Variance error

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

Estimations based on statistics can be made for total accuracy, precision, and reproducibility of results related to the sampling procedure being applied. Statistical error is expressed in terms of variance. Total samphng error is the sum of error variance from each step of the process. However, discussions herein will take into consideration only step (I)—mechanical extraction of samples. Mechanical-extracdion accuracy is dependent on design reflecding mechanical and statistical factors in carrying out efficient and practical collection of representative samples S from a bulk quantity B,... 

Example 1 Sample Quantity for Composition Quality Control Testing An example is sampling for quality control of a 1,000 metric ton (VFg) trainload of-Ks in (9.4 mm) nominal top-size bentonite. The specification requires silica to be determined with an accuracy of plus or minus three percent for two standard errors (s.e.). With one s.e. of 1.5 percent, V is 0.000225 (one s.e. weight fraction of 0.015 squared). The problem to be solved is thus calculating weight of sample to determine sihca with the specified error variance. 

The two-sided confidence intervals for the coefficients b and b, w hen and are random variables having t distributions with (n - 2) degrees of freedom and error variances of... 

On the other hand, MCCC considers the influence of the variation of one parameter on model output in the context of simultaneous variations of all other parameters. In this situation, is smaller than 1 in absolute value and its size depends on the relative importance of the variation of model output due to the parameter of interest and the variation of model output given by the sum total of all sources (namely, the variability in all structural parameter values plus the error variance). 

Total error variance may not be constant with conversion ... 

It has been implicitly assumed in the procedure described above that all experimental points were independent on each other and were determined with the same error variance, i.e. they had the same uncertainty. Performing replications for each observation, the error variance for each point can be calculated. The function SSre.< to be minimized can be then modified as follows ... 

A number of replications under at least one set of operating conditions must be carried out to test the model adequacy (or lack of fit of the model). An estimate of the pure error variance is then calculated from ... 

There is a plethora of model adequacy tests that the user can employ to decide whether the assumed mathematical model is indeed adequate. Generally speaking these tests are based on the comparison of the experimental error variance estimated by the model to that obtained experimentally or through other means. 

Let us first consider models that have only one measured variable (w=l). We shall consider two cases. One where we know precisely the value of the experimental error variance and the other when we have an estimate of it. Namely, there is quantifiable uncertainty in our estimate of the experimental error variance. 

In this case we assume that we know precisely the value of the standard experimental error in the measurements (of). Using Equation 11.2 we obtain an estimate of the experimental error variance under the assumption that the model is adequate. Therefore, to test whether the model is adequate we simply need to test the hypothesis... 

The error in variables method can be simplified to weighted least squares estimation if the independent variables are assumed to be known precisely or if they have a negligible error variance compared to those of the dependent variables. In practice however, the VLE behavior of the binary system dictates the choice of the pairs (T,x) or (T,P) as independent variables. In systems with a... 

In the last section of the paper, we discuss a Bayesian approach to the treatment of experimental error variances, and its first limited implementation to obtain MaxEnt distributions from a fit to noisy data. 

The calculations discussed in the previous section fit the noise-free amplitudes exactly. When the structure factor amplitudes are noisy, it is necessary to deal with the random error in the observations we want the probability distribution of random scatterers that is the most probable a posteriori, in view of the available observations and of the associated experimental error variances. 

In this section we briefly discuss an approximate formalism that allows incorporation of the experimental error variances in the constrained maximisation of the Bayesian score. The problem addressed here is the derivation of a likelihood function that not only gives the distribution of a structure factor amplitude as computed from the current structural model, but also takes into account the variance due to the experimental error. 

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

We have for now implemented a drastic simplification, whereby the likelihood function is taken equal to the error-free likelihood, but to the variance parameter Z2 appearing in the latter function the experimental error variance is added ... 

Perhaps the most challenging part of analyzing free energy errors in FEP or NEW calculations is the characterization of finite sampling systematic error (bias). The perturbation distributions / and g enable us to carry out the analysis of both the finite sampling systematic error (bias) and the statistical error (variance). 

Here, N is the number of data points (time bins), yt the measured intensity in time bin i, of the measurement error (variance) for yh xt the time position of bin i, and / the theoretical function describing the decay. 

Certain assumptions underly least squares computations such as the independence of the unobservable errors ef, a constant error variance, and lack of error in the jc s (Draper and Smith, 1998). If the model represents the data adequately, the residuals should possess characteristics that agree with these basic assumptions. The analysis of residuals is thus a way of checking that one or more of the assumptions underlying least squares optimization is not violated. For example, if the model fits well, the residuals should be randomly distributed about the value of y predicted by the model. Systematic departures from randomness indicate that the model is unsatisfactory examination of the patterns formed by the residuals can provide clues about how the model can be improved (Box and Hill, 1967 Draper and Hunter, 1967). 

Detection of changes in the error variance (usually assumed to be constant). 

Estimation of Measurement Error Variances from Process Data... 

Chen, J., Bandoni, A., and Romagnoli, J. A. (1997). Robust estimation of measurement error variance/ covariance from process sampling data. Comput. Chem. Eng. 21, 593-600. 

Darouach, M., Ragot, R., Zasadzinski, M., and Krzakala, G. (1989). Maximum likelihood estimator of measurement error variances in data reconciliation. IFAC, AIPAC Symp. 2, 135-139. 

ESTIMATION OF MEASUREMENT ERROR VARIANCES FROM PROCESS DATA... 

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