Errors propagation matrix

If we assume that the residuals in Equation 2.35 (e,) are normally distributed, their covariance matrix ( ,) can be related to the covariance matrix of the measured variables (COV(sy.,)= LyJ through the error propagation law. Hence, if for example we consider the case of independent measurements with a constant variance, i.e. 

Having an estimate (through the error propagation law) of the covariance matrix L, we can obtain the ML parameter estimates by minimizing the objective function,... 

In practice, the full matrix analysis is rarely applicable because of spectral overlap and because of the global error propagation. In full matrix analysis all the elements are interconnected and the error in one volume element propagates into all cross-relaxation rates. This property is not favorable in practical situations in which a part of the spectrum may be ill-defined although a good portion of the spectrum is of a satisfactory quality. Then, the more favorable analysis is localized, i.e., errors are confined within respective cross-relaxation rates. However, such analysis is possible only on data in which spin diffusion is not dominant. 

Lorber, A. (1986), Error propagation and figures of merit for quantification by solving matrix equations, Anal. Chem., 58,1167. 

Standard deviations in unit-cell parameters may be calculated analytically by error propagation. In these programs, however, the Jacobian of the transformation from Sj,. .., s6 to unit-cell parameters and volume is evaluated numerically and used to transform the variance-covariance matrix of Si,. .., s6 into the variances of the cell parameters and volume from which standard deviations are calculated. If suitable standard deviations are not obtained for certain of the unit cell parameters, it is easy to program the computer to measure additional reflections which strongly correlate with the desired parameters, and repeat the final calculations with this additional data. 

Classical, semi-classical, and quantum mechanical procedures have been developed to rationalize and predict the rates of electron transfer. In summary, the observed rate of a self-exchange reaction can be calculated as a function of interatomic distances, force constants, electronic coupling matrix element, and solvent parameters. These model parameters are either calculated, estimated, or determined by experiment, in each case with a corresponding standard deviation. Error propagation immediately demonstrates that calculated rates have error ranges of roughly two orders of magnitude, independent of the level of sophistication in the numerical procedures. 

Uncertainty can be assessed in various ways, and often a combination of procedures is necessary. In principle, uncertainty can be judged directly from measurement comparisons or indirectly from an analysis of individual error sources according to the law of error propagation ( error budget )- Measurement comparison may consist of a method comparison study with a reference method based on patient samples according to the principles outlined previously or by measurement of certified matrix reference materials (CRMs). 

Using this approach, calibration can be performed knowing only the one component of interest in the system. Interfering compounds only have to be present, not quantified. They are implicitly modeled with this approach. The major implication of this technique is that application of the sensor array in remote environments is better facilitated. A restriction that this model imposes is that the number of sensors must be less than the number of calibration samples in order to perform the generalized inverse. This method usually has more error propagation due to the instability of the R matrix inversion. Collinearity plays an important role in this case. 

An important consideration has been omitted in [3-5], which are devoted to this approach, and that is the usually rather large experimental errors associated with microarray measurements. It is important to know how such errors propagate in the calculations and their effects on the proper identification of the connectivity matrix. A simple example worked out in detail in section 12.5 shows the possible multiplicative effects of such errors in nonlinear kinetic equations. Until this problem is addressed, the approach of linearization must be viewed with caution. 

If one is interested in the standard uncertainties of quantities that are derived from refined parameters such as bond lengths and angles, rtab instructions (see Chapter 2) can be added to trigger the calculation of their values and their estimated standard uncertainties as derived by error propagation (which is based on the fuU variance-covariance matrix of the problem). For details see Example 10.3.2. 

We found that limiting steps in X so that the resulting steps in a and 3 are less than 2 was sufficient. The linear errors in calculation of the propagator matrix were removed by computing the derivative matrices at the middle of each step in X. Thus,... 

Accurate Determination of the Charge-Bond-Order Matrix and Error Propagation ... 

Prom this formula we see how the global error of a multistep method is built up. There is in every step a (local) contribution M , which is of the size of the local residual. Therefore, a main task is to control the integration in such a way that this contribution is kept small. The effect of these local residuals on the global error is influenced by n h)- The local effects can be damped or amplified depending on the properties of the propagation matrix n( )- This leads to the discussion of the stability properties of the method and its relation to the stability of the problem. 

The local sensitivity matrix S shows the effect of a unit change of parameter values on the model results. This can provide useful information on the relative influence of parameters close to their nominal values and it may also be useful to estimate how uncertainty in these parameter values can propagate to predictive uncertainty in model outputs. The normalised sensitivity matrix S shows the effect of a unit relative (e.g. 1 %) change of the parameters. If we assume that the uncertainty of the parameters is known and can be characterised by the covariance matrix then local uncertainty analysis is based on the application of the Gaussian error propagation rule ... 

The Kalman filter algorithm is initialized using an initial state estimate vector X o i and its error covariance matrix P[oi i], both assumed known. Hereafter, it propagates by computing state estimates X i +in recursively from the following equation ... 

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