With the technical advances in neuroscience experiments in the past few decades, we have seen a massive expansion in our ability to record neural activity. These advances enable neuroscientists to analyze more complex neural coding and communication properties, and at the same time, raise new challenges for analyzing neural spiking data, which keeps growing in scale, dimension, and complexity. This thesis proposes several new statistical methods that advance statistical analysis approaches for neural spiking data, including sequential Monte Carlo (SMC) methods for efficient estimation of neural dynamics from membrane potential threshold crossings, state-space models using multimodal observation processes, and goodness-of-fit analysis methods for neural marked point process models. In a first project, we derive a set of iterative formulas that enable us to simulate trajectories from stochastic, dynamic neural spiking models that are consistent with a set of spike time observations. We develop a SMC method to simultaneously estimate the parameters of the model and the unobserved dynamic variables from spike train data. We investigate the performance of this approach on a leaky integrate-and-fire model. In another project, we define a semi-latent state-space model to estimate information related to the phenomenon of hippocampal replay. Replay is a recently discovered phenomenon where patterns of hippocampal spiking activity that typically occur during exploration of an environment are reactivated when an animal is at rest. This reactivation is accompanied by high frequency oscillations in hippocampal local field potentials. However, methods to define replay mathematically remain undeveloped. In this project, we construct a novel state-space model that enables us to identify whether replay is occurring, and if so to estimate the movement trajectories consistent with the observed neural activity, and to categorize the content of each event. The state-space model integrates information from the spiking activity from the hippocampal population, the rhythms in the local field potential, and the rat's movement behavior. Finally, we develop a new, general time-rescaling theorem for marked point processes, and use this to develop a general goodness-of-fit framework for neural population spiking models. We investigate this approach through simulation and a real data application.

Inflation is one of the most important macroeconomic variables. It is vital that policy makers receive accurate forecasts of inflation so that they can adjust their monetary policy to attain stability in the economy which has been shown to lead to economic growth. The purpose of this study is to model inflation and evaluate if applying the Kalman filter to SARIMA models lead to higher forecast accuracy compared to just using the SARIMA model. The Box-Jenkins approach to SARIMA modelling is used to obtain well-fitted SARIMA models and then to use a subset of observations to estimate a SARIMA model on which the Kalman filter is applied for the rest of the observations. These models are identified and then estimated with the use of monthly inflation for Luxembourg, Mexico, Portugal and Switzerland with the target to use them for forecasting. The accuracy of the forecasts are then evaluated with the error measures mean squared error (MSE), mean average deviation (MAD), mean average percentage error (MAPE) and the statistic Theil's U. For all countries these measures indicate that the Kalman filtered model yield more accurate forecasts. The significance of these differences are then evaluated with the Diebold-Mariano test for which only the difference in forecast accuracy of Swiss inflation is proven significant. Thus, applying the Kalman filter to SARIMA models with the target to obtain forecasts of monthly inflation seem to lead to higher or at least not lower predictive accuracy for the monthly inflation of these countries.

In the present paper we study switching state space models from a Bayesian point of view. For estimation, the model is reformulated as a hierarchical model. We discuss various MCMC methods for Bayesian estimation, among them unconstrained Gibbs sampling, constrained sampling and permutation sampling. We address in detail the problem of unidentifiability, and discuss potential information available from an unidentified model. Furthermore the paper discusses issues in model selection such as selecting the number of states or testing for the presence of Markov switching heterogeneity. The model likelihoods of all possible hypotheses are estimated by using the method of bridge sampling. We conclude the paper with applications to simulated data as well as to modelling the U.S./U.K. real exchange rate. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Target tracking is a broad subject area extensively studied in many engineering disciplines. In this thesis, target tracking implies the temporal estimation of target features such as the target's direction-of-arrival (DOA), the target's boundary pixels in a sequence of images, and/or the target's position in space. For multiple target tracking, we have introduced a new motion model that incorporates an acceleration component along the heading direction of the target. We have also shown that the target motion parameters can be considered part of a more general feature set for target tracking, e.g., target frequencies, which may be unrelated to the target motion, can be used to improve the tracking performance. We have introduced an acoustic multiple-target tracker using a flexible observation model based on an image tracking approach by assuming that the DOA observations might be spurious and that some of the DOAs might be missing in the observation set. We have also addressed the acoustic calibration problem from sources of opportunity such as beacons or a moving source. We have derived and compared several calibration methods for the case where the node can hear a moving source whose position can be reported back to the node. The particle filter, as a recursive algorithm, requires an initialization phase prior to tracking a state vector. The Metropolis-Hastings (MH) algorithm has been used for sampling from intractable multivariate target distributions and is well suited for the initialization problem. Since the particle filter only needs samples around the mode, we have modified the MH algorithm to generate samples distributed around the modes of the target posterior. By simulations, we show that this mode hungry algorithm converges an order of magnitude faster than the original MH scheme. Finally, we have developed a general framework for the joint state-space tracking problem. A proposal strategy for joint state-space tracking using the particle filters is defined by carefully placing the random support of the joint filter in the region where the final posterior is likely to lie. Computer simulations demonstrate improved performance and robustness of the joint state-space when using the new particle proposal strategy.

We define a subclass of dynamic linear models with unknown hyperparameters called d-inverse-gamma models. We then approximate the marginal p.d.f.s of the hyperparameter and the state vector by the data augmentation algorithm of Tanner/Wong. We prove that the regularity conditions for convergence hold. A sampling based scheme for practical implementation is discussed. Finally, we illustrate how to obtain an iterative importance sampling estimate of the model likelihood. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Sequential Monte Carlo (SMC) and Forward Filtering Backward Sampling (FFBS) are the two most often seen algorithms for Bayesian state space models analysis. Various results regarding the applicability has been either claimed or shown. It is said that SMC would excel under nonlinear, non-Gaussian situations, and less computationally expansive. On the other hand, it has been shown that with techniques such as Grid approximation (Hore et al. 2010), FFBS based methods would do no worse, though still can be computationally expansive, but provide more exact information. The purpose of this report to compare the two methods with simulated data sets, and further explore whether there exist some clear criteria that may be used to determine a priori which methods would suit the study better. / text

It is suggested to discriminate between different state space models for a given time series by means of a Bayesian approach which chooses the model that minimizes the expected loss. Practical implementation of this procedures requires a fully Bayesian analysis for both the state vector and the unknown hyperparameters which is carried out by Markov chain Monte Carlo methods. Application to some non-standard situations such as testing hypotheses on the boundary of the parameter space, discriminating non-nested models and discrimination of more than two models is discussed in detail. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Le modèle RiskMetrics développé par la Banque JP Morgan suite à l'amendement des accords de Bâle de 1988 a été érigé comme mesure de risque financier pour faire face aux importantes perturbations ayant affecté les marchés bancaires internationaux. Communément appelé Value at Risk, il a été admis par l'ensemble des organes et institutions financiers comme une mesure de risque cohérente. Malgré sa popularité, elle est le sujet de beaucoup de controverses. En effet, les paramètres d'estimation du système RiskMetrics sont supposés fixes au cours du temps ce qui est contraire aux caractéristiques des marchés financiers. Deux raisons valables permettent de justifier cette instabilité temporelle : * la présence d'agents hétérogènes fait qu'on n'analyse plus la VaR en se focalisant sur une seule dimension temporelle mais plutôt sur des fréquences de trading (nous recourons pour cela à la méthode Wavelet). * la structure des séries financières qui d'habitude est affectée par les phénomènes de crash, bulle etc. Ceux-ci peuvent être considérés comme des variables cachées qu'on doit prendre en compte dans l'évaluation du risque. Pour cela, nous recourons à la modélisation espace d'états et au filtre de Kalman. Nous savons d'emblée que les performances de la VaR s'évaluent en recourant au test de backtesting. Celui-ci repose sur la technique de régression roulante qui montre une faille évidente : Nous ne pouvons pas connaitre le processus gouvernant la variation des paramètres, il n'y a pas endogénéisation de la dynamique de ceux-ci. Pour apporter une solution à ce problème, nous proposons une application du filtre de Kalman sur les modèles VaR et WVaR. Ce filtre, par ses fonctions corrige de manière récursive les paramètres dans le temps. En ces termes nous définissons une mesure de risque dit SVaR qui en réalité est la VaR obtenue par une actualisation des paramètres d'estimation. Elle permet une estimation précise de la volatilité qui règne sur le marché financier. Elle donne ainsi la voie à toute institution financière de disposer de suffisamment de fonds propres pour affronter le risque de marché. / The RiskMetrics model developed by the bank JP Morgan following the amendment of Basel accords 1988 was erected as a measure of financial risk to deal with important disturbances affecting international banking markets. Commonly known as Value at Risk, it was accepted by all bodies and financial institutions to be a coherent risk measure. Despite its popularity, it is the subject of many controversies. Indeed, the estimation parameters of RiskMetrics are assumed to be fixed over time, which is contrary to the characteristics of financial markets. Two valid reasons are used to justify temporal instability : *Due to the presence of heterogenous agents the VaR is not analysed by focusing on a single temporal dimension but rather on trading frequencies (we use Wavelet method for it). *The structure of financial time series wich is usually affected by the crash bubble phenomenons and so on. These can be considered as hidden variables that we must take into account in the risk assessment. For this, we use state space modeling and kalman filter. We immediately know that performances of the VaR are evaluated using backtesting test. This is based on the technique of rolling regression wich shows an obvious break : We can not know the processes governing the variation of parameters; there is no endogeneisation dynamics thereof. To provide a solution to this problem, we propose an application of the kalman filter on VaR and WVaR models. This filter recursively corrects by its functions the parameters of time. In these terms we define a risk measure called SVaR wich in realitity is the VaR obtained by updating estimation parameters. It provides an accurate estimate of the volatility existing in the financial market. It thus gives way to any financial institution to have enough capital to face market risk.

This work describes a framework for simultaneous estimation and modeling (SEAM) of dynamic systems using non-Gaussian belief fusion by first presenting the relevant fundamental formulations, then building upon these formulations incrementally towards a more general and ubiquitous framework. Multi-Gaussian belief fusion (MBF) is introduced as a natural and effective method of fusing non-Gaussian probability distribution functions (PDFs) in arbitrary dimensions efficiently and with no loss of accuracy. Construction of some multi-Gaussian structures for potential use in MBF is addressed. Furthermore, recursive Bayesian estimation (RBE) is developed for linearized systems with uncertainty in model parameters, and a rudimentary motion model correction stage is introduced. A subsequent improvement to motion model correction for arbitrarily non-Gaussian belief is developed, followed by application to observation models. Finally, SEAM is generalized to fully nonlinear and non-Gaussian systems. Several parametric studies were performed on simulated experiments in order to assess the various dependencies of the SEAM framework and validate its effectiveness in both estimation and modeling. The results of these studies show that SEAM is capable of improving estimation when uncertainty is present in motion and observation models as compared to existing methods. Furthermore, uncertainty in model parameters is consistently reduced as these parameters are updated throughout the estimation process. SEAM and its constituents have potential uses in robotics, target tracking and localization, state estimation, and more. / Doctor of Philosophy / The simultaneous estimation and modeling (SEAM) framework and its constituents described in this dissertation aim to improve estimation of signals where significant uncertainty would normally introduce error. Such signals could be electrical (e.g. voltages, currents, etc.), mechanical (e.g. accelerations, forces, etc.), or the like. Estimation is accomplished by addressing the problem probabilistically through information fusion. The proposed techniques not only improve state estimation, but also effectively "learn" about the system of interest in order to further refine estimation. Potential uses of such methods could be found in search-and-rescue robotics, robust control algorithms, and the like. The proposed framework is well-suited for any context where traditional estimation methods have difficulty handling heightened uncertainty.

<p>Transcriptional regulation has been studied intensively in recent decades. One important aspect of this regulation is the interaction between regulatory proteins, such as transcription factors (TF) and nucleosomes, and the genome. Different high-throughput techniques have been invented to map these interactions genome-wide, including ChIP-based methods (ChIP-chip, ChIP-seq, etc.), nuclease digestion methods (DNase-seq, MNase-seq, etc.), and others. However, a single experimental technique often only provides partial and noisy information about the whole picture of protein-DNA interactions. Therefore, the overarching goal of this dissertation is to provide computational developments for jointly modeling different experimental datasets to achieve a holistic inference on the protein-DNA interaction landscape. </p><p>We first present a computational framework that can incorporate the protein binding information in MNase-seq data into a thermodynamic model of protein-DNA interaction. We use a correlation-based objective function to model the MNase-seq data and a Markov chain Monte Carlo method to maximize the function. Our results show that the inferred protein-DNA interaction landscape is concordant with the MNase-seq data and provides a mechanistic explanation for the experimentally collected MNase-seq fragments. Our framework is flexible and can easily incorporate other data sources. To demonstrate this flexibility, we use prior distributions to integrate experimentally measured protein concentrations. </p><p>We also study the ability of DNase-seq data to position nucleosomes. Traditionally, DNase-seq has only been widely used to identify DNase hypersensitive sites, which tend to be open chromatin regulatory regions devoid of nucleosomes. We reveal for the first time that DNase-seq datasets also contain substantial information about nucleosome translational positioning, and that existing DNase-seq data can be used to infer nucleosome positions with high accuracy. We develop a Bayes-factor-based nucleosome scoring method to position nucleosomes using DNase-seq data. Our approach utilizes several effective strategies to extract nucleosome positioning signals from the noisy DNase-seq data, including jointly modeling data points across the nucleosome body and explicitly modeling the quadratic and oscillatory DNase I digestion pattern on nucleosomes. We show that our DNase-seq-based nucleosome map is highly consistent with previous high-resolution maps. We also show that the oscillatory DNase I digestion pattern is useful in revealing the nucleosome rotational context around TF binding sites. </p><p>Finally, we present a state-space model (SSM) for jointly modeling different kinds of genomic data to provide an accurate view of the protein-DNA interaction landscape. We also provide an efficient expectation-maximization algorithm to learn model parameters from data. We first show in simulation studies that the SSM can effectively recover underlying true protein binding configurations. We then apply the SSM to model real genomic data (both DNase-seq and MNase-seq data). Through incrementally increasing the types of genomic data in the SSM, we show that different data types can contribute complementary information for the inference of protein binding landscape and that the most accurate inference comes from modeling all available datasets. </p><p>This dissertation provides a foundation for future research by taking a step toward the genome-wide inference of protein-DNA interaction landscape through data integration.</p> / Dissertation