Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or "particle") representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example.

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A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking

Any series of observations ordered along a single dimension, such as time, may be thought of as a time series. The emphasis in time series analysis is on studying the dependence among observations at different points in time. What distinguishes time series analysis from general multivariate analysis is precisely the temporal order imposed on the observations. Many economic variables, such as GNP and its components, price indices, sales, and stock returns are observed over time. In addition to being interested in the contemporaneous relationships among such variables, we are often concerned with relationships between their current and past values, that is, relationships over time.

Time Series Analysis

计量经济分析 = Econometric analysis

Monte Carlo methods are revolutionizing the on-line analysis of data in fields as diverse as financial modeling, target tracking and computer vision. These methods, appearing under the names of bootstrap filters, condensation, optimal Monte Carlo filters, particle filters and survival of the fittest, have made it possible to solve numerically many complex, non-standard problems that were previously intractable. This book presents the first comprehensive treatment of these techniques, including convergence results and applications to tracking, guidance, automated target recognition, aircraft navigation, robot navigation, econometrics, financial modeling, neural networks, optimal control, optimal filtering, communications, reinforcement learning, signal enhancement, model averaging and selection, computer vision, semiconductor design, population biology, dynamic Bayesian networks, and time series analysis. This will be of great value to students, researchers and practitioners, who have some basic knowledge of probability. Arnaud Doucet received the Ph. D. degree from the University of Paris-XI Orsay in 1997. From 1998 to 2000, he conducted research at the Signal Processing Group of Cambridge University, UK. He is currently an assistant professor at the Department of Electrical Engineering of Melbourne University, Australia. His research interests include Bayesian statistics, dynamic models and Monte Carlo methods. Nando de Freitas obtained a Ph.D. degree in information engineering from Cambridge University in 1999. He is presently a research associate with the artificial intelligence group of the University of California at Berkeley. His main research interests are in Bayesian statistics and the application of on-line and batch Monte Carlo methods to machine learning. Neil Gordon obtained a Ph.D. in Statistics from Imperial College, University of London in 1993. He is with the Pattern and Information Processing group at the Defence Evaluation and Research Agency in the United Kingdom. His research interests are in time series, statistical data analysis, and pattern recognition with a particular emphasis on target tracking and missile guidance.

/pdf/sequential-monte-carlo-methods-in-practice-3kfxeebn8e.pdf

Sequential Monte Carlo methods in practice

In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the deterministic filtering literatures these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.

/pdf/on-sequential-monte-carlo-sampling-methods-for-bayesian-3bit26usft.pdf

On sequential Monte Carlo sampling methods for Bayesian filtering

High frequency financial data allows us to learn more about volatility, volatility of volatility and jumps. One of the key techniques developed in the literature in recent years has been bipower variation and its multipower extension, which estimates time-varying volatility robustly to jumps. We improve the scope and efficiency of multipower variation by the use of a more sophisticated exploitation of high frequency data. This suggests very significant improvements in the power of jump tests. It also yields efficient estimates of the integrated variance of the continuous part of a semimartingale. The paper also shows how to extend the theory to the case where there is microstructure in the observations and derive the first nonparametric high frequency estimator of the volatility of volatility. A fundamental device in the paper is a new type of result showing path-by-path (strong) approximation between multipower and the (unobserved) RV based on the continuous part of the process.

Efficient and feasible inference for the components of financial variation using blocked multipower variation

Bojinov & Shephard (2019) defined potential outcome time series to nonparametrically measure dynamic causal effects in time series experiments. Four innovations are developed in this paper: "instrumental paths," treatments which are "shocks," "linear potential outcomes" and the "causal response function." Potential outcome time series are then used to provide a nonparametric causal interpretation of impulse response functions, generalized impulse response functions, local projections and LP-IV.

Econometric analysis of potential outcomes time series: instruments, shocks, linearity and the causal response function

In this paper we extend Rydberg-Shephard's activity, direction and size decomposition of trade-by-trade price movements to the multivariate case. We illustrate our ideas using a bivariate modelling problem --- modelling the evolution of the prices of Ford and GM shares. Throughout we use the continuous record of trades made in the first five months of 1997 on the New York Stock Exchange (NYSE).

https://papers.ssrn.com/sol3/Delivery.cfm/99091723.pdf?abstractid=182248&mirid=1&type=2

Modelling Trade-By-Trade Price Movements of Multiple Assets Using Multivariate Compound Poisson Processes

This article proposes a novel model of financial prices where (i) prices are discrete; (ii) prices change in continuous time; (iii) a high proportion of price changes are reversed in a fraction of a second. Our model is analytically tractable and directly formulated in terms of the calendar time and price impact curve. The resulting cadlag price process is a piecewise constant semimartingale with finite activity, finite variation, and no Brownian motion component. We use moment-based estimations to fit four high-frequency futures datasets and demonstrate the descriptive power of our proposed model. This model is able to describe the observed dynamics of price changes over three different orders of magnitude of time intervals. Supplementary materials for this article are available online.

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Neil Shephard

Papers

Efficient and feasible inference for the components of financial variation using blocked multipower variation

Econometric analysis of potential outcomes time series: instruments, shocks, linearity and the causal response function

Modelling Trade-By-Trade Price Movements of Multiple Assets Using Multivariate Compound Poisson Processes

Continuous time analysis of fleeting discrete price moves

Auxiliary variable based particle filters