Nonparametric Uncertainty Quantification for Stochastic Gradient Flows

This paper presents a nonparametric statistical modeling method for quantifying uncertainty in stochastic gradient systems with isotropic diffusion. The central idea is to apply the diffusion maps algorithm to a training data set to produce a stochastic matrix whose generator is a discrete approximation to the backward Kolmogorov operator of the underlying dynamics. The eigenvectors of this stochastic matrix, which we will refer to as the diffusion coordinates, are discrete approximations to the eigenfunctions of the Kolmogorov operator and form an orthonormal basis for functions defined on the data set. Using this basis, we consider the projection of three uncertainty quantification (UQ) problems (prediction, filtering, and response) into the diffusion coordinates. In these coordinates, the nonlinear prediction and response problems reduce to solving systems of infinite-dimensional linear ordinary differential equations. Similarly, the continuous-time nonlinear filtering problem reduces to solving a system of infinite-dimensional linear stochastic differential equations. Solving the UQ problems then reduces to solving the corresponding truncated linear systems in finitely many diffusion coordinates. By solving these systems we give a model-free algorithm for UQ on gradient flow systems with isotropic diffusion. We numerically verify these algorithms on a 1-dimensional linear gradient flow system where the analytic solutions of the UQ problems are known. We also apply the algorithm to a chaotically forced nonlinear gradient flow system which is known to be well approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11

Addressing the Data Recency Problem in Collaborative Filtering Systems

Recommender systems are being widely applied in many E-commerce sites to suggest products, services, and information items to potential users. Collabora-tive filtering systems, the most successful recommender system technology to date, help people make choices based on the opinions of other people. While collaborative filtering systems have been a substantial success, there are sev-eral problems that researchers and commercial applications have identified: the early rater problem, the sparsity problem, and the large scale problem. Moreover, existing collaborative filtering systems do not consider data re-cency. For this reason, if a user\u27s preferences have changed over time, the sys-tems might not recognize it quickly. This thesis studies how to apply data re-cency to collaborative filtering systems to get more predictive accuracy. We define the data recency problem as the negative impact of old data on the pre-dictive accuracy of collaborative filtering systems. In order to mitigate this shortcoming, the combinations of time-based forgetting mechanisms, pruning and non-pruning strategies and linear and kernel functions, are utilized to ap-ply weights. A clustering technique is employed to detect the user\u27s changing preferences. We apply our research approach to the DeliBook dataset. The goal of our experiments is to show that our algorithm that incorporates tempo-ral factors provides better recommendations than existing methods

Distributed Kalman Filters over Wireless Sensor Networks: Data Fusion, Consensus, and Time-Varying Topologies

Kalman filtering is a widely used recursive algorithm for optimal state estimation of linear stochastic dynamic systems. The recent advances of wireless sensor networks (WSNs) provide the technology to monitor and control physical processes with a high degree of temporal and spatial granularity. Several important problems concerning Kalman filtering over WSNs are addressed in this dissertation. First we study data fusion Kalman filtering for discrete-time linear time-invariant (LTI) systems over WSNs, assuming the existence of a data fusion center that receives observations from distributed sensor nodes and estimates the state of the target system in the presence of data packet drops. We focus on the single sensor node case and show that the critical data arrival rate of the Bernoulli channel can be computed by solving a simple linear matrix inequality problem. Then a more general scenario is considered where multiple sensor nodes are employed. We derive the stationary Kalman filter that minimizes the average error variance under a TCP-like protocol. The stability margin is adopted to tackle the stability issue. Second we study distributed Kalman filtering for LTI systems over WSNs, where each sensor node is required to locally estimate the state in a collaborative manner with its neighbors in the presence of data packet drops. The stationary distributed Kalman filter (DKF) that minimizes the local average error variance is derived. Building on the stationary DKF, we propose Kalman consensus filter for the consensus of different local estimates. The upper bound for the consensus coefficient is computed to ensure the mean square stability of the error dynamics. Finally we focus on time-varying topology. The solution to state consensus control for discrete-time homogeneous multi-agent systems over deterministic time-varying feedback topology is provided, generalizing the existing results. Then we study distributed state estimation over WSNs with time-varying communication topology. Under the uniform observability, each sensor node can closely track the dynamic state by using only its own observation, plus information exchanged with its neighbors, and carrying out local computation

Adaptive filtering techniques for gravitational wave interferometric data: Removing long-term sinusoidal disturbances and oscillatory transients

It is known by the experience gained from the gravitational wave detector proto-types that the interferometric output signal will be corrupted by a significant amount of non-Gaussian noise, large part of it being essentially composed of long-term sinusoids with slowly varying envelope (such as violin resonances in the suspensions, or main power harmonics) and short-term ringdown noise (which may emanate from servo control systems, electronics in a non-linear state, etc.). Since non-Gaussian noise components make the detection and estimation of the gravitational wave signature more difficult, a denoising algorithm based on adaptive filtering techniques (LMS methods) is proposed to separate and extract them from the stationary and Gaussian background noise. The strength of the method is that it does not require any precise model on the observed data: the signals are distinguished on the basis of their autocorrelation time. We believe that the robustness and simplicity of this method make it useful for data preparation and for the understanding of the first interferometric data. We present the detailed structure of the algorithm and its application to both simulated data and real data from the LIGO 40meter proto-type.Comment: 16 pages, 9 figures, submitted to Phys. Rev.

A minimum variance filter for discrete time linear systems with parametric uncertainty

A minimum variance filter for a class of discrete time systems with additive as well as multiplicative noise is investigated in this paper. We extend the results from recent work by Ponomareva and Date to account for multiplicative noise in the measurement equation. More importantly, we provide an interpretation of the multiplicative noise in both transition and measurement equations in terms of parameter perturbations in a linear additive model. The utility of the proposed filtering algorithm is demonstrated through numerical simulation experiments using models from academic literature where the parameters are estimated from real data

An Adaptive Identification and Prediction Algorithm for the Real-Time Forecasting of Hydrologic Time Series

In order to achieve the effective control of water resources systems, one must know the future behavior of the inputs to that particular system. Because of the uncertainties inherent in water resources processes, the prediction algorithm, to be constructed, should include stochastic elements, too. Moreover, the algorithm should be recursive to avoid cumbersome computations and to be able for real-time forecasting. In this paper we present a method which is applicable for both linear and nonlinear hydrologic systems having not completely time-invariant properties. The algorithms are based on the state space description of the processes involved and utilize the Kalman stochastic filtering technique. Due to the unknown nature of noise processes, the basic algorithms were changed to be adaptive. Using the algorithms the joint handling of water quantity and quality data becomes feasible