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. KálmánRelated topics
calculus control differential dual duality duplicate early equation error estimation extend field filter filtering great illustrated infinite interest matrix principle problem proof relating role several significance smoothing solution stationary statistics theory type variance canonical covariance hamiltonian optimal side-by-sideRelated quotes
The implications of Descartes' analytic reformulation of geometry are obvious. Not only did the new method make possible a systematic investigation of known curves, but, what is of infinitely greater significance, it potentially created a whole universe of geometric forms beyond conception by the synthetic method.
Descartes also saw that his method applies equally as well to surfaces... but he did not develop this. With the extension to surfaces, there was no reason why geometry should stop with equations in three variables; and the generalization to systems of equations in any finite number of variables was readily made in the nineteenth century. Finally, in the twentieth century, the farthest extension possible in this direction led to spaces of a non-denumerable infinity of dimensions. ...The path from Descartes to the creators of higher space is straight and clear; the remarkable thing is that it was not traveled earlier than it was.
René Descartes
The approach that dominates organizational theory, teaching, and practice for most of the twentieth century looked at organizations from the top-down, starting with a view of the CEO as the "leader" who shapes the organization's strategy, structure, culture, and performance potential. The nature of work and the role of the workforce enter the analysis much later, after considerations of technology and organization design have been considered. However, if the key source of value in the twenty-first-century organization is to be derived from the workforce itself, an inversion of the dominant approach will be needed. The new perspective will start not at the top of the organization, but at but at the front lines, with people and the work itself - which is where value is created. Such an inversion will lead to a transformation in the management and organization of work workers, and knowledge. This transformation was signalled by McGregor, but we must go further.
Douglas McGregor
When we examine this suggestion, we see that it is no more than a formal acknowledgement of a problem, the problem of how (by what institutional arrangement, by what organization of affairs) the equilibrium prices are to be discovered. Repeated trial and error, while the market stands in suspense awaiting the outcome, is not a practical resort. The number of distinct trials, even if confined to discrete steps of price and quantity, would be so immense that the necessary 'market day' would extend beyond human life-times... [The] theoretical ideal applies to mutually isolated days or moments, each to be treated as perfectly self-contained and looking to no yesterday and no tomorrow. But the real market is dealing with goods inherited from yesterday, and in means of production whose products will not be ready till tomorrow. Meanwhile the non-economic circumstances are changing and rendering each successive equilibrium obsolete.
G. L. S. Shackle
Philosophers of science have repeatedly demonstrated that more than one theoretical construction can always be placed upon a given collection of data. History of science indicates that, particularly in the early developmental stages of a new paradigm, it is not even very difficult to invent such alternates. But that invention of alternates is just what scientists seldom undertake except during the pre-paradigm stage of their science's development and at very special occasions during its subsequent evolution. So long as the tools a paradigm supplies continue to prove capable of solving the problems it defines, science moves fastest and penetrates most deeply through confident employment of those tools. The reason is clear. As in manufacture so in science-retooling is an extravagance to be reserved for the occasion that demands it. The significance of crises is the indication they provide that an occasion for retooling has arrived.
Thomas Samuel Kuhn
In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.
Carl Friedrich Gauss
