Linear Least-Squares Smoothing and Prediction, with Applications

1241 1242 1243 1249 1258 1263 1267 1274 1274 1282 1285 1288 1289 1292 1293 1293 For a number of years, beginning roughly at the end of World War II, there has been a growing interest in theories of optimum smoothing and prediction. Much of the work has been concerned with optimum smoothing and prediction of the linear least-squares sort tipplied to statistically stationary time series -- a subject which is both attractive to mathematicians and important in various engineering problems. This paper describes techniques for solving smoothing and prediction problems of the stationary, linear, least-squares sort using a circuit theory point of view. It avoids the more difficult mathematics of the very general, completely rigorous treatments, but maintains sufficient generality for many engineering applications. It develops general techniques in terms of specific engineering problems, which are of real interest in themselves and may also serve as patterns to be followed in solving other problems. Among the problems considered explicitly are the following: classical smoothing and prediction problems solved by Wiener,1 Kolmogoroff,2 Zadeh and Ragazzini,3 etc.; the simultaneous use of different instruments, with different error spectra, for the observation of single physical variables; applications of the matheviatics of data smoothing to circuit design problems which do not actually involve data smoothing as such. The general techniques described here have been developed over a period of years.