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)
Contents
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
Estimating x(t) at time
where the hat (^) denotes an estimate, N is the number of samples in the data window,
The
The combined terms
As in the case of coefficients in conventional series expansions, the
Fractional order estimator
As described in, the MFOE can be written more efficiently as
Perhaps the most useful MFOE tracking estimator is the simple fractional order estimator (FOE) where
where the scalar fraction
The mean-square error (MSE) from the FOE applied to an accelerating target is
The first term on the right of the equal sign is the FOE target location estimator variance
Setting the derivative of the MSE with respect to
where
The optimal FOE is then very simply
Substituting the optimal FOE into the MSE yields the minimum MSE:
Although not obvious, the
Application of the FOE
Since a target's future location is generally of more interest than where it is or has been, consider one-step prediction. Normalized with respect to measurement noise variance, the MSE for equally spaced samples reduces for the predicted position to
where N is the number of samples in the non-recursive sliding data window. Note that the first term on the right of the equal sign is the variance from estimating the first coefficient (position); the second term is the variance from estimating the 2nd coefficient (velocity); and the 3rd term with
Estimator variances obviously increase exponentially with unit order increases. In the absence of process noise, the KF yields variances equivalent to these. (A derivation of the variance from a 1st degree polynomial corresponding to
Kalman filter tuning
Tuning the KF consists of a trade-off between measurement noise and process noise to minimize the estimation error. The KF process noise serves two roles: First, its covariance is sized to account for the maximum expected target acceleration. Second, process noise covariance establishes an effective recursive data window (analogous to the non-recursive sliding data window), described by Brookner as the Kalman filter memory.
Contrary to process noise covariance as a single independent parameter in the KF serving two roles, the FOE has the advantage of two separate independent parameters: one for acceleration and the other for sizing the sliding data window. Therefore, as opposed to being limited to just two tuning parameters (process and measurement noises) as is the KF, the FOE includes three independent tuning parameters: measurement noise variance, the assumed maximum deterministic target acceleration (for simplicity both target acceleration and measurement noise are included in the ratio of the single parameter
Consider tuning a 2nd order predictor applied to the simple and practical tracking example in to minimize the MSE when the target acceleration is
Setting
where the first term on the right of the equal sign is the normalized 2nd order one-step prediction variance and the second term is the normalized bias squared from acceleration. This MSE is plotted as a function of N in Figure 1 along with both the variance and bias squared.
Clearly, only integer order steps are possible in a non-recursive estimator. However, for use in approximating the tuned 2nd order KF, this MSE plot is stepped in tenths of a unit to show more precisely where the minimum occurs. The minimum MSE of 4.09 occurs at N = 2.9. The tuned KF can be approximated by sizing the process noise covariance in the KF such that the effective recursive data window—i.e., the Kalman filter memory—matches N = 2.9 in Figure 1 (i.e.,
FOE as a multiple-model estimator
The FOE can be viewed as a non-recursive multiple-model (MM) estimator composed of 2nd and 3rd order estimator models with the fraction
Note that this differs from the one-step prediction MSE in that the signs within the parentheses containing N are reversed. The higher order pattern continues here also. Normalized with respect to the measurement noise variance, the minimum position MSE reduces for equally spaced samples to
where
in
A plot of the position
Consider again the scenario of, in this case as the target maneuvers. After traveling at a constant velocity, the target accelerates at
As the acceleration varies from zero to maximum, the MSE is automatically adjusted (no external tinkering or adaptivity) between the variance at
Choosing N = 4 at the knee of the
Since trackers encounter greatest difficulties and often lose track during target maneuvers at maximum acceleration, the much smoother
IMM compared with the optimal FOE
The 4-point FOE in Figure 4 yields much smoother MSE transitions than the IMM (as well as the KF) in the parallel 1 Hz case of. It produces no error spikes or volatility as do the 8-point FOE and the IMM. In this example only 4 multiplies, 3 adds, and a window shift are required to implement the 4-point FOE, significantly few operations than required by the IMM or KF. Similar comparisons of several additional MMs from the literature with the optimal FOE are made in
Of the KF based MMs, the interacting MM (IMM) is generally considered the state-of-the-art tracking model and usually the method of choice. Since two model IMMs are most often used, consider the following two models: 2nd and 3rd order KFs. The estimated IMM state equation is the sum of the 2nd order KF times the model probability
where
As in the case of the FOE, this suggests a more descriptive estimate equal to the sum of the 2nd order KF plus the difference between the 3rd and 2nd order KFs times
In this formulation the difference between the 3rd and 2nd order KFs effectively augments the 2nd order KF with a fraction of the estimated target acceleration as a function of
One major difference between the IMM and optimal FOE is that the IMM is not optimum. The IMM model probabilities and interpolation are based on likelihoods and ad hoc transition probabilities with no mechanism for minimizing the MSE. Of course, not being optimum at any time increment k, the IMM cannot achieve the optimal FOE accuracy shown in Figure 2.
Moveover, the IMM
In order to make a reasonable comparison of the IMM with the FOE, reference constructs a non-recursive IMM analogy (IMMA). It includes
The
Two significant points of interest stand out as shown by the vertical lines. First, the largest deviation of the ideal
Included in Figure 6 for reference are a curve of the 3rd order variance, 2nd order variance, and the 2nd order MSE. The large deviation of
Furthermore, the MSE is exacerbated in the non-ideal IMMA by adaptivity, as shown in Figure 7 where the IMMA from noisy
Indeed, not only is the noisy IMMA MSE larger than the 3rd order variance (by nearly a factor of two at the worst point), once the noisy IMMA MSE exceeds the 3rd order variance, it does not drop below as does the ideal IMMA. In contrast, the optimal FOE MSE (i.e.,
This analysis compellingly suggests that adaptivity significantly degrades IMM accuracy rather than improving it. Of course, this should not come as a surprise since for
These analyses reveal the incredible and disconcerting lack of tracking literature that addresses fundamentals (e.g., optimal IMM interpolation,
Deficiencies and oversights in the Kalman filter
Comparisons of the KF with the derivation, analysis, design, and implementation of MFOE have uncovered a number of deficiencies and oversights in the KF that are overcome by the MFOE. They are reported and discussed in.