The Delusion of Maneuver Detection

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    THE DELUSION OF MANEUVER OR ACCELERATION

    DETECTION

    J. W. Bell,bellco@infowest.com

    October 27, 2012

    ABSTRACT

    The purpose of this note is to demonstrate by example that a smooth transition occurs in

    the MSE (or RMSE) of a tracking estimator when a target accelerates; and that major spikes

    actually occur only when the target stops accelerating. As a result, it appears to be a delusion to

    believe that a maneuver or increase in acceleration can be effectively detected and used.

    KEY WORDS

    Tracking Filter, Tracking, Kalman Filter, Optimal Filter, Optimal Estimation, Mean Square

    Estimation, Least Squares Method, Minimum Mean Square Error

    mailto:bellco@infowest.commailto:bellco@infowest.commailto:bellco@infowest.commailto:bellco@infowest.com
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    EXAMPLE SCENARIO

    Consider the following scenario: Target has constant velocity for 10 scans, accelerates at

    3g for 10 scans, travels at constant velocity for 10 scans, decelerates at 1.5g for 10 scans, and

    ends up traveling at a constant velocity for the final 10 scans.

    Assume 5-point one step prediction. As we saw in Example 5 of [1], the sample period at

    which the MSE of a 5-point 2nd

    order predictor applied to a target with a maximum acceleration

    of 3g's crosses over the a 5-point 3rd

    order predictor variance (MSE) is c = 0.235 sec.

    Consider the optimal 5-point FOE matching that sample period of c = 0.235 for

    comparison with the standard 5-point 2nd

    and 3rd

    order predictors.

    The optimal FOE 5-point one step prediction weights are wopt= [0.1 -0.35 -0.3 0.25, 1.3].

    The 3rd

    order 5-point one step predictor weights are w3 = [0.6 -0.6 -0.8 0.0 1.8], and the 2nd

    order 5-point one step predictor weights are w2 = [-0.4 -0.1 0.2 0.5 0.8].

    Figure 1 shows the plots of the one step predicted RMSEs (used rather than the MSE to

    reduce the scale of the plot) comprised of the theoretical variance (based on a measurement noise

    variance of unity) plus the actual bias squared. Since the variance remains constant, the only

    variation in the plots is due to changes in the bias. Track initiation is ignored for simplicity.

    Therefore, the first sample in the plots actually start on the 5th

    scan; and the acceleration begins

    on the 6th sample (10th scan in the plot).

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    ANALYSIS AND DISCUSSION

    The most striking feature of these plots is that all of the estimators make a smooth RMSE

    transition when the target begins to accelerate at the 6th

    sample (or decelerate at the 26th

    sample).

    In fact, major transition spikes occur only at the end of acceleration at the 15th

    sample (or

    deceleration at the 35th

    sample).

    Thus, in this case a maneuver detector or acceleration estimator would be expected to

    kick in only after the target has stopped accelerating (or decelerating). Therefore, not only would

    adaptivity increase the error as suggested in [2], it would cause more havoc by calling for an

    estimator matched to an acceleration which no longer existsa major double whammy.

    Figure 1

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    Other things to note are as follows:

    (1) The 2nd

    order RMSE does equal the 3rd

    order RMSE for max acceleration of 3g as designed.

    (2) The 3rd

    Order RMSE has double spikes for both acceleration and deceleration.

    (3) Both 2nd

    and 3rd

    order RMSEs have spike widths of 4 samples for both acceleration and

    deceleration.

    (4) Optimal predictor RMSE has both lower spikes than 2nd

    and 3rd

    order predictors and spike

    widths of only 2 samples in both caseshalf that of both the 2

    nd

    and 3

    rd

    order predictors.

    FORMULAS AND NUMBERS

    Normalized 3rd

    order predictor variance:)2)(1(

    699||

    22

    3

    NNN

    NN = 4.6, for N = 5.

    15.2RMSE . This we see pretty much everywhere in Figure 1 except for the large spikes as

    well as the small blip at N=7 during acceleration and the almost imperceptible blip at N=27

    during deceleration.

    Normalized 2nd

    order predictor variance:)1(

    24|| 22

    NN

    N = 1.1, for N = 5.

    05.1RMSE . This occurs everywhere there is no acceleration or deceleration: between N = 1 to

    5, N = 20 to 25, and N = 40 to 45.

    Normalized 2nd order predictor bias squared at worst case acceleration of 3gs:

    2

    3

    22

    2T

    a

    = 3.5 where

    2

    2

    36

    )2)(1(

    NNT =49 and

    )4)(1(

    180

    2 22

    22

    NNN

    a= 1/14

    for N = 5. This occurs between N = 10 to 15.

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    Normalized 2nd

    order predictor mean square error (MSE) at worst case:

    2

    3

    22

    2

    22

    || Ta

    MSE

    = 4.6 for N=5. This matches the 3rd order variance at maximum

    acceleration of 3gs as designed. Again 15.2RMSE . This occurs between N = 10 to 15.

    Normalized FOE predictor MSE:

    2

    33

    22

    2

    3

    2

    3

    2

    2 )1(2

    )|||(|

    fT

    afMSE ,

    wheref3 is the interpolation factor (or fraction) between the 2nd

    and 3rd

    order predictors. For the

    optimum at worst case of 3gs, this reduces to )|||(| 23,32

    2 optopt fMSE , where

    2

    2

    2

    3

    2

    3 |||||| .

    By design the optimal FOE in this case is based on the optimal fraction, or interpolation

    factor off3opt = 0.5. Therefore, 85.2)6.41.1(5.0)|||(|5.02

    3

    2

    2 MSE . 69.1RMSE .

    This occurs also at maximum acceleration between N = 10 to 15.

    On the other hand, if there is no bias, the FOE predictor MSE reduces to the normalized

    variance: )|||(|var 232322 fiancenomalizedMSE , which can be written as

    975.1)1.136.4(4

    1)||3|(|

    4

    1)|||(|

    4

    1|| 22

    2

    3

    2

    2

    2

    3

    2

    2

    2 xMSE p .

    40.1RMSE . This occurs during no acceleration or deceleration: N = 1 to 5, N = 20 to 25, and

    N=40 to 45.

    CONCLUSION

    Detecting the onset of a target maneuver or acceleration is shown in this example to be

    problematic. Spikes in the RMSE at the end of a maneuver or acceleration suggest that a change

    in the tracking estimator based on maneuver or acceleration detection is likely to cause more

    havoc by introducing the change after the fact. Therefore, not only would adaptivity increase the

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    error as suggested in [2], it would call for an estimator matched to an acceleration which no

    longer existsa major double whammy.

    REFERENCES

    [1] Bell, J. W., The Fall acy Of Using A 2ndOrder Estimator On Acceleration,

    http://site.infowest.com/personal/j/jeffbell/The2ndOrderEstimatorFallacy.pdf

    [2] Bell, J. W., Tracking Estimator Adaptivity: What Pri ce?,

    http://site.infowest.com/personal/j/jeffbell/WhatPriceAdaptivity.pdf

    http://site.infowest.com/personal/j/jeffbell/The2ndOrderEstimatorFallacy.pdfhttp://site.infowest.com/personal/j/jeffbell/The2ndOrderEstimatorFallacy.pdfhttp://site.infowest.com/personal/j/jeffbell/WhatPriceAdaptivity.pdfhttp://site.infowest.com/personal/j/jeffbell/WhatPriceAdaptivity.pdfhttp://site.infowest.com/personal/j/jeffbell/WhatPriceAdaptivity.pdfhttp://site.infowest.com/personal/j/jeffbell/The2ndOrderEstimatorFallacy.pdf