Contours uncertainty

Consider a Nyquist contour for the nominal open-loop system Gm(iLu)C(iuj) with the model uncertainty given by equation (9.119). Let fa( ) be the bound of additive uncertainty and therefore be the radius of a disk superimposed upon the nominal Nyquist contour. This means that G(iuj) lies within a family of plants 7r(C(ja ) e tt) described by the disk, defined mathematically as... 

Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. 

The use of contour maps provides a good estimate of the uncertainty in optimized rate constants. 

Figure 6.17 shows some diagnostic plots for the optimal model resulting from the analysis of the BIC values in Figure 6.16. The left picture visualizes the result of the classification, the middle picture shows the uncertainty of the classification by symbol size, and the right picture shows contour lines for the clusters. The result of model-based clustering fits much better to the visually evident groups than the result from /t-means clustering (shown in Figure 6.9).

The uncertainty contour in Figure 12.24 is lumpy because the uncertainty depends not only on the distance from the center of the design, but also on the distance from points at which the other experiments have been carried out. For some... 

Assuming the same s] as was used to draw Figure 12.24, the uncertainty surface for the rotatable design of Equation 12.81 is shown in Figure 12.26. The uncertainty depends only on the distance from the center of the design i.e., the contours of constant uncertainty are circular about the center of the design. 

This allocation of experiments has the effect of making the normalized uncertainty and normalized information contours more axially symmetric (the design isn t quite rotatable there are still only four mirror-image planes of reflection symmetry). However, because no experiments are now being carried out at the center point, the amount of uncertainty is greater there (and the amount of information is smaller there). The overall effect is to provide a normalized information surface that looks like a slightly square-shaped volcano. 

Raw data gathered from seismic surveys must be processed to compensate for and to remove a variety of distortions unwanted noises created by weathered near-surface rocks, normal time delays, and echoing by rebounding acoustic waves—to provide the clearest possible image of the strata below. Computers can restore these distortions in a fraction of the time that was formerly required to adjust the data painstakingly by hand. Advanced techniques not only permit presentations in three dimensions, but also in color, and to create contour maps and models of subterranean features. However, even with the use of sophisticated tools, there remains a large measure of uncertainty. History has shown repeatedly that a prospective area rejected by one petroleum firm has been accepted by another and proved to be successful. 

Wyckoff et al. (62) have provided a preliminary coordinate list of all nonhydrogen atoms in RNase-S. Along with the list is a series of notations on the quality of the map and the fit of the atomic model to the electron density contours. The following comments concerning group accessibilities are based on this coordinate list, but detailed interpretations must be made with caution in view of the uncertainties in many parts of the structure. 

Fig. 1.1.1 Schematic illustration of the probability density, x(R,t) 2, associated with a chemical reaction, A + BC — AB + C. The contour lines represent the potential energy surface (see Chapter 3), and the probability density is shown at two times before the reaction where only reactants are present, and after the reaction where products as well as reactants are present. The arrows indicate the direction of motion associated with the relative motion of reactants and products. (Note that, due to the finite uncertainty in the A-B distance, Rab, there is some uncertainty in the initial relative translational energy of A + BC.)...

Because of the uncertainty concerning both the location and intensity of correlations in HMBC contour plots, cross sections should be taken through individual chemical shifts on both the H and C axes. 

H decoupling for FLOCK typically is performed as the C signal is acquired and is accomplished with the WALTZ sequence (Section 5-8). FLOCK data are presented in either the phase-sensitive or the absolute-value mode. Because of uncertainty concerning both the location and intensity of correlations in FLOCK contour plots, cross sections should be taken through individual chemical shifts on both the H and C axes, as with HMBC spectra. 

Figure A3.13.7. Continuation of the time evolution for the CH chromophore in CHF after 90 fs of irradiation (see also figure A3,13.6). Distances between the contour lines are 10, 29, 16 and 9 x 10 u pm in the order of the four images shown. The averaged energy of the wave packet corresponds to 9200 cm (roughly 6300 cm absorbed) with a quantum mechanical uncertainty of 5700 cm (from [97]).

Figure A3.13.il. Illustration of the time evolution of reduced two-dimensional probability densities I I and I I for the excitation of CHD between 50 and 70 fs (see [154] for further details). The full curve is a cut of the potential energy surface at the momentary absorbed energy corresponding to 3000 cm during the entire time interval shown here ( 6000 cm , if zero point energy is included). The dashed curves show the energy uncertainty of the time-dependent wave packet, approximately 500 cm . Left-hand side excitation along the x-axis (see figure A3.13.5). The vertical axis in the two-dimensional contour line representations is...

The contours in the above figures were calculated assuming the refractive index of the particles n = 1.7 + 0.15 /. Figure 18 shows how the contours for broaden into bands should there be uncertainty in the value assumed... 

The optimal parameter vector 0 can be found by minimizing the objective function J. Furthermore, the updated PDF allows one to quantify the uncertainty of the estimation, e.g., to calculate the standard deviation or the contours with equal probability density. 

Figure 3.17 shows the contours of the marginal PDF of Ki and A 3. It is not surprising that there is negative correlation on the estimates of these two parameters sinee the equivalent linear stiffness of the system is contributed by the combination of Ki and A 3. In this case, the uncertainty for K3 is significantly larger than that of K since the level of the excitation is mild and the nonlinear cubic term is difficult to be identified. 

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