Probabilistic considerations

This leads us to the other hand, which, it should be obvious, is that we feel that Chemometrics should be considered a subfield of Statistics, for the reasons given above. Questions currently plaguing us, such as How many MLR/PCA/PLS factors should I use in my model , Can I transfer my calibration model (or more importantly and fundamentally How can I tell if I can transfer my calibration model ), may never be answered in a completely rigorous and satisfactory fashion, but certainly improvements in the current state of knowledge should be attainable, with attendant improvements in the answers to such questions. New questions may arise which only fundamental statistical/probabilistic considerations may answer one that has recently come to our attention is, What is the best way to create a qualitative (i.e., identification) model, if there may be errors in the classifications of the samples used for training the algorithm ... 

To put equation 44-6 into a usable form under the conditions we wish to consider, we could start from any of several points of view the statistical approach of Hald (see [10], pp. 115-118), for example, which starts from fundamental probabilistic considerations and also derives confidence intervals (albeit for various special cases only) the mathematical approach (e.g., [11], pp. 550-554) or the Propagation of Uncertainties approach of Ingle and Crouch ([12], p. 548). In as much as any of these starting points will arrive at the same result when done properly, the choice of how to attack an equation such as equation 44-6 is a matter of familiarity, simplicity and to some extent, taste. 

The kernels of these integral equations, which are derived from simple probabilistic considerations, represent up to the factor 1 the product of two factors. The first of them, wa(r]), is equal to the fraction of a-th type blocks, whose lengths exceed rj. The second one, Vap(rj), is the rate with which an active center located on the end of a growing block of monomeric units M with length r) switches from a-th type to /i-lh type under the transition of this center from phase a into phase /3. The right-hand side of Eq. 74 comprises items equal to the product of the rate of initiation Ia of a-th type polymer chains and the Dirac delta function <5( ). 

The logic and rigor of statistics also reside at the heart of modern science. Statistical evaluation is the first step in considering whether a body of measurements allows one to discriminate among rival models for a biochemical process. Beyond their value in experimental sciences, probabilistic considerations also help us to formulate theories about the behavior of molecules and particles and to conceptualize stochastic and chaotic events. 

The first attempts to seek a closed expression for n(t) did not include an analysis of pair correlation of defects, even at the level of pair densities, and seem at least to be ambiguous. Such an approach based on simple probabilistic considerations was first used in [24] (see also [108]). Since here we are not treating explicitly the relationship between two-particle and higher-particle densities, it is difficult to make a correct estimate of the effect of overlap of the forbiden volumes of several closely-lying defects. This leads to the need to introduce some a priori assumptions. A characteristic example is [24], where the implicit assumption of a chaotic distribution of defects through the reaction volume (along with partially taking it into account) led to a physically incorrect result - the absence of an effect of saturation of concentration doses (see Table 7.6 below and comments to it). 

In the above-mentioned paper [25, 26, 28], which assumes the existence of steady-state concentration of accumulated defects in a one-dimensional model, and which takes the distribution of A and B defects separately in the form of periodically distributed clusters (groups) of identical dimension, it was found from simple probabilistic considerations that the mean number of defects per cluster is... 

An interesting idea has been proposed where probabilistic considerations are largely taken into account [36-38]. However, none of these models has found (published) applications in the field of Ps formation. 

The observed length dependence was consistent with the probabilistic considerations of defect distribution. The probability of encountering a defect being lower in a shorter fiber, therefore, could account for its greater strength. 

Hoeg, K., and Tang, W. H. (1977), Probabilistic considerations in the foundation engineering for offshore structures. Proceedings of the 2nd International Conference on Structural Safety and Reliability, Mimich, Germany, pp. 267-296. 

The development of these stress criteria for graphite is based on probabilistic considerations and includes a damage model which equates damage (failure) with the inability of the graphite component to accommodate safe shutdown by insertion of control rods or RSC material. Based on the risk contributions from graphite damage to the overall plant risk envelope, reliability allocations for the structural components are obtained. These allocations are then expressed in terms of non-probabilistic structural requirements such as stress-to-strength limits. 

Recently there are efforts to fix the times between inspections of functionality and integrity on the basis of probabilistic considerations (cf. Chap. 9). This is known as risk-based inspection or risk-based maintenance. 

Design basis accidents (DBAs) are those accidents chosen by the deterministic method or with the help of probabilistic considerations, in order to design all the plant systems, but particularly the safety ones. Some of the following considerations are of interest for DBAs and for the other accidents. Most of the quoted data are taken from examples of typical 1000 MWe pressurized plants. 

The assignment of SSCs to safety classes should be based on national approaches and should appropriately credit deterministic and probabilistic considerations as well as engineering judgement. 

Probabilistic analysis has been a hallmark of CANDU safety philosophy since its inception Canada remains one of the few countries whose regulations are based largely on probabilistic considerations. Probabilistic safety assessments have been performed for the Darlington station by Ontario Hydro, and for CANDU 6 in a joint study by AECL and the Dutch research organization KEMA. Both studies reached a similar conclusion e.g. because of the redundant heat removal paths in CANDU, and redundant shutdown systems, the predicted severe core damage frequency is less than about 4x10-6 events/year. 

For the seismic design of SSCs, external events such as floods or fires assumed to occur at the site as a consequence of an earthquake should be taken into account. They should be defined on the basis of probabilistic considerations. These loadings as a consequence of an earthquake should be combined with either SL-1 or SL-2 loadings, with due account taken of event timing and duration. 

We have thus derived the second-order product density equation for the pure breakage process based purely on probabilistic considerations. It could also have been derived by averaging the master density equation (7.2.5), using the definition (7.1.11). The boundary conditions for the second-order product density can be obtained in much the same manner as the first-order product density by accounting for the formation of nuclei. Thus, recognizing the symmetry of the product density, we have the boundary condition... 

Also simple is the method where the (precisely unknown) state (/n ) of inventory n at time ty, i preceding time is replaced by its estimate (/ (.,), and only the measured value s (4) is reconciled at time t. in the daily balancing. Disregarding the error in (4, ) (and the corresponding probabilistic considerations), the constraint equation (11.2.11) at time ty. can be written in the form... 

Over the last 15 years improved awareness of wave impact induced failures has focused attention on the need to account for the dynamic response of maritime structures to wave impact loads. Also, the attention was focused on the probabilistic consideration in the design of breakwaters.The most relevant collaborative research projects carried out in the recent past in the European Union on wave loads on caisson breakwaters and seawalls include ... 

In some cases probabilistic considerations on the probabUity of the load combination and its associated risk can justify less stringent acceptance criteria... 

Generally speaking, an event which may be at the origin of severe consequences requires 2a + b LODs to be rejected into residual risk. Limiting events should be emphasised because they are Beyond Design Basis events with a bounding character which have not been chosen for purely probabilistic considerations. It is usually considered that a medium line is required to reject limiting events into residual risk. 

Rules for safety analysis The rules for safety analysis result from probabilistic considerations, but most of the time these rules can be set up in a deterministic way which in many cases may be quite conservative. In the Design Basis faults are analysed with very stringent rules ... 

It was also noted that at sufficiently high temperatures diffusion and hydrolysis may occur simultaneously (Abeysinghe et al. 1982) and that hydrolysis lowers Tg (Zhou and Lucas 1999). A theoretical model that employs probabilistic considerations for chain scissions by chemically reacted water, and thereby weight loss, had successfully predicted such combined weight gain/weight loss data (Xiao and Shanahan 1997). Those data correspond to curve D in Fig. 4.2. 

However, to make the final decision and choose one from the preliminary set of options, specified inputs must be established, namely the standards and good practices, operating experience, deterministic and probabilistic considerations, organizational and security systems and other factors such as research or economic insights. Relative importance of each element depends upon the decision to be made and should be weighted qualitatively. This process leads to evaluation and reduction the preliminary set of options. Finally one of them should be chosen, implemented and monitored. Moreover, if the performance of just implemented decision is not satisfactory corrective actions should be undertaken and the hst of option needs to be redefined (IAEA 2011). 

Including probabilistic considerations in the assessment is particularly useful in the evaluation of decision alternatives and strategies. A typical application in the past was the assessment of modifications with respect to in-service inspection intervals. 

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