Journal ArticleDOI
The innovations approach to detection and estimation theory
Thomas Kailath
- Vol. 58, Iss: 5, pp 680-695
TLDR
Seven applications to linear and nonlinear least-squares estimation, Gaussian and non-Gaussian detection problems, solution of Fredholm integral equations, and the calculation of mutual information, will be described.Abstract:
Given a stochastic process, its innovations process will be defined as a white Gaussian noise process obtained from the original process by a causal and causally invertible transformation. The significance of such a representation, when it exists, is that statistical inference problems based on observation of the original process can be replaced by simpler problems based on white noise observations. Seven applications to linear and nonlinear least-squares estimation. Gaussian and non-Gaussian detection problems, solution of Fredholm integral equations, and the calculation of mutual information, will be described. The major new results are summarized in seven theorems. Some powerful mathematical tools will be introduced, but emphasis will be placed on the considerable physical significance of the results.read more
Citations
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Journal ArticleDOI
Mutual information and minimum mean-square error in Gaussian channels
TL;DR: In this article, the authors showed that the mutual information with respect to the signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input statistics.
Book
Stochastic systems : estimation, identification, and adaptive control
P. R. Kumar,Pravin Varaiya +1 more
TL;DR: The mathematics of filtering and ee/ise 556: stochastic systems fall 2013 usc search identification and system parameter estimation 1991 gbv is described.
Posted Content
Mutual Information and Minimum Mean-square Error in Gaussian Channels
TL;DR: A new formula is shown that connects the input-output mutual information and the minimum mean-square error (MMSE) achievable by optimal estimation of the input given the output, which has an unexpected consequence in continuous-time nonlinear estimation.
Book
The statistical theory of linear systems
E. J. Hannan,Manfred Deistler +1 more
TL;DR: In this paper, the authors discuss the development of a rather complete inferential theory for ARMAX models and discuss the asymptotic properties of these estimators without assuming the data to be Gaussian and also discuss the basis of the assumptions that appear to be minimal.
Journal ArticleDOI
A view of three decades of linear filtering theory
TL;DR: Developments in the theory of linear least-squares estimation in the last thirty years or so are outlined and particular attention is paid to early mathematica[ work in the field and to more modern developments showing some of the many connections between least-Squares filtering and other fields.
References
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Book
Probability, random variables and stochastic processes
TL;DR: This chapter discusses the concept of a Random Variable, the meaning of Probability, and the axioms of probability in terms of Markov Chains and Queueing Theory.
Book
Probability, random variables, and stochastic processes
TL;DR: In this paper, the meaning of probability and random variables are discussed, as well as the axioms of probability, and the concept of a random variable and repeated trials are discussed.
Journal ArticleDOI
New Results in Linear Filtering and Prediction Theory
R. E. Kalman,R. S. Bucy +1 more
TL;DR: The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results and properties of the variance equation are of great interest in the theory of adaptive systems.