Fisher information matrix

An analytical description of the photon-economy and additive noise could be carried out by the estimation of the Fisher-information matrix of the used estimators [34],... 

Furthermore, under symplectic transformations, it is relatively easy to show, using the Hessian formula for calculating the Fisher information matrix, that the measurement covariance matrix transforms as... 

Using the notation of experimental design, F represents the extended design matrix, where the elements of its k x I row-vectors, f, are known functions of x. The matrix (FT) is the Fisher information matrix and its inverse, (FT)-1, is the dispersion matrix of the regression coefficients. 

Provided the estimator is unbiased, this expression indicates that the diagonal elements of the inverse information matrix, the Cramer-Rao bound (CRB), give the highest attainable variance of the unknown parameters. For zero-mean noise and a correct model r, no difference is expected between the values of the estimates and the true values. These minimal variances depend on the pdf of the signal, because the Fisher information matrix I is given by... 

Retout, S. Duffull, S.B. Mentre, E. Development and implementation of the population fisher information matrix... 

An analysis based upon the Fisher information matrix showed that the process model is highly sensitive to the Henry coefficients, mass transfer resistances and reaction rate. These parameters are, therefore, re-estimated online at every cycle. Figure 9.8 compares the concentration profiles collected in the recycling line with the simulated ones. The parameters were initialized with the values given in Table 9.1. 

S. Retout, F. Mentre, and R. Bruno, Fisher information matrix for non-linear mixed-effects models evaluation and application for optimal design of enoxaparin population pharmacokinetics. StatMed 21 2623-2639 (2002). 

PFIM-4 Population Fisher information matrix D-optimai design (4 sampies/subject) 0.5, 4, 50.5, and 60... 

S. Retout and E. Mentre, Eurther developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics. J Biop harm Stat 13 209-227 (2003). 

In previous work we have discussed and analyzed how Fisher Information can be used to quantify the performance of an electronic nose (Sanchez-Montands and Pearce 2001, Pearce and Sdnchez-Montanes 2003). Basically, the Fisher Information Matrix (FIM) F is a square and symmetric matrix of i x i components, where s is the number of individual compounds whose concentration we are interested to estimate. In order to calculate F we should first calculate the individual FIMs for each sensor f. 

Importantly, the Fisher Information matrix F is closely related to the discrimination ability of the system, which is why we consider it in this context. For instance, it can be demonstrated that in a two-alternative forced choice discrimination between two stimuli (i.e. the system has to determine which of two possible complex odours ci and C2 is being presented), the optimal probability of error P(e) that can be achieved using linear sensors is P(e) = 0.5-[l-erf( 0.5 )] with k = Vi- 5c - F -5c and 5c = C2-C1. 

The spatio-temporal Fisher Information Matrix was ealeulated for the miero-ehannel responding to toluene and ethanol odorants, sueh as that shown in Figure 5.3. Then the trace of the inverse of this matrix was ealeulated whieh corresponds to the optimal square error that any method estimating the individual eoncentrations eould obtain. 

Proof This will be shown by deriving the Fisher information matrix for the prediction error method. 

In the Fisher approach, the Fisher information matrix /, which is the inverse of the lower bound of the covariance matrix, is treated as a function of the design variables and usually the determinant of / (this is called D-optimal design) is maximized in order to maximize precision of parameter estimates, and thus numerical identifiability. 

ABSTRACT This paper illustrates the main features of the MUlti-STAte DEgradation Process analysis Tool (MUSTADEPT), a new software tool which allows quantitatively describing the evolution of the degradation process of an industrial equipment, modeled as a discrete-state transport process. Two different, complementary approaches are offered. One is based on statistics, specifically the Maximum Likelihood Estimation (MLE) technique to estimate the parameters of the degradation process model and the Fisher Information Matrix for evaluating the uncertainty associated to the estimates. The other approach relies on information elicited from experts and describes and propagates the associated uncertainty within the DSTE framework. In both cases, the probabilities that the component occupies the different degradation states over time are estimated with the associated uncertainties. 

The procedure implemented in MUSTADEPT to exploit the maintenance inspection records uses the MLE technique to estimate the stochastic transition parameters, and a bootstrap-based technique to estimate the Fisher Information Matrix, which allows evaluating the uncertainty in the MLE values. 

Spall, J.C. 2009. Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings. Journal... 

Implement a technique based on the Fisher Information Matrix to estimate the confidence intervals on the estimated parameters ... 

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