The Probability Scale

Sometimes it is necessary to plot on a log-probability graph but the probability scale is not available as such in a spreadsheet. The probability scale can be calculated using the right-hand side of equation (4.19). The probability scale, therefore, will be proportional to erf (2Cx-l). where C is a percentage figure for which the scale is required. 

The mathematical term, erf xr), is a tabulated integral, which may be obtained from any good mathematical book of tabulated transcendental functions. 

The probability scale is frequently used in decanter work and it is useful to know how to create such a scale when a ready-made one is not available. 

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Figures 62.8, 62.9, 62.10 show the data for generator fan failure plotted on exponential, normal and log normal hazard paper respectively. The exponential plot is a reasonably straight line which indicates that the failure rate is relatively constant over the range of the data. It should be noted that the reason the probability scale on the exponential hazard plot is crossed out is because that is not the proper way to plot data. (This will be discussed later.) The normal plot is curved concave upward which...

For any distribution, the cumulative hazard function and the cumulative distribution junction are connected by a simple relationship. The probability scale for the cumulative distribution function appears on the horizontal axis at the top of hazard paper and is determined from that relationship. Thus, the line fitted to data on hazard paper... 

Suppose, for example, that an estimate based on a Wei-bull fit to the fan data is desired of the fifth percentile of the distribution of time to fan failure. Enter the Weibull plot. Figure 62.6, on the probability scale at the chosen percentage point, 5 per cent. Go vertically down to the fitted line and then horizontally to the time scale where the estimate of the percentile is read and is 14,000 hours. 

An estimate of the probability of failure before some chosen specific time is obtained by the following. Suppose that an estimate is desired of the probability of fan failure before 100,000 hours, based on a Weibull fit to the fan data. Enter the Weibull plot on the vertical time scale at the chosen time, 100,000 hours. Go horizontally to the fitted line and then up to the probability scale where the estimate of the probability of failure is read and is 38 per cent. In other words, an estimated 38 per cent of the fans will fail before they run for 100,000 hours. 

To clarify the use of probabilities let us consider the following treatment of illustrative data. A sample that has low concentrations of fingerprint elements has ratios of these elements to zinc that are at the high end of the probability scale. Another sample that has high concentrations of the same elements has ratios that are at the low end of the probability scale as shown by the following two randomly selected Brazil samples ... 

This step involves calibration of the apparatus which will serve as a reference. It consists of analysing the greatest number possible (minimum 50) wines or must samples containing different and accurately known concentrations of each analyte. The concentration points should be uniformly distributed over the probable scale of measure for each analyte. The matrices should mimic as accurately as possible the wines and musts destined for analysis using that particular instrument. For each calibration sample, a measurement is carried out at a maximum number of wavelengths in the infra-red. Multi-linear regression is then carried out on the results which enables the following relationship to be established ... 

The use of probability plots is of value when the arithmetic or geometric mean is required, since these values may be read directly from the 50% point on a logarithmic probability plot. By definition, the size corresponding to the 50% point on the probability scale is the geometric mean diameter. The geometric standard deviation is given (for % LTSS) by ... 

Figure 2.27 Model calculation, (a) Probability of the electron residing on the initial donor site for a 572-atom lattice with periodic boundary conditions and rms interatomic lattice coupling between the donor and lattice of 1/56 (black line), 1/28 (blue line), and 1/14 (dotted line). The 1/e decay in the probability scales approximately quadratically with coupling. The inset depicts the r = 0 boundary condition with the electron (darkened circle) localised on the dye (donor) molecule (b) For comparison the back electron-transfer process is modelled assuming the electron is one lattice site removed (as shown in inset) from the dye and the dye-surface atom coupling is 1/3 the rms lattice coupling (very strong coupling to the dye). Once the wavefunction has localised in the solid-state there is little chance of back electron transfer from the band states.

Data plotted as (100 — Z) vs. I on linear probability paper using the probability scale for (100 - Z)... 

Visman and Van Krevelen (VI) replotted the data of Coulson and Maitra on linear probability paper, using 100 — Z (wrhich they called the per cent unmixed) on the probability scale, and time t, on the arith-... 

Another transformation which is commonly employed by statisticians is the logit transformation. Suppose we are interested in looking at the effect of a treatment on the probability of survival of patients over a given time period. The probability of survival will lie between 0 and 1 for any patient. If we use the probability scale to make our analysis we may come to some conclusion such as (say) the effect of treatment is to increase the probability of survival by 0.23. Suppose, however, that we now wish to apply the treatment to a type of patient whose probability of survival without treatment we believe to be 0.86. Applying our treatment estimate would lead to the nonsensical conclusion that her chances of survival were now 0.86 + 0.23 = 1.09 This can be avoided if, instead of using a scale like the probability scale, which is bounded by 0 and 1, we use a scale which, although related to it, is not so bounded. An example of such a scale is the logit scale and it is defined by... 

Fig. 5 Probability map for sodium cations. The figure is centered in the middle of a cage, with six D8Rs surrounding it. The framework is shown in black with the A1 atoms in green, a The probability scale is such that even locations that are very infrequendy visited are shown in blue. These blue regions reveal that the cations explore within their site and are thus to some extent mobile, b Most visited locations (the 0.1 max probability contour is shown)... 

The evaluation of MS/CV is demonstrated in Figure 2.18a with the sieve analysis data from Table 2.13. The cumulative undersizes (or oversizes if preferred) are plotted on the probability scale, the sieve aperture sizes on the arithmetic scale. If the data between about 10 and 90 per cent lie on a straight line, the MS/CV method can be applied. The data in Figure 2.18a comply with this requirement. Thus the median size is 870 pm. The coefficient of variation can be deduced as follows. 

Similarly, the probability scales are, in many instances, either broadly defined subjective word descriptions or, if quantitative, calculated guesses with no supporting calculations. In order to produce probability numbers, assumptions may frequently be necessary about failure rates, the number of units in the system, and the expected service life of the unit or the system. These... 

The Latin hypercube sampling technique (LHS Helton and Davis 2003) can be employed in order to reduce the number of simulations, Ns, in addition to achieving an acceptable level of accuracy for the statistical characteristics of response. The LHS is a special type of MC simulation that uses the stratification of the theoretical CDFs of uncertain parameters. Stratification divides the CDF curve into Ns equal intervals on the probability scale (i.e., 0.0 to 1.0). A sample is then randomly drawn from each interval or stratification of the input CDFs based on the... 

These tables show that the traditional FMEA uses five scales and scores of one to ten, to measure the probability of occurrence, severity and the probability of detection. Though this simplifies the computation, converting the probability into another scoring system, and then finding the multiplication of factor scores may cause problems. From Tables 7.1 and 7.3 it can be seen that the relation between and the probability scale is non-linear, while it is linear for that between Sd and the probability scale. 

Figure 10.1 is intended to put the probability scale into perspective. Note that P is used to indicate the probability of occurrence of the event. P is dimensionless, failure rates are not. 

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