Modelling of nonlinear filtering Poisson time series

© Published under licence by IOP Publishing Ltd.In this article, algorithms of non-linear filtering of Poisson time series are tested using statistical modelling. The objective is to find a representation of a time series as a wavelet series with a small number of non-linear coefficients, which allows distinguishing statistically significant details. There are well-known efficient algorithms of non-linear wavelet filtering for the case when the values of a time series have a normal distribution. However, if the distribution is not normal, good results can be expected using the maximum likelihood estimations. The filtration is studied according to the criterion of maximum likelihood by the example of Poisson time series. For direct optimisation of the likelihood function, different stochastic (genetic algorithms, annealing method) and deterministic optimization algorithms are used. Testing of the algorithm using both simulated series and empirical data (series of rare words frequencies according to the Google Books Ngram data were used) showed that filtering based on the criterion of maximum likelihood has a great advantage over well-known algorithms for the case of Poisson series. Also, the most perspective methods of optimisation were selected for this problem

Signal processing with Levy information

Levy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Levy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a "message", then under the transformed measure the original Levy process takes on the character of an "information process". In this paper we develop a theory of such Levy information processes. The underlying Levy process, which we call the fiducial process, represents the "noise type". Each such noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverse Gaussian, and normal inverse Gaussian type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations.Comment: 27 pages. Version to appear in: Proc. R. Soc. London

Extended Object Tracking: Introduction, Overview and Applications

This article provides an elaborate overview of current research in extended object tracking. We provide a clear definition of the extended object tracking problem and discuss its delimitation to other types of object tracking. Next, different aspects of extended object modelling are extensively discussed. Subsequently, we give a tutorial introduction to two basic and well used extended object tracking approaches - the random matrix approach and the Kalman filter-based approach for star-convex shapes. The next part treats the tracking of multiple extended objects and elaborates how the large number of feasible association hypotheses can be tackled using both Random Finite Set (RFS) and Non-RFS multi-object trackers. The article concludes with a summary of current applications, where four example applications involving camera, X-band radar, light detection and ranging (lidar), red-green-blue-depth (RGB-D) sensors are highlighted.Comment: 30 pages, 19 figure

A filtering approach to tracking volatility from prices observed at random times

This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process $S=(S_{t})_{t\geq0}$ is given by $$dS_{t}=r(\theta_{t})S_{t}dt+v(\theta_{t})S_{t}dB_{t},$$ where $B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function, and $\theta=(\theta_{t})_{t\geq0}$ is a c\'{a}dl\'{a}g strong Markov process. The random process $\theta$ is unobservable. We assume also that the asset price $S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<....$ This is an appropriate assumption when modelling high frequency financial data (e.g., tick-by-tick stock prices). In the above setting the problem of estimation of $\theta$ can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process $(\tau_{k},\log S_{\tau_{k}})$. While quite natural, this problem does not fit into the standard diffusion or simple point process filtering frameworks and requires more technical tools. We derive a closed form optimal recursive Bayesian filter for $\theta_{t}$, based on the observations of $(\tau_{k},\log S_{\tau_{k}})_{k\geq1}$. It turns out that the filter is given by a recursive system that involves only deterministic Kolmogorov-type equations, which should make the numerical implementation relatively easy

Dynamic Variation in Sexual Contact Rates in a Cohort of HIV-Negative Gay Men

Human immunodeficiency virus (HIV) transmission models that include variability in sexual behavior over time have shown increased incidence, prevalence, and acute-state transmission rates for a given population risk profile. This raises the question of whether dynamic variation in individual sexual behavior is a real phenomenon that can be observed and measured. To study this dynamic variation, we developed a model incorporating heterogeneity in both between-person and within-person sexual contact patterns. Using novel methodology that we call iterated filtering for longitudinal data, we fitted this model by maximum likelihood to longitudinal survey data from the Centers for Disease Control and Prevention's Collaborative HIV Seroincidence Study (1992–1995). We found evidence for individual heterogeneity in sexual behavior over time. We simulated an epidemic process and found that inclusion of empirically measured levels of dynamic variation in individual-level sexual behavior brought the theoretical predictions of HIV incidence into closer alignment with reality given the measured per-act probabilities of transmission. The methods developed here provide a framework for quantifying variation in sexual behaviors that helps in understanding the HIV epidemic among gay men