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

[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.

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Information Theory, Inference and Learning Algorithms

Today's Web-enabled deluge of electronic data calls for automated methods of data analysis. Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach. The coverage combines breadth and depth, offering necessary background material on such topics as probability, optimization, and linear algebra as well as discussion of recent developments in the field, including conditional random fields, L1 regularization, and deep learning. The book is written in an informal, accessible style, complete with pseudo-code for the most important algorithms. All topics are copiously illustrated with color images and worked examples drawn from such application domains as biology, text processing, computer vision, and robotics. Rather than providing a cookbook of different heuristic methods, the book stresses a principled model-based approach, often using the language of graphical models to specify models in a concise and intuitive way. Almost all the models described have been implemented in a MATLAB software package--PMTK (probabilistic modeling toolkit)--that is freely available online. The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.

Machine Learning : A Probabilistic Perspective

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.

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Sequential Monte Carlo methods in practice

In target tracking systems using radars on moving platforms the locations of these platforms is available from GPS based estimates. However, these estimated locations are subject to errors that are, typically, stationary autocorrelated random processes, i.e., slowly varying biases. In situations where there are no known-location targets to estimate these biases, the next best recourse is to use targets of opportunity at fixed but unknown locations. It is shown that these biases can be estimated in such a scenario, i.e., they meet the complete observability condition. Following this, the achievable accuracy for a generic scenario is evaluated. It is shown that accurate georegistration can be obtained even with a small number of measurements.

Mobile radar bias estimation using unknown location targets

Experimental set-up and procedures to test and validate battery fuel gauge algorithms

In this paper we compare the performance of three kinematic state models, i.e., the White Noise Acceleration (WNA), the Wiener process acceleration (WPA), and the Keplerian State (KPS) model, for the tracking of earth orbiting space objects (SOs). The three models considered are all simplified approximate models for the motion of Earth orbiting SOs and are not suitable for the prediction of target tracks for long time periods. However, for track updates with new measurements coming at a high rate, such simplified motion models can be effectively used with small or no loss in estimation accuracy. For the KPS model, we use a novel mixed-coordinate SO tacking (McSOT) filter, where the target state space is defined in the Cartesian, i.e., Earth-Central Inertial (ECI), coordinates for track representation and updates, while the track propagation is done in the Keplerian Coordinates. It is shown that when the measurement accuracy is high, the McSOT filter with the KPS model, which has the highest complexity among the three, is able to achieve significantly better estimation accuracy than the filters with the WNA and WPA models. The WPA model is able to achieve better tracking accuracy than the WNA model at the cost of moderate increase of algorithm complexity. On the other hand, when the measurement accuracy is low, the filters with the WNA and WPA models which operate solely in the Cartesian coordinates, i.e., the Earth-Central Inertial (ECI) coordinates, is more robust than the McSOT filter with the KPS model.

Comparison of three approximate kinematic models for space object tracking

The subject of this paper is the application of stochastic control theory to resource allocation under uncertainty. In these problems it is assumed that the results of a given allocation of resources are not known with certainty, but that a limited number of experiments can be performed to reduce the uncertainty. The problem is to develop a policy for performing experiments and allocating resources on the basis of the outcome of the experiments such that a performance index is optimized. The problem is first analyzed using the basic stochastic dynamic programming approach. A computationally practical algorithm for obtaining an approximate solution is then developed. This algorithm preserves the "closed-loop" feature of the dynamic programming solution in that the resulting decision policy depends both on the results of past experiments and on the statistics of the outcomes of future experiments. In other words, the present decision takes into account the value of future information. The concepts are discussed in the context of the general problem of allocating resources to repair machines where it is possible to perform a limited number of diagnostic experiments to learn more about potential failures. Illustrative numerical results are given.

Application of stochastic control theory to resource allocation under uncertainty

A realistic stochastic control problem for hybrid systems with Markovian jump parameters can have the switching parameters in both the state and measurement equations. Furthermore, both the system state and the jump states are, in general, not perfectly observed. Currently there are only two existing controllers for this problem. One is based upon a heuristic multiple model partitioning (MMP) and hypothesis pruning. The other utilizes the entire future tree of models, and is called the Full-Tree (FT) controller. The performance of the latter is significantly superior to the former and their complexities are similar. In this paper we present a new stochastic control algorithm for stochastic systems with Markovian jump parameters. This control algorithm is derived through the use of stochastic dynamic programming and is designed to be used for realistic stochastic control problems, i.e., with noisy state observations. This new scheme, which is based upon the interaction of r (the number of models) model-conditioned Riccati equations, has a natural parallel to implementation. The state estimation and model identification is done via the recently developed Interacting Multiple Model algorithm. Simulation results show that a substantial reduction in cost can be obtained by this new control algorithm over the MMP scheme. Furthermore, the performance of the new algorithm is shown to be practically the same as that of the FT scheme even though the new scheme, which has a fixed amount of computations at each step of the recursion, is much simpler than both the MMP and FT algorithms.

Yaakov Bar-Shalom

Papers

Mobile radar bias estimation using unknown location targets

Experimental set-up and procedures to test and validate battery fuel gauge algorithms

Comparison of three approximate kinematic models for space object tracking

Application of stochastic control theory to resource allocation under uncertainty

Control of Discrete-Time Hybrid Stochastic Systems