The multi-fractional order estimator (MFOE) is a straightforward, practical, and flexible alternative to the Kalman filter (KF) for tracking targets. The MFOE is focused strictly on simple and pragmatic fundamentals along with the integrity of mathematical modeling. Like the KF, the MFOE is based on the least squares method (LSM) invented by Gauss and the orthogonality principle at the center of Kalman's derivation. Optimized, the MFOE yields better accuracy than the KF and subsequent algorithms such as the extended KF and the interacting multiple model (IMM). The MFOE is an expanded form of the LSM, which effectively includes the KF and ordinary least squares (OLS) as subsets (special cases). OLS is revolutionized in for application in econometrics. The MFOE also intersects with signal processing, estimation theory, economics, finance, statistics, and the method of moments. The MFOE offers two major advances: (1) minimizing the mean squared error (MSE) with fractions of estimated coefficients (useful in target tracking) and (2) describing the effect of deterministic OLS processing of statistical inputs (of value in econometrics)
Description
Consider equally time spaced noisy measurement samples of a target trajectory described by
where n represents both the time samples and the index; the polynomial describing the trajectory is of degree J-1; and is zero mean, stationary, white noise (not necessarily Gaussian) with variance .
Estimating x(t) at time with the MFOE is described by
where the hat (^) denotes an estimate, N is the number of samples in the data window, is the time of the desired estimate, and the data weights are
The are orthogonal polynomial coefficient estimators. (a function detailed in) projects the estimate of the polynomial coefficient to the desired estimation time . The MFOE parameter 0≤≤1 can apply a fraction of the projected coefficient estimate.
The combined terms effectively constitute a novel set of expansion functions with coefficients . The MFOE can be optimized at time as a function of the s for given measurement noise, target dynamics, and non-recursive sliding data window size, N. However, for all , the MFOE reduces and is equivalent to the KF in the absence of process noise, and to the standard polynomial LSM.
As in the case of coefficients in conventional series expansions, the s typically decrease monotonically as higher order terms are included to match complex target trajectories. For example, in the s monotonically decreased in the MFOE from to , where for m ≧ 6. The MFOE in consisted of five point, 5th order processing of composite real (but altered for declassification) cruise missile data. A window of only 5 data points provided excellent maneuver following; whereas, 5th order processing included fractions of higher order terms to better approximate the complex maneuvering target trajectory. The MFOE overcomes the long-ago rejection of terms higher than 3rd order because, taken at full value (i.e., ), estimator variances increase exponentially with linear order increases. (This is elucidated below in the section "Application of the FOE".)
Fractional order estimator
As described in, the MFOE can be written more efficiently as where the estimator weights of order m are components of the estimating vector . By definition and . The angle brackets and comma denote the inner product, and the data vector comprises noisy measurement samples .
Perhaps the most useful MFOE tracking estimator is the simple fractional order estimator (FOE) where and for all m > 3, leaving only . This is effectively an FOE of fractional order , which linear interpolates between the 2nd and 3rd order estimators described in) as
where the scalar fraction is the linear interpolation factor, the vector , and (which comprises the components ) is the vector estimator of the 3rd polynomial coefficient (a is acceleration and Δ is the sample period). The vector is the acceleration estimator from .
The mean-square error (MSE) from the FOE applied to an accelerating target is
, where for any vector , .
The first term on the right of the equal sign is the FOE target location estimator variance composed of the 2nd order location estimator variance and part of the variance from the 3rd order acceleration estimator as determined by the interpolation factor squared . The second term is the bias squared from the 2nd order target location estimator as a function of acceleration in .
Setting the derivative of the MSE with respect to equal to zero and solving yields the optimal :
where , as defined in.