376 research outputs found
A Constrained Hybrid Cramér-Rao Bound for Parameter Estimation
In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. Numerous works have shown the versatility of deterministic constrained CrameÌr-Rao bound for estimation performance analysis and design of a system of measurement. However in many systems both random and non-random parameters may occur simultaneously. In this communication, we propose a constrained hybrid lower bound which take into account of equality constraint on deterministic parameters. The usefulness of the proposed bound is illustrated with an application to radar Doppler estimation
BornĂ© de CramĂ©r-Rao sous contraintes pour lâestimation simultanĂ©e de paramĂštres alĂ©atoires et non alĂ©atoires
In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both
non-random and random parameters. On the other hand, numerous works have shown the versatility of deterministic constrained Cramér-Rao
bound for estimation performance analysis and design of a system of measurement. In this communication, we propose a constrained hybrid
lower bound which takes into account equality constraints on deterministic parameters. The proposed bound is then compared to previous bounds
of the literature. Finally, the usefulness of the proposed bound is illustrated with an application to radar Doppler estimation
Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications
Inferring information from a set of acquired data is the main objective of
any signal processing (SP) method. In particular, the common problem of
estimating the value of a vector of parameters from a set of noisy measurements
is at the core of a plethora of scientific and technological advances in the
last decades; for example, wireless communications, radar and sonar,
biomedicine, image processing, and seismology, just to name a few. Developing
an estimation algorithm often begins by assuming a statistical model for the
measured data, i.e. a probability density function (pdf) which if correct,
fully characterizes the behaviour of the collected data/measurements.
Experience with real data, however, often exposes the limitations of any
assumed data model since modelling errors at some level are always present.
Consequently, the true data model and the model assumed to derive the
estimation algorithm could differ. When this happens, the model is said to be
mismatched or misspecified. Therefore, understanding the possible performance
loss or regret that an estimation algorithm could experience under model
misspecification is of crucial importance for any SP practitioner. Further,
understanding the limits on the performance of any estimator subject to model
misspecification is of practical interest. Motivated by the widespread and
practical need to assess the performance of a mismatched estimator, the goal of
this paper is to help to bring attention to the main theoretical findings on
estimation theory, and in particular on lower bounds under model
misspecification, that have been published in the statistical and econometrical
literature in the last fifty years. Secondly, some applications are discussed
to illustrate the broad range of areas and problems to which this framework
extends, and consequently the numerous opportunities available for SP
researchers.Comment: To appear in the IEEE Signal Processing Magazin
Optimized Quantification of Spin Relaxation Times in the Hybrid State
Purpose: The analysis of optimized spin ensemble trajectories for relaxometry
in the hybrid state.
Methods: First, we constructed visual representations to elucidate the
differential equation that governs spin dynamics in hybrid state. Subsequently,
numerical optimizations were performed to find spin ensemble trajectories that
minimize the Cram\'er-Rao bound for -encoding, -encoding, and their
weighted sum, respectively, followed by a comparison of the Cram\'er-Rao bounds
obtained with our optimized spin-trajectories, as well as Look-Locker and
multi-spin-echo methods. Finally, we experimentally tested our optimized spin
trajectories with in vivo scans of the human brain.
Results: After a nonrecurring inversion segment on the southern hemisphere of
the Bloch sphere, all optimized spin trajectories pursue repetitive loops on
the northern half of the sphere in which the beginning of the first and the end
of the last loop deviate from the others. The numerical results obtained in
this work align well with intuitive insights gleaned directly from the
governing equation. Our results suggest that hybrid-state sequences outperform
traditional methods. Moreover, hybrid-state sequences that balance - and
-encoding still result in near optimal signal-to-noise efficiency. Thus,
the second parameter can be encoded at virtually no extra cost.
Conclusion: We provide insights regarding the optimal encoding processes of
spin relaxation times in order to guide the design of robust and efficient
pulse sequences. We find that joint acquisitions of and in the
hybrid state are substantially more efficient than sequential encoding
techniques.Comment: 10 pages, 5 figure
A Fresh Look at the Bayesian Bounds of the Weiss-Weinstein Family
International audienceMinimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the WeissâWeinstein family. Among this family, we have Bayesian CramĂ©r-Rao bound, the BobrovskyâMayerWolfâZakaĂŻ bound, the Bayesian Bhattacharyya bound, the BobrovskyâZakaĂŻ bound, the ReuvenâMesser bound, and the WeissâWeinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, MayerâWolf, and ZakaĂŻ. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the ReuvenâMesser bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the ReuvenâMesser bound, the BobrovskyâZakaĂŻ bound, and the Bayesian CramĂ©râRao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem
Cram\'er-Rao bound-informed training of neural networks for quantitative MRI
Neural networks are increasingly used to estimate parameters in quantitative
MRI, in particular in magnetic resonance fingerprinting. Their advantages over
the gold standard non-linear least square fitting are their superior speed and
their immunity to the non-convexity of many fitting problems. We find, however,
that in heterogeneous parameter spaces, i.e. in spaces in which the variance of
the estimated parameters varies considerably, good performance is hard to
achieve and requires arduous tweaking of the loss function, hyper parameters,
and the distribution of the training data in parameter space. Here, we address
these issues with a theoretically well-founded loss function: the Cram\'er-Rao
bound (CRB) provides a theoretical lower bound for the variance of an unbiased
estimator and we propose to normalize the squared error with respective CRB.
With this normalization, we balance the contributions of hard-to-estimate and
not-so-hard-to-estimate parameters and areas in parameter space, and avoid a
dominance of the former in the overall training loss. Further, the CRB-based
loss function equals one for a maximally-efficient unbiased estimator, which we
consider the ideal estimator. Hence, the proposed CRB-based loss function
provides an absolute evaluation metric. We compare a network trained with the
CRB-based loss with a network trained with the commonly used means squared
error loss and demonstrate the advantages of the former in numerical, phantom,
and in vivo experiments.Comment: Xiaoxia Zhang, Quentin Duchemin, and Kangning Liu contributed equally
to this wor
Multi-Axis Identifiability Using Single-Surface Parameter Estimation Maneuvers on the X-48B Blended Wing Body
The problem of parameter estimation on hybrid-wing-body type aircraft is complicated by the fact that many design candidates for such aircraft involve a large number of aero- dynamic control effectors that act in coplanar motion. This fact adds to the complexity already present in the parameter estimation problem for any aircraft with a closed-loop control system. Decorrelation of system inputs must be performed in order to ascertain individual surface derivatives with any sort of mathematical confidence. Non-standard control surface configurations, such as clamshell surfaces and drag-rudder modes, further complicate the modeling task. In this paper, asymmetric, single-surface maneuvers are used to excite multiple axes of aircraft motion simultaneously. Time history reconstructions of the moment coefficients computed by the solved regression models are then compared to each other in order to assess relative model accuracy. The reduced flight-test time required for inner surface parameter estimation using multi-axis methods was found to come at the cost of slightly reduced accuracy and statistical confidence for linear regression methods. Since the multi-axis maneuvers captured parameter estimates similar to both longitudinal and lateral-directional maneuvers combined, the number of test points required for the inner, aileron-like surfaces could in theory have been reduced by 50%. While trends were similar, however, individual parameters as estimated by a multi-axis model were typically different by an average absolute difference of roughly 15-20%, with decreased statistical significance, than those estimated by a single-axis model. The multi-axis model exhibited an increase in overall fit error of roughly 1-5% for the linear regression estimates with respect to the single-axis model, when applied to flight data designed for each, respectively
Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond
Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form MarkovâBayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed âGaussian conjugacyâ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity
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