We are knowledgeable about various historical approaches, their assumptions, their derivation, and their historical evolution.
	We have gone through the rigorous supporting mathematics, yet we summarize the results in a clear straightforward manner, expressed as simply as possible.
	We are familiar with the application constraints associated with utilizing these algorithms and are aware of what application constraints are usually actively in force.

[1] Kerr, T. H., “Poseidon Improvement Studies: Real-Time Failure Detection in the SINS\ESGM,” TASC Report TR-418-20, Reading, MA, June 1974 (Confidential) for Navy, SP-2413 (Jerome “Jerry” Katz).

 

[2] Kerr, T. H., “A Two Ellipsoid Overlap Test for Real-Time Failure Detection and Isolation by Confidence Regions,” Proceedings of IEEE Conference on Decision and Control, Phoenix, AZ, December 1974.

 

[3] Kerr, T. H., “Failure Detection in the SINS\ESGM System,” TASC Report TR-528-3-1, Reading, MA, July 1975 (Confidential) for Navy, SP-2413 (Jerome Katz).

 

[4] Kerr, T. H., “Improving ESGM Failure Detection in the SINS\ESGM System (U),” TASC Report TR-678-3-1, Reading, MA, October 1976 (Confidential) for Navy, SP-2413 (Jerome Katz).

 

[5] Kerr, T. H., “Failure Detection Aids for Human Operator Decisions in a Precision Inertial Navigation System Complex,” Proceedings of Symposium on Applications of Decision Theory to Problems of Diagnosis and Repair, Keith Womer (editor), Wright-Patterson AFB, OH: AFIT TR 76-15, AFIT\EN, Oct. 1976, sponsored by the local Dayton Chapter of the American Statistical Association, Fairborn, Ohio, pp. 98-127, June 1976.

 

[6] Kerr, T. H., “Real-Time Failure Detection: A Static Nonlinear Optimization Problem that Yields a Two Ellipsoid Overlap Test,” Journal of Optimization Theory and Applications, Vol. 22, No. 4, pp. 509-535, August 1977.

 

[7] Kerr, T. H., “Statistical Analysis of a Two Ellipsoid Overlap Test for Real-Time Failure Detection,” IEEE Transactions on Automatic Control, Vol. AC-25, No. 4, pp. 762-773, August 1980.

 

[8] Kerr, T. H., “False Alarm and Correct Detection Probabilities Over a Time Interval for Restricted Classes of Failure Detection Algorithms,” IEEE Transactions on Information Theory, Vol. IT-28, No. 4, pp. 619-631, July 1982.

 

[9] Kerr, T. H., “Examining the Controversy Over the Acceptability of SPRT and GLR Techniques and Other Loose Ends in Failure Detection,” Proceedings of the American Control Conference, San Francisco, CA, pp. 966-977, 22-24 June 1983.

 

[10] Carlson, N. A., Kerr, T. H., Sacks, J. E., “Integrated Navigation Concept Study,” Intermetrics Report No. IR-MA-321, 15 June 1984, for ITT (Nutley, NJ) for ICNIA (Wright Patterson AFB).

 

[11] Kerr, T. H., “Decentralized Filtering and Redundancy Management Failure Detection for Multi-Sensor Integrated Navigation Systems,” Proceedings of the National Technical Meeting of the Institute of Navigation (ION), San Diego, CA, 15-17 January 1985.

 

[12] Kerr, T. H., “Decentralized Filtering and Redundancy Management for Multisensor Navigation,” IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-23, No. 1, pp. 83-119, Jan. 1987 (minor corrections appear on p. 412 of May and on p. 599 of July 1987 issues of same journal). This won 1988 M. Barry Carlton Award and $3000 honorarium for Outstanding Paper to appear in IEEE Aerospace and Electronic Systems Transactions in ’87 [for more details, please see page 822 of Vol. 24, No. 6, Nov. 1988 of the aforementioned journal].

[13] Kerr, T. H., “Comments on ‘A Chi-Square Test for Fault Detection in Kalman Filters’,” IEEE Transactions on Automatic Control, Vol. AC-35, No. 11, pp. 1277-1278, November 1990.

[14] Kerr, T. H., “A Critique of Several Failure Detection Approaches for Navigation Systems,” IEEE Transactions on Automatic Control, Vol. AC-34, No. 7, pp. 791-792, July 1989.

[15] Kerr, T. H., “On Duality Between Failure Detection and Radar\Optical Maneuver Detection,” IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-25, No. 4, pp. 581-583, July 1989.

[16] Kerr, T. H., “Comments on ‘An Algorithm for Real-Time Failure Detection in Kalman Filters’,” IEEE Trans. on Automatic Control, Vol. 43, No. 5, pp. 682-683, May 1998.  

[17] Kerr, T. H., “Comments on ‘Determining if Two Solid Ellipsoids Intersect’,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 28, No. 1, pp. 189-190, Jan.-Feb. 2005. (Offers a simpler implementation and reminds readers that real-time embedded applications do not usually have MatLab algorithms available, as otherwise required for implementing the algorithm without the simplification that we provide.)

[18] Kerr, T. H., “Integral Evaluation Enabling Performance Trade-offs for Two Confidence Region-Based Failure Detection,”  AIAA Journal of Guidance, Control, and Dynamics, Vol. 29, No. 3, pp. 757-762, May-June 2006 (click here to download a 229KB version of this paper). or Click here.

The failure event detection approach, developed above by Thomas H/ Kerr III, is independently endorsed in: 

[19] Brumback, B. D., Srinath, M. D., “A Chi-Square Test for Fault-Detection in Kalman Filters,” IEEE Trans. on Automatic Control, Vol. 32, No. 6, pp. 532-554, June 1987.

and also independently endorsed in:

[20] Uwe D. Hanbeck, “Recursive Nonlinear Set-Theoretic Estimation Based on Pseudo-Ellipsoids,” Proceedings of the IEEE Conference on Multisensor Fusion and Integration for Intelligent Systems, pp. 159–164 (MFI’ 2001), Baden–Baden.
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.304.8080  
https://www.researchgate.net/publication/3955109_Recursive_nonlinear_set-theoretic_estimation_based_on_pseudo_ellipsoids   

[21] Kerr, T. H., “Extending Decentralized Kalman Filtering (KF) to 2-D for Real-Time Multisensor Image Fusion and\or Restoration,” Signal Processing, Sensor Fusion, and Target Recognition V, Proceedings of SPIE Conference, Vol. 2755, Orlando, FL, pp. 548-564, 8-10 April 1996. 

The above diagram is encountered in Kerr’s original development of the Two Confidence Region approach to Failure Detection for a submarine navigation application.

Again, this diagram is encountered in Kerr’s original development of the Two Confidence Region approach to Failure Detection for a submarine navigation application.

Two straight shooting, “no-holes-barred” exposes follow below with the corresponding references that are cited here appearing underneath these discussions:

Prof. Yaakov Bar-Shalom (University of Connecticut)

(Click here for more on what UCONN has pursued for defense) 

http://www.ieee.org/about/awards/bios/picard_recipients.html

Please click here to download a more detailed account of the current status of GLR and  IMM within a 213KByte pdf file.

Icelandic: “Ef satt skal segja.”

Status of Interactive Multiple Model (IMM) Parallel Bank-of-Kalman filters Approach for Nonlinear Applications

Since we have a working perspective into many other aspects of Kalman filtering including its generalizations to approximate nonlinear estimation [Ref. 4]; and its related concerns-including having found and exposed and corrected historical fallacies [Ref. 5], even those relating to random or stochastic processes [Ref.  26]; we use this forum to also point out perceived weaknesses that have not been widely publicized hithertofore that we, as specialist in this area, perceive to exist in several other alternative approaches to failure detection. Such considerations arise in reducing mere theory to a final practical implementation instead of continuing to dwell on ideal starting points of an event detection approach without considering the realities of the constraints that exist in real implementations (one such being the standard use of reduced-order filters, where filter residuals are no longer ideally white and unbiased [Ref. 27] thus foiling or corrupting the original approach of [Ref. 38] which explicitly relies on whiteness of residuals as a gauge of normal unfailed behavior, as is also relied upon by GLR [Ref. 16]). Gaussian confidence regions still persist as ellipsoidal Gaussians when reduced-order filters are inserted in the application and certain reduced-order filters still avail exact covariances on-line in real-time [Ref. 37] so CR2 is therefore robust with respect to this aspect for mechanizations using this or other similar reduced-order filter formulations. Moreover, Gaussian confidence regions arise even when the pdf’s are from the more general exponential family in situations where the important conditional and marginal distributions are still Gaussian [Ref. 28, Chaps. 1-4].

While, by now, it is routine to consider the generalization of Kalman filter estimation techniques from mere linear systems (for which Kalman filters are optimal estimators [Ref. 1], [Ref. 2], [Ref. 3]) to nonlinear systems (for which Extended Kalman filters or Iterated Extended Kalman filters [Ref. 4] are frequently useful, tractable approximate estimators for nonlinear filtering [Ref. 5, Sec. 12]), as also discussed in [Ref. 1], [Ref. 2, Vol. 2, 1982], [Ref. 3]. Similar ideas should successfully generalize each of the Kalman filters arising in the bank-of-Kalman filters that occur in Interactive Multiple Model (IMM) mechanizations as IMM is generalized beyond the exclusively linear case for which it was originally rigorously derived as a two level approximation (even in the purely linear case), where the sojourn times and Markov chain transition probabilities are a new contrivance within IMM, useful by providing additional parameters for tuning to better match potential application situations by keeping alternative models more actively viable than they had been for the Magill bank-of-Kalman-filters (1965); however, the associated IMM probability calculations are more suspect in an attempted generalization to the nonlinear case. Specifically, in each of the following three references [Ref. 6, before Eq. 4], [Ref. 7, after Eq. 2], [Ref. 8, after Eq. 6], “the critical mixture is assumed to be a sum of Gaussians, then the prior pdf is a Gaussian mixture and can be approximated (via moment matching) with a single Gaussian....” (While sums of Gaussian random variables or sums of Gaussian random processes are always Gaussian, that is not the issue or situation here where the topic instead is whether the resulting pdf of the output is a weighted sum of the Gaussian pdf’s called a “Gaussian mixture”, as claimed. Moreover, it has been asserted that the resulting Gaussian mixture  consisting of a weighting of  several Gaussian pdf’s can be subsumed as one Gaussian.) 

Our objections to this aspect is four-fold:

1. For nonlinear systems, the estimates outputted by an EKF are not Gaussian in general (unlike the assumption);

2. There are already existing analytic results [Ref. 9] which caution that a single subsuming Gaussian pdf is not always possible (nor usually possible) even if the individual participating pdf’s were in fact Gaussian when the means of the various contributing pdf’s are not in close enough proximity, as gauged by the spread of the associated covariances. This topic has been an issue since the historically well-known Gaussian-Sums approach of [Ref. 10], [Ref. 11] which also used a bank-of-Kalman-filters structure (which also did not match “expectations”, so to speak ). Indeed, nonlinear filtering situations frequently exhibit multimodal output estimates as a fact of life, as discussed in [Ref. 12];

3. The “moment matching” called for in [Ref. 6], [Ref. 7], [Ref. 8] is not explained there nor is there an opportunity to do so within the algorithm for each time-step k, as needed;

4. It is not clarified what is to be matched in “moment matching” by what and to what and by what gauge will it be determined that it matches closely enough.

Nothing about these aspects has been explained in the three references cited above, which is why we raise the issue by pointing it out here now.  Please see Ref. [40] below and our comments in red attached to it. Go to Top

Please click here for a more detailed account in a 112KByte pdf file.

Status of Residual-Based Failure Detection or Maneuver Detection when Quiescent Condition of Whiteness and Unbiasedness is Jeopardized by use of Reduced Order Kalman Filters in Actual Mechanization

We now seek to point out apparent weaknesses that have not been widely publicized or even acknowledged hithertofore that we, as specialist in this area, perceive to exist in several alternative approaches to  failure detection (being a special case of event detection). Such considerations arise in reducing mere theory to a final practical implementation instead of continuing to dwell on ideal starting points of the original formulation of an event detection approach without explicitly considering the realities of the constraints that exist in implementation within the actual applications. One such prevalent constraint being the standard use of reduced-order sub-optimal filters, where filter residuals are no longer ideally white and unbiased (specifically, filter residuals are white and unbiased if and only if the system and sensor model used in the Kalman filter are identical to what exists for the actual system or in it’s truth model representation used in the simulation, otherwise the residuals are either nonwhite or biased or both) thus degrading or corrupting the original idealized aspects of many detection approaches which explicitly rely on an assumption of “whiteness and unbiasness of residuals” as a gauge of normal unfailed behavior.

The “whiteness of Kalman filter residuals” is also relied upon in another approach to a failure detection formulation using the Generalized Likelihood Ratio (GLR), where, again, reduced-order filter usage introduces bias and non-whiteness of the associated filter residuals even in the nominally unfailed situation. Such effects introduce ambiguity into the algorithmic decision of whether to declare that “a failure has occurred” or to declare that “no failure is present” since now the situation is less of a dichotomy for the decision algorithm after “the water has been muddied” by the use of a reduced-order filter, as historically required in most applications (where similar issues also arise for use of reduced-order observers in application environments where noise is relatively less significant). Are the Kalman filter residuals now non-white and biased because of a failure occurring or because of the standard use of a reduced-order filter in the particular application? Such obscuring effects are consequentially time-varying when the associated navigation (NAV) filter structures which provoke or aggravate them are similarly time-varying).

The Generalized Likelihood Ratio (GLR) approach to event detection, where maximum likelihood estimates of unknown parameters are utilized within the ratio of the H₁ pdf to the H₀ pdf in lieu of not knowing the actual requisite parameters of the mixed hypothesis since they are in fact unknown, is presented and developed by Davenport and Root [Ref. 29]. Root went further [Ref. 30] to investigate applicability of GLR techniques in the radar detection problem of resolving closely spaced targets in a background of either known arbitrary correlated Gaussian noise or in Gaussian white noise. However, Root [Ref. 30] obtained explicit criteria that could be applied to indicate conditions under which one could expect to not resolve two known signals (of unknown amplitudes and parameters) and additionally pointed out a difficulty of using GLR for this purpose.

McAulay and Denlinger [Ref. 31] advocated use of GLR in conjunction with a Kalman filter in decision-directed adaptive control applications. Finally, Stuller [Ref. 32] defined an M-ary GLR test that ostensibly overcame Root’s original objections [Ref. 30] to GLR for this type of application. ([Ref. 32] also provides a limited history of GLR developments for radar, excepting no mention of [Ref. 31], which possibly eluded him as being so recent. The use of GLR for failure detection was pioneered by Willsky and Jones [Ref. 16] using an identical GLR formulation as presented by McAulay and Denlinger [Ref. 31]. While both Willsky and McAulay claim optimality of the GLR, they never explicitly specify a criteria by which it may be judged optimal nor do they supply a proof or reference where such a claim is demonstrated (specifically, [Ref. 31] references the proof to be in a English translation of an identified German textbook but diligent follow-up on our part here revealed no such substantiation located there).

While use of GLR has potential in many detection situations, it is not without its drawbacks that are frequently overlooked:

	Selin found that some of the unknown parameters (such as unknown relative carrier phase) must also be estimated in order to maximize the a posteriori probability in the estimation of two similar signals in white Gaussian noise [Ref. 33];
	Selin further identified four standard caveats [Ref. 34, p. 106] associated with use of a maximum likelihood estimate of the unknown parameters in a likelihood ratio (as occurs within GLR);
	The GLR is not a Uniformly Most Powerful (UMP) test [Ref. 35, p. 92];
	There are cases where use of GLR can give bad results [Ref. 35, p. 96];
	That use of a Maximum likelihood estimate (MLE) is not necessarily statistically consistent in general is explicitly demonstrated in a counterexample in [Ref. 36, p. 146].

GLR is again being advocated for use in radar applications but those that do so appear to ignore the historical objections for use of GLR in these types of applications as well as the explicit counterexamples in [Ref. 39, 968 ff, App. A, pp. 973-974] that, apparently, have never been refuted. The new version of GLR (called “Ed Kelly’s GLR”) is of a different form than used by the others mentioned above [Ref. 21] and is apparently a pseudo-GLR but useful none-the-less for radar tracking but evidently somewhat lacking in its present form for failure detection since it ignores the underlying onset time of the detection event of interest. Use of the Entropy Maximization (E-M) algorithm may placate Selin’s and Roots’ concerns above but is a large computational burden that may defy a real-time implementation.

References:

1. Jazwinski, A. H., Stochastic Processes and Filtering Theory, Academic Press, N.Y., 1970.
2. Maybeck, P. S., Stochastic Models, Estimation, and Control, Part I, Academic Press, N.Y., 1979.
3. Gelb, A. (Ed.), Applied Optimal Estimation, MIT Press, Cambridge, MA, 1974.
4. Kerr, T. H., “Streamlining Measurement Iteration for EKF Target Tracking,” IEEE Trans. on AES, Vol. 27, No. 2, pp. 408-420, Mar. 1991 (minor correction in Nov. 1991).
5. Kerr, T. H., “Exact Methodology for Testing Linear System Software Using Idempotent Matrices and Other Closed-Form Analytic Results,” Proceedings of SPIE, Session 4473: Tracking Small Targets, San Diego, CA, pp. 142-168, 29 Jul.-3 Aug. 2001.
6. Bar-Shalom, Y., Chang, K. C. and Blom, H. A. P., “Tracking a Maneuvering Target Using Input Estimation Versus the Interactive Multiple Model,” IEEE Trans. on AES, Vol. 25, No. 2, pp. 296-300, Mar. 1989.
7. Mehra, R., Rago, C., Seereeram, S., “Autonomous Failure Detection, Identification, and Fault-tolerant Estimation with Aerospace Applications,” Proceedings of IEEE Aerospace Conference, Part 2, Vol. 2, Aspen, CO, pp. 133-138, 21-28 Mar. 1998.
8. Rago, C., Prasanth, R., Mehra, R. K., Fortenbaugh, R., “Failure Detection and Identification and Fault-Tolerant Control using IMM-KF with Applications to the Eagle-Eye UAV,” Proceedings of the 37^th IEEE Conference on Decision and Control, Part 4, Vol. 4, Tampa, FL, pp. 4208-4213, 16-18 Sept. 1998.
9. Konstantellos, A. C., “Unimodality Conditions for Gaussian Sums,” IEEE Trans. on Automatic Control, Vol. 25, No. 4, pp. 838-839, Aug. 1980.
10. Alspack, D. L., and Sorenson, H. W., “Nonlinear Bayesian Estimation Using Gaussian Sum Approximations,” IEEE Trans. on Automatic Control, Vol. 16, No. 4, pp. 439-448, Aug. 1972.
11. Alspack, D. L., “A Gaussian Sums Approach to the Multitarget Identification Tracking Problem,” Automatica, Vol. 11, pp. 285-296, 1975.
12. Netto, N. L., Andrade, L., Gimeno, L., and Mendes, M. J., “On the Optimal and Suboptimal Nonlinear Filtering Problem for Discrete-Time Systems,” IEEE Trans. on Automatic Control, Vol. 23, No. 6, pp. 1062-1067, Dec. 1978.
13. Kerr, T. H., “Critique of Some Neural Network Architectures and Claims for Control and Estimation,” IEEE Transactions on Aerospace and Electronic Systems, Vol.34, No. 2, pp. 406-419, Apr. 1998.
14. Kerr, T. H., “Decentralized Filtering and Redundancy Management for Multisensor Navigation,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 23, No. 1, pp. 83-119, Jan. 1987 (corrections: May, p. 412; Jul., p. 599). This won 1988 M. Barry Carlton Award and $2000 honorarium for Outstanding Paper to appear in IEEE Aerospace and Electronic Systems Transactions in ’87 [for more details, please see page 822 of Vol. 24, No. 6, Nov. 1988 of the aforementioned journal].
15. Kerr, T. H., “A Critique of Several Failure Detection Approaches for Navigation Systems,” IEEE Transactions on Automatic Control, Vol. 34, No. 7, pp. 791-792, Jul. 1989.
16. Willsky, A. S., Jones, H. L., “A Generalized Likelihood Ratio Approach to Detection and Estimation of Jumps in Linear Systems,” IEEE Transactions on Automatic Control, Vol. 21, No. 1, pp. 108-112, Feb. 1976.
17. Kerr, T. H., and Chin, L., “A Stable Decentralized Filtering Implementation for JTIDS RelNav,” Proceedings of IEEE Position, Location, and Navigation Symposium (PLANS), Atlantic City, NJ, pp. 318-329, 8-11 Dec. 1980.
18. Kerr, T. H., “Stability Conditions for the RelNav Community as a Decentralized Estimator-Final Report,” Intermetrics, Inc. Report No. IR-480, Cambridge, MA, 10 Aug. 1980.
19. Kerr, T.H., and Chin, L., “The Theory and Techniques of Discrete-Time Decentralized Filters,” in Advances in the Techniques and Technology in the Application of Nonlinear Filters and Kalman Filters, edited by C. T. Leondes, NATO Advisory Group for Aerospace Research and Development, AGARDograph No. 256, Noordhoff International Publishing, Lieden, pp. 3-1 to 3-39, 1981.
20. Kerr, T. H., “On Duality Between Failure Detection and Radar/Optical Maneuver Detection,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 25, No. 4, pp. 581-583, Jul. 1989.
21. Kerr, T. H., “New Lamps for Old: a Shell Game for Generalized Likelihood Use in Radar? Or this isn’t your father’s GLR!,” Proceedings of SPIE, Session 4473: Tracking Small Targets, San Diego, CA, pp. 476-483, 29 Jul.-3 Aug. 2001.
22. Kerr, T. H., “Status of CR-Like Lower bounds for Nonlinear Filtering,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 25, No. 5, pp. 590-601, Sept. 1989 (Author’s reply: Vol. 26, No. 5, pp. 896-898, Sept. 1990).
23. Kerr, T. H., UEWR Design Notebook-Section 2.3: Track Analysis, TeK Associates, Lexington, MA, (for XonTech, Hartwell Rd, Lexington, MA), XonTech Report No. D744-10300, 29 Mar. 1999.
24. Kerr, T. H., “Emulating Random Process Target Statistics (using MSF),” IEEE Transactions on Aerospace and Electronic Systems, Vol. 30, No. 2, pp. 556-577, Apr. 1994.
25. Kerr, T. H., “Fallacies in Computational Testing of Matrix Positive Definiteness/Semidefiniteness,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 26, No. 2, pp. 415-421, Mar. 1990.
26. Kerr, T. H., “Vulnerability of Recent GPS Adaptive Antenna Processing (and all STAP/SLC) to Statistically Non-Stationary Jammer Threats,” Proceedings of SPIE, Session 4473: Tracking Small Targets, San Diego, CA, pp. 62-73, 29 Jul.-3 Aug. 2001.
27. Boozer, D. D., McDaniel, W. I., “On Innovation Sequence Testing of the Kalman Filter,” IEEE Transactions on Automatic Control, Vol. 17, No. 1, pp. 158-160, Feb. 1972.
28. Arnold, B. C., Castillo, E., Sarabia, J. M., Conditional Specification of Statistical Models, Springer-Verlag, NY, 1999.
29. Davenport, W. B., Root, W. L., Introduction to the Theory of Random Signals in Noise, McGraw-Hill, NY, 1958.
30. Root, W. L., “Radar Resolution of Closely Spaced Targets,” IRE Transactions on Military Electronics, Vol. MIL-6, pp. 197-204, Apr. 1962.
31. McAuley, R. J., Denlinger, E., “A Decision-Directed Adaptive Tracker,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 9, No. 2, pp. 229-236, Mar. 1973.
32. Stuller, J. A., “Generalized Likelihood Signal Resolution,” IEEE Transactions on Information Theory, Vol. 21, No. 3, May 1975, pp. 276-282.
33. Selin, I., “Estimation of the Relative Delay of Two Similar Signals of Unknown Phases in Gaussian Noise,” IEEE Transactions on Information Theory, Vol. 10, pp. 189-191, Jul. 1964.
34. Selin, I., Detection Theory, Princeton University Press, Princeton, NJ, 1965.
35. Van Trees, H. L., Detection, Estimation, and Modulation Theory, Part I: Detection, Estimation and Linear Modulation Theory, John Wiley and Sons, Inc., NY, 1968.
36. Patel, J. K., Kapadia, C. H., Owen, D. B., Handbook of Statistical Distributions, Marcel Dekker, Inc. NY, 1976.
37. Kerr, T. H., “Computational Techniques for the Matrix Pseudoinverse in Minimum Variance Reduced-Order Filtering and Control,” in Control and Dynamic Systems-Advances in Theory and Applications, Vol. XXVIII: Advances in Algorithms and computational Techniques for Dynamic Control Systems, Part 1 of 3, C. T. Leondes (Ed.), Academic Press, NY, pp. 57-107, 1988.
38. Mehra, R. K., Peschon, J., “An Innovations Approach to Fault Detection and Diagnosis in Dynamic Systems,” Automatica, Vol. 7, No. 5, pp. 637-640, 1971.
39. Kerr, T. H., “The Controversy Over Use of SPRT and GLR Techniques and Other Loose-Ends in Failure Detection,” Proceedings of American Control Conference, Vol. 3, San Francisco, CA, pp. 966-977, 22-24 Jun. 1983.                 15_pub_AUTO.pdf 

40. Bar-Shalom, Y., Challa, S., Blom, H. A. P., “IMM Estimator Versus Optimal Estimator for Hybrid Systems,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 41, No. 3, pp. 986-991, July 2005. [If M(k) were in fact known beforehand, then {x(k), M(k)} would, in fact, be a Markov process as these authors rely on in their proofs. However, just how the system will switch is unknown to the user or engineer a priori or even immediately after the evolution in time of the system of interest having this potential structure. Notice that the authors need to assume that the underlying multiple model systems have a linear structure before a Gaussian sums mixture model can be invoked. When models are nonlinear, probability density function calculations are infinite dimensional in general (i.e., the actual calculations for the associated probabilities, while tractable, cannot be accomplished in real-time since the indicated integrations to be performed are infinite-dimensional in all but the simplest of cases). You can write expressions that involve integral evaluations but you are SOL for real-time computations of the required answers. This is essentially Bucy’s representation theorem of 40 years ago.]  

41. He, T., Ben-David, S., Tong, L., “Nonparametric Change Detection and Estimation in Large-Scale Sensor Networks,” IEEE Trans. on Signal Processing, Vol. 54, No. 4, pp. 1204-1217, April 2006. 

42. Nyberg, M., Frisk, E., “Residual Generation for Fault Diagnosis of Systems Described by Linear Differential-Algebraic Equations,” IEEE Trans. on Automatic Control, Vol. 51, No. 12, pp. 1995-1999, December 2006.

43. Stoica, P., “Performance Evaluation of Some Methods for Off-Line Detection of Changes in Autoregessive Signals,” Selected Papers on Performance Evaluation of Signal and Image Processing Systems, Firooz Sadjadi (editor), SPIE Milestone Series, 5, No. 4, pp. 134-136, 1999. [T. H. Kerr’s comment: An assertion within this paper is that existing fault detection or change detection schemes involve monitoring test statistics that consist of quadratic forms of sub-collections of states that are Gaussianly distributed with zero means, and that as a consequence, the test statistic has a Chi-square distribution. We object in that while this statement is almost true for many different approaches, it is not true for CR2 nor for GLR. We at TeK Associates have worked out the statistics in detail and published them in the open peer-reviewed literature for CR2 for the transient time-varying situation that faced us. If we had the luxury of waiting until the time-invariant steady-state was reached (which was impossibly long and never arrived for the submarine navigation application for which CR2 was developed), then the distribution of CR2 is also asymptotically chi-square with m-degrees-of-freedom, where m is dimension of the number of states being simultaneously monitored. For GLR, the GLR test statistic itself is the maximum of several different quadratic forms of Gaussianly distributed states so it is the maximum of several different Chi-squared entities which results in something that is not chi-squared even if the situation were so benign as to have an identical chi-squared distribution in common (which it does not). Again the maximization operation alters the distribution of the output in a predictable way (via A. Papoulis’ textbook) but its calculation is not  tractable and has yet to be done.]

44. Cheng, Q., Varshney, P. K., Michels, J. H., Belcastro, C. M., “Fault Detection in Dynamic Systems via Decision Fusion,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 44, No. 1, pp. 227-242, Jan. 2008.

45. Tudoroiu, N., Khorasani, K., “Satellite Fault Diagnosis using a Bank of Interacting Kalman Filters,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 43, No. 4, pp. 1334-1350, Oct. 2007.    

46. Chan, Y. K., Edsinger, R. W., “A Correlated Random Number Generator and Its Use to Estimate False Alarm Rates,” IEEE Trans. on Automatic control, June 1981. 

47. Chien, T. T., “An Adaptive Technique for a Redundant-Sensor Navigation System,” Report No. T-560, C. S. Draper Laboratory, Cambridge, MA, 1972. 

48. Owens, A. B., Empirical Likelihood, (QA276.8.094 Hayden at MIT), 2001.

49. Niels Haering, Niels da Vitoria Lobo, Visual Event Detection, Kluwer Academic Publishing, Boston, MA, 2001.

50. Rife, J., “Influence of GNSS Integrity Monitoring on Undetected Fault Probabilities for Single and Multiple Fault Events,” Navigation: Journal of the Institute of Navigation, Vol. 56, No. 4, pp. 275-288, January 2009. 

51. Roecker, J. A., “On Combining Multidimensional Target Location Ellipsoids,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 27, No. 1, pp. 176-177, January 1991. 

52. Uwe D. Hanbeck, “Recursive Nonlinear Set-Theoretic Estimation Based on Pseudo-Ellipsoids,” Proceedings of the IEEE Conference on Multisensor Fusion and Integration for Intelligent Systems, pp. 159–164 (MFI’ 2001), Baden–Baden.
    http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.304.8080  
    https://www.researchgate.net/publication/3955109_Recursive_nonlinear_set-theoretic_estimation_based_on_pseudo_ellipsoids 
    https://core.ac.uk/display/22665895 
    https://core.ac.uk/display/24506648
    http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.3779
    Their patent abandoned by Siemens AG (perhaps because of prior art: me): https://patents.google.com/patent/US20060234722A1/en 

53. Fuqiang You, Hualu Zhang, Fuli Wang, “A new set-membership estimation method based on zonotopes and ellipsoids,” Transactions of the Institute of Measurement and Control, Vol. 40, issue 7, pp. 2091-2099, Article first published online: 27 July 2016; Issue published: 1 April 2018.

54. Grocholsky B., Stump E., Shiroma P.M., Kumar V., “Control for Localization of Targets Using Range-Only Sensors,” in Khatib, O., Kumar, V., Rus, D. (eds), Experimental Robotics, Springer Tracts in Advanced Robotics Series, Vol 39. Springer, Berlin, Heidelberg, pp. 191-200, 2008. 

55. Ashok K. Rao, Yih-Fang Huang, and Soura Dasgupta, “ARMA Parameter Estimation Using a Novel Recursive Estimation Algorithm with Selective Updating,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 38, No. 3, Mar. 1990.
    https://www3.nd.edu/~huang/papers/IEEE-T-SP-Mar1990-RH.pdf 

56. Also related to Set Estimation: https://en.wikipedia.org/wiki/Set_estimation

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