Homogeneous error structure

Assumes a homoscedastic error structure (common or homogeneous variance regardless of response). The random error is the same for all observations. 

If 0 = 0, sf is not dependent on the magnitude of the y values, and w = K for all data points. This is the case for an error that is constant throughout the data (homogeneous or constant error). Thus, if the error structure is homogeneous, weighting of the data is not required. A value... 

Figure 3.7. Mean residual analysis for the experimental data set. The pattens obtained suggest a homogeneous, or constant, error structure in the data.

These residuals will be referred to as mean residuals. It is important to realize that the criterion used to judge whether a weighted regression analysis should be carried out is the error structure of the experimental data, not the error structure of the fit of the model to the data. The mean-residuals plot depicted in Fig. 3.7 suggests that the error stmcture of the data is homogeneous, or constant. This being the case, weighting is not necessary. A more quantitative analysis of the error structure of... 

Fig. 38. Substrate dependence of the PS/PMMA domain structure spin-cast from THF a-c on SiOx d-f on octadecyl mercaptane (ODM) [352]. The SFM images have lateral dimensions of 14x14 pm2. a,d as spin cast b after immersion in cyclohexane to remove PS. e after immersion in acetic acid to remove the PMMA-rich phase. The cross-sections (c,f), which were recorded along the lines in (a,b,c,d), reveal the vertical distribution of the PS (dark grey) and PMMA (light grey) phases. The error bar in (c) indicates the accuracy of the superposition procedure. PMMA preferentially adsorbs on the more polar SiOx surface to form a homogeneous layer next to the substrate. On the ODM a PS/PMMA bilayer is observed. Courtesy of U. Steiner...

Besides the hypothesis of spatially homogeneous processes in this stochastic formulation, the particle model introduces a structural heterogeneity in the media through the scarcity of particles when their number is low. In fact, the number of differential equations in the stochastic formulation for the state probability keeps track of all of the particles in the system, and therefore it accounts for the particle scarcity. The presence of several differential equations in the stochastic formulation is at the origin of the uncertainty, or stochastic error, in the process. The deterministic version of the model is unable to deal with the stochastic error, but as stated in Section 9.3.4, that is reduced to zero when the number of particles is very large. Only in this last case can the set of Kolmogorov differential equations be adequately approximated by the deterministic formulation, involving a set of differential equations of fixed size for the states of the process. 

E-fS ES ES. The standard deviation of the distribution, (Atopen ) = 8.3 2ms, reflects the distribution bandwidth. For the individual T4 lysozyme molecules examined under the same enz unatic reaction conditions, we found that the first and second moments of the single-molecule topen distributions are homogeneous, within the error bars. The hinge-bending motion allows sufficient structural flexibility for the enzyme to optimize its domain conformation the donor fluorescence essentially reaches the same intensity in each turnover, reflecting the domain conformation reoccurrence. The distribution with a defined first moment and second moment shows typical oscillatory conformational motions. The nonequilibrium conformational motions in forming the active enzymatic reaction intermediate states intrinsically define a recurrence of the essentially similar potential surface for the enzymatic reaction to occur, which represents a memory effect in the enzymatic reaction conformational dynamics [12,41,42]. 

In order to meet the ever-increasing danands for enantiopure compounds, heterogene-ons, homogeneous and enzymatic catalysis evolved independently in the past. Although all three approaches have yielded industrially viable processes, the latter two are the most widely used and can be regarded as complanentary in many respects. Despite the progress in structural, computational and mechanistic smdies, however, to date there is no universal recipe for the optimization of catalytic processes. Thus, a trial-and-error approach remains predominant in catalyst discovery and optimization. 

---