Adaptive variance estimator

Consider now robustness. If the estimators A are computed from independent response variables then, as noted in Section 1, the estimators have equal variances and are usually at least approximately normal. Thus the usual assumptions, that estimators are normally distributed with equal variances, are approximately valid and we say that there is inherent robustness to these assumptions. However, the notion of robust methods of analysis for orthogonal saturated designs refers to something more. When making inferences about any effect A, all of the other effects At (k i) are regarded as nuisance parameters and robust means that the inference procedures work well, even when several of the effects ft are large in absolute value. Lenth s method is robust because the pseudo standard error is based on the median absolute estimate and hence is not affected by a few large absolute effect estimates. The method would still be robust even if one used the initial estimate 6 of op, rather than the adaptive estimator 6L, for the same reason. 

Figure 4 shows performance of the proposed AUKF for open-loop system. As described, it s assumed that the induced drift in sensor noises are not known a priori to the filter and variances of measurement noises are estimated recursively by the adaptive estimation algorithm. 

Data in these studies were generated from a so-called giant rat study in our laboratory. Animals were sacrificed to obtain serial blood and tissue samples. Each point represents the measurement from one individual rat and data from all these different rats were analyzed together to obtain a time prohle as though it came from one giant rat. A naive pooled data analysis approach was therefore employed for all model fittings using ADAPT II software (21). The maximum likelihood method was used with the variance model specified as V(a, 6, h) = (j Y(d, where V a, 9, ti) is the variance for the ith point, Y 6, t,) is the ith predicted value from the dynamic model, 9 represents the estimated structural parameters, and oi and 02 are the variance parameters that were estimated. 

As discussed in Sec. 12.3.1, we constructed the class of nonstationary Gaussian processes such, that they become locally stationary for small scales (see Eq. (12.8)). Hence, when the averaging kernel A(.) is adapted to the process variability on each scale, it includes more and more reproducing kernels, such that the variance of the spectral estimate vanishes in the limit of small scales. Then, the following relation for the variance Far a (a) of the averaged wavelet spectral estimate holds ... 

The error-in-variables method was used to estimate the reactivity ratios. This method was developed by Reilly et al. (57, 58), and it was first applied for the determination of reactivity ratios by O Driscoll, Reilly, and co-workers (59, 60). In this work, a modified version by MacGregor and Sutton (61) adapted by Gloor (62) for a continuous stirred tank reactor was used. The error-in-variables method shows two important advantages compared to the other common methods for the determination of copolymer reactivity ratios, which are statistically incorrect, as for example, Fineman-Ross (63) or Kelen-Tiidos (64). First, it accounts for the errors in both dependent and independent variables the other estimation methods assume the measured values of monomer concentration and copolymer composition have no variance. Second, it computes the joint confidence region for the reactivity ratios, the area of which is proportional to the total estimation error. 

Fig. 5.25 Normalised estimates of first-order contributions to the overall variance of predicted butane mole fraction at 750 K calculated using first-order local sensitivities (grey) and the global HDMR method (black). Both are derived from a model describing the oxidation of n-butane in a jet stirred reactor (residence time of 6 s, atmospheric pressure, stoichiometric mixtures containing 4 % (mol) -butane diluted in helium). EXGAS notation is used. Adapted with permission from Cord et al. (2012). Copyright (2012) American Chemical Society...

This section discussed a variety of simulation techniques to estimate structural reliability. Topics such as Monte Carlo sampling, importance sampling, and stratified sampling were discussed in detail, and some other advanced topics such as adaptive sampling, variance... 

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