There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Modern Applied Statistics With S

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

On robust estimation of the location parameter

Praise for the Third Edition: "This is one of the best books available. Its excellent organizational structure allows quick reference to specific models and its clear presentation . . . solidifies the understanding of the concepts being presented."IIE Transactions on Operations EngineeringThoroughly revised and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fourth Edition continues to present the basic statistical principles that are necessary to analyze the probabilistic nature of queues. Rather than presenting a narrow focus on the subject, this update illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research.This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics of the Fourth Edition include:Retrial queuesApproximations for queueing networksNumerical inversion of transformsDetermining the appropriate number of servers to balance quality and cost of serviceEach chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results. In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations. New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site.With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.

Fundamentals of Queueing Theory

This article presents a new estimation method for the parameters of a time series model. We consider here composite Gaussian processes that are the sum of independent Gaussian processes which, in turn, explain an important aspect of the time series, as is the case in engineering and natural sciences. The proposed estimation method offers an alternative to classical estimation based on the likelihood, that is straightforward to implement and often the only feasible estimation method with complex models. The estimator furnishes results as the optimization of a criterion based on a standardized distance between the sample wavelet variances (WV) estimates and the model-based WV. Indeed, the WV provides a decomposition of the variance process through different scales, so that they contain the information about different features of the stochastic model. We derive the asymptotic properties of the proposed estimator for inference and perform a simulation study to compare our estimator to the MLE and the LSE with...

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Wavelet-variance-based estimation for composite stochastic processes

In this article, we investigate two different algorithms for the integration of GPS with redundant MEMS-IMUs. Firstly, the inertial measurements are combined in the observation space to generate a synthetic set of data which is then integrated with GPS by the standard algorithms. In the second approach, the method of strapdown navigation needs to be adapted in order to account for the redundant measurements. Both methods are evaluated in experiments where redundant MEMS-IMUs are fixed in different geometries: orthogonally-redundant and skew-redundant IMUs. For the latter configuration, the performance improvement using a synthetic IMU is shown to be 30% on the average. The extended mechanization approach provides slightly better results (about 45% improvement) as the systematic errors of the individual sensors are considered separately rather than their fusion when forming compound measurements. The maximum errors are shown to be reduced even by a factor of 2.

/pdf/redundant-mems-imu-integrated-with-gps-for-performance-2p6mmc9lwr.pdf

Redundant MEMS-IMU integrated with GPS for performance assessment in sports

This research studies the reduction and the estimation of the noise level within a redundant configuration of low-cost (MEMS-type) inertial measurement units (IMUs). Firstly, independent observations between units and sensors are assumed and the theoretical decrease in the system noise level is analyzed in an experiment with four MEMS-IMU triads. Then, more complex scenarios are presented in which the noise level can vary in time and for each sensor. A statistical method employed for studying the volatility of financial markets (GARCH) is adapted and tested for the usage with inertial data. This paper demonstrates experimentally and through simulations the benefit of direct noise estimation in redundant IMU setups.

/pdf/noise-reduction-and-estimation-in-multiple-micro-electro-2p4j64y2z8.pdf

Noise reduction and estimation in multiple micro-electro-mechanical inertial systems

Although experimental results have demonstrated that redundant MEMS-IMUs integrated with GPS are an efficient way to improve navigation performances, the precise relationship between the number of sensors employed and the accuracy enhancement remains unclear. This article aims at demonstrating, with the help of simulations, that multiple MEMS-IMU systems can be designed according to specifications. This enables to better define the relationship between the number of sensors employed and the accuracy improvement as well as to ascertain the precise number of sensors needed to fulfill the system's requirements. This proves to be highly helpful in designing navigation systems for applications that require a specific precision. This article also aims at demonstrating the impact of sensors' orientation on the system performances. To achieve this, a new method based on partial redundancies is introduced to formalize the determination of optimal geometry of multi-IMU systems. It shows that, when dealing with IMU triads, the optimality of such systems is independent of the geometry between them. This result, moreover, presents important practical implications since it demonstrates that complicated geometries, traditionally employed in such systems, can be avoided. Additionally, it also proves that navigation performances obtained by simulations with a certain number of sensors are valid independently from orientation amongst these sensors.

Improving Accuracy with Multiple Sensors: Study of Redundant MEMS-IMU/GPS Configurations

The integration of observations issued from a satellite-based system (GNSS) with an inertial navigation system (INS) is usually performed through a Bayesian filter such as the extended Kalman filter (EKF). The task of designing the navigation EKF is strongly related to the inertial sensor error modeling problem. Accelerometers and gyroscopes may be corrupted by random errors of complex spectral structure. Consequently, identifying correct error-state parameters in the INS/GNSS EKF becomes difficult when several stochastic processes are superposed. In such situations, classical approaches like the Allan variance (AV) or power spectral density (PSD) analysis fail due to the difficulty of separating the error processes in the spectral domain. For this purpose, we propose applying a recently developed estimator based on the generalized method of wavelet moments (GMWM), which was proven to be consistent and asymptotically normally distributed. The GMWM estimator matches theoretical and sample-based wavelet variances (WVs), and can be computed using the method of indirect inference. This article mainly focuses on the implementation aspects related to the GMWM, and its integration within a general navigation filter alibration procedure. Regarding this, we apply the GMWM on error signals issued from MEMS-based inertial sensors by building and estimating composite stochastic processes for which classical methods cannot be used. In a first stage, we validate the resulting models using AV and PSD analyses and then, in a second stage, we study the impact of the resulting stochastic models design in terms of positioning accuracy using an emulated scenario with statically observed error signatures. We demonstrate that the GMWM-based calibration framework enables to estimate complex stochastic models in terms of the resulting navigation accuracy that are relevant for the observed structure of errors.

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Stéphane Guerrier

Papers

Wavelet-variance-based estimation for composite stochastic processes

Redundant MEMS-IMU integrated with GPS for performance assessment in sports

Noise reduction and estimation in multiple micro-electro-mechanical inertial systems

Improving Accuracy with Multiple Sensors: Study of Redundant MEMS-IMU/GPS Configurations

Generalized Method of Wavelet Moments for Inertial Navigation Filter Design