Independent set

The set of parameters given in Table 11.1 is by no means unique, since any independent set of combinations of the given parameters will serve equally well. The particular set adopted will depend primarily on the purpose for which they are to be used. Thus, if we are interested in the dependence of the effectiveness factor on one particular physical variable, it is obviously convenient to choose the dimensionless parameters in such a way that all but one are independent of this variable. A plot of the effectiveness factor against one dimensionless parameter will then summarize the desired information. 

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

This results in sensible numbers for the class boundaries and allows for comparisons with other series of observations that have different extrema, for example (.375,. 892) or (.25, 1.11). Strict adherence to the theory-inspired rule would have yielded class boundaries. 327,. 4833,. 6395,. 7958, and. 952, with the extreme values being exactly on the edges of the graph. That this is impractical is obvious, mainly because the class boundaries depend on the stochastic element innate in Xmin resp. x ax- Program HISTO, option (Scale), allows for an independent setting of a subdivided range R with C bins of width R/C, and lower/upper limits on the graph. 

Now the equations derived from Kirchoff s first law are essentially material balances around each of (N — 1) vertices. As an alternative, balances could also be drawn up around groups of such vertices. Is there a special way of grouping the vertices, which will yield a particularly advantageous formulation Also, as we have noted, the selection of cycles is not unique, but the cycles must be independent. How can we generate an independent set of cycles Are some of these independent sets more fundamental than others If so, how many fundamental sets are there To answer these questions we must explore further the properties of a graph. 

Node weights in a two-dimensional SOM. Each node has its own independent set of weights. 

As these rules are somewhat difficult to understand in the abstract, we will suggest an example.8 Let us assume that there are three independent sets of programmes, such as spending on education, health care and roads. Within each of them there are several mutually exclusive options (only one of them will be implemented) with varying costs and benefits. The steps to be taken are as follows ... 

Note that for the calculations of precision and standard deviation (equations 38-1 through 38-4), the numerator expression is given as 2(n — 1). This expression is used due to the 2 times error contribution from independent errors found in each independent set (i.e., X and Y) of results. 

For the IRT 3000 a simultaneous calibration concept [6], that is shown schematically in Fig. 3.48, has been developed. At two different ambient temperatures two measurements at different blackbody temperatures are performed with completely assembled thermometers connected to external computers. This results in four independent sets of output values of both ambient and thermopile sensors (see Eq. 6 and 7) ... 

Fig. 1.15. Three-level calculations with full ID vibrational motion of the excitation of Hj as a function of wavelength. Also included are two independent sets of experimental data for wavelength scans from 490 to 520 nm...

For a stationary (time-independent) set of wavefunctions, the time derivative on the right hand side vanishes and the equation obviously transforms back into the previous one. Now the system is perturbed by an external perturbation Hp, which is chosen to have a well-defined temporal periodicity ... 

As noted earlier, it is possible to use JVe < Ns by choosing a linearly independent set of higher-order moments. For example, the multi-environment models discussed in Section 5.10 use jVe = 2-4. 

The Bs are constants indicating how many carbon atoms are attached to the observed C [5 = 1 one adjacent C (i.e., C is CH3) S = 2 C is CH2 S = 3 C is CH S = 4 C is quaternary] and thereby contain a-parameters. For the remaining parameters there are independent sets for each 5 as well. ASM denotes the steric configuration at the adjacent carbon atoms, where M is defined analogously to S for example, A23 signifies a contribution from the methine to the methylene atom under consideration. Consequently, the ASM s replace the p-... 

A third relevant reaction is also presented below. However, this reaction is simply a combination of the other two. Of the three reactions, any two can be utilized as an independent set of reactions for analysis, and should be chosen for the user s convenience. Here we have chosen the steam reforming and the shift reactions. 

In Figure 6.10 we compare these two expressions obtained from the independent sets of experimental data. At > 0.2, the two graphs are... 

It is interesting to note that if the matrix in Eqs. (40) and (48) is neglected, we obtain two independent sets of equations in which TE and TM modes of the slices do not mutually couple. They represent semivectorial approximations to the fully vectorial problem. 

Figure 6.4 shows the relationship between (the least squares estimate of Pq) nd zero for the data in Figure 6.1. In this example, the parameter po has been estimated on the basis of only two experimental results if another independent set of two experiments were carried out on the same system, we would very likely obtain different values for and y,4 and thus obtain a different estimate for Pq from the second set of experiments. It is for this reason that the estimation of Po is usually subject to uncertainty.

Figure 6.5 Five gaussian curves, each estimated from an independent set of two experiments on the same system, showing uncertainty of estimating and...

The estimation of Pq is therefore doubly uncertain, first because the value of p is not known with certainty, and second because the exact distribution of estimates for Po is unknown. Figure 6.5 illustrates the problem each curve represents a pair of estimates of bg and s obtained from an independent set of two experiments on the same system. 

Typically, the final part of QSAR model development is the model validation [17, 18], when the predictive power of the model is tested on an independent set of compounds. In essence, predictive power is one of the most important characteristics of QSAR models. It can be defined as the ability of a model to predict accurately the target property (e.g., biological activity) of compounds that were not used for model development. The typical problem of QSAR modeling is that at the time of the model development a researcher only has, essentially, training set molecules, so predictive ability can only be characterized by statistical characteristics of the training set model and not by true external validation. 

External resources may he needed if the incident investigation work exceeds site capahilities. These resources could include corporate personnel or experts from outside the company. (The team leader may also he external if the incident is major since the leader s independence sets the tone for the investigation.) Company business unit leaders should confer with the team leader to determine whether external assistance is recommended. Factors to consider include significant offsite consequences such as environmental impact or product quality concerns. A team of trained specialists should formally investigate any process incident that could significantly affect the business. At the lower end of the scale, if a near miss or minor incident occurs that has no potential for significant consequences, local supervision or front-line personnel normally may perform the investigation without outside assistance. 

If we don t have one stated value, but two independent sets of data (e.g. two analytical results from different laboratories or methods) we have to use the two-sample t-test, because we have to consider the dispersion of both data sets. In the same way as above we have to look carefully, what our question is it may be two-tailed (are the results significantly different ) or one-tailed (is the result from method A significantly lower than that from method B )... 

This test compares two independent sets of data (anaiyticai resuKs)... 

Note that these recommendations relate to confirmatory claims and statements, for post hoc exploratory investigations there are no restrictions in this multiplicity sense. Any findings arising from such analyses, however, cannot be viewed as confirmatory unless the finding can be clearly replicated in an independent setting. 

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