Cramer-Rao Analysis
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Cramér-Rao Lower Bound (CRLB) Analysis & Evaluation

Key Benefits:

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 typical applications utilizing these algorithms and are aware of what application constraints are usually actively in force.

Capabilities:

We have a facility with analytic manipulations that enabled the following:
insights in obtaining new theoretical results at the point were they were needed to expand and improve the existing theory, as conveyed below in references [1], [2], and applied in [3]-[8]. Also see the recent [9] by others that cite us. Please see a somewhat recent IEEE review that we authored in this CRLB area.  Here and in what follows below, Thomas H. Kerr IIIs comments and annotations are in a different color font to make it easier for readers to distinguish (and, perhaps, to ignore).

An interesting result due to A. J. Stam [24, 1959] is the derivation of Weyl-Heisenberg uncertainty principle in physics using a specific version of the CRB. Further applications in physics of CRB and Fisher information as a concept underlying the well known physical theories can be found in the book by B. Roy Frieden, “Physics from Fisher Information” [25].

Recent developments are the Quantum Cramér-Rao Bound in the estimation of manifolds in Quantum Physics, by Brody and Houghston [26] and the concept of Cramér-Rao Functional based on Cramér-Rao Bound by E. Mayor Wolf [27, 1990]. CRB has been extended to estimation of “manifolds” as “complexified and intrinsic” CRB and used in signal processing.

Click here to obtain a detailed 266Kilobyte resume for Thomas H. Kerr III emphasizing only Target Tracking for strategic Early Warning Radar (UEWR).

Historical Account of our experience therein:

[1] Kerr, T. H., “A New Multivariate Cramer-Rao Inequality for Parameter Estimation (Application: Input Probing Function Specification),” Proceedings of IEEE Conference on Decision and Control, Phoenix, AZ, pp. 97-103, Dec. 1974.

[2] 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 (Authors reply in Vol. 26, No. 5, pp. 896-898, Sept. 1990).

[3] Kerr, T. H., “Cramér-Rao Lower Bound Implementation and Analysis for NMD Radar Target Tracking,” TeK Associates Technical Report No. 97-101 (for MITRE), Lexington, MA, 26-30 Oct. 1997.

[4] Kerr, T. H., “NMD White Paper on Designated Action Item,” MITRE, Bedford, MA, Jan. 1998.

[5] Kerr, T. H., “Cramér-Rao Lower Bound Implementation and Analysis: CRLB Target Tracking Evaluation Methodology for NMD Radars,” MITRE Technical Report, Contract No. F19628-94-C-0001, Project No. 03984000-N0, Bedford, MA, Feb. 1998.

[6] Kerr, T. H., “Developing Cramér-Rao Lower Bounds to Gauge the Effectiveness of UEWR Target Tracking Filters,” Proceedings of AIAA\BMDO Technology Readiness Conference and Exhibit, Colorado Springs, CO, 3-7 Aug. 1998.

[7] 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.

[8] Kerr, T. H., A Critical View of Some New and Older approaches to EWR Target Tracking: A Summary and Endorsement of Kalman Filter-Related Techniques, a tutorial submitted to IEEE Aerospace and Electronic Systems, (yet to be submitted).

[9] Hue, C., Le Cadre, J.-P., Perez, P., “Posterior Cramér-Rao Bounds for Multi-Target Tracking,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 42, No. 1, pp. 37-49, January 2006.

[10] Gault, S., Hachem, W., Ciblat, P., “Joint Sampling Clock Offset and Channel Estimation for OFDM Signals: Cramér-Rao Bound and Algorithm, IEEE Trans. on Signal Processing, Vol. 54, No. 5, pp. 1875-1885, May 2006.

[11] Yetik, I. S., Nehorai, A., “Performance Bounds on Image Registration,IEEE Trans. on Signal Processing, Vol. 54, No. 5, pp. 1737-1749, May 2006.

[12] Zou, Q., Lin, Z., Ober, R. J., “The Cramér-Rao Lower Bound for Bilinear Systems,IEEE Trans. on Signal Processing, Vol. 54, No. 5, pp. 1666-1680, May 2006.

[13] Eldar, Y., “Uniformly Improving the Cramér-Rao Bound and Maximum-Likelihood Estimation, IEEE Trans. on Signal Processing, Vol. 54, No. 8, pp. 2943-2956, Aug. 2006.

[14] Brehard, T., Le Cadre, J.-P., “Closed-Form Posterior Cramér-Rao Bounds for Bearings-Only Tracking,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 42, No. 4, pp. 1198-1223, Oct. 2006.

[15] Au-Yueng, C. K., Wong, K. T., “CRB: Sinusoid-Sources' Estimation using Collocated Dipoles/Loops,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 45, No. 1, pp. 94-109, Jan. 2009.

[16] Kay, S., Xu, C., “CRLB via  the Characteristic Function with Application to the K-Distribution,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 44, No. 3, pp. 1161-1168, Jul. 2008.

[17] Smith, S. T., “Statistical Resolution Limits and the Complexified Cramér-Rao Bound,IEEE Trans. on Signal Processing, Vol. 53, No. 5, pp. 1597-1609, May 2005.

[18] Smith, S. T., “Covariance, Subspace, and Intrinsic Cramér-Rao Bounds,IEEE Trans. on Signal Processing, Vol. 53, No. 5, pp. 1610-1630, May 2005.

[19] Gini, F., Regianini, R., Mengali, U., The Modified Cramér-Rao Lower Bound in Vector Parameter Estimation,IEEE Trans. on Signal Processing, Vol. 46, No. 1, pp. 52-60, Jan. 1998.   

[20] Buzzi, S., Lops, M., Sardellitti, S., “Further Results on Cramér-Rao Bounds for Parameter Estimation in Long-Code DS/CDMA Systems, IEEE Trans. on Signal Processing, Vol. 53, No. 3, pp. 1216- 1221, Mar. 2005.

[21] Branko Ristic, “Cramer Rao Bounds for Target Tracking,International Conference on Sensor Networks and Information Processing, 6 Dec. 2005.

[22] Boers, Y., Driessen, H., “A Note on Bounds for Target Tracking with Pd < 1,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 45, No. 2, pp. 640-646, Apr. 2009.

[23] Bobrovsky, Wolf, and Zakai, Annals of Statistics, Vol. 15, pp. 1421-1438, 1987.

[24] Stam,  A. J., Information and Control, Vol. 2, pp. 101-112, 1959.

[25] Frieden, B. Roy, Physics from Fisher Information, Cambridge University Press, 1998.

[26] Brody, D. C. and Hughston, L. P., Proceedings of the  Royal Society, Vol. 454, pp. 2445-2475, 1998.

[27] Wolf, E. Mayor, Annals of  Probability, Vol. 18, pp. 840-850, 1990.

[28] http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_cramer_rao.htm 

[29] http://www.scholarpedia.org/article/Cramér-Rao_bound 

[30] http://www.dauwels.com/Papers/BCRB.pdf  

[31] I. Rapoport and Y. Oshman, “A Cramér-Rao-type estimation lower bound for systems with measurement faults,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 50, No. 9, pp. 1234–1245, Sept. 2005.

[32] R. M. Taylor, Jr., B. P. Flanagan, and J. A. Uber, “Computing the recursive posterior Cramér-Rao bound for a nonlinear nonstationary system,” in IEEE International Conference on Acoustics, Speech, and Signal Processeing (ICASSP),  Vol. 6, pp. 673–676, Apr. 2003.

[33] F. E. Daum, “Bounds on track purity for multiple target tracking,” in Proceedings of the 28th Conference on Decision and Control, Tampa, FL, pp. 1423–1424, Dec. 1989.

[40] F. E. Daum, “Bounds on performance for multiple target tracking,” IEEE Transactions on Automatic Control, Vol. 35, No. 4, pp. 443–446, Apr. 1990.

[41] F. E. Daum, “A Cramér-Rao bound for multiple target tracking,” in Proceedings of SPIE, Signal and Data Processing of Small Targets, Vol. 1481, Orlando, FL, pp. 341–343, Apr. 1991.

[42] X. Zhang, P. Willett, and Y. Bar-Shalom, “The
Cramér-Rao bound for dynamic target tracking with measurement origin uncertainty,” in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, pp. 3428–3433, Dec. 2002.

[43] M. L. Hernandez, A. D. Marrs, N. J. Gordon, S. R. Maskell, and C. M. Reed, “Cramér-Rao bounds for non-linear filtering with measurement origin uncertainty,” in Proceedings of the 5th International Conference on Information Fusion, Vol. 1, pp. 18–25, Jul. 2002.

[44] X. Zhang, P. Willett, and Y. Bar-Shalom, “Dynamic Cramér-Rao bound for target tracking in clutter,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 41, No. 5, pp. 1154–1167, Oct. 2005.

[45] N. Bergman, “Recursive Bayesian estimation: Navigation and tracking applications,” Ph.D. dissertation, Linkoping University, SE-581 83 Linkoping, Sweden, April 1999.

[46] N. Bergman, Sequential Monte Carlo Methods in Practice, New York: Springer, New York, 2001 (Chapter: Posterior Cramér-Rao Bounds for Sequential Estimation, pp. 321–338).

[47] Justin Dauwels, “Computing Bayesian Cramer-Rao bounds," http://www.dauwels.com/Papers/BCRB.pdf
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.399.3443 
dauweks@isi.ee.ethz.ch
 

[48] Justin Dauwels, Sascha, Korl, “A Numerical Method to Compute Cramer-Rao-Type Bounds for Challenging Estimation Problems,”
http://www.dauwels.com/Papers/ICASSP2006.pdf

[49]  CORE https://core.ac.uk/display/23680005  (several different papers coauthored by others, as, perhaps, offshoots)

[50] Frank R. Kschischang , Brendan J. Frey , Hans-Andrea Loeliger, “Factor Graphs and the Sum-Product Algorithm,” IEEE Transactions on Information Theory, 1998.
Abstract: A factor graph is a bipartite graph that expresses how a “global” function of many variables factors into a product of "local" functions. Factor graphs subsume many other graphical models including Bayesian networks, Markov random fields, and Tanner graphs. Following one simple computational rule, the sum-product algorithm operates in factor graphs to compute---either exactly or approximately---various marginal functions by distributed message-passing in the graph. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative “turbo” decoding algorithm, Pearl's belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform algorithms.

[51] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.1570&rank=1&q=cyclic%20factor%20graph&osm=&ossid= 
Abstract: A factor graph is a bipartite graph that expresses how a “global” function of many variables factors into a product of "local" functions. Factor graphs subsume many other graphical models including Bayesian networks, Markov random fields, and Tanner graphs. Following one simple computational rule, the sum-product algorithm operates in factor graphs to compute---either exactly or approximately---various marginal functions by distributed message-passing in the graph. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative “turbo” decoding algorithm, Pearl’s belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform algorithms. [Thomas H. Kerr's comment here: In the 1960's, it was demonstrated that standard signal flow graph theory using “nodes” and “branches” was equivalent to the already familiar block diagrams of linear time-invariant dynamic control systems, as handled in the frequency domain. Later came Petri-nets. Wikimedia: A Petri net, also known as a place/transition (PT) net, is one of several mathematical modeling languages for the description of distributed systems. It is a class of discrete event dynamic system. A Petri net is a directed bipartite graph, in which the nodes represent transitions (i.e., events that may occur, represented by bars) and places (i.e., conditions, represented by circles). The directed arcs describe which places are pre- and/or post-conditions for which transitions (signified by arrows). Like industry standards such as UML activity diagramsBusiness Process Model and Notation and EPCs, Petri nets offer a graphical notation for stepwise processes that include choice, iteration, and concurrent execution. Unlike these standards, Petri nets have an exact mathematical definition of their execution semantics, with a well-developed mathematical theory for process analysis.
Now we have factor graphs as somewhat of a natural progression along these lines.]

How to Handle Jamming Considerations:

[52]  Myers, L., Improved Radio Jamming Techniques: Electronic Guerilla Warfare, ISBN 0873645200, Paladin Press, Boulder, CO, 1989.

 [53] Chapter 8 of Bar-Shalom, Y., Blair, W. D., (Eds.), Multitarget-Multisensor Tracking Applications and Advances, Vol. III, Artech House Inc., Boston, MA, 2000.

Gaussian Confidence Region or uncertainty ellipsoid (associated with target tracking) goes from being a pancake (at the horizon) to being a football (after sufficient radar returns have accrued). When knowledge of targets trajectory is sufficiently accurate, then the target can be successfully intercepted.

Tom Kerr and Eli Brookner (Click here to view info about Elis recent book.)

For more about Eli, please click: http://www.raytheon.com/newsroom/feature/ids_brookner06/ 

  http://www.google.com/patents/about?id=zsR3AAAAEBAJ&dq=parachute+camera&ie=ISO-8859-1

http://ieee-aess.org/main/index.php?option=com_content&view=article&id=57&Itemid=66    http://pipl.com/directory/name/Brookner/Eli

http://www.ieee.org/about/awards/bios/picard_recipients.html                            http://www.mwjournal.com/Journal/Print.asp?Id=AR_5352

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