DATA SMOOTHING AND TREND ESTIMATION
INSTITUTE FOR DEFENSE ANALYSES ALEXANDRIA VA RESEARCH AND ENGINEERING SUPPORT DIV
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Explicit formulas are presented for estimating position, velocity, and acceleration in low-order polynomial trends, based on least-squares smoothing of sampled data accompanied by statistically uncorrelated measurement errors. Formulas are also given for interpolation and prediction of position and velocity. Expressions for the variances and covariances of consistent position, velocity, and acceleration estimates are given, and the systematic errors accruing from use of a trend estimation basis which is one order lower than the actual trend are presented. One interesting result is that the normalized correlation between the errors in an estimate of current position and those in an estimate of current velocity approaches 12 the square root of 3 when the number of measurements in the estimates becomes large. Finally, the problem of implementing real-time least-squares estimation and prediction formulas in practical systems is discussed. It is concluded that arithmetic execution time requirements can be relaxed by generating certain sums recursively, and that data storage requirements can frequently be eased by collapsing the raw data into short-term-average samples.
- Statistics and Probability