arXiv subjects

Sourabh Bhattacharya

Publications and source records attributed to Sourabh Bhattacharya.

At least 73 records · Page 4Linked to original sources

arXiv2017

Nonparametric Dynamic State Space Modeling of Observed Circular Time Series with Circular Latent States: A Bayesian Perspective

Circular time series has received relatively little attention in statistics and modeling complex circular time series using the state space approach is non-existent in the literature. In this article we introduce a flexible Bayesian nonparametric approach to state space modeling of observed circular time series where even the latent states are circular random variables. Crucially, we assume that the forms of both observational and evolutionary functions, both of which are circular in nature, are unknown and time-varying. We model these unknown circular functions by appropriate wrapped Gaussian processes having desirable properties. We develop an effective Markov chain Monte Carlo strategy for implementing our Bayesian model, by judiciously combining Gibbs sampling and Metropolis-Hastings methods. Validation of our ideas with a simulation study and two real bivariate circular time series data sets, where we assume one of the variables to be unobserved, revealed very encouraging performance of our model and methods. We finally analyse a data consisting of directions of whale migration, considering the unobserved ocean current direction as the latent circular process of interest. The results that we obtain are encouraging, and the posterior predictive distribution of the observed process correctly predicts the observed whale movement.

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arXiv2017

A Brief Tutorial on Transformation Based Markov Chain Monte Carlo and Optimal Scaling of the Additive Transformation

We consider the recently introduced Transformation-based Markov Chain Monte Carlo (TMCMC) (Dutta and Bhattacharya (2014)), a methodology that is designed to update all the parameters simultaneously using some simple deterministic transformation of a onedimensional random variable drawn from some arbitrary distribution on a relevant support. The additive transformation based TMCMC is similar in spirit to random walk Metropolis, except the fact that unlike the latter, additive TMCMC uses a single draw from a onedimensional proposal distribution to update the high-dimensional parameter. In this paper, we first provide a brief tutorial on TMCMC, exploring its connections and contrasts with various available MCMC methods. Then we study the diffusion limits of additive TMCMC under various set-ups ranging from the product structure of the target density to the case where the target is absolutely continuous with respect to a Gaussian measure; we also consider the additive TMCMC within Gibbs approach for all the above set-ups. These investigations lead to appropriate scaling of the one-dimensional proposal density. We also show that the optimal acceptance rate of additive TMCMC is 0.439 under all the aforementioned set-ups, in contrast with the well-established 0.234 acceptance rate associated with optimal random walk Metropolis algorithms under the same set-ups. We also elucidate the ramifications of our results and clear advantages of additive TMCMC over random walk Metropolis with ample simulation studies and Bayesian analysis of a real, spatial dataset with which 160 unknowns are associated.

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arXiv2017

On Geometric Ergodicity of Additive and Multiplicative Transformation Based Markov Chain Monte Carlo in High Dimensions

Recently Dutta and Bhattacharya (2013) introduced a novel Markov Chain Monte Carlo methodology that can simultaneously update all the components of high dimensional parameters using simple deterministic transformations of a one-dimensional random variable drawn from any arbitrary distribution defined on a relevant support. The methodology, which the authors refer to as Transformation-based Markov Chain Monte Carlo (TMCMC), greatly enhances computational speed and acceptance rate in high-dimensional problems. Two significant transformations associated with TMCMC are additive and multiplicative transformations. Combinations of additive and multiplicative transformations are also of much interest. In this work we investigate geometric ergodicity associated with additive and multiplicative TMCMC, along with their combinations, and illustrate their efficiency in practice with simulation studies.

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arXiv2016

Towards a Framework for Tracking Multiple Targets: Hybrid Systems meets Computational Geometry

We investigate a variation of the art gallery problem in which a team of mobile guards tries to track an unpredictable intruder in a simply-connected polygonal environment. In this work, we use the deployment strategy for diagonal guards originally proposed in [1]. The guards are confined to move along the diagonals of a polygon and the intruder can move freely within the environment. We define critical regions to generate event-triggered strategies for the guards. We design a hybrid automaton based on the critical regions to model the tracking problem. Based on reachability analysis, we provide necessary and sufficient conditions for tracking in terms of the maximal controlled invariant set of the hybrid system. We express these conditions in terms of the critical curves to find sufficient conditions for n/4 guards to track the mobile intruder using the reachability analysis.

cs.RO↗

arXiv2016

Partitioning Strategies and Task Allocation for Target-tracking with Multiple Guards in Polygonal Environments

This paper presents an algorithm to deploy a team of {\it free} guards equipped with omni-directional cameras for tracking a bounded speed intruder inside a simply-connected polygonal environment. The proposed algorithm partitions the environment into smaller polygons, and assigns a guard to each partition so that the intruder is visible to at least one guard at all times. Based on the concept of {\it dynamic zones} introduced in this paper, we propose event-triggered strategies for the guards to track the intruder. We show that the number of guards deployed by the algorithm for tracking is strictly less than $\lfloor {\frac{n}{3}} \rfloor$ which is sufficient and sometimes necessary for coverage. We derive an upper bound on the speed of the mobile guard required for successful tracking which depends on the intruder's speed, the road map of the mobile guards, and geometry of the environment. Finally, we extend the aforementioned analysis to orthogonal polygons, and show that the upper bound on the number of guards deployed for tracking is strictly less than $\lfloor {\frac{n}{4}} \rfloor$ which is sufficient and sometimes necessary for the coverage problem.

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arXiv2016

On the Optimal Policies for Visibility-Based Target Tracking

In this paper, we investigate a pursuit-evasion game in which a mobile observer tries to track a target in an environment containing obstacles. We formulate the game as an optimal control problem with state inequality constraint in a simple environment. We show that for some initial conditions, there are two different regimes in the optimal strategy of the pursuer depending on whether the state-constraint is activated. We derive the equations that characterize the switching time between the two regimes. The pursuer's optimal tracking strategy in a simple environment is further extended to a general environment with multiple polygonal obstacles. We propose techniques to construct a "pursuit field" based on the optimal solutions to guide the motion of the observer in a general environment.

math.OC↗

arXiv2016

On Asymptotics Related to Classical Inference in Stochastic Differential Equations with Random Effects

Delattre et al. (2013) considered n independent stochastic differential equations (SDEs), where in each case the drift term is associated with a random effect, the distribution of which depends upon unknown parameters. Assuming the independent and identical (iid) situation the authors provide independent proofs of weak consistency and asymptotic normality of the maximum likelihood estimators (MLEs) of the hyper-parameters of their random effects parameters. In this article, as an alternative route to proving consistency and asymptotic normality in the SDE set-up involving random effects, we verify the regularity conditions required by existing relevant theorems. In particular, this approach allowed us to prove strong consistency under weaker assumption. But much more importantly, we further consider the independent, but non-identical set-up associated with the random effects based SDE framework, and prove asymptotic results associated with the MLEs.

math.ST↗

arXiv2016

On Bayesian Asymptotics in Stochastic Differential Equations with Random Effects

Delattre et al. (2013) investigated asymptotic properties of the maximum likelihood estimator of the population parameters of the random effects associated with n independent stochastic differential equations (SDEs) assuming that the SDEs are independent and identical (iid). In this article, we consider the Bayesian approach to learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution in the iid set-up as well as when the SDEs are independent but non-identical.

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arXiv2015

Bayesian Nonparametric Dynamic State Space Modeling with Circular Latent States

State space models are well-known for their versatility in modeling dynamic systems that arise in various scientific disciplines. Although parametric state space models are well studied, nonparametric approaches are much less explored in comparison. In this article we propose a novel Bayesian nonparametric approach to state space modeling assuming that both the observational and evolutionary functions are unknown and are varying with time; crucially, we assume that the unknown evolutionary equation describes dynamic evolution of some latent circular random variable. Based on appropriate kernel convolution of the standard Wiener process we model the time-varying observational and evolutionary functions as suitable Gaussian processes that take both linear and circular variables as arguments. Additionally, for the time-varying evolutionary function, we wrap the Gaussian process thus constructed around the unit circle to form an appropriate circular Gaussian process. We show that our process thus created satisfies desirable properties. For the purpose of inference we develop an MCMC based methodology combining Gibbs sampling and Metropolis-Hastings algorithms. Applications to a simulated data set, a real wind speed data set and a real ozone data set demonstrated quite encouraging performances of our model and methodologies.

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arXiv2015

Bayesian Nonparametric Estimation of Milky Way Model Parameters Using a New Matrix-Variate Gaussian Process Based Method

In this paper we develop an inverse Bayesian approach to find the value of the unknown model parameter vector that supports the real (or test) data, where the data comprises measurements of a matrix-variate variable. The method is illustrated via the estimation of the unknown Milky Way feature parameter vector, using available test and simulated (training) stellar velocity data matrices. The data is represented as an unknown function of the model parameters, where this high-dimensional function is modelled using a high-dimensional Gaussian Process (${\cal GP}$). The model for this function is trained using available training data and inverted by Bayesian means, to estimate the sought value of the model parameter vector at which the test data is realised. We achieve a closed-form expression for the posterior of the unknown parameter vector and the parameters of the invoked ${\cal GP}$, given test and training data. We perform model fitting by comparing the observed data with predictions made at different summaries of the posterior probability of the model parameter vector. As a supplement, we undertake a leave-one-out cross validation of our method.

stat.AP↗

arXiv2015

Particle Swarm Optimization Based Source Seeking

Signal source seeking using autonomous vehicles is a complex problem. The complexity increases manifold when signal intensities captured by physical sensors onboard are noisy and unreliable. Added to the fact that signal strength decays with distance, noisy environments make it extremely difficult to describe and model a decay function. This paper addresses our work with seeking maximum signal strength in a continuous electromagnetic signal source with mobile robots, using Particle Swarm Optimization (PSO). A one to one correspondence with swarm members in a PSO and physical Mobile robots is established and the positions of the robots are iteratively updated as the PSO algorithm proceeds forward. Since physical robots are responsive to swarm position updates, modifications were required to implement the interaction between real robots and the PSO algorithm. The development of modifications necessary to implement PSO on mobile robots, and strategies to adapt to real life environments such as obstacles and collision objects are presented in this paper. Our findings are also validated using experimental testbeds.

math.OC↗

arXiv2014

On Single Variable Transformation Approach to Markov Chain Monte Carlo

Random Walk Metropolis Hastings (RWMH) algorithm, is quite inefficient in high dimensions because of its abysmally slow acceptance rate. The slow acceptance rate results from the fact that RWMH separately updates each coordinate of the chain at every step. Dutta and Bhattacharya (2013) proposed a new technique called Transformation based Markov Chain Monte Carlo (TMCMC) aimed at overcoming these problems. This method updates all co-ordinates at a time- ensuring stable acceptance in all dimensions. We have shown here that geometric ergodicity is achieved for sub-exponential targets for two versions of TMCMC- the additive and the additive-multiplicative hybrid TMCMC schemes. Also, we obtain the optimal scaling by maximizing the diffusion speed of the limiting time-scaled diffusion process for TMCMC. We show that the optimal acceptance rate is 0.439 for TMCMC which is almost twice as large as RWMH (0.234). We observe that convergence to stationarity for TMCMC is faster than RWMH but the mixing property in RWMH is relatively better. However TMCMC is more robust with respect to scaling and dimensionality. This is attested by simulation runs on Gaussian and nearest neighbor models.

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arXiv2014

Minimum Distance Estimation of Milky Way Model Parameters and Related Inference

We propose a method to estimate the location of the Sun in the disk of the Milky Way using a method based on the Hellinger distance and construct confidence sets on our estimate of the unknown location using a bootstrap based method. Assuming the Galactic disk to be two-dimensional, the sought solar location then reduces to the radial distance separating the Sun from the Galactic center and the angular separation of the Galactic center to Sun line, from a pre-fixed line on the disk. On astronomical scales, the unknown solar location is equivalent to the location of us earthlings who observe the velocities of a sample of stars in the neighborhood of the Sun. This unknown location is estimated by undertaking pairwise comparisons of the estimated density of the observed set of velocities of the sampled stars, with densities estimated using synthetic stellar velocity data sets generated at chosen locations in the Milky Way disk according to four base astrophysical models. The "match" between the pair of estimated densities is parameterized by the affinity measure based on the familiar Hellinger distance. We perform a novel cross-validation procedure to establish a desirable "consistency" property of the proposed method.

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arXiv2014

A Note on the Misuse of the Variance Test in Meteorological Studies

The erroneous assumption "for all distributions for which the theoretical variance can be computed independently from parameters estimated by any method different from the method of moments" has been used in the case of fitting the gamma distribution to a rainfall data by Mooley (1973) which was followed by several researchers. We show that the asymptotic distribution of the test statistic is generally not even comparable to any central chi-square distribution. We also describe a method for checking the validity of the asymptotic distribution for a class of distributions.

stat.AP↗

arXiv2014

Bayesian Inference in Nonparametric Dynamic State-Space Models

We introduce state-space models where the functionals of the observational and the evolutionary equations are unknown, and treated as random functions evolving with time. Thus, our model is nonparametric and generalizes the traditional parametric state-space models. This random function approach also frees us from the restrictive assumption that the functional forms, although time-dependent, are of fixed forms. The traditional approach of assuming known, parametric functional forms is questionable, particularly in state-space models, since the validation of the assumptions require data on both the observed time series and the latent states; however, data on the latter are not available in state-space models. We specify Gaussian processes as priors of the random functions and exploit the "look-up table approach" of \ctn{Bhattacharya07} to efficiently handle the dynamic structure of the model. We consider both univariate and multivariate situations, using the Markov chain Monte Carlo (MCMC) approach for studying the posterior distributions of interest. In the case of challenging multivariate situations we demonstrate that the newly developed Transformation-based MCMC (TMCMC) of \ctn{Dutta11} provides interesting and efficient alternatives to the usual proposal distributions. We illustrate our methods with a challenging multivariate simulated data set, where the true observational and the evolutionary equations are highly non-linear, and treated as unknown. The results we obtain are quite encouraging. Moreover, using our Gaussian process approach we analysed a real data set, which has also been analysed by \ctn{Shumway82} and \ctn{Carlin92} using the linearity assumption. Our analyses show that towards the end of the time series, the linearity assumption of the previous authors breaks down.

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arXiv2014

Adaptive-Rate Compressive Sensing Using Side Information

We provide two novel adaptive-rate compressive sensing (CS) strategies for sparse, time-varying signals using side information. Our first method utilizes extra cross-validation measurements, and the second one exploits extra low-resolution measurements. Unlike the majority of current CS techniques, we do not assume that we know an upper bound on the number of significant coefficients that comprise the images in the video sequence. Instead, we use the side information to predict the number of significant coefficients in the signal at the next time instant. For each image in the video sequence, our techniques specify a fixed number of spatially-multiplexed CS measurements to acquire, and adjust this quantity from image to image. Our strategies are developed in the specific context of background subtraction for surveillance video, and we experimentally validate the proposed methods on real video sequences.

cs.CV↗

arXiv2013

An Improved Bayesian Semiparametric Model for Palaeoclimate Reconstruction: Cross-validation Based Model Assessment

Fossil-based palaeoclimate reconstruction is an important area of ecological science that has gained momentum in the backdrop of the global climate change debate. The hierarchical Bayesian paradigm provides an interesting platform for studying such important scientific issue. However, our cross-validation based assessment of the existing Bayesian hierarchical models with respect to two modern proxy data sets based on chironomid and pollen, respectively, revealed that the models are inadequate for the data sets. In this paper, we model the species assemblages (compositional data) by the zero-inflated multinomial distribution, while modelling the species response functions using Dirichlet process based Gaussian mixtures. This modelling strategy yielded significantly improved performances, and a formal Bayesian test of model adequacy, developed recently, showed that our new model is adequate for both the modern data sets. Furthermore, combining together the zero-inflated assumption, Importance Resampling Markov Chain Monte Carlo (IRMCMC) and the recently developed Transformation-based Markov Chain Monte Carlo (TMCMC), we develop a powerful and efficient computational methodology.

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arXiv2013

Clustering Categorical Time Series into Unknown Number of Clusters: A Perfect Simulation based Approach

Pamminger and Fruwirth-Schnatter (2010) considered a Bayesian approach to model-based clustering of categorical time series assuming a fixed number of clusters. But the popular methods for selecting the number of clusters, for example, the Bayes Information Criterion (BIC), turned out to have severe problems in the categorical time series context. In this paper, we circumvent the difficulties of choosing the number of clusters by adopting the Bayesian semiparametric mixture model approach introduced by Bhattacharya (2008), who assume that the number of clusters is a random quantity, but is bounded above by a (possibly large) number of clusters. We adopt the perfect simulation approach of Mukhopadhyay and Bhattacharya (2012) for posterior simulation for completely solving the problems of convergence of the underlying Markov chain Monte Carlo (MCMC) approach. Importantly, within our main perfect simulation algorithm, there arose the necessity to simulate perfectly from the joint distribution of a set of continuous random variables with log-concave full conditional densities. We propose and develop a novel and efficient perfect simulation methodology for joint distributions with log-concave full conditionals. This perfect sampling methodology is of independent interest as well since in a very large and important class of Bayesian applications the full conditionals turn out to be log-concave. We will consider application of our model and methodology to the Austrian wage mobility data, also analysed by Pamminger and Fruwirth-Schnatter (2010), and adopting the methods developed in Mukhopadhyay et al. (2011), Mukhopadhyay et al. (2012), will obtain the posterior modes of clusterings and also the desired highest posterior distribution credible regions of the posterior distribution of clusterings.

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