Filtering Quotes
I remember that night. It was late at night-in the daytime one was bothered with the gaping, silly students-and I worked then sometimes till dawn. It came suddenly, splendid and complete in my mind. I was alone; the laboratory was still, with the tall lights burning brightly and silently. In all my great moments I have been alone. 'One could make an animal-a tissue-transparent! One could make it invisible! All except the pigments-I could be invisible!' I said, suddenly realising what it meant to be an albino with such knowledge. It was overwhelming. I left the filtering I was doing, and went and stared out of the great window at the stars. 'I could be invisible!' I repeated.To do such a thing would be to transcend magic. And I beheld, unclouded by doubt, a magnificent vision of all that invisibility might mean to a man-the mystery, the power, the freedom. Drawbacks I saw none.
H. G. Wells
In order to protect our jobs, our visas, our profits, our chances of receiving good grades, our sanity, we pretend not to see, we sanitise our own perception, filtering out the pain, pretending that it is not here but out there, far away, in Africa, in Russia, a hundred years ago, in an otherness that, by being alien, cleanses our own experience of all negativity.
John Holloway
At present, a nonspecialist might well regard the Wiener-Kolmogorov theory of filtering and prediction [1, 2] as "classical' - in short, a field where the techniques are well established and only minor improvements and generalizations can be expected.
That this is not really so can be seen convincingly from recent results of Shinbrot [3], Stceg [4], Pugachev [5, 6], and Parzen [7]. Using a variety of methods, these investigators have solved some long-stauding problems in nonstationary filtering and prediction theory. We present here a unified account of our own independent researches during the past two years (which overlap with much of the work [3-71 just mentioned), as well as numerous new results. We, too, use time-domain methods, and obtain major improvements and generalizations of the conventional Wiener theory. In particular, our methods apply without modification to multivariate problems.
Rudolf E. Kálmán
A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side.
Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.
Rudolf E. Kálmán
