Rule of errors

Applying the rules of error propagation and considering that x, is a function of o,-, cf x) can be approximated by the following equations ... 

The covariance between two fractions can also be calculated using the rules of error propagation, this leads to ... 

D is the Debye-Huckel term in molal units and / , the ionic strength converted to molal units by using the conversion factors listed in [76BAE/MES] (p.439). The following list gives the details of this calculation. The resulting uncertainties in log p are obtained based on the rules of error propagation as described in Section C.6.2. 

Side chain generation is often a source of error. It will be most reliable if certain rules of thumb are obeyed. Start with structurally conserved side chains and hold them fixed. Then look at the energy and entropy of rotamers for the remaining side chains. Conventional conformation search techniques are often used to place each side chain. 

Order-of-magnitude estimate (ratio estimate). Rule-of-thumb method based on cost data for previous similar types of plant probable error within 10 to 50 percent. 

Fig. 10.8 Seven set fuzzy input windows for error (e) and rate of change of error ice). Assume that a certain rule in the rulebase is given by equation (10.22) OR IF e is A AND ce is B THEN u = C...

Now you can reconsider the material balance equations by adding those additional factors identified in the previous step. If necessary, estimates of unaccountable losses will have to be calculated. Note that, in the case of a relatively simple manufacturing plant, preparation of a preliminary material-balance system and its refinement (Steps 14 and 15) can usefully be combined. For more-complex P2 assessments, however, two separate steps are likely to be more appropriate. An important rule to remember is that the inputs should ideally equal the outputs - but in practice this will rarely be the case. Some judgment will be required to determine what level of accuracy is acceptable, and we should have an idea as to what the unlikely sources of errors are (e.g., evaporative losses from outside holding ponds may be a materials loss we cannot accurately account for). In the case of high concentrations of hazardous wastes, accurate measurements are needed to develop cost-effective waste-reduction options. It is possible that the material balance for a number of unit operations will need to be repeated. Again, continue to review, refine, and, where necessary, expand your database. The compilation of accurate and comprehensive data is essential for a successful P2 audit and subsequent waste-reduction action plan. Remember - you can t reduce what you don t know is therel... 

The precondition for the use of the normal distribution in estimating the random error is that adequate reliable estimates are available for the parame-rcrs ju. and cr. In case of a repeated measurement, the estimates are calculated using Eqs. (12.1) and (12,3). When the sample size iiicrease.s, the estimates m and s approach the parameters /c and cr. A rule of rhumb is that when s 30. the normal distribution can be osecl,... 

In order to cope with a complex environment, people make extensive use of rules or assumptions. This rule based mode of operation is normally very efficient. However, errors will arise when the underlying assumptions required by the rules are not fulfilled. Chapter 2 discusses the causes of these rule based errors in detail. 

An influential classification of the different types of information processing involved in industrial tasks was developed by J. Rasmussen of the Rise Laboratory in Denmark. This scheme provides a useful framework for identifying the types of error likely to occur in different operational situations, or within different aspects of the same task where different types of information processing demands on the individual may occur. The classification system, known as the skill-, rule-, knowledge-based (SRK) approach is described in a... 

Once certain categories of error have been ruled out, the analyst decides whether or not any of the errors in the remaining applicable categories could occur within the task, subtask, or task step being evaluated. 

Semi-open formulas are used when the problem exists at only one limit. At the closed end of the integration, the weights from the standard closed-type formulas are used and at the open end, the weights from open formulas are used. (Weights for closed and open formulas of various orders of error may be found in standard numerical methods texts.) Given a closed extended trapezoidal rule of one order higher than the preceding formula, i.e.. 

Theorem 4-9 is stronger than the converse theorems that have previously appeared in the literature.9 These other theorems deal with the probability of error on a decoded sequence. Proving that the probability of error on a sequence of length N approaches 1 as N -+ oo does not prove that the error probability per source digit is bounded away from zero. If each sequence error corresponds to only one source digit error, then the source digit error rate would be 1/N times the sequence error rate. Thus theorems about the sequence error rate do not rule out the possibility of reliable communication for large N. 

For equiprobable source letters, the decoding rule of Eq. (4-85) thus chooses the most likely source letter conditional on the received sequence. If no source letter is more probable than all others, we assume that a decoding error is made. 

Given a particular set of code words, and given the decoding rule of Eq. (4-85), the probability of decoding error can in principle be calculated. Let Ym be the set of channel output sequences that satisfy Eq. (4-85) for each m, 1 <, m <, M. Then the probability of decoding error for Hie source letter um is... 

In Fig. 1 there is indicated the division of the nine outer orbitals into these two classes. It is assumed that electrons occupying orbitals of the first class (weak interatomic interactions) in an atom tend to remain unpaired (Hund s rule of maximum multiplicity), and that electrons occupying orbitals of the second class pair with similar electrons of adjacent atoms. Let us call these orbitals atomic orbitals and bond orbitals, respectively. In copper all of the atomic orbitals are occupied by pairs. In nickel, with ou = 0.61, there are 0.61 unpaired electrons in atomic orbitals, and in cobalt 1.71. (The deviation from unity of the difference between the values for cobalt and nickel may be the result of experimental error in the cobalt value, which is uncertain because of the magnetic hardness of this element.) This indicates that the energy diagram of Fig. 1 does not change very much from metal to metal. Substantiation of this is provided by the values of cra for copper-nickel alloys,12 which decrease linearly with mole fraction of copper from mole fraction 0.6 of copper, and by the related values for zinc-nickel and other alloys.13 The value a a = 2.61 would accordingly be expected for iron, if there were 2.61 or more d orbitals in the atomic orbital class. We conclude from the observed value [Pg.347]

Figure 4.28. Correlation graph for file PROFILE.dat. The facts that (a) 23 out of 55 combinations yield probabilities of error below p = 0.04 (42% expected due to chance alone =8%) and (b) that they fall into a clear pattern makes it highly probable that the peak areas [%] of the corresponding chromatograms follow a hidden set of rules. This is borne out by plotting the vectors two by two. Because a single-sided test is used, p cannot exceed 0.5.

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