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polynomial least squares
In mathematical statistics, polynomial least squares refers to a broad range of statistical methods for estimating an underlying polynomial that describes observations. These methods include polynomial regression, curve fitting, linear regression, least squares, ordinary least squares, simple linear regression, linear least squares, approximation theory and method of moments. Polynomial least squares has applications in radar trackers, estimation theory, signal processing, statistics, and econometrics.
Two common applications of polynomial least squares methods are approximating a low-degree polynomial that approximates a complicated function and estimating an assumed underlying polynomial from corrupted (also known as "noisy") observations. The former is commonly used in statistics and econometrics to fit a scatter plot with a first degree polynomial (that is, a line). The latter is commonly used in target tracking in the form of Kalman filtering, which is effectively a recursive implementation of polynomial least squares.〔Sorenson, H. W., Least-squares estimation: Gauss to Kalman, IEEE Spectrum, July, 1970.〕〔Bell, J. W., Simple Disambiguation Of Orthogonal Projection In Kalman’s Filter Derivation, Proceedings of the International Conference on Radar Systems, Glasgow, UK. October, 2012.〕〔Bell, J. W., A Simple Kalman Filter Alternative: The Multi-Fractional Order Estimator, IET-RSN, Vol. 7, Issue 8, October 2013.〕 Estimating an assumed underlying deterministic polynomial can be used in econometrics as well.〔()〕 In effect, both applications produce average curves as generalizations of the common average of a set of numbers, which is equivalent to zero degree polynomial least squares.〔〔〔Papoulis, A., Probability, RVs, and Stochastic Processes, McGraw-Hill, New York, 1965〕
In the above applications, the term "approximate" is used when no statistical measurement or observation errors are assumed, as when fitting a scatter plot. The term "estimate", derived from statistical estimation theory, is used when assuming that measurements or observations of a polynomial are corrupted.
==Polynomial least squares estimate of a deterministic first degree polynomial corrupted with observation errors==
Assume the deterministic first degree polynomial equation '''' with unknown coefficients and as follows:
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