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

We present a new adaptive technique for segmenting a sequence of images and tracking the moving objects contained therein. The algorithm is illustrated on a biomedical problem, namely, the tracking of a group of fibroblast (tissue) cells whose motion is induced by an external electric field using phase contrast micrographs. Because of their proximity to one another and their motion characteristics, in addition to the nature of the images, the objects cannot be segmented accurately: they appear to merge/overlap and split. An adaptive fine segmentation technique, which can handle large objects that "appear" to have merged or split, is presented. The combined segmentation/tracking is formulated as a global optimization problem where the merging/splitting and the motion characteristics of the objects determine the final results. The motion parameters of the objects are estimated using a multiassignment algorithm (a scheme that associates segmented regions to tracks) combined with a modified version of a tracking filter (motion estimator), known as the probabilistic data association filter. Analytical expressions quantifying the accuracy of the segmentation scheme are also presented. The technique is also applicable to tracking people, ground vehicles and ballistic missiles, where the targets can be closely-spaced.

Combined segmentation and tracking of overlapping objects with feedback

The problem of target track detection and time delay estimation using cross-correlation of wideband signals has been analyzed. The concepts of target detection and (accurate) localization (TD&L), target detection (TD, with a possibly smeared measurement), and false detection (FD), have been defined in terms of resolution cell exceedances. Receiver operating curves relating to P/sub TD&L/ to P/sub FD/ and P/sub TD/ to P/sub FD/ have been presented. A generalized maximum likelihood estimate of the time delay based on the main peak and spillover cells of the cross-correlation has been presented. The accuracy of this estimate has been derived using the Cramer-Rao lower bound in terms of the resolution cell, signal-to-noise ratio (SNR), and the number of samples per observation interval. This accuracy has been verified via simulations for SNR as low as -6 dB for a 128-sample window. >

Analysis of wide-band cross-correlation for target detection and time delay estimation

The scheduling of track dwells to minimize radar energy and time with an agile beam radar is considered. Typically, a radar scheduler picks a waveform pulsewidth to achieve a desired signal-to-noise (SNR) level, and this is called the nominal tracking SNR. In theory, the combination of high quality measurements and good kinematic modeling in the tracker yields longer track revisit periods, thus minimizing total radar time allocated to the track. However, high quality measurements are at the cost of higher tracking SNR, thus higher energy. In this study, the trade between higher energy waveforms and radar time is further investigated. The role of those higher energy waveforms with more accurate measurements in the performance of the Interacting Multiple Model (IMM) estimator is also investigated. Using simulation studies, the optimal tracking SNR levels that minimize radar time and energy are investigated.

Optimizing radar signal to noise ratio for tracking maneuvering targets

Hidden Markov Models (HMMs) are powerful statistical techniques with many applications, and in this paper they are used for modeling asymmetric threats. The observations generated by such HMMs are generally cluttered with observations that are not related to the HMM. In this paper a Bernoulli filter is proposed, which processes cluttered observations and is capable of detecting if there is an HMM present, and if so, estimate the state of the HMM. Results show that the proposed filter is capable of detecting and estimating an HMM except in circumstances where the probability of observing the HMM is lower than the probability of receiving a clutter observation.

A Bernoulli filter approach to detection and estimation of hidden Markov models using cluttered observation sequences

In this paper we present the development of a Variable Structure Interacting Multiple Model (VS-IMM) estimator for tracking groups of ground targets on constrained paths using Moving Target Indicator (MTI) reports obtained from an airborne sensor. The targets are moving along a highway, with varying inter-visibility (obscuration) due to changing terrain conditions. In addition, the roads can branch, merge or cross. Some of the targets may also move in an open field. This constrained motion estimation problem is handled using an IMM estimator with varying mode sets depending on the topography. The number of models in the IMM estimator, their types and their parameters are modified adaptively, in real-time, based on the estimated position of the target and the corresponding road/visibility conditions. This topography-based variable structure mechanism eliminates the need for carrying all the possible models throughout the entire tracking period as in the standard IMM estimator, improving performance and reducing computational load.

Yaakov Bar-Shalom

Papers

Combined segmentation and tracking of overlapping objects with feedback

Analysis of wide-band cross-correlation for target detection and time delay estimation

Optimizing radar signal to noise ratio for tracking maneuvering targets

A Bernoulli filter approach to detection and estimation of hidden Markov models using cluttered observation sequences

Ground Target tracking with topography-based Variable Structure IMM estimator