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Error Propagation (error + propagation)
Selected AbstractsError propagation in multiscale approaches to the elasticity of polycrystalsPHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 12 2008Martin Friák Abstract The error propagation properties of the Voigt, Reuss, Voigt,Reuss,Hill,Gilvarry, and Hershey schemes for the determination of the integral elastic response of texture free polycrystalline aggregates with cubic structure were studied. The sensitivity of the homogenized polycrystalline shear modulus was tested (i) analytically on the partial derivatives of the shear modulus with respect to the individual elastic constants within extremal Voigt and Reuss schemes, and (ii) numerically for all four methods. The sensitivity of the Hershey shear modulus on the input parameters, the single crystalline elastic constants B, C ,, C44, is shown to be within the limiting values found for the Voigt and Reuss schemes. This conclusion is illustrated numerically on a set of five cubic materials with very different physical properties. The influence of the bulk modulus was found to be approximately two orders of magnitude smaller than that of C , and C44. The Hershey modulus was also found to be non-linear, asymmetric, and strongly dependent on the level of the elastic anisotropy of the studied system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Error propagation from prime variables into specific rates and metabolic fluxes for mammalian cells in perfusion cultureBIOTECHNOLOGY PROGRESS, Issue 4 2009Chetan T. Goudar Abstract Error propagation from prime variables into specific rates and metabolic fluxes was quantified for high-concentration CHO cell perfusion cultivation. Prime variable errors were first determined from repeated measurements and ranged from 4.8 to 12.2%. Errors in nutrient uptake and metabolite/product formation rates for 5,15% error in prime variables ranged from 8,22%. The specific growth rate, however, was characterized by higher uncertainty as 15% errors in the bioreactor and harvest cell concentration resulted in 37.8% error. Metabolic fluxes were estimated for 12 experimental conditions, each of 10 day duration, during 120-day perfusion cultivation and were used to determine error propagation from specific rates into metabolic fluxes. Errors of the greater metabolic fluxes (those related to glycolysis, lactate production, TCA cycle and oxidative phosphorylation) were similar in magnitude to those of the related greater specific rates (glucose, lactate, oxygen and CO2 rates) and were insensitive to errors of the lesser specific rates (amino acid catabolism and biosynthesis rates). Errors of the lesser metabolic fluxes (those related to amino acid metabolism), however, were extremely sensitive to errors of the greater specific rates to the extent that they were no longer representative of cellular metabolism and were much less affected by errors in the lesser specific rates. We show that the relationship between specific rate and metabolic flux error could be accurately described by normalized sensitivity coefficients, which were readily calculated once metabolic fluxes were estimated. Their ease of calculation, along with their ability to accurately describe the specific rate-metabolic flux error relationship, makes them a necessary component of metabolic flux analysis. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009 [source] Non-linearity and error in modelling soil processesEUROPEAN JOURNAL OF SOIL SCIENCE, Issue 1 2001T. M. Addiscott Summary Error in models and their inputs can be propagated to outputs. This is important for modelling soil processes because soil properties used as parameters commonly contain error in the statistical sense, that is, variation. Model error can be assessed by validation procedures, but tests are needed for the propagation of (statistical) error from input to output. Input error interacts with non-linearity in the model such that it contributes to the mean of the output as well as its error. This can lead to seriously incorrect results if input error is ignored when a non-linear model is used, as is demonstrated for the Arrhenius equation. Tests for non-linearity and error propagation are suggested. The simplest test for non-linearity is a graph of the output against the input. This can be supplemented if necessary by testing whether the mean of the output changes as the standard deviation of the input increases. The tests for error propagation examine whether error is suppressed or exaggerated as it is propagated through the model and whether changes in the error in one input influence the propagation of another. Applying these tests to a leaching model with rate and capacity parameters showed differences between the parameters, which emphasized that statements about non-linearity must be for specific inputs and outputs. In particular, simulations of mean annual concentrations of solute in drainage and concentrations on individual days differed greatly in the amount of non-linearity revealed and in the way error was propagated. This result is interpreted in terms of decoherence. [source] Geoelectric dimensionality in complex geological areas: application to the Spanish Betic ChainGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2004Anna Martí SUMMARY Rotational invariants of the magnetotelluric impedance tensor may be used to obtain information on the geometry of underlying geological structures. The set of invariants proposed by Weaver et al. (2000) allows the determination of a suitable dimensionality for the modelling of observed data. The application of the invariants to real data must take into account the errors in the data and also the fact that geoelectric structures in the Earth will not exactly fit 1-D, 2-D or simple 3-D models. In this work we propose a method to estimate the dimensionality of geoelectric structures based on the rotational invariants, bearing in mind the experimental error of real data. A data set from the Betic Chain (Spain) is considered. We compare the errors of the invariants estimated by different approaches: classical error propagation, generation of random Gaussian noise and bootstrap resampling, and we investigate the matter of the threshold value to be used in the determination of dimensionality. We conclude that the errors of the invariants can be properly estimated by classical error propagation, but the generation of random values is better to ensure stability in the errors of strike direction and distortion parameters. The use of a threshold value between 0.1 and 0.15 is recommended for real data of medium to high quality. The results for the Betic Chain show that the general behaviour is 3-D with a disposition of 2-D structures, which may be correlated with the nature of the crust of the region. [source] Sequential integrated inversion of refraction and wide-angle reflection traveltimes and gravity data for two-dimensional velocity structuresGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2000Rosaria Tondi A new algorithm is presented for the integrated 2-D inversion of seismic traveltime and gravity data. The algorithm adopts the ,maximum likelihood' regularization scheme. We construct a ,probability density function' which includes three kinds of information: information derived from gravity measurements; information derived from the seismic traveltime inversion procedure applied to the model; and information on the physical correlation among the density and the velocity parameters. We assume a linear relation between density and velocity, which can be node-dependent; that is, we can choose different relationships for different parts of the velocity,density grid. In addition, our procedure allows us to consider a covariance matrix related to the error propagation in linking density to velocity. We use seismic data to estimate starting velocity values and the position of boundary nodes. Subsequently, the sequential integrated inversion (SII) optimizes the layer velocities and densities for our models. The procedure is applicable, as an additional step, to any type of seismic tomographic inversion. We illustrate the method by comparing the velocity models recovered from a standard seismic traveltime inversion with those retrieved using our algorithm. The inversion of synthetic data calculated for a 2-D isotropic, laterally inhomogeneous model shows the stability and accuracy of this procedure, demonstrates the improvements to the recovery of true velocity anomalies, and proves that this technique can efficiently overcome some of the limitations of both gravity and seismic traveltime inversions, when they are used independently. An interpretation of field data from the 1994 Vesuvius test experiment is also presented. At depths down to 4.5 km, the model retrieved after a SII shows a more detailed structure than the model obtained from an interpretation of seismic traveltime only, and yields additional information for a further study of the area. [source] A step-wise incremented Fourier series model for chemical reactivity predictionJOURNAL OF CHEMOMETRICS, Issue 6-7 2006Saloua Saidane Abstract In this paper, chemical reactivity is modeled as a time series of events defined by a reactant's concentration decay measured at consecutive discrete time periods. Since traditional time series techniques such as ARIMA and current Artificial Neural Networks require large data sets that are typically not available for chemical reactions, we developed a Step Wise Incremented Fourier Series (SWIFS) algorithm to model and predict nonlinear short time series. The application of SWIFS to experimental data from first- and second-order reactions produced a significant improvement in prediction accuracy over traditional integrated rate laws. In forward-time prediction, SWIFS has achieved significantly higher prediction accuracy with first- and second-order chemical reactions data. SWIFS also proved more robust in terms of error propagation caused by the effect of the size of the estimation set. The proposed SWIFS model also outperformed rate law models in backwards-time prediction. The ability of SWIFS to provide high accuracy in predicting chemical reactions may have beneficial implications on the efficiency of industrial production of chemicals as well as on the effective control of hazardous materials degradation. Copyright © 2007 John Wiley & Sons, Ltd. [source] Noise propagation and error estimations in multivariate curve resolution alternating least squares using resampling methodsJOURNAL OF CHEMOMETRICS, Issue 7-8 2004Joaquim Jaumot Abstract Different approaches for the calculation of prediction intervals of estimations obtained in multivariate curve resolution using alternating least squares optimization methods are explored and compared. These methods include Monte Carlo simulations, noise addition and jackknife resampling. Obtained results allow a preliminary investigation of noise effects and error propagation on resolved profiles and on parameters estimated from them. The effect of noise on rotational ambiguities frequently found in curve resolution methods is discussed. This preliminary study is shown for the resolution of a three-component equilibrium system with overlapping concentration and spectral profiles. Copyright © 2004 John Wiley & Sons, Ltd. [source] An analysis of uncertainty in non-equilibrium and equilibrium geothermobarometryJOURNAL OF METAMORPHIC GEOLOGY, Issue 9 2004J. R. ASHWORTH Abstract In statistically optimised P,T estimation, the contributions to overall uncertainty from different sources are represented by ellipses. One source, for a diffusion-controlled reaction at non-equilibrium, is diffusion modelling of the reaction texture. This modelling is used to estimate ratios, Q, between free-energy differences, ,G, of reactions among mineral end-members, to replace the equilibrium condition ,G = 0. The associated uncertainty is compared with those already inherent in the equilibrium case (from end-member data, activity models and mineral compositions). A compact matrix formulation is introduced for activity coefficients, and their partial derivatives governing error propagation. The non-equilibrium example studied is a corona reaction with the assemblage Grt,Opx,Cpx,Pl,Qtz. Two garnet compositions are used, from opposite sides of the corona. In one of them, affected by post-reaction Fe, Mg exchange with pyroxene, the problem of reconstructing the original composition is overcome by direct use of ratios between chemical-potential differences, given by the diffusion modelling. The number of geothermobarometers in the optimisation is limited by near-degeneracies. Their weightings are affected by strong correlations among Q ratios. Uncertainty from diffusion modelling is not large in comparison with other sources. Overall precision is limited mainly by uncertainties in activity models. Hypothetical equilibrium P,T are also estimated for both garnet compositions. By this approach, departure from equilibrium can be measured, with statistical uncertainties. For the example, the result for difference from equilibrium pressure is 1.2 ± 0.7 kbar. [source] Factorized approach to nonlinear MPC using a radial basis function modelAICHE JOURNAL, Issue 2 2001Sharad Bhartiya A new computationally efficient approach for nonlinear model predictive control (NMPC) presented here uses the factorability of radial basis function (RBF) process models in a traditional model predictive control (MPC) framework. The key to the approach is to formulate the RBF process model that can make nonlinear predictions across a p-step horizon without using future unknown process measurements. The RBF model avoids error propagation from use of model predictions us input in a recursive or iterative manner. The resulting NMPC formulation using the RBF model provides analytic expressions for the gradient and Hessian of the controller's objective function in terms of RBF network parameters. Solution of the NMPC optimization problem is simplifed significantly by factorization of the RBF model output into terms containing only known and unknown parts of the process. [source] Error propagation from prime variables into specific rates and metabolic fluxes for mammalian cells in perfusion cultureBIOTECHNOLOGY PROGRESS, Issue 4 2009Chetan T. Goudar Abstract Error propagation from prime variables into specific rates and metabolic fluxes was quantified for high-concentration CHO cell perfusion cultivation. Prime variable errors were first determined from repeated measurements and ranged from 4.8 to 12.2%. Errors in nutrient uptake and metabolite/product formation rates for 5,15% error in prime variables ranged from 8,22%. The specific growth rate, however, was characterized by higher uncertainty as 15% errors in the bioreactor and harvest cell concentration resulted in 37.8% error. Metabolic fluxes were estimated for 12 experimental conditions, each of 10 day duration, during 120-day perfusion cultivation and were used to determine error propagation from specific rates into metabolic fluxes. Errors of the greater metabolic fluxes (those related to glycolysis, lactate production, TCA cycle and oxidative phosphorylation) were similar in magnitude to those of the related greater specific rates (glucose, lactate, oxygen and CO2 rates) and were insensitive to errors of the lesser specific rates (amino acid catabolism and biosynthesis rates). Errors of the lesser metabolic fluxes (those related to amino acid metabolism), however, were extremely sensitive to errors of the greater specific rates to the extent that they were no longer representative of cellular metabolism and were much less affected by errors in the lesser specific rates. We show that the relationship between specific rate and metabolic flux error could be accurately described by normalized sensitivity coefficients, which were readily calculated once metabolic fluxes were estimated. Their ease of calculation, along with their ability to accurately describe the specific rate-metabolic flux error relationship, makes them a necessary component of metabolic flux analysis. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009 [source] | |||||||||||||