Linear Prediction Problem Research Articles

Discussion of the Paper by Dr Brown, Professor Durbin and Mr Evans

Discussion of the Paper by Dr Brown, Professor Durbin and Mr Evans

Turbine Inlet Gas Temperature Measurement System Using a Fluidic Temperature Sensor

05-06-69.1,1969, Grumman Aerospace Corp., Bethpage, N.Y. Joseph, P. D. and Tou, J. T., On Linear Control Theory, AIEE Transactions, Vol. 80, Pt. n, Sept. 1961, pp. 193-196. Wonham, W. M., On Pole Assignment in Multi-input Controlable Linear Systems, IEEE Transactions on Automatic Control, Vol. AC-12, Dec. 1967, pp. 660-665. Kalman, R. E., A New Approach to Linear Filtering and Prediction Problems, Transactions of the ASME Journal of Basic Engineering, Vol. 82D, March 1960, pp. 33-45. Widnal, W. S., The Minimum-Time Thrust Vector Control Law in the Apollo Lunar Module, Automatica, Vol. 6, No. 5, Sept. 1970, pp. 661-672. Cox, K., A Case Study of the Apollo Lunar Module Digital Autopilot, Proceedings of the IEEE Group on Automatic Control, 1969 JACC Case Studies in Systems Control, University of Colorado, Boulder, Colo., Aug. 1969.

The use of operator factorization for linear control and estimation

The linear filtering, prediction and smoothing problems as well as the linear quadratic control problems can very generally be formulated as operator equations using basic linear algebra. The equations are of Fredholm type II, and they are difficult to solve directly. It is shown how the operator can be factorized into two Volterra operators using a matrix Riccati equation. Recursive solution of these triangular operator equations is then obtained by two initial value differential equations. The proofs of smoothing and optimal control under known disturbances are in this way especially clear and simple.

Formulation of a Nonlinear Predictor

The classical linear prediction problem has received much attention both theoretically and empirically in the past few decades and a great deal is known about it (though there remain important open questions). See, e.g., discussions and lists of references in Parzen (1969) and Rozanov (1967). The nonlinear prediction problem is much more difficult and much less is known about it. See, e.g., Masani and Wiener (1959). We reduce a certain type of nonlinear prediction problem to the multiple linear prediction problem, see Wiener (1956) and Van Ness (1966). We can then take advantage of theoretical results and computational techniques which have been developed to solve the linear problem. The methods are applied to both real and generated data using an iterative method due to Parzen and fast Fourier transform techniques on a small computer (an IBM 1130).

Multiplicity and representation theory of generalized random processes

Let ( D) be the Schwartz space of infinitely differentiable scalar functions on the line, with compact supports, and (Ω, Σ, P) be a fixed probability space. Let X : (D) → L 2(Ω, Σ, P) be a purely nondeterministic generalized random process (g.r.p.) in the sense of Itô with zero mean functional. A multiplicity representation theorem for X is obtained as a result of the Hellinger-Hahn theory. The representation can be assumed to be proper canonical. Thus each g.r.p. determines a unique cardinal number N ≤ ℵ o , termed the multiplicity of the g.r.p. X. As a corollary, every stationary g.r.p. has multiplicity one. A class of harmonizable g.r.p.'s of multiplicity one can be constructed to include all the stationary g.r.p.'s. There exist g.r.p.'s of any prescribed multiplicity. The related linear least-squares prediction problem is obtained.

An Iterative Linear Prediction Method

AbstractLet ξ(t) be a (possibly non stationary) stochastic process that we want to predict. When ξ(t) has mean zero and known covariance function a method is given for constructing a sequence of predictors, each predictor being a finite linear combination of observed process values. Asymptotic convergence to the optimal linear predictor is proved and very rapid convergence is indicated by an example.Then the unbiased linear prediction problem is shown to be equivalent to conventional prediction of a transformed process with mean zero and thus solvable by the iterative method.Prediction when the mean is replaced by a sum of functions with random coefficients is touched upon.In a numerical study is shown how varying knowledge about the process mean can affect the prediction errors.

On the Prediction of Non-Stationary Processes

Summary We consider the problem of linear least-squares prediction for a class of non-stationary processes which possess “evolutionary spectral representations”. It is shown that, under certain conditions, such processes admit moving-average representations in terms of time-dependent coefficients. This feature enables us to develop a close analogue of the Wiener–Kolmogorov approach to the corresponding problem for stationary processes.

Multiconditional terminal control of trajectories

The problem of controlling a multistage vehicle trajectory is examined in ligh of prescribed terminal constraints on position, velocity and flight time. The basic approach, guidance on extrapolated error, is applicable to inertial and command guidance systems. Command guidance is emphasized in the paper. Kalman's approach to the discrete linear filtering and prediction problem is employed to minimize the influence of radar tracking inaccuracies on the extrapolated errors. The results of a digital simulation of the system are summarized.

A New Approach to Linear Filtering and Prediction Problems

The classical filtering and prediction problem is re-examined using the Bode-Shannon representation of random processes and the “state-transition” method of analysis of dynamic systems. New results are: (1) The formulation and methods of solution of the problem apply without modification to stationary and nonstationary statistics and to growing-memory and infinite-memory filters. (2) A nonlinear difference (or differential) equation is derived for the covariance matrix of the optimal estimation error. From the solution of this equation the co-efficients of the difference (or differential) equation of the optimal linear filter are obtained without further calculations. (3) The filtering problem is shown to be the dual of the noise-free regulator problem. The new method developed here is applied to two well-known problems, confirming and extending earlier results. The discussion is largely self-contained and proceeds from first principles; basic concepts of the theory of random processes are reviewed in the Appendix.

On the linear prediction problem for certain stochastic processes

On the linear prediction problem for certain stochastic processes

On Prediction in Stationary Time Series

In time series analysis there are two lines of approach, here called the functional and the stochastic. In the former case, the given time series is interpreted as a mathematical function, in the latter case as a random specimen out of a universe of mathematical functions. The close relation between the two approaches is in section 2 shown to amount to a genuine isomorphism. Considering the problem of prediction from this viewpoint, the author gives in sections 3-4 the functional equivalence of his earlier theorem on the decomposition of a stationary stochastic process with a discrete time parameter (see [9], theorem 7). In section 5 the decomposition theorem is applied to the problem of linear prediction. Finally in section 6 a few comments are made. Since various aspects of the isomorphism in question are known, this paper might be regarded as essentially expository.