Deterministic Parameter Estimation Research Articles

Optimal Power Allocation for Secure Estimation of Multiple Parameters

Optimal power allocation for secure estimation of multiple deterministic parameters is investigated under a total power constraint. The goal is to minimize the Cramer-Rao lower bound (CRLB) at an intended receiver while keeping estimation errors at an eavesdropper above specified target levels. To that end, an optimization problem is formulated by considering measurement models involving linear transformation of the parameter vector and additive Gaussian noise. Although the proposed optimization problem is nonconvex, it is decomposed into convex sub-problems by utilizing the structure of the secrecy constraints. Then, optimal solutions to the sub-problems are characterized via optimization theoretic approaches. An algorithm utilizing that characterization is developed to obtain the optimal solution of the proposed problem.

Stochastic properties of bond-slip parameters at fibre pull-out

This paper is a continuation of our work on modelling and parameter identification in fibre reinforced concrete with steel fibres. The main characteristic of this material in failure is that fibres do not break but get pulled-out of the concrete matrix and the bond between the fibre and the matrix (bond-slip law) strongly influences the behaviour in failure. We are presenting results of a series of experiments of pulling-out of a fibre from a concrete block and their comparison with Monte Carlo simulations. The model for Monte Carlo simulation is based on the fibre bundle model with stochastic and deterministic parameters that are determined from experimental results. The devised inverse procedure enables successful estimation of both stochastic and deterministic parameters simultaneously. We have demonstrated that the averaging procedure can be applied on the experimental results and the averaged data could then be used for parameter estimation. Novel application of the factor analysis on experimental results demonstrates that it is equivalent to the averaging procedure.

Fusion-Based Deterministic and Stochastic Parameters Estimation for a Lithium-Polymer Battery Model

In this study, a vehicle localization technique was employed to determine the required quantities in the identification of battery models by considering the behavior of multiple batteries instead of data from a single battery. In previous studies, a plant (e.g., a battery, motor, super-capacitor, or fuel cell) was identified based on a single piece of data. However, such an approach is disadvantageous in that it neglects the effect of process and measurement noise and assumes that the parameters obtained using data from a single plant are identical for all plants of the same type. First, deterministic parameter estimation (DPE), particle swarm optimization (PSO), and teaching-learning-based optimization (TLBO) were initially applied to estimate the battery model parameters using data from a single battery. Second, a fusion-based approach was used to address the process and measurement noise problems through an adaptive unscented Kalman filter algorithm. With this approach, maximum likelihood estimation was employed to fuse multiple-battery data streams to enable the DPE, PSO, and TLBO to recalculate the model parameters based on filtered and fused quantities. A comparison between the experimental results and model outputs obtained using the aforementioned methods for parameter estimation indicated that the proposed multiple-battery approach enhances the accuracy of several identification methods. In contrast, it requires a high computational effort.

Optimized Transmission for Parameter Estimation in Wireless Sensor Networks

A central problem in analog wireless sensor networks is to design the gain or phase-shifts of the sensor nodes (i.e. the relaying configuration) in order to achieve an accurate estimation of some parameter of interest at a fusion center, or more generally, at each node by employing a distributed parameter estimation scheme. In this paper, by using an over-parametrization of the original design problem, we devise a cyclic optimization approach that can handle tuning both gains and phase-shifts of the sensor nodes, even in intricate scenarios involving sensor selection or discrete phase-shifts. Each iteration of the proposed design framework consists of a combination of the Gram-Schmidt process and power method-like iterations, and as a result, enjoys a low computational cost. Along with formulating the design problem for a fusion center, we further present a consensus-based framework for decentralized estimation of deterministic parameters in a distributed network, which results in a similar sensor gain design problem. The numerical results confirm the computational advantage of the suggested approach in comparison with the state-of-the-art methods—an advantage that becomes more pronounced when the sensor network grows large.

Constructing a Statistical Model of Displacement Magnifying Structure Based on Modeling Validation

Numerical simulation results for the optimum design are often different from the experimental results because of simulation errors resulting from the idealization and modeling errors. Less accurate simulation will make the optimization lower confidence. On the natural frequency test of displacement magnifying structure (DMS) which we have obtained by the optimum design, the experimental result had a large gap from the simulation results. In this study, the statistical model is constructed to quantify the gap that is considered as uncertainties. Full factorial numerical integration method that is one of the techniques of uncertainty propagation is adopted to construct the statistical model. The statistical model is used to quantify the effect of the uncertainty on the structural performance such as the reliability concerning the natural frequency of DMS. That differs from the deterministic model parameter estimation method such as finite element updating, that is difficult to quantify the effect of the uncertainties. The proposed method is adopted to construct the statistical model of DMS.

The Risk-Unbiased Cramér–Rao Bound for Non-Bayesian Multivariate Parameter Estimation

How accurately can one estimate a deterministic parameter subject to other unknown deterministic model nuisance parameters? The most popular answer to this question is given by the Cramer–Rao bound (CRB). The main assumption behind the derivation of the CRB is local unbiased estimation of all model parameters. The foundations of this paper rely on doubting this assumption. Generally, in multivariate parameter estimation, each parameter in its turn can be treated as a single parameter of interest, whereas the other model parameters are treated as nuisance, as their misknowledge interferes with the estimation of the parameter of interest. This approach is utilized in this paper to provide a fresh look at deterministic parameter estimation. A new Cramer–Rao (CR) type bound is derived without assuming unbiased estimation of the nuisance parameters. Rather than that, we apply Lehmann's concept of unbiasedness for a risk that measures the distance between the estimator and the locally best unbiased estimator, which assumes perfect knowledge of the nuisance parameters. The proposed risk-unbiased CRB (RUCRB) is proven to be asymptotically attainable by the maximum likelihood estimator while being tighter than the conventional CRB. Furthermore, simulations verify the asymptotic achievability of the RUCRB by the maximum likelihood estimator for an array processing problem.

Pharmacodynamic modeling of propofol-induced general anesthesia in young adults.

Target controlled infusion (TCI) of intraveneous anesthetics can assist clinical practitioners to provide improved care for General Anesthesia (GA). Pharmacokinetic/Pharmacodynamic (PK/PD) models help in relating the anesthetic drug infusion to observed brain activity inferred from electroencephalogram (EEG) signals. The parameters in popular population PK/PD models for propofol-induced GA (Marsh and Schnider models) are either verified based on proprietary functions of the EEG signal which are difficult to correlate with the neurophysiological models of anesthesia, or the marker itself needs to be estimated simultaneously with the PD model. Both these factors make these existing paradigms challenging to apply in real-time context where a patient-specific tuning of parameters is desired. In this work, we propose a simpler EEG marker from frequency domain description of EEG and develop two corresponding PK/PD modeling approaches which differ in whether they use existing population-level PK models (approach 1) or not (approach 2). We use a simple deterministic parameter estimation approach to identify the unknown PK/PD model parameters from an existing human EEG data-set. We infer that both approaches 1 and 2 yield similar and reasonably good fits to the marker data. This work can be useful in developing patient-specific TCI strategies to induce GA.

On Lower Bounds for Nonstandard Deterministic Estimation

We consider deterministic parameter estimation and the situation where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. Unfortunately, in the general case, this marginalization is mathematically intractable, which prevents from using the known standard deterministic lower bounds (LBs) on the mean squared error (MSE). Actually the general case can be tackled by embedding the initial observation space in a hybrid one where any standard LB can be transformed into a modified one fitted to nonstandard deterministic estimation, at the expense of tightness however. Furthermore, these modified LBs (MLBs) appears to include the submatrix of hybrid LBs which is an LB for the deterministic parameters. Moreover, since in the nonstandard estimation, maximum likelihood estimators (MLEs) can be no longer derived, suboptimal nonstandard MLEs (NSMLEs) are proposed as being a substitute. We show that any standard LB on the MSE of MLEs has a nonstandard version lower bounding the MSE of NSMLEs. We provide an analysis of the relative performance of the NSMLEs, as well as a comparison with the MLBs for a large class of estimation problems. Last, the general approach introduced is exemplified, among other things, with a new look at the well-known Gaussian complex observation models.

Dual substrate limitation modeling and implications for mainstream deammonification

Substrate limitation occurs frequently in wastewater treatment and knowledge about microbial behavior at limiting conditions is essential for the use of biokinetic models in system design and optimization. Monod kinetics are well-accepted for modeling growth rates when a single substrate is limiting, but several models exist for treating two or more limiting substrates simultaneously. In this study three dual limitation models (multiplicative, minimum, and Bertolazzi) were compared based on experiments using nitrite-oxidizing bacteria (limited by dissolved oxygen and nitrite) and ANaerobic AMMonia-OXidizing bacteria or Aanammox (limited by ammonium and nitrite) within mixed liquor from deammonification pilots. A deterministic likelihood-based parameter estimation followed by Bayesian inference was used to estimate model-specific parameters. The minimum model outperformed the other two by a slight margin in three separate analyses. 1) Parameters estimated using the minimum model were closest to parameters estimated from single limitation batch tests. 2) Among simulations based on each model's own estimated parameters, the minimum model best described the experimental observations. 3) Among simulations based on parameters estimated from single limitation, the minimum model best described the experimental observations. The dual substrate model selected among the three studied can effect a 75% process performance variation based on simulations of a full-scale mainstream deammonification system.

Improvements in deterministic error modeling and calibration of inertial sensors and magnetometers

We consider the deterministic modeling, calibration, and model parameter estimation of two commonly employed inertial measurement units based on real test data acquired from a flight motion simulator. Each unit comprises three tri-axial devices: an accelerometer, a gyroscope, and a magnetometer. We perform the deterministic error modeling and calibration of accelerometers based on an improved measurement model, and the technique we propose for gyroscopes lowers costs by eliminating the need for additional sensors and relaxing the test bed requirement. We present an extended measurement model for magnetometers that reduces calibration errors by modeling orientation-dependent hard-iron errors in a gimbaled angular position-control machine. While we employ the model-based Levenberg-Marquardt optimization algorithm for the parameter estimation of accelerometers and magnetometers, we use a model-free evolutionary optimization algorithm (particle swarm optimization) for estimating the calibration parameters of gyroscopes. Errors are considerably reduced as a result of proper modeling and calibration.

Improved deterministic measurement model for consumer‐grade accelerometers

Deterministic error modelling, calibration and model parameter estimation of consumer-grade accelerometers is considered and improvement to the traditionally used measurement model is proposed. Calibration experiments on a flight motion simulator are performed for experimental verification. Model parameters are estimated using the Levenberg-Marquardt optimisation algorithm. Residual errors are considerably reduced as a result of the improved measurement model.

Simulation-free estimation of reaction propensities in cellular reactions and gene signaling networks

Classical reaction kinetics based on deterministic rates laws are not valid for the description of cellular events in which the small number of molecules introduces stochasticity with discrete instead of continuous state transitions. Stochastic models are suitable for simulating transcriptional and translational events inside biological cells, but are impractical for solving inverse problems, which aim to estimate unknown reaction propensities from experimental observations. We introduce a new mathematical framework of Ito stochastic differential equations for the modeling of discrete cellular events and the robust and consistent parameter estimation of cellular dynamics where classical reaction kinetics is invalid. The results supported by case studies on gene expression in B. subtilis cells and viral gene transcription and translation inside non-lytic viral cells demonstrate that the proposed methodology performs as reliable as the gold standard Gillespie algorithm for simulating cellular events. More importantly, the new Ito process framework is ideal for estimating unknown reaction propensities from data as readily as in deterministic parameter estimation by using the novel ‘SPE – simulation free parameter estimation’ approach. Also, the computation time for the stochastic differential equation models is significantly low when compared to discrete event simulations.

On the Accuracy and Resolvability of Vector Parameter Estimates

In this paper we address the problem of fundamental limitations on resolution in deterministic parameters estimation. We introduce a definition of resolvability based on probability and incorporating a requirement for accuracy unlike most existing definitions. Indeed in many application the key problem is to obtain distributions of estimates that are not only distinguishable but also accurate and compliant with a required precision. We exemplify the proposed definition with estimators that produce normal estimates, as in the conditional model for which the Gaussianity and efficiency of maximum likelihood estimators (MLEs) in the asymptotic region of operation (in terms of signal-to-noise ratio and/or in large number of snapshots) is well established, even for a single snapshot. In order to measure the convergence in distribution, we derive a simple test allowing to check whether the conditional MLEs operate in the asymptotic region of operation. Last, we discuss the resolution of two complex exponentials with closely spaced frequencies and compare the results obtained with the ones provided by the various statistical resolution limit released in the open literature.

Statistics of the MLE and Approximate Upper and Lower Bounds—Part I: Application to TOA Estimation

In nonlinear deterministic parameter estimation, the maximum likelihood estimator (MLE) is unable to attain the Cramér-Rao lower bound at low and medium signal-to-noise ratios (SNRs) due the threshold and ambiguity phenomena. In order to evaluate the achieved mean-squared error (MSE) at those SNR levels, we propose new MSE approximations (MSEA) and an approximate upper bound by using the method of interval estimation (MIE). The mean and the distribution of the MLE are approximated as well. The MIE consists in splitting the a priori domain of the unknown parameter into intervals and computing the statistics of the estimator in each interval. Also, we derive an approximate lower bound (ALB) based on the Taylor series expansion of noise and an ALB family by employing the binary detection principle. The accuracy of the proposed MSEAs and the tightness of the derived approximate bounds are validated by considering the example of time-of-arrival estimation.

Maximum-Likelihood Parameter Estimation for Detecting Local Concentration from a Carbon Nanotube-based Sensor

This paper proposes an optimal parameter estimation method for a stochastic process involving monomolecular adsorption and desorption occurring at the nano-scale. Recently, several carbon nanotube sensors that can selectively detect target molecules at a trace concentration level have been developed. These sensors make use of light intensity changes mediated by the adsorption or desorption phenomena on their surfaces. However, the molecular-level events are inherently stochastic, posing a challenge for optimal estimation. The stochastic behavior is modeled by the chemical master equation (CME), which contains a set of ordinary differential equations describing the time evolution of probabilities for the possible adsorption states. Given the inherent stochastic nature of the underlying phenomena, rigorous stochastic estimation based on the CME should outperform deterministic parameter estimation formulated based on the continuum model. Motivated by this expectation, we formulate the maximum likelihood estimation (MLE) based on an analytical solution of the relevant CME for both the constant and time-varying parameter cases. The performance of the MLE and the deterministic least squares are compared using data generated by kinetic Monte Carlo (KMC) simulations of the stochastic process.

AN OVERVIEW OF INVERSE MATERIAL IDENTIFICATION WITHIN THE FRAMEWORKS OF DETERMINISTIC AND STOCHASTIC PARAMETER ESTIMATION

This work investigates the problem of parameter estimation within the frameworks of deterministic and stochastic parameter estimation methods. For the deterministic methods, we look at constrained and unconstrained optimization approaches. For the constrained optimization approaches we study three different formulations: L 2 , error in constitutive equation method (ECE), and the modified error in constitutive equation (MECE) method. We investigate these formulations in the context of both Tikhonov and total variation (TV) regularization. The constrained optimization approaches are compared with an unconstrained nonlinear least-squares (NLLS) approach. In the least-squares framework we investigate three different formulations: standard, MECE, and ECE. With the stochastic methods, we first investigate Bayesian calibration, where we use Monte Carlo Markov chain (MCMC) methods to calculate the posterior parameter estimates. For the Bayesian methods, we investigate the use of a standard likelihood function, a likelihood function that incorporates MECE, and a likelihood function that incorporates ECE. Furthermore, we investigate the maximum a posteriori (MAP) approach. In the MAP approach, parameters’ full posterior distribution are not generated via sampling; however, parameter point estimates are computed by searching for the values that maximize the parameters’ posterior distribution. Finally, to achieve dimension reduction in both the MCMC and NLLS approaches, we approximate the parameter field with radial basis functions (RBF). This transforms the parameter estimation problem into one of determining the governing parameters for the RBF.

General Classes of Performance Lower Bounds for Parameter Estimation—Part I: Non-Bayesian Bounds for Unbiased Estimators

In this paper, a new class of lower bounds on the mean square error (MSE) of unbiased estimators of deterministic parameters is proposed. Derivation of the proposed class is performed by projecting each entry of the vector of estimation error on a Hilbert subspace of L2. This Hilbert subspace contains linear transformations of elements in the domain of an integral transform of the likelihood-ratio function. The integral transform generalizes the traditional derivative and sampling operators, which are applied on the likelihood-ratio function for computation of performance lower bounds, such as Cramer-Rao, Bhattacharyya, and McAulay-Seidman bounds. It is shown that some well-known lower bounds on the MSE of unbiased estimators can be derived from this class by modifying the kernel of the integral transform. A new lower bound is derived from the proposed class using the kernel of the Fourier transform. In comparison with other existing bounds, the proposed bound is computationally manageable and provides better prediction of the threshold region of the maximum-likelihood estimator, in the problem of single tone estimation.

Clutter metric based on the Cramer-Rao lower bound on automatic target recognition

This is a performance evaluation on the implementation of the Cramer-Rao lower bound (CRLB) for background clutter measurement on automatic target recognition (ATR). In essence the background clutter evaluation problem for ATR is consistent with the deterministic parameter estimation problem. Thus, useful concepts and theories of deterministic parameter estimation can be introduced into the investigation of background clutter. In this paper, the CRLB is employed as a metric for clutter measurement. Requirements needed for this application are analyzed, and the approach for obtaining the CRLB of a scene image is produced. The flexibility of the CRLB metric is analyzed. Discussion and comparison are made on the relationship between the CRLB metric and the Sims signal-to-clutter metric. Finally, we illustrate how this metric defines the potential for false alarms by determining the correspondence level between a target and background through the application of the CRLB.

Consensus in Ad Hoc WSNs With Noisy Links—Part I: Distributed Estimation of Deterministic Signals

We deal with distributed estimation of deterministic vector parameters using ad hoc wireless sensor networks (WSNs). We cast the decentralized estimation problem as the solution of multiple constrained convex optimization subproblems. Using the method of multipliers in conjunction with a block coordinate descent approach we demonstrate how the resultant algorithm can be decomposed into a set of simpler tasks suitable for distributed implementation. Different from existing alternatives, our approach does not require the centralized estimator to be expressible in a separable closed form in terms of averages, thus allowing for decentralized computation even of nonlinear estimators, including maximum likelihood estimators (MLE) in nonlinear and non-Gaussian data models. We prove that these algorithms have guaranteed convergence to the desired estimator when the sensor links are assumed ideal. Furthermore, our decentralized algorithms exhibit resilience in the presence of receiver and/or quantization noise. In particular, we introduce a decentralized scheme for least-squares and best linear unbiased estimation (BLUE) and establish its convergence in the presence of communication noise. Our algorithms also exhibit potential for higher convergence rate with respect to existing schemes. Corroborating simulations demonstrate the merits of the novel distributed estimation algorithms.

Computational methods for stochastic modeling and parameter fitting of models based on Hodgkin-Huxley formalism

Event Abstract Back to Event Computational methods for stochastic modeling and parameter fitting of models based on Hodgkin-Huxley formalism Antti Saarinen1*, Antti Ylipää1, Olli Yli-Harja1 and Marja-Leena Linne1 1 Tampere University of Technology, , Finland Stochasticity plays a crucial role in shaping the dynamic behavior of a neuron. Therefore, it is important to take this stochasticity into account when modeling, for example, the electroresponsiveness of a neuron. In this work, we demonstrate the use of stochastic differential equations (SDEs) in neuronal modeling. We introduce an SDE model of a single cerebellar granule cell (Saarinen et al. PLoS Computational Biology, 2008) where the description of the ion channel kinetics incorporate the natural stochasticity observed in the dynamic behavior of these channels. We present also ways of incorporating stochasticity to the well-known Hodgkin-Huxley (HH) model of action potential generation in a squid axon and a methodology for fitting these stochastic models to irregular current-clamp data. These data contain subthreshold oscillations, spontaneous action potentials, and clusters of action potentials, among other stochastic features. This kind of fitting and use of irregular data have not been possible with the existing deterministic models and deterministic parameter estimation techniques. We use a Bayesian estimation technique which relies on transforming the probability distributions of the estimation problem into distributions which are easy to sample. This transformation allows us to use Sequential Monte Carlo (SMC) techniques to draw samples from the desired posterior distributions. Based on these samples, a Maximum Likelihood (ML) estimation technique in utilized to produce ML estimates for the selected model parameters; these include maximal conductances of ionic currents and the intensity of random fluctuations in the current-clamp data. We show that we are able to obtain accurate ML estimates for the selected model parameters based on the learning data. These data have been corrupted with different levels of measurement noise. The approximation of the likelihood function allows us to study also the sensitivity of the model parameters and effects of the changes in their values to the model behavior. The sharper the peak is in the likelihood, around the correct parameter value, the more sensitive is the model behavior with respect to value of that parameter. In conclusion, the use of SDEs in neuronal modeling enables more accurate reproduction of the irregular electrophysiological activity of a neuron. The presented method offers also an attractive way to perform parameter estimation in situations where it has not been possible with the deterministic models and methods. Our approach provides a comprehensive set of analytical tools to analyze model behavior and the estimation problem, since the theory of stochastic calculus has been widely studied in the field of mathematics. However, our estimation method is, in its present form, rather computationally heavy. Therefore, we are studying the possibilities of distributed computing, including GRID technology, in our studies. In the future, other methods of approximating the likelihood and speeding up the computations will be considered. In spite of the heavy computing, SMC methods offer a set of methods which are very flexible, relatively easy to implement, parallelizable, and applicable in very general settings. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Saarinen A, Ylipää A, Yli-Harja O and Linne M (2008). Computational methods for stochastic modeling and parameter fitting of models based on Hodgkin-Huxley formalism. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.032 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Antti Saarinen, Tampere University of Technology,, Tampere, Finland, antti.saarinen@cs.tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Antti Saarinen Antti Ylipää Olli Yli-Harja Marja-Leena Linne Google Antti Saarinen Antti Ylipää Olli Yli-Harja Marja-Leena Linne Google Scholar Antti Saarinen Antti Ylipää Olli Yli-Harja Marja-Leena Linne PubMed Antti Saarinen Antti Ylipää Olli Yli-Harja Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.