribution by accepting or rejecting those candidate samples as the new state of the chain. On the other hand, IS techniques draw samples from a simple proposal density and then assign them suitable weights that measure their quality in some appropriate way. In this paper, we perform a thorough review of MC methods for the estimation of static parameters in signal processing applications. A historical note on the development of MC schemes is also provided, followed by the basic MC method and a brief description of the rejection sampling (RS) algorithm, as well as three sections describing many of the most relevant MCMC and IS algorithms, and their combined use. Finally, five numerical examples (including the estimation of the parameters of a chaotic system, a localization problem in wireless sensor networks and a spectral analysis application) are provided in order to demonstrate the performance of the described approaches.
State‐space models are an increasingly common and important tool in the quantitative ecologists’ armoury, particularly for the analysis of time‐series data. This is due to both their flexibility and ...intuitive structure, describing the different individual processes of a complex system, thus simplifying the model specification step.
State‐space models are composed of two processes (a) the system (or state) process that describes the dynamics of the true underlying state of the system over time; and (b) the observation process that links the observed data with the current true state of the system at that time. Specification of the general model structure consists of considering each distinct ecological process within the system and observation processes, which are then automatically combined within the state‐space structure.
There is typically a trade‐off between the complexity of the model and the associated model‐fitting process. Simpler model specifications permit the application of simpler model‐fitting tools; whereas more complex model specifications, with nonlinear dynamics and/or non‐Gaussian stochasticity often require more sophisticated model‐fitting algorithms to be applied.
We provide a brief overview of general state‐space models before focusing on the different model‐fitting tools available. In particular for different general state‐space model structures we discuss established model‐fitting tools that are available. We also offer practical guidance for choosing a specific fitting procedure.
Blind Kalman Filtering for Short-Term Load Forecasting Sharma, Shalini; Majumdar, Angshul; Elvira, Victor ...
IEEE transactions on power systems,
2020-Nov., 2020-11-00, 20201101, 2020-11, Letnik:
35, Številka:
6
Journal Article
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In this work we address the problem of short-term load forecasting. We propose a generalization of the linear state-space model where the evolution of the state and the observation matrices is ...unknown. The proposed blind Kalman filter algorithm proceeds via alternating the estimation of these unknown matrices and the inference of the state, within the framework of expectation-maximization. A mini-batch processing strategy is introduced to allow on-the-fly forecasting. The experimental results show that the proposed method outperforms the state-of-the-art techniques by a considerable margin, both on load profile estimation and peak load forecast problems.
Understanding animal movement and behaviour can aid spatial planning and inform conservation management. However, it is difficult to directly observe behaviours in remote and hostile terrain such as ...the marine environment. Different underlying states can be identified from telemetry data using hidden Markov models (HMMs). The inferred states are subsequently associated with different behaviours, using ecological knowledge of the species. However, the inferred behaviours are not typically validated due to difficulty obtaining ‘ground truth’ behavioural information. We investigate the accuracy of inferred behaviours by considering a unique data set provided by Joint Nature Conservation Committee. The data consist of simultaneous proxy movement tracks of the boat (defined as visual tracks as birds are followed by eye) and seabird behaviour obtained by observers on the boat. We demonstrate that visual tracking data is suitable for our study. Accuracy of HMMs ranging from 71% to 87% during chick‐rearing and 54% to 70% during incubation was generally insensitive to model choice, even when AIC values varied substantially across different models. Finally, we show that for foraging, a state of primary interest for conservation purposes, identified missed foraging bouts lasted for only a few seconds. We conclude that HMMs fitted to tracking data have the potential to accurately identify important conservation‐relevant behaviours, demonstrated by a comparison in which visual tracking data provide a ‘gold standard’ of manually classified behaviours to validate against. Confidence in using HMMs for behavioural inference should increase as a result of these findings, but future work is needed to assess the generalisability of the results, and we recommend that, wherever feasible, validation data be collected alongside GPS tracking data to validate model performance. This work has important implications for animal conservation, where the size and location of protected areas are often informed by behaviours identified using HMMs fitted to movement data.
Behaviours inferred from telemetry data using hidden Markov models (HMMs) are not typically validated due to difficulty obtaining ‘ground truth’ behavioural information. Our study is unique in that we investigate the accuracy of HMM‐inferred behaviours by considering movement (visual tracks) and observed behavioural data sets recorded at the same time frequency provided by Joint Nature Conservation Committee. We show that HMMs fitted to tracking data can accurately identify important conservation‐relevant behaviours.
Bayesian models have become very popular over the last years in several fields such as signal processing, statistics, and machine learning. Bayesian inference requires the approximation of ...complicated integrals involving posterior distributions. For this purpose, Monte Carlo (MC) methods, such as Markov Chain Monte Carlo and importance sampling algorithms, are often employed. In this work, we introduce the theory and practice of a Compressed MC (C-MC) scheme to compress the statistical information contained in a set of random samples. In its basic version, C-MC is strictly related to the stratification technique, a well-known method used for variance reduction purposes. Deterministic C-MC schemes are also presented, which provide very good performance. The compression problem is strictly related to the moment matching approach applied in different filtering techniques, usually called as Gaussian quadrature rules or sigma-point methods. C-MC can be employed in a distributed Bayesian inference framework when cheap and fast communications with a central processor are required. Furthermore, C-MC is useful within particle filtering and adaptive IS algorithms, as shown by three novel schemes introduced in this work. Six numerical results confirm the benefits of the introduced schemes, outperforming the corresponding benchmark methods. A related code is also provided. (The code is provided athttp://www.lucamartino.altervista.org/CMC_CODE_pub_EX1.zip)
State-space models (SSM) are central to describe time-varying complex systems in countless signal processing applications such as remote sensing, networks, biomedicine, and finance to name a few. ...Inference and prediction in SSMs are possible when the model parameters are known, which is rarely the case. The estimation of these parameters is crucial, not only for performing statistical analysis, but also for uncovering the underlying structure of complex phenomena. In this paper, we focus on the linear-Gaussian model, arguably the most celebrated SSM, and particularly in the challenging task of estimating the transition matrix that encodes the Markovian dependencies in the evolution of the multi-variate state. We introduce a novel perspective by relating this matrix to the adjacency matrix of a directed graph, also interpreted as the causal relationship among state dimensions in the Granger-causality sense. Under this perspective, we propose a new method called GraphEM based on the well sounded expectation-maximization (EM) methodology for inferring the transition matrix jointly with the smoothing/filtering of the observed data. We propose an advanced convex optimization solver relying on a consensus-based implementation of a proximal splitting strategy for solving the M-step. This approach enables an efficient and versatile processing of various sophisticated priors on the graph structure, such as parsimony constraints, while benefiting from convergence guarantees. We demonstrate the good performance and the interpretable results of GraphEM by means of two sets of numerical examples.
Optimized Population Monte Carlo Elvira, Victor; Chouzenoux, Emilie
IEEE transactions on signal processing,
2022, Letnik:
70
Journal Article
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Adaptive importance sampling (AIS) methods are increasingly used for the approximation of distributions and related intractable integrals in the context of Bayesian inference. Population Monte Carlo ...(PMC) algorithms are a subclass of AIS methods, widely used due to their ease in the adaptation. In this paper, we propose a novel algorithm that exploits the benefits of the PMC framework and includes more efficient adaptive mechanisms, exploiting geometric information of the target distribution. In particular, the novel algorithm adapts the location and scale parameters of a set of importance densities (proposals). At each iteration, the location parameters are adapted by combining a versatile resampling strategy (i.e., using the information of previous weighted samples) with an advanced optimization-based scheme. Local second-order information of the target distribution is incorporated through a preconditioning matrix acting as a scaling metric onto a gradient direction. A damped Newton approach is adopted to ensure robustness of the scheme. The resulting metric is also used to update the scale parameters of the proposals. We discuss several key theoretical foundations for the proposed approach. Finally, we show the successful performance of the proposed method in three numerical examples, involving challenging distributions.
The Effective Sample Size (ESS) is an important measure of efficiency of Monte Carlo methods such as Markov Chain Monte Carlo (MCMC) and Importance Sampling (IS) techniques. In the IS context, an ...approximation ESS^ of the theoretical ESS definition is widely applied, involving the inverse of the sum of the squares of the normalized importance weights. This formula, ESS^, has become an essential piece within Sequential Monte Carlo (SMC) methods, to assess the convenience of a resampling step. From another perspective, the expression ESS^ is related to the Euclidean distance between the probability mass described by the normalized weights and the discrete uniform probability mass function (pmf). In this work, we derive other possible ESS functions based on different discrepancy measures between these two pmfs. Several examples are provided involving, for instance, the geometric mean of the weights, the discrete entropy (including the perplexity measure, already proposed in literature) and the Gini coefficient among others. We list five theoretical requirements which a generic ESS function should satisfy, allowing us to classify different ESS measures. We also compare the most promising ones by means of numerical simulations.
State-space models (SSMs) are a powerful statistical tool for modelling time-varying systems via a latent state. In these models, the latent state is never directly observed. Instead, a sequence of ...data points related to the state are obtained. The linear-Gaussian state-space model is widely used, since it allows for exact inference when all model parameters are known, however this is rarely the case. The estimation of these parameters is a very challenging but essential task to perform inference and prediction. In the linear-Gaussian model, the state dynamics are described via a state transition matrix. This model parameter is known to be particularly hard to estimate, since it encodes the between-step relationships of the state elements, which are never observed. In many real-world applications, this transition matrix is sparse since not all state components directly affect all other state components. However, most contemporary parameter estimation methods do not exploit this feature. In this work, we take a fully probabilistic approach and propose SpaRJ, a novel simulation method that obtains sparse samples from the posterior distribution of the transition matrix of a linear-Gaussian state-space model. We exploit the sparsity of the latent space by uncovering its underlying structure. Our proposed method is the first algorithm to provide a fully Bayesian quantification of the sparsity in the model. SpaRJ belongs to the family of reversible jump Markov chain Monte Carlo methods. Our method obtains sparsity via exploring a set of models that exhibit differing sparsity patterns in the transition matrix. The algorithm implements a new set of transition kernels that are specifically tailored to efficiently explore the space of sparse matrices. Moreover, we also design new effective rules to explore transition matrices within the same level of sparsity. This novel methodology has strong theoretical guarantees and efficiently explores sparse subspaces, which unveils the latent structure of the data generating process, thereby enhancing interpretability. The excellent performance of SpaRJ is showcased in a synthetic example with dimension 144 in the parameter space, and in a numerical example with real data.
We design a sequential Monte Carlo scheme for the dual purpose of Bayesian inference and model selection. We consider the application context of urban mobility, where several modalities of transport ...and different measurement devices can be employed. Therefore, we address the joint problem of online tracking and detection of the current modality. For this purpose, we use interacting parallel particle filters, each one addressing a different model. They cooperate for providing a global estimator of the variable of interest and, at the same time, an approximation of the posterior density of each model given the data. The interaction occurs by a parsimonious distribution of the computational effort, with online adaptation for the number of particles of each filter according to the posterior probability of the corresponding model. The resulting scheme is simple and flexible. We have tested the novel technique in different numerical experiments with artificial and real data, which confirm the robustness of the proposed scheme.
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