Forward Filtering Backward Sampling Research Articles

Forecasting Intra-individual Changes of Affective States Taking into Account Inter-individual Differences Using Intensive Longitudinal Data from a University Student Dropout Study in Math

The longitudinal process that leads to university student dropout in STEM subjects can be described by referring to (a) inter-individual differences (e.g., cognitive abilities) as well as (b) intra-individual changes (e.g., affective states), (c) (unobserved) heterogeneity of trajectories, and d) time-dependent variables. Large dynamic latent variable model frameworks for intensive longitudinal data (ILD) have been proposed which are (partially) capable of simultaneously separating the complex data structures (e.g., DLCA; Asparouhov et al. in Struct Equ Model 24:257–269, 2017; DSEM; Asparouhov et al. in Struct Equ Model 25:359–388, 2018; NDLC-SEM, Kelava and Brandt in Struct Equ Model 26:509–528, 2019). From a methodological perspective, forecasting in dynamic frameworks allowing for real-time inferences on latent or observed variables based on ongoing data collection has not been an extensive research topic. From a practical perspective, there has been no empirical study on student dropout in math that integrates ILD, dynamic frameworks, and forecasting of critical states of the individuals allowing for real-time interventions. In this paper, we show how Bayesian forecasting of multivariate intra-individual variables and time-dependent class membership of individuals (affective states) can be performed in these dynamic frameworks using a Forward Filtering Backward Sampling method. To illustrate our approach, we use an empirical example where we apply the proposed forecasting method to ILD from a large university student dropout study in math with multivariate observations collected over 50 measurement occasions from multiple students (N = 122). More specifically, we forecast emotions and behavior related to dropout. This allows us to predict emerging critical dynamic states (e.g., critical stress levels or pre-decisional states) 8 weeks before the actual dropout occurs.

Bayesian Multiple Changepoint Detection for Stochastic Models in Continuous Time

A multiple changepoint model in continuous time is formulated as a continuous-time hidden Markov model, defined on a countable infinite state space. The new formulation of the multiple changepoint model allows the model complexities, i.e. the number of changepoints, to accrue unboundedly upon the arrivals of new data. Inference on the number of changepoints and their locations is based on a collapsed Gibbs sampler. We suggest a new version of forward-filtering backward-sampling (FFBS) algorithm in continuous time for simulating the full trajectory of the latent Markov chain, i.e. the changepoints. The FFBS algorithm is based on a randomized time-discretization for the latent Markov chain through uniformization schemes, combined with a discrete-time version of FFBS algorithm. It is shown that, desirably, both the computational cost and the memory cost of the FFBS algorithm are only quadratic to the number of changepoints. The new formulation of the multiple changepoint models allows varying scale of run lengths of changepoints to be characterized. We demonstrate the methods through simulations and a real data example for earthquakes.

Bayesian multiple changepoints detection for Markov jump processes

A Bayesian multiple changepoint model for the Markov jump process is formulated as a Markov double chain model in continuous time. Inference for this type of multiple changepoint model is based on a two-block Gibbs sampling scheme. We suggest a continuous-time version of forward-filtering backward-sampling (FFBS) algorithm for sampling the full trajectories of the latent Markov chain via inverse transformation. We also suggest a continuous-time version of Viterbi algorithm for this Markov double chain model, which is viable for obtaining the MAP estimation of the set of locations of changepoints. The model formulation and the continuous-time version of forward-filtering backward-sampling algorithm and Viterbi algorithm can be extended to simultaneously monitor the structural breaks of multiple Markov jump processes, which may have either variable transition rate matrix or identical transition rate matrix. We then perform a numerical study to demonstrate the methods. In numerical examples, we compare the performance of the FFBS algorithm via inverse transformation with the FFBS algorithm via uniformization. The two algorithms performs well in different cases. These models are potentially viable for modelling the credit rating dynamics in credit risk management.

Bayesian Inference on Dynamic Linear Models of Day-to-Day Origin-Destination Flows in Transportation Networks

Estimation of origin–destination (OD) demand plays a key role in successful transportation studies. In this paper, we consider the estimation of time-varying day-to-day OD flows given data on traffic volumes in a transportation network for a sequence of days. We propose a dynamic linear model (DLM) in order to represent the stochastic evolution of OD flows over time. DLMs are Bayesian state-space models which can capture non-stationarity. We take into account the hierarchical relationships between the distribution of OD flows among routes and the assignment of traffic volumes on links. Route choice probabilities are obtained through a utility model based on past route costs. We propose a Markov chain Monte Carlo algorithm, which integrates Gibbs sampling and a forward filtering backward sampling technique, in order to approximate the joint posterior distribution of mean OD flows and parameters of the route choice model. Our approach can be applied to congested networks and in the case when data are available on only a subset of links. We illustrate the application of our approach through simulated experiments on a test network from the literature.

Expectation-Maximization Tensor Factorization for Practical Location Privacy Attacks

Abstract Location privacy attacks based on a Markov chain model have been widely studied to de-anonymize or de-obfuscate mobility traces. An adversary can perform various kinds of location privacy attacks using a personalized transition matrix, which is trained for each target user. However, the amount of training data available to the adversary can be very small, since many users do not disclose much location information in their daily lives. In addition, many locations can be missing from the training traces, since many users do not disclose their locations continuously but rather sporadically. In this paper, we show that the Markov chain model can be a threat even in this realistic situation. Specifically, we focus on a training phase (i.e. mobility profile building phase) and propose Expectation-Maximization Tensor Factorization (EMTF), which alternates between computing a distribution of missing locations (E-step) and computing personalized transition matrices via tensor factorization (M-step). Since the time complexity of EMTF is exponential in the number of missing locations, we propose two approximate learning methods, one of which uses the Viterbi algorithm while the other uses the Forward Filtering Backward Sampling (FFBS) algorithm. We apply our learning methods to a de-anonymization attack and a localization attack, and evaluate them using three real datasets. The results show that our learning methods significantly outperform a random guess, even when there is only one training trace composed of 10 locations per user, and each location is missing with probability 80% (i.e. even when users hardly disclose two temporally-continuous locations).

Convergence of sequential quasi-Monte Carlo smoothing algorithms

Gerber and Chopin [J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 509–579] recently introduced Sequential quasi-Monte Carlo (SQMC) algorithms as an efficient way to perform filtering in state–space models. The basic idea is to replace random variables with low-discrepancy point sets, so as to obtain faster convergence than with standard particle filtering. Gerber and Chopin (2015) describe briefly several ways to extend SQMC to smoothing, but do not provide supporting theory for this extension. We discuss more thoroughly how smoothing may be performed within SQMC, and derive convergence results for the so-obtained smoothing algorithms. We consider in particular SQMC equivalents of forward smoothing and forward filtering backward sampling, which are the most well-known smoothing techniques. As a preliminary step, we provide a generalization of the classical result of Hlawka and Muck [Computing (Arch. Elektron. Rechnen) 9 (1972) 127–138] on the transformation of QMC point sets into low discrepancy point sets with respect to non uniform distributions. As a corollary of the latter, we note that we can slightly weaken the assumptions to prove the consistency of SQMC.

Dynamic generalized extreme value modeling via particle filters

ABSTRACTThis article addresses how particle filters compare to MCMC methods for posterior density approximations of a model that allows for a dynamic state with fixed parameters and where the observation equation is nonlinear. This is a problem that was not been well studied in the specialized literature. We prove that these state and parameter estimations can be achieved via particle filter methods without the need of more expensive Forward Filtering Backward Sampling (FFBS) simulation. Estimation of a time-varying extreme value model via the generalized extreme value distribution is considered using these particle filter methods and compared to a MCMC algorithm that involves a variety of Metropolis-Hastings steps. We illustrate and compare the different methodologies with simulated data and some minimum daily stock returns occurring monthly from January 4, 1990 to December 28, 2007 using the Tokyo Stock Price Index (TOPIX).

Efficient Gibbs sampling for Markov switching GARCH models

Efficient simulation techniques for Bayesian inference on Markov-switching (MS) GARCH models are developed. Different multi-move sampling techniques for Markov switching state space models are discussed with particular attention to MS-GARCH models. The multi-move sampling strategy is based on the Forward Filtering Backward Sampling (FFBS) approach applied to auxiliary MS-GARCH models. A unified framework for MS-GARCH approximation is developed and this not only encompasses the considered specifications, but provides an avenue to generate new variants of MS-GARCH auxiliary models. The use of multi-point samplers, such as the multiple-try Metropolis and the multiple-trial metropolized independent sampler, in combination with FFBS, is considered in order to reduce the correlation between successive iterates and to avoid getting trapped by local modes of the target distribution. Antithetic sampling within the FFBS is also suggested to further improve the sampler’s efficiency. The simulation study indicates that the multi-point and multi-move strategies can be more efficient than other MCMC schemes, especially when the MS-GARCH is not strongly persistent. Finally, an empirical application to financial data shows the efficiency and effectiveness of the proposed estimation procedure.

Efficient Gibbs Sampling for Markov Switching GARCH Models

We develop efficient simulation techniques for Bayesian inference on switching GARCH models. Our contribution to existing literature is manifold. First, we discuss different multi-move sampling techniques for Markov Switching (MS) state space models with particular attention to MS-GARCH models. Our multi-move sampling strategy is based on the Forward Filtering Backward Sampling (FFBS) applied to an approximation of MS-GARCH. Another important contribution is the use of multi-point samplers, such as the Multiple-Try Metropolis (MTM) and the Multiple Trial Metropolize Independent Sampler, in combination with FFBS for the MS-GARCH process. In this sense we extend to the MS state space models the work of So (2006) on efficient MTM sampler for continuous state space models. Finally, we suggest to further improve the sampler efficiency by introducing the antithetic sampling of Craiu and Meng (2005) and Craiu and Lemieux (2007) within the FFBS. Our simulation experiments on MS-GARCH model show that our multi-point and multi-move strategies allow the sampler to gain efficiency when compared with single-move Gibbs sampling.

Factor Selection in Hedge Fund Replication Dynamic Models

While the use of dynamic factor models for Hedge Fund Replication (HFR) has proven to be superior to standard OLS methodologies (e.g., Roncalli and Weisang, 2009a,b), current factor selection methodology in these dynamic settings by means of direct PCA-based estimation of the factors (e.g., Darolles and Mero, 2008) provides results having poor interpretability and small, yet significant, improvement over more naive selection methodologies. In this paper, I contend on the one hand that this type of direct factor estimation is doomed to be at best biased and unreliable in finite samples as indicated by results from random matrix theory (Harding, 2008). On the other hand, forward filtering-backward sampling (FFBS) algorithm and reversible-jump MCMC allow considering a factor selection methodology using observable instruments as potential factors hence solving both problems of identification and interpretability. I provide here a detailed algorithm and the results of its application to a simple and classic replication example.