Covariance matrix

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In probability theory and statistics, a covariance matrix (also known as dispersion matrix) is a matrix whose element in the i, j position is the covariance between the i and j elements of a random vector. (Wikipedia.org)






Conferences related to Covariance matrix

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2018 24th International Conference on Pattern Recognition (ICPR)

ICPR will be an international forum for discussions on recent advances in the fields of Pattern Recognition, Machine Learning and Computer Vision, and on applications of these technologies in various fields

  • 2016 23rd International Conference on Pattern Recognition (ICPR)

    ICPR'2016 will be an international forum for discussions on recent advances in the fields of Pattern Recognition, Machine Learning and Computer Vision, and on applications of these technologies in various fields.

  • 2014 22nd International Conference on Pattern Recognition (ICPR)

    ICPR 2014 will be an international forum for discussions on recent advances in the fields of Pattern Recognition; Machine Learning and Computer Vision; and on applications of these technologies in various fields.

  • 2012 21st International Conference on Pattern Recognition (ICPR)

    ICPR is the largest international conference which covers pattern recognition, computer vision, signal processing, and machine learning and their applications. This has been organized every two years by main sponsorship of IAPR, and has recently been with the technical sponsorship of IEEE-CS. The related research fields are also covered by many societies of IEEE including IEEE-CS, therefore the technical sponsorship of IEEE-CS will provide huge benefit to a lot of members of IEEE. Archiving into IEEE Xplore will also provide significant benefit to the all members of IEEE.

  • 2010 20th International Conference on Pattern Recognition (ICPR)

    ICPR 2010 will be an international forum for discussions on recent advances in the fields of Computer Vision; Pattern Recognition and Machine Learning; Signal, Speech, Image and Video Processing; Biometrics and Human Computer Interaction; Multimedia and Document Analysis, Processing and Retrieval; Medical Imaging and Visualization.

  • 2008 19th International Conferences on Pattern Recognition (ICPR)

    The ICPR 2008 will be an international forum for discussions on recent advances in the fields of Computer vision, Pattern recognition (theory, methods and algorithms), Image, speech and signal analysis, Multimedia and video analysis, Biometrics, Document analysis, and Bioinformatics and biomedical applications.

  • 2002 16th International Conference on Pattern Recognition


2017 10th Global Symposium on Millimeter-Waves (GSMM)

The main theme of the symposium is Millimeter-Wave and Terahertz Sensing and Communications. It covers millimeter- wave and THz antennas, circuits, devices, systems and applications.

  • 2018 11th Global Symposium on Millimeter Waves (GSMM)

    The main theme of the GSMM2018 is Millimeter-wave Propagation: Hardware, Measurements and Systems. It covers millimeter-wave and THz devices, circuits, systems, and applications, with a special focus on mmWave propagation. The conference will include keynote talks, technical sessions, panels, and exhibitions on the listed topics.

  • 2016 Global Symposium on Millimeter Waves (GSMM) & ESA Workshop on Millimetre-Wave Technology and Applications

    The main theme of the conference is millimeter-wave and terahertz sensing and communications and the conference covers different topics related to millimeter-wave and terahertz technologies, such as: antennas and propagation, passive and active devices, radio astronomy, earth observation and remote sensing, communications, wireless power transfer, integration and packaging, photonic systems, and emerging technologies.

  • 2015 Global Symposium On Millimeter Waves (GSMM)

    The main theme of the GSMM 2015 is “Future Millimeter-wave and Terahertz Wireless and Wireline”. It will cover all emerging and future millimeter wave and terahertz software and hardware aspects ranging from communicating devices, circuits, systems and applications to passive and active sensing and imaging technologies and applications. The GSMM 2015 will feature world-class keynote speeches, technical sessions, panel discussions and industrial exhibitions in the following (but not limited to) topics listed below.In addition to the regular program, the GSMM 2015 will organize a unique industrial forum for presenting and discussing future wireless technologies and trends including 5G and Terahertz Wireless Systems.

  • 2012 5th Global Symposium on Millimeter Waves (GSMM 2012)

    The aim of the conferences is to bring together people involved in research and development of millimeter-wave components, equipment and systems, and to explore common areas.

  • 2009 Global Symposium On Millimeter Waves (GSMM 2009)

    The GSMM2009 will be held in Sendai, Japan from April 20 to April 22, 2009. The GSMM2009 is the second international conference in its name after the three conferences of TSMMW, MINT-MIS, and MilliLab Workshop on Millimeter-wave Technology and Applications were integrated into GSMM (Global Symposium on Millimeter Waves) in 2007. The main theme of the GSMM2009 is "Millimeter Wave Communications at Hand" and it will focus on millimeter wave devices and systems to realize Giga-bit wireless applications. The

  • 2008 Global Symposium On Millimeter Waves (GSMM 2008)

    Frequency Management and Utilization, Millimeter-Wave Communication Systems, Devices and Circuit Technologies, Wireless Access Systems, Mobile Access Systems, Satellite Communications, LANs and PANs, Home Link Systems, Photonics, Antennas and Propagation, Phased Array Antennas, Signal Processing, Wearable Devices and Systems, Automotive Radars and Remote Sensing, Supporting and Related Technologies


2017 13th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD)

International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD) is a premier international forum for scientists and researchers to present the state of the art of data mining and intelligent methods inspired from nature, particularly biological, linguistic, and physical systems, with applications to computers, circuits, systems, control, robotics, communications, and more.

  • 2016 12th International Conference on Natural Computation and 13th Fuzzy Systems and Knowledge Discovery (ICNC-FSKD)

    International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD) is a premier international forum for scientists and researchers to present the state of the art of data mining and intelligent methods inspired from nature, particularly biological, linguistic, and physical systems, with applications to computers, circuits, systems, control, robotics, communications, and more.

  • 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

    FSKD is an international forum on fuzzy systems and knowledge discovery. Specific topics include fuzzy sets, Bioinformatics and Bio-Medical Informatics, Genomics, Proteomics, Big Data, Databases and Applications, Semi-Structured/Unstructured Data Mining, Multimedia Mining, Web and Text Data Mining, Graphic Model Discovery, Data Warehousing and OLAP, Pattern Recognition and Diagnostics, etc..

  • 2014 11th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

    FSKD is an international forum on fuzzy systems and knowledge discovery. Specific topics include fuzzy sets, rough sets, Statistical methods, Parallel/ Distributed data mining, KDD Process and human interaction, Knowledge management, Knowledge visualization, Reliability and robustness, Knowledge Discovery in Specific Domains, High dimensional data, Temporal data, Data streaming, Scientific databases, Semi-structured/unstructured data, Multimedia, Text, Web and the Internet, Graphic model discovery, Software warehouse and software mining, Data engineering, Communications and networking, Software engineering, Distributed systems and computer hardware

  • 2013 10th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

    FSKD is an international forum on fuzzy systems and knowledge discovery. Specific topics include fuzzy sets, rough sets, Statistical methods, Parallel/ Distributed data mining, KDD Process and human interaction, Knowledge management, Knowledge visualization, Reliability and robustness, Knowledge Discovery in Specific Domains, High dimensional data, Temporal data, Data streaming, Scientific databases, Semi-structured/unstructured data, Multimedia, Text, Web and the Internet, Graphic model discovery, Software warehouse and software mining, Data engineering, Communications and networking, Software engineering, Distributed systems and computer hardware, etc.

  • 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

    FSKD is an international forum on fuzzy systems and knowledge discovery. Specific topics include fuzzy theory and foundations; stability of fuzzy systems; fuzzy methods and algorithms; fuzzy image, speech and signal processing; multimedia; fuzzy hardware and architectures; data mining.

  • 2011 Eighth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

    FSKD is an international forum on fuzzy systems and knowledge discovery. Specific topics include fuzzy theory and foundations; stability of fuzzy systems; fuzzy methods and algorithms; fuzzy image, speech and signal processing; multimedia; fuzzy hardware and architectures; data mining.

  • 2010 Seventh International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

    FSKD is an international forum on fuzzy systems and knowledge discovery. Specific topics include fuzzy theory and foundations; stability of fuzzy systems; fuzzy methods and algorithms; fuzzy image, speech and signal processing; multimedia; fuzzy hardware and architectures; data mining.

  • 2007 International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

    FSKD '07 covers all aspects of fuzzy systems and knowledge discovery, including recent theoretical advances and interesting applications, for example, fuzzy theory and models, mathematical foundation of fuzzy systems, fuzzy image/signal processing, fuzzy control and robotics, fuzzy hardware and architectures, fuzzy systems and the internet, fuzzy optimization and modeling, fuzzy decision and support, classification, clustering, statistical methods, knowledge etc.


2017 14th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS)

AVSS is the premier annual international conference in the field of video and signal-based surveillance that brings together experts from academia, industry, and government to advance theories, methods, systems, and applications related to surveillance.


2017 14th International Multi-Conference on Systems, Signals & Devices (SSD)

The International Multi-Conference on Systems, Signals and Devices 2017 is a forum for researchers and specialists in different fields of electrical engineering departments from leading research centers and universities around the world to present their research results and to share experiences with other attendees. On behalf of International Multi-Conference on Systems, Signals & Devices Organizing Committee, it is our great pleasure to extend a warm invitation to you to participate in SSD'17. Since 2001, SSD has grown substantially from a brand new conference with a strong vision to the SSD of today, dedicated to the advancement of electrical engineering research fields and their practice.

  • 2016 13th International Multi-Conference on Systems, Signals & Devices (SSD)

    SSD is a renowned scientic event that includes five scheduled conferences covering almost allfields of electrical and electronics engineeringnamely: Systems Automation and Controls (SAC), Communication, Signal Processing & Information Technologies (CSP), Sensors, Circuits and Instrumentation Systems (SCI), Micro and Nano Electronic Systems (MiNE) and Power Systems and Smart Energies (PSE).The event is a forum for researchers and practitioners to discuss the latest findings in electricaland electronics engineering research. It provides a serious opportunity for participants to promote networking between scientists and universities.

  • 2015 12th International Multi-Conference on Systems, Signals & Devices (SSD)

    SSD is a renowned scientic event that includes five scheduled conferences covering almost all fields of electrical and electronics engineeringnamely: Systems Analysis and Automatic Control (SAC), Power Electrical Systems (PES), Communication and Signal Processing (CSP), Sensors, Circuits andInstrumentation Systems (SCI), Computers & Information Technology (CIT).The event is a forum for researchers and practitioners to discuss the latest findings in electrical and electronics engineering research. It provides a seriousopportunity for participants to promote networking between scientists and universities.

  • 2014 11th International Multi-Conference on Systems, Signals & Devices (SSD)

    The International Multi-Conference on Systems, Signals and Devices 2014 is a forum for researchers and specialists in different fields of electrical engineering departments from leading research centers and universities around the world to present their research results and to share experiences with other attendees. SSD14 has four main topics: i) System Analysis and Automatic Control, ii) Power Electrical Systems, iii) Communication and Signal Processing, and iv) Sensors Circuits and Instrumentation Systems. The conference combines plenary sessions, keynote talks, academic presentations, discussion forums and academic-industry interaction.

  • 2013 10th International Multi-Conference onSystems, Signals & Devices (SSD)

    The 2013 International Multi-Conference on Systems, Signals and Devices is a forum for researchers and specialists in different fields of electrical engineering departments from leading research centers and universities around the world to present their research results and to share experiences with other attendees. It is the 10th multi -conference since the founding of SSD in 2001 whichis supported by international organizations such as IEEE, TSS and different scientific journals.

  • 2012 IEEE 9th International Multi-Conference on Systems, Signals and Devices (SSD)

    The International Multi-Conference on Systems, Signals and Devices 2012 is a forum for researchers and specialists in different fields of electrical engineering departments from leading reserach centers and universities around the world to present their research results and to share experiences with other attendees. It is the 9th multi-conference since the founding of SSD in 2001 which is supported by international organizations such as IEEE, TSS and different scientific journals.

  • 2011 8th International Multi-Conference on Systems, Signals and Devices (SSD)

    The SSD conference is a multi conference covering most topics in the the field of electrical engineering. It is celebrations its tenth birth day in 2011.

  • 2010 7th International Multi-Conference on Systems, Signals and Devices (SSD)

    The event consists of four specialized conferences that cover a wide spectrum of fields in electrical and electronics engineering and as follows Systems Analysis and Automatic Control Power Electrical Systems Communication and Signal Processing Sensors, Circuits and Instrumentation Systems

  • 2009 6th International Multi-Conference on Systems, Signals and Devices (SSD)

    The International Conference SSD 09 is a forum for specialists to present their research results and to share experiences with other attendees coming from all over the world. It is the 6th conference since the founding of SSD in 2001. SSD is supported by international organizations such as IEEE, TSS and different scientific journals. SSD 09 includes keynote lectures by eminent scientists as well as oral and poster sessions. All papers are peer reviewed on the basis of full manuscripts. SSD participants hav

  • 2008 5th International Multi-Conference on Systems, Signals and Devices (SSD)

    The International Multi-Conference on Systems signals & Devices is an annual scientific event that includes four scheduled conferences covering almost fields of electrical and electronics engineering: (1)Systems Analysis and Automatic Control, (2)Power Electrical Systems, (3)Communication and Signal Processing (4)Sensors, Circuits and Instrumentation Systems.


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Periodicals related to Covariance matrix

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Audio, Speech, and Language Processing, IEEE Transactions on

Speech analysis, synthesis, coding speech recognition, speaker recognition, language modeling, speech production and perception, speech enhancement. In audio, transducers, room acoustics, active sound control, human audition, analysis/synthesis/coding of music, and consumer audio. (8) (IEEE Guide for Authors) The scope for the proposed transactions includes SPEECH PROCESSING - Transmission and storage of Speech signals; speech coding; speech enhancement and noise reduction; ...


Automatic Control, IEEE Transactions on

The theory, design and application of Control Systems. It shall encompass components, and the integration of these components, as are necessary for the construction of such systems. The word `systems' as used herein shall be interpreted to include physical, biological, organizational and other entities and combinations thereof, which can be represented through a mathematical symbolism. The Field of Interest: shall ...


Biomedical Engineering, IEEE Transactions on

Broad coverage of concepts and methods of the physical and engineering sciences applied in biology and medicine, ranging from formalized mathematical theory through experimental science and technological development to practical clinical applications.


Communications Letters, IEEE

Covers topics in the scope of IEEE Transactions on Communications but in the form of very brief publication (maximum of 6column lengths, including all diagrams and tables.)


Communications, IEEE Transactions on

Telephone, telegraphy, facsimile, and point-to-point television, by electromagnetic propagation, including radio; wire; aerial, underground, coaxial, and submarine cables; waveguides, communication satellites, and lasers; in marine, aeronautical, space and fixed station services; repeaters, radio relaying, signal storage, and regeneration; telecommunication error detection and correction; multiplexing and carrier techniques; communication switching systems; data communications; and communication theory. In addition to the above, ...


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Most published Xplore authors for Covariance matrix

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Xplore Articles related to Covariance matrix

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On the relationship between identification and local tests

B. Delyon; A. Benveniste Proceedings of the 36th IEEE Conference on Decision and Control, 1997

Convergence rates and related central limit theorems have been the subject of numerous papers. In Benveniste et al. (1987) a systematic link was first established between system identification, and model validation or testing for small changes. Similarities and relations are discussed in Benveniste et al. (1990), and this so-called local approach has proved very successful in practical applications. In this ...


A more direct proof of the minimum variance property of the linear weighted least-squares estimator

B. Fang IEEE Transactions on Automatic Control, 1969

A simple and direct proof of the minimum variance property of the linear weighted least-squares estimator is presented.


Beamspace virtual-ESPRIT

E. Gonen; J. M. Mendel Conference Record of The Twenty-Ninth Asilomar Conference on Signals, Systems and Computers, 1995

A beamspace direction finding and waveform recovery method, beamspace virtual- ESPRIT algorithm (beamspace-VESPA) is presented which is applicable to arbitrary and unknown arrays provided the array contains an identical response sensor pair. The proposed method works with arbitrary beamspace transformations. An experiment demonstrating the method is provided.


Blind adaptive multiuser detection with averaging for cellular systems

D. Das; M. K. Varunasi 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060), 2000

We consider blind adaptive multiuser detection in correlated waveform multiple access (CWMA)-based cellular radio networks. A common stochastic approximation (SA) based framework is proposed from which three blind adaptive algorithms for linear MMSE detection are obtained. Two of them coincide with previously proposed algorithms and the third is shown to be best suited for implementation at a base station. Improvement ...


Bias and resolution of the vector space methods in the presence of coherent planewaves

V. Shahmirian; S. Kesler ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing, 1987

The high resolution vector space methods of angle-of-arrival estimation which are based on the eigendecomposition of the covariance matrix of the received signal are known to break down when the incoming signals are fully correlated. In order to alleviate this problem, the covariance matrix is preprocessed using spatial smoothing averaging. An alternative method which is based on Singular Value Decomposition ...


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Educational Resources on Covariance matrix

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eLearning

On the relationship between identification and local tests

B. Delyon; A. Benveniste Proceedings of the 36th IEEE Conference on Decision and Control, 1997

Convergence rates and related central limit theorems have been the subject of numerous papers. In Benveniste et al. (1987) a systematic link was first established between system identification, and model validation or testing for small changes. Similarities and relations are discussed in Benveniste et al. (1990), and this so-called local approach has proved very successful in practical applications. In this ...


A more direct proof of the minimum variance property of the linear weighted least-squares estimator

B. Fang IEEE Transactions on Automatic Control, 1969

A simple and direct proof of the minimum variance property of the linear weighted least-squares estimator is presented.


Beamspace virtual-ESPRIT

E. Gonen; J. M. Mendel Conference Record of The Twenty-Ninth Asilomar Conference on Signals, Systems and Computers, 1995

A beamspace direction finding and waveform recovery method, beamspace virtual- ESPRIT algorithm (beamspace-VESPA) is presented which is applicable to arbitrary and unknown arrays provided the array contains an identical response sensor pair. The proposed method works with arbitrary beamspace transformations. An experiment demonstrating the method is provided.


Blind adaptive multiuser detection with averaging for cellular systems

D. Das; M. K. Varunasi 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060), 2000

We consider blind adaptive multiuser detection in correlated waveform multiple access (CWMA)-based cellular radio networks. A common stochastic approximation (SA) based framework is proposed from which three blind adaptive algorithms for linear MMSE detection are obtained. Two of them coincide with previously proposed algorithms and the third is shown to be best suited for implementation at a base station. Improvement ...


Bias and resolution of the vector space methods in the presence of coherent planewaves

V. Shahmirian; S. Kesler ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing, 1987

The high resolution vector space methods of angle-of-arrival estimation which are based on the eigendecomposition of the covariance matrix of the received signal are known to break down when the incoming signals are fully correlated. In order to alleviate this problem, the covariance matrix is preprocessed using spatial smoothing averaging. An alternative method which is based on Singular Value Decomposition ...


More eLearning Resources

IEEE-USA E-Books

  • Covariances in Computer Vision and Machine Learning

    <p>Covariance matrices play important roles in many areas of mathematics, statistics, and machine learning, as well as their applications. In computer vision and image processing, they give rise to a powerful data representation, namely the covariance descriptor, with numerous practical applications.</p> <p>In this book, we begin by presenting an overview of the {it finite- dimensional covariance matrix} representation approach of images, along with its statistical interpretation. In particular, we discuss the various distances and divergences that arise from the intrinsic geometrical structures of the set of Symmetric Positive Definite (SPD) matrices, namely Riemannian manifold and convex cone structures. Computationally, we focus on kernel methods on covariance matrices, especially using the Log-Euclidean distance.</p> <p>We then show some of the latest developments in the generalization of the finite-dimensional covariance matrix representati n to the {it infinite-dimensional covariance operator} representation via positive definite kernels. We present the generalization of the affine-invariant Riemannian metric and the Log-Hilbert-Schmidt metric, which generalizes the Log Euclidean distance. Computationally, we focus on kernel methods on covariance operators, especially using the Log-Hilbert-Schmidt distance. Specifically, we present a two-layer kernel machine, using the Log-Hilbert- Schmidt distance and its finite-dimensional approximation, which reduces the computational complexity of the exact formulation while largely preserving its capability. Theoretical analysis shows that, mathematically, the approximate Log-Hilbert-Schmidt distance should be preferred over the approximate Log- Hilbert-Schmidt inner product and, computationally, it should be preferred over the approximate affine-invariant Riemannian distance.</p> <p>Numerical experiments on image classification demonstrate significant improvements o the infinite-dimensional formulation over the finite-dimensional counterpart. Given the numerous applications of covariance matrices in many areas of mathematics, statistics, and machine learning, just to name a few, we expect that the infinite-dimensional covariance operator formulation presented here will have many more applications beyond those in computer vision.</p>

  • Vector Radiative Transfer

    Chapter 2 introduces the vector radiative transfer (VRT) theory of random media. It provides the scattering, absorption and extinction coefficients and the phase matrix of non-spherical scatterers in natural media. The first-order Mueller matrix solution of VRT for vegetation canopy model is derived. Polarization indexes, eigen-analysis and entropy are presented. Statistics of multi-look SAR images and covariance matrix are also investigated.

  • Analysis of Contour Motions

    A reliable motion estimation algorithm must function under a wide range of conditions. One regime, which we consider here, is the case of moving objects with contours but no visible texture. Tracking distinctive features such as corners can disambiguate the motion of contours, but spurious features such as T-junctions can be badly misleading. It is difficult to determine the reliability of motion from local measurements, since a full rank covariance matrix can result from both real and spurious features. We propose a novel approach that avoids these points altogether, and derives global motion estimates by utilizing information from three levels of contour analysis: edgelets, boundary fragments and contours. Boundary fragment are chains of orientated edgelets, for which we derive motion estimates from local evidence. The uncertainties of the local estimates are disambiguated after the boundary fragments are properly grouped into contours. The grouping is done by constructing a graphical model and marginalizing it using importance sampling. We propose two equivalent representations in this graphical model, reversible switch variables attached to the ends of fragments and fragment chains, to capture both local and global statistics of boundaries. Our system is successfully applied to both synthetic and real video sequences containing high-contrast boundaries and textureless regions. The system produces good motion estimates along with properly grouped and completed contours.

  • Evaluating Multisensor Classification Performance with Bayesian Networks

    This chapter presents a new analytical approach for quantifying the long¿¿¿run performance of a multisensor, discrete¿¿¿state classification system, under the assumption of independent, asynchronous measurements, and consolidates earlier research. It addresses the problem of evaluating the classification performance of both single¿¿¿ and multisensor classification systems, where one or more sensors observe repeated measurements of a target's features/attributes and compute posterior probability estimates to aid in target identification. The chapter develops a new analytical approach for off¿¿¿line evaluation of the long¿¿¿run classification performance of a single¿¿¿ or a multisensor system for the case of independent sensor measurement. It defines a new approach to quantify the classification performance based on the global classification matrix (GCM) of a classification system. The GCM is analogous to the covariance matrix used in kinematic performance evaluation and is an off¿¿¿line measure that does not depend on the actual sensor measurements.

  • Covariance, Subspace, and Intrinsic CramrRao Bounds

    Cram¿¿¿r-Rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists. The frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail. The set of subspaces, called the Grassmann manifold, and the set of covariance (positive-definite Hermitian) matrices have no fixed coordinate system associated with them and do not possess a vector space structure, both of which are required for deriving classical Cram¿¿¿r-Rao bounds. Intrinsic versions of the Cram¿¿¿r-Rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators. In the example of covariance matrix estimation, closed-form expressions for both the intrinsic and flat bounds are derived and compared with the root-mean-square error (RMSE) of the sample covariance matrix (SCM) estimator for varying sample support _K_. The accuracy bound on unbiased covariance matrix estimators is shown to be about (10/log 10)n/_K_ 1/2 dB, where n is the matrix order. Remarkably, it is shown that from an intrinsic perspective, _the SCM is a biased and inefficient estimator_ and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice. The RMSE of the standard method of estimating subspaces using the singular value decomposition (SVD)is compared with the intrinsic subspace Cram¿¿¿r-Rao bound derived in closed form by varying both the signal-to-noise ratio (SNR) of the unknown _p_-dimensional subspace and the sample support. In the simplest case, the Cram¿¿¿r-Rao bound on subspace estimation accuracy is shown to be about (_p_(_n_ - _p_)1/2 _K_-1/2SN-1/2 rad for _p_-dimensional subspace s. It is seen that the SVD-based method yields accuracies very close to the Cram¿¿¿r-Rao bound, establishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace. The analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere, such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation.

  • Some Probability and Stochastic Convergence Fundamentals

    This chapter contains sections titled: * Notations and Definitions * The Covariance Matrix of a Function of a Random Variable * Sample Variables * Mixing Random Variables * Preliminary Example * Definitions of Stochastic Limits * Interrelations between Stochastic Limits * Properties of Stochastic Limits * Laws of Large Numbers * Central Limit Theorems * Properties of Estimators * Cram¿¿r-Rao Lower Bound * How to Prove Asymptotic Properties of Estimators? * Pitfalls * Preliminary Example¿¿-¿¿Continued * Properties of the Noise after a Discrete Fourier Transform * Exercises * Appendixes

  • MeanSquared Error and Threshold SNR Prediction of MaximumLikelihood Signal Parameter Estimation With Estimated Colored Noise Covariances

    An interval error-based method (MIE) of predicting mean squared error (MSE) performance of maximum-likelihood estimators (MLEs) is extended to the case of signal parameter estimation requiring intermediate estimation of an unknown colored noise covariance matrix; an intermediate step central to adaptive array detection and parameter estimation. The successful application of MIE requires good approximations of two quantities: 1) interval error probabilities and 2) asymptotic (SNR ?> ?> local MSE performance of the MLE. Exact general expressions for the pairwise error probabilities that include the effects of signal model mismatch are derived herein, that in conjunction with the Union Bound provide accurate prediction of the required interval error probabilities. The Cram¿¿r-Ran Bound (CRB) often provides adequate prediction of the asymptotic local MSE performance of MLE. The signal parameters, however, are decoupled from the colored noise parameters in the Fisher Information Matrix for the deterministic signal model, rendering the CRB incapable of reflecting loss due to colored noise covariance estimation. A new modification of the CRB involving a complex central beta random variable different from, but analogous to the Reed, Mallett, and Brennan beta loss factor provides a working solution to this problem, facilitating MSE prediction well into the threshold region with remarkable accuracy.

  • Normal Equations

    This chapter contains sections titled: * Mean-Square Error Criterion * Minimization by Differentiation * Minimization by Completion of Squares * Minimization of the Error Covariance Matrix * Optimal Linear Estimator

  • Estimation with Unknown Noise Model - Standard Solutions

    This chapter contains sections titled: * Introduction * Discussion of the Disturbing Noise Assumptions * Properties of the ML Estimator Using a Sample Covariance Matrix * Properties of the GTLS Estimator Using a Sample Covariance Matrix * Properties of the BTLS Estimator Using a Sample Covariance Matrix * Properties of the SUB Estimator Using a Sample Covariance Matrix * Identification in the Presence of Nonlinear Distortions * Illustration and Overview of the Properties * Identification of Parametric Noise Models * Identification in Feedback * Appendixes

  • The CramrRao Estimation Error Lower Bound Computation for Deterministic Nonlinear Systems

    For condouous-time nonlinear deterministie system models with diserete nonlinear measuremeuts in additive Gaussian white noise,the extended Kalman filter (EKF) convariance propagation equations _linearized about the true unknown trajectory_ provide the Cram¿¿r-Rao lower bound to the estimadon error covariance matrix. A useful application is establishing the optimum filter performance for a given nonlinear estimation problem by developing a simulation of the nonlinear system and an EKF linearized about the true trajectory.



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