The Shiryayev sequential probability ratio test (SSPRT) focuses on the detection in a series of conditionally independent measurements by noting the change in the probability density function of the measurements. Some results have recently been reported for fault detection using SSPRT [63, 93].
The optimality of SSPRT is to minimize an expected cost at each time step. This cost includes the measurement cost and the cost due to a terminal decision error by false alarm or
miss detection. Since a Bayesian framework is used, it is necessary to define prior information. This includes the a priori probabilities of Hi (i= 0,1) and the transition probabilities of H0
toH1fromk−1 tok(which can be time variant or invariant). When a binary hypothesis test
is performed, the transition probabilities degenerate into a single number, usually assumed to be time-invariant for simplicity.
The decision rule of the SSPRT is obtained by defining the posterior probability ratio
Pk. Letpk =P{n≤k|zk}denote the posterior probability that a change occurs (at unknown
time n) by time k given the available measurements zk, and
Pk =4 pk 1−pk , PT =4 pT 1−pT (5.5)
where the choice of a preset threshold pT is related to the desirable decision error rate. The
SSPRT becomes
1) AcceptH1 (declare a change) ifPk≥PT. The stopping time is ˆn = min{k :Pk≥PT}.
2) Continue the test (k 7→k+ 1) if Pk < PT.
Calculatingpkis the key to SSPRT. Fortunately, it can be done recursively. Letpi0denote
the prior probability of hypothesis Hi being true, π the transition probability from H0 to
H1, and φik
4
=P(n ≤k+ 1|zk) (i= 0,1). The recursive form of the posterior probability of
Hi is given by p1k = φ 1 k−1f zk|H1, zk−1 P1 i=0φik−1f(zk|Hi, zk−1) , p0k = 1−p1k φ1k−1 =p1k−1+π 1−p1k−1, φ0k−1 = 1−φ1k−1
Then, the test statistic of the SSPRT is Pk=4 p1 k p0 k = f(zk|H1, z k−1) f(zk|H0, zk−1) Pk−1+π 1−π , P0 4 = p 1 0 p0 0
Note that no reset mechanism is necessary for the SSPRT due to the nature of its problem formulation that determines when a disruption ofH1is true. Further, ifπis zero, the SSPRT
becomes the SPRT. This makes sense in that SPRT assumes all data relate to one of the two hypotheses.
Note that the above SPRT-based procedures assume that measurements are independent. However, measurements are correlated in many practical problems, such as the target track- ing problem. In this case the above test still works provided the sequence hlki of marginal
likelihood ratios is independent, where
lκ =
f(˜zκ|H1, zκ−1)
f(˜zκ|H0, zκ−1)
Fortunately, this is approximately the case since the measurement residual sequence is ap- proximately Gaussian distributed and weakly coupled under some conditions [44]. Thus measurement residuals, instead of measurements, should be used to compute likelihood ra- tios.
Chapter 6
Sequential Detection of Target
Maneuvers
6.1
Introduction and Related Research
Maneuvering target tracking (MTT) is an important problem complicated by the fact that accelerations are generally unknown and that structural variations may also exist as the target moves into and out of the maneuvering mode. Neither accelerations nor possible structural changes are available directly through measurements in practice. In general, the MTT problem is a hybrid estimation problem since it involves discrete mode or parameter estimation as well as continuous state estimation. The decision-based techniques for MTT, which appeared after the decision free adaptive Kalman filter techniques based on a single model, have become quite popular and have been studied extensively in the literature [6, 8, 12, 50]. In decision-based approaches, the state estimation is based on a hard decision on the target motion model which is made by the maneuver detector. Therefore making reliable
and timely decisions is key to these approaches for satisfactory state estimation. Many such algorithms and techniques have been developed to detect maneuvers [50].
Target maneuvers consist of maneuver onset and termination, which are observable as changes appearing in the measurements. The problem of detecting maneuvers thus can be classified as maneuver onset detection and termination detection. Onset detection algorithms can be categorized based on which test they utilize: the chi-square based test and the likelihood ratio based test. Techniques based on the chi-square test include measurement residual based and input estimate based detectors. Those based on likelihood ratio tests include the generalized likelihood ratio (GLR) based and marginalized likelihood ratio based (MLR) detectors. These algorithms are widely used in MTT applications but there are no comprehensive references available for performance comparison to our knowledge. Thus, this chapter will first focus on the onset detection performance comparison for six existing maneuvering detection algorithms in various scenarios:
• Measurement residual based chi-square detector (MR)
• Input estimate based chi-square detector (IE)
• Input estimate based Gaussian significance detector (IEG)
• Generalize likelihood ratio test detector (GLR)
• Marginalized likelihood ratio detector (MLR)
• Cumulative sum based detector (CUSUM)
but is also less important [50]. Up to date maneuver termination is rarely defined by any rigorous problem formulation and few methods have been presented in the literature.
Besides aforementioned traditional detection algorithms that are based on batch pro- cessing, another class of statistical tests that can be applied to maneuver detection is the sequential tests for change point detection, as pointed out in [50, 82]. Sequential detection procedures have been successfully applied to fault detection (e.g., [63, 93]) but not to target maneuver detection, to our knowledge, except for a quickest detector in [97]. Some min-max based solutions have been proposed in [75, 76, 77] with applications to navigation system integrity monitoring. It optimizes the worst case situation with given decision error rates.
In this chapter, we consider detecting a target’s maneuver as a binary hypothesis testing problem (H0: no maneuver; H1: with maneuver). Once a target starts maneuvering, it
should be detected as quickly as possible under certain constraints such as decision errors. Two target maneuver onset detectors based on CUSUM and SSPRT tests are developed by using a likelihood marginalization technique to cope with the difficulty that target maneuver accelerations are unknown. The proposed approach essentially utilizes a priori information about the maneuver accelerations in typical tracking engagements and thus allows improve- ment of the detection performance, especially for normal accelerations, as compared with two widely used maneuver detectors.
This chapter is organized as follows. In Section 2, the problem of maneuvering target detection is formulated as binary composite hypothesis testing. Traditional maneuver detec- tors are briefly discussed in Section 3 and are compared in Section 4 over various scenarios. Motivation of sequential detection of target maneuvers is presented in Section 5. Sequential procedures based on CUSUM and SSPRT are derived by means of likelihood marginalization
using two typical prior models of maneuvers in Section 6. The performance of the proposed maneuver detectors is evaluated by simulations and compared with that of two widely used detectors in Section 7. A summary is provided in Section 8.