Sequential Monte Carlo Estimation Research Articles

A point mass proposal method for Bayesian state-space model fitting

State-space models (SSMs) are commonly used to model time series data where the observations depend on an unobserved latent process. However, inference on the model parameters of an SSM can be challenging, especially when the likelihood of the data given the parameters is not available in closed-form. One approach is to jointly sample the latent states and model parameters via Markov chain Monte Carlo (MCMC) and/or sequential Monte Carlo approximation. These methods can be inefficient, mixing poorly when there are many highly correlated latent states or parameters, or when there is a high rate of sample impoverishment in the sequential Monte Carlo approximations. We propose a novel block proposal distribution for Metropolis-within-Gibbs sampling on the joint latent state and parameter space. The proposal distribution is informed by a deterministic hidden Markov model (HMM), defined such that the usual theoretical guarantees of MCMC algorithms apply. We discuss how the HMMs are constructed, the generality of the approach arising from the tuning parameters, and how these tuning parameters can be chosen efficiently in practice. We demonstrate that the proposed algorithm using HMM approximations provides an efficient alternative method for fitting state-space models, even for those that exhibit near-chaotic behavior.

Vegetation responses to variations in climate: A combined ordinary differential equation and sequential Monte Carlo estimation approach

Vegetation responses to variation in climate are a current research priority in the context of accelerated shifts generated by climate change. However, the interactions between environmental and biological factors still represent one of the largest uncertainties in projections of future scenarios, since the relationship between drivers and ecosystem responses has a complex and nonlinear nature. We aimed to develop a model to study the vegetation's primary productivity dynamic response to temporal variations in climatic conditions as measured by rainfall, temperature and radiation. Thus, we propose a new way to estimate the vegetation response to climate via a non-autonomous version of a classical growth curve, with a time-varying growth rate and carrying capacity parameters according to climate variables. With a Sequential Monte Carlo Estimation to account for complexities in the climate-vegetation relationship to minimize the number of parameters. The model was applied to six key sites identified in a previous study, consisting of different arid and semiarid rangelands from North Patagonia, Argentina. For each site, we selected the time series of MODIS NDVI, and climate data from ERA5 Copernicus hourly reanalysis from 2000 to 2021. After calculating the time series of the a posteriori distribution of parameters, we analyzed the explained capacity of the model in terms of the linear coefficient of determination and the parameters distribution variation. Results showed that most rangelands recorded changes in their sensitivity over time to climatic factors, but vegetation responses were heterogeneous and influenced by different drivers. Differences in this climate-vegetation relationship were recorded among different cases: (1) a marginal and decreasing sensitivity to temperature and radiation, respectively, but a high sensitivity to water availability; (2) high and increasing sensitivity to temperature and water availability, respectively; and (3) a case with an abrupt shift in vegetation dynamics driven by a progressively decreasing sensitivity to water availability, without any changes in the sensitivity either to temperature or radiation. Finally, we also found that the time scale, in which the ecosystem integrated the rainfall phenomenon in terms of the width of the window function used to convolve the rainfall series into a water availability variable, was also variable in time. This approach allows us to estimate the connection degree between ecosystem productivity and climatic variables. The capacity of the model to identify changes over time in the vegetation-climate relationship might inform decision-makers about ecological transitions and the differential impact of climatic drivers on ecosystems.

Conditionally Exponential Prior in Focal Near- and Far-Field EEG Source Localization via Randomized Multiresolution Scanning (RAMUS)

In this paper, we focus on the inverse problem of reconstructing distributional brain activity with cortical and weakly detectable deep components in non-invasive Electroencephalography. We consider a recently introduced hybrid reconstruction strategy combining a hierarchical Bayesian model to incorporate a priori information and the advanced randomized multiresolution scanning (RAMUS) source space decomposition approach to reduce modelling errors, respectively. In particular, we aim to generalize the previously extensively used conditionally Gaussian prior (CGP) formalism to achieve distributional reconstructions with higher focality. For this purpose, we introduce as a hierarchical prior, a general exponential distribution, which we refer to as conditionally exponential prior (CEP). The first-degree CEP corresponds to focality enforcing Laplace prior, but it also suffers from strong depth bias, when applied in numerical modelling, making the deep activity unrecoverable. We sample over multiple resolution levels via RAMUS to reduce this bias as it is known to depend on the resolution of the source space. Moreover, we introduce a procedure based on the physiological a priori knowledge of the brain activity to obtain the shape and scale parameters of the gamma hyperprior that steer the CEP. The posterior estimates are calculated using iterative statistical methods, expectation maximization and iterative alternating sequential algorithm, which we show to be algorithmically similar and to have a close resemblance to the iterative ell _1 and ell _2 reweighting methods. The performance of CEP is compared with the recent sampling-based dipole localization method Sequential semi-analytic Monte Carlo estimation (SESAME) in numerical experiments of simulated somatosensory evoked potentials related to the human median nerve stimulation. Our results obtained using synthetic sources suggest that a hybrid of the first-degree CEP and RAMUS can achieve an accuracy comparable to the second-degree case (CGP) while being more focal. Further, the proposed hybrid is shown to be robust to noise effects and compares well with the dipole reconstructions obtained with SESAME.

On assigning probabilities to new hypotheses

• Knowledge mining on varying sets of compound hypotheses is inspected. • Bayesian learning is equipped with technique allowing to extend the set of hypotheses. • The technique copes with learning with less data than the amount of parameters. • The use of data forecasters as a knowledge source in parameter estimation is refined. • A rich set of interesting related research and application problems is presented. The paper proposes the way how to assign a proper prior probability to a new, generally compound, hypothesis. To this purpose, it uses the minimum relative-entropy principle and a forecaster-based knowledge transfer. Methodologically, it opens a way towards enriching the standard Bayesian framework by the possibility to extend the set of models during learning without the need to restart. The presented use scenarios concern: (a) creating new hypotheses, (b) learning problems with an insufficient amount of data, and (c) sequential Monte Carlo estimation. They indicate a strong application potential of the proposed technique. Related interesting open research problems are listed.

Sequential Monte Carlo Filters with Parameters Learning for Commodity Pricing Models

In this article, an estimation methodology based on the sequential Monte Carlo algorithm is proposed, thatjointly estimate the states and parameters, the relationship between the prices of futures contracts and the spot prices of primary products is determined, the evolution of prices and the volatility of the historical data of the primary market (Gold and Soybean) are analyzed. Two stochastic models for an estimate the states and parameters are considered, the parameters and states describe physical measure (associated with the price) and risk-neutral measure (associated with the markets to futures), the price dynamics in the short-term through the reversion to the mean and volatility are determined, while that in the long term through markets to futures. Other characteristics such as seasonal patterns, price spikes, market dependent volatilities, and non-seasonality can also be observed. In the methodology, a parameter learning algorithm is used, specifically, three algorithms are proposed, that is the sequential Monte Carlo estimation (SMC) for state space modelswith unknown parameters: the first method is considered a particle filter that is based on the sampling algorithm of sequential importance with resampling (SISR). The second implemented method is the Storvik algorithm [19], the states and parameters of the posterior distribution are estimated that have supported in low-dimensional spaces, a sufficient statistics from the sample of the filtered distribution is considered. The third method is (PLS) Carvalho’s Particle Learning and Smoothing algorithm [31]. The cash prices of the contracts with future delivery dates are analyzed. The results indicate postponement of payment, the future prices on different maturity dates with the spot price are highly correlated. Likewise, the contracts with a delivery date for the last periods of the year 2017, the spot price lower than the prices of the contracts with expiration date for 12 and 24 months is found, opposite occurs in the contracts with expiration date for 1 and 6 months.

Bayesian Allocation Model: Marginal Likelihood-Based Model Selection for Count Tensors

In this article, we introduce a dynamic generative model, the Bayesian allocation model (BAM), for modeling count data. BAM covers various probabilistic nonnegative tensor factorization (NTF) and topic models under one general framework. In BAM, allocations are made using a Bayesian network, whose conditional probability tables can be integrated out analytically. We show that, when allocations are viewed as sequential, the resulting marginal process is a special type of Polya urn process, which we name as Polya-Bayes process, an integer valued self-reinforcing process. Exploiting the Polya urn construction, we develop a novel sequential Monte Carlo (SMC) algorithm for marginal likelihood estimation in BAM, leading to a unified scoring method for discrete variable Bayesian networks with hidden nodes, including various NTF and topic models. The SMC estimator for marginal likelihood has the remarkable property of being unbiased in contrast to variational algorithms which are generally biased. We also demonstrate how our novel SMC-based likelihood estimation can be integrated within a Markov chain Monte Carlo algorithm for a principled and correct (in terms of respecting the true posterior distribution) Bayesian model selection and hyperparameter estimation for BAM. We provide several numerical examples, both on artificial and real datasets, that demonstrate the performance of the algorithms for various data regimes.

Automated Neuron Reconstruction from 3D Fluorescence Microscopy Images Using Sequential Monte Carlo Estimation

Microscopic images of neuronal cells provide essential structural information about the key constituents of the brain and form the basis of many neuroscientific studies. Computational analyses of the morphological properties of the captured neurons require first converting the structural information into digital tree-like reconstructions. Many dedicated computational methods and corresponding software tools have been and are continuously being developed with the aim to automate this step while achieving human-comparable reconstruction accuracy. This pursuit is hampered by the immense diversity and intricacy of neuronal morphologies as well as the often low quality and ambiguity of the images. Here we present a novel method we developed in an effort to improve the robustness of digital reconstruction against these complicating factors. The method is based on probabilistic filtering by sequential Monte Carlo estimation and uses prediction and update models designed specifically for tracing neuronal branches in microscopic image stacks. Moreover, it uses multiple probabilistic traces to arrive at a more robust, ensemble reconstruction. The proposed method was evaluated on fluorescence microscopy image stacks of single neurons and dense neuronal networks with expert manual annotations serving as the gold standard, as well as on synthetic images with known ground truth. The results indicate that our method performs well under varying experimental conditions and compares favorably to state-of-the-art alternative methods.

Time-Varying Modeling of Land Cover Change Dynamics Due to Forest Fires

Seasonal variations in land cover are commonly represented using a constant frequency cosine model with time-varying parameters. As frequency represents the constant annual vegetation growth cycle, the model is not adequate to represent dynamics such as sudden changes in land cover and subsequent regrowth. In this paper, we present a new model to capture time-varying changes in the vegetation growth cycle and detect abrupt changes in land cover due to forest fires. We also design a sequential Monte Carlo estimation approach of the time-varying frequency in the proposed nonlinear model using the particle filter (PF). We further propose a binary hypothesis land cover change detector that is based on a dissimilarity measure between windowed time-series observed during the same month of consecutive years. Experiments show that the PF estimation can detect change with lower delay than the existing approaches. Unsupervised mapping of the fire severity from the model parameter estimates is also developed.

Estimation of agent-based models using sequential Monte Carlo methods

Estimation of agent-based models is currently an intense area of research. Recent contributions have to a large extent resorted to simulation-based methods mostly using some form of simulated method of moments estimation (SMM). There is, however, an entire branch of statistical methods that should appear promising, but has to our knowledge never been applied so far to estimate agent-based models in economics and finance: Markov chain Monte Carlo methods designed for state space models or models with latent variables. This latter class of models seems particularly relevant as agent-based models typically consist of some latent and some observable variables since not all the characteristics of agents would mostly be observable. Indeed, one might often not only be interested in estimating the parameters of a model, but also to infer the time development of some latent variable. However, agent-based models when interpreted as latent variable models would be typically characterized by non-linear dynamics and non-Gaussian fluctuations and, thus, would require a computational approach to statistical inference. Here we resort to Sequential Monte Carlo (SMC) estimation based on a particle filter. This approach is used here to numerically approximate the conditional densities that enter into the likelihood function of the problem. With this approximation we simultaneously obtain parameter estimates and filtered state probabilities for the unobservable variable(s) that drive(s) the dynamics of the observable time series. In our examples, the observable series will be asset returns (or prices) while the unobservable variables will be some measure of agents’ aggregate sentiment. We apply SMC to two selected agent-based models of speculative dynamics with somewhat different flavor. The empirical application to a selection of financial data includes an explicit comparison of the goodness-of-fit of both models.

IMM Bernoulli Filter for Cooperative Object Tracking in Road Traffic

Cooperative multi-model state estimation is considered in road traffic scenario. Our goal is to detect and track a maneuvering object based on vehicular radar measurements. To handle the motion model switching of the navigating object we used the interacting Multiple Model (IMM) estimator with the Bernoulli filter (BF). We modified a recently derived IMM-BF, realized in a Gaussian mixture approximation to fulfil the requirements of a multi-sensor object tracking estimator with moving sensors. Our proposed method is implemented as a Sequential Monte Carlo estimator that allows nonlinear motion and sensor models and is capable of functioning effectively in a cooperative manner.

Structured filtering

A major challenge facing existing sequential Monte Carlo methods for parameter estimation in physics stems from the inability of existing approaches to robustly deal with experiments that have different mechanisms that yield the results with equivalent probability. We address this problem here by proposing a form of particle filtering that clusters the particles that comprise the sequential Monte Carlo approximation to the posterior before applying a resampler. Through a new graphical approach to thinking about such models, we are able to devise an artificial-intelligence based strategy that automatically learns the shape and number of the clusters in the support of the posterior. We demonstrate the power of our approach by applying it to randomized gap estimation and a form of low circuit-depth phase estimation where existing methods from the physics literature either exhibit much worse performance or even fail completely.

Particle rejuvenation of Rao-Blackwellized sequential Monte Carlo smoothers for conditionally linear and Gaussian models

This paper focuses on sequential Monte Carlo approximations of smoothing distributions in conditionally linear and Gaussian state spaces. To reduce Monte Carlo variance of smoothers, it is typical in these models to use Rao-Blackwellization: particle approximation is used to sample sequences of hidden regimes while the Gaussian states are explicitly integrated conditional on the sequence of regimes and observations, using variants of the Kalman filter/smoother. The first successful attempt to use Rao-Blackwellization for smoothing extends the Bryson-Frazier smoother for Gaussian linear state space models using the generalized two-filter formula together with Kalman filters/smoothers. More recently, a forward-backward decomposition of smoothing distributions mimicking the Rauch-Tung-Striebel smoother for the regimes combined with backward Kalman updates has been introduced. This paper investigates the benefit of introducing additional rejuvenation steps in all these algorithms to sample at each time instant new regimes conditional on the forward and backward particles. This defines particle-based approximations of the smoothing distributions whose support is not restricted to the set of particles sampled in the forward or backward filter. These procedures are applied to commodity markets which are described using a two-factor model based on the spot price and a convenience yield for crude oil data.

Gaussian process regression for monitoring and fault detection of wastewater treatment processes.

Monitoring and fault detection methods are increasingly important to achieve a robust and resource efficient operation of wastewater treatment plants (WWTPs). The purpose of this paper was to evaluate a promising machine learning method, Gaussian process regression (GPR), for WWTP monitoring applications. We evaluated GPR at two WWTP monitoring problems: estimate missing data in a flow rate signal (simulated data), and detect a drift in an ammonium sensor (real data). We showed that GPR with the standard estimation method, maximum likelihood estimation (GPR-MLE), suffered from local optima during estimation of kernel parameters, and did not give satisfactory results in a simulated case study. However, GPR with a state-of-the-art estimation method based on sequential Monte Carlo estimation (GPR-SMC) gave good predictions and did not suffer from local optima. Comparisons with simple standard methods revealed that GPR-SMC performed better than linear interpolation in estimating missing data in a noisy flow rate signal. We conclude that GPR-SMC is both a general and powerful method for monitoring full-scale WWTPs. However, this paper also shows that it does not always pay off to use more sophisticated methods. New methods should be critically compared against simpler methods, which might be good enough for some scenarios.

Valuation of barrier options using sequential Monte Carlo

Sequential Monte Carlo (SMC) methods have successfully been used in many applications in engineering, statistics and physics. However, these are seldom used in financial option pricing literature and practice. This paper presents SMC method for pricing barrier options with continuous and discrete monitoring of the barrier condition. Under the SMC method, simulated asset values rejected due to barrier condition are re-sampled from asset samples that do not breach the barrier condition improving the efficiency of the option price estimator; while under the standard Monte Carlo many simulated asset paths can be rejected by the barrier condition making it harder to estimate option price accurately. We compare SMC with the standard Monte Carlo method and demonstrate that the extra effort to implement SMC when compared with the standard Monte Carlo is very little while improvement in price estimate can be significant. Both methods result in unbiased estimators for the price converging to the true value as $1/\sqrt{M}$, where $M$ is the number of simulations (asset paths). However, the variance of SMC estimator is smaller and does not grow with the number of time steps when compared to the standard Monte Carlo. In this paper we demonstrate that SMC can successfully be used for pricing barrier options. SMC can also be used for pricing other exotic options and also for cases with many underlying assets and additional stochastic factors such as stochastic volatility; we provide general formulas and references.

Battery state estimation with a self-evolving electrochemical ageing model

The parameters for most battery state estimation techniques are usually calibrated during the development stage before vehicle production. However different usages of battery lead to different ageing processes, and end up with different model parameters. As a result, an estimation algorithm based on pre-calibrated parameters may not generate accurate estimates. This paper presents a new advanced methodology to address this challenge. First, we employ a comprehensive electrochemical model for lead acid batteries that were developed recently. The battery model explicitly characterizes the electric and thermal behavior of the battery, as well as the evolution of major battery failure modes. Second, we apply a two-time-scale model-based estimation method, where the micro time-scale algorithm estimates battery SOC and SOH, and the macro time-scale algorithm tracks the changes of the battery being monitored and updates the parameters of the battery model accordingly. The algorithms at both time scales use particle filter, a sequential Monte-Carlo estimation technique. Several datasets based on real vehicle driving profiles are used to validate the proposed algorithms. The experiment results show that the proposed estimation scheme is effective, due to both the careful selection of battery model and the tailored filtering technique.

On the behaviour of low cost laser scanners in HW/SW particle filter SLAM applications

Particle filters (PFs) are computationally intensive sequential Monte Carlo estimation methods with applications in the field of mobile robotics for performing tasks such as tracking, simultaneous localization and mapping (SLAM) and navigation, by dealing with the uncertainties and/or noise generated by the sensors as well as with the intrinsic uncertainties of the environment. However, the application of PFs with an important number of particles has traditionally been difficult to implement in real-time applications due to the huge number of operations they require. This work presents a hardware implementation on FPGA (field programmable gate arrays) of a PF applied to SLAM which aims to accelerate the execution time of the PF algorithm with moderate resource. The presented system is evaluated for different sensors including a low cost Neato XV-11 laser scanner sensor. First the system is validated by post processing data provided by a realistic simulation of a differential robot, equipped with a hacked Neato XV-11 laser scanner, that navigates in the Robot@Factory competition maze. The robot was simulated using SimTwo, which is a realistic simulation software that can support several types of robots. The simulator provides the robot ground truth, odometry and the laser scanner data. Then the proposed solution is further validated on standard laser scanner sensors in complex environments. The results achieved from this study confirmed the possible use of low cost laser scanner for different robotics applications which benefits in several aspects due to its cost and the increased speed provided by the SLAM algorithm running on FPGA.

Tracking Neural Modulation Depth by Dual Sequential Monte Carlo Estimation on Point Processes for Brain-Machine Interfaces.

Classic brain-machine interface (BMI) approaches decode neural signals from the brain responsible for achieving specific motor movements, which subsequently command prosthetic devices. Brain activities adaptively change during the control of the neuroprosthesis in BMIs, where the alteration of the preferred direction and the modulation of the gain depth are observed. The static neural tuning models have been limited by fixed codes, resulting in a decay of decoding performance over the course of the movement and subsequent instability in motor performance. To achieve stable performance, we propose a dual sequential Monte Carlo adaptive point process method, which models and decodes the gradually changing modulation depth of individual neuron over the course of a movement. We use multichannel neural spike trains from the primary motor cortex of a monkey trained to perform a target pursuit task using a joystick. Our results show that our computational approach successfully tracks the neural modulation depth over time with better goodness-of-fit than classic static neural tuning models, resulting in smaller errors between the true kinematics and the estimations in both simulated and real data. Our novel decoding approach suggests that the brain may employ such strategies to achieve stable motor output, i.e., plastic neural tuning is a feature of neural systems. BMI users may benefit from this adaptive algorithm to achieve more complex and controlled movement outcomes.

The metabolic network of Clostridium acetobutylicum: Comparison of the approximate Bayesian computation via sequential Monte Carlo (ABC-SMC) and profile likelihood estimation (PLE) methods for determinability analysis

The Gram-positive bacterium Clostridium acetobutylicum is an anaerobic endospore-forming species which produces acetone, butanol and ethanol via the acetone–butanol (AB) fermentation process, leading to biofuels including butanol. In previous work we looked to estimate the parameters in an ordinary differential equation model of the glucose metabolism network using data from pH-controlled continuous culture experiments. Here we combine two approaches, namely the approximate Bayesian computation via an existing sequential Monte Carlo (ABC-SMC) method (to compute credible intervals for the parameters), and the profile likelihood estimation (PLE) (to improve the calculation of confidence intervals for the same parameters), the parameters in both cases being derived from experimental data from forward shift experiments. We also apply the ABC-SMC method to investigate which of the models introduced previously (one non-sporulation and four sporulation models) have the greatest strength of evidence. We find that the joint approximate posterior distribution of the parameters determines the same parameters as previously, including all of the basal and increased enzyme production rates and enzyme reaction activity parameters, as well as the Michaelis–Menten kinetic parameters for glucose ingestion, while other parameters are not as well-determined, particularly those connected with the internal metabolites acetyl-CoA, acetoacetyl-CoA and butyryl-CoA. We also find that the approximate posterior is strongly non-Gaussian, indicating that our previous assumption of elliptical contours of the distribution is not valid, which has the effect of reducing the numbers of pairs of parameters that are (linearly) correlated with each other. Calculations of confidence intervals using the PLE method back this up. Finally, we find that all five of our models are equally likely, given the data available at present.

Assessment of type II diabetes mellitus using irregularly sampled measurements with missing data.

Diabetes mellitus is one of the leading diseases in the developed world. In order to better regulate blood glucose in a diabetic patient, improved modelling of insulin-glucose dynamics is a key factor in the treatment of diabetes mellitus. In the current work, the insulin-glucose dynamics in type II diabetes mellitus can be modelled by using a stochastic nonlinear state-space model. Estimating the parameters of such a model is difficult as only a few blood glucose and insulin measurements per day are available in a non-clinical setting. Therefore, developing a predictive model of the blood glucose of a person with type II diabetes mellitus is important when the glucose and insulin concentrations are only available at irregular intervals. To overcome these difficulties, we resort to online sequential Monte Carlo (SMC) estimation of states and parameters of the state-space model for type II diabetic patients under various levels of randomly missing clinical data. Our results show that this method is efficient in monitoring and estimating the dynamics of the peripheral glucose, insulin and incretins concentration when 10, 25 and 50% of the simulated clinical data were randomly removed.

A Class of Fast Exact Bayesian Filters in Dynamical Models With Jumps

We address the statistical filtering problem in dynamical models with jumps. When a particular application is adequately modeled by linear and Gaussian probability density functions with jumps, a usual method consists in approximating the optimal Bayesian estimate [in the sense of the minimum mean square error (MMSE)] in a linear and Gaussian jump Markov state space system (JMSS). Practical solutions include algorithms based on numerical approximations or on sequential Monte Carlo (SMC) methods. In this paper, we propose a class of alternative methods which consists in building statistical models which, locally, similarly model the problem of interest, but in which the computation of the MMSE estimate can be be computed exactly (without numerical nor SMC approximations) and at a computational cost which is linear in the number of observations.