Probability Plots

Fig. 18.4. Survival probability plotted on "Weibull probability" axes for samples of volume Vq. This is just Fig. 1 8.3(b) plotted with axes that straighten out the lines of constant m.

The procedure is to fit the population frequency curve as a straight line using the sample moments and parameters of the proposed probability function. The data are then plotted by ordering the data from the largest event to the smallest and using the rank (i) of the event to obtain a probability plotting position. Two of the more common formulas are Weibull... 

Graphical analysis of failure data is most commonly plotted using probability. However, in order to understand the hazard plotting method presented here, is not necessary to understand probability plotting. While it is difficult to utilize probability plotting for multiply-censored data, it is... 

This section will cover the potential difficulties that may be encountered when using hazard and probability plotting paper. It will also look at how to use the plotting paper for the most effective results. Much of the discussion applies equally to both hazard or probability plotting, especially where good plotting techniques are concerned. 

The behavior of the failure rate as a function of time can be gaged from a hazard plot. If data are plotted on exponential hazard paper, the derivative of the cumulative hazard function at some time is the instantaneous failure rate at that time. Since time to failure is plotted as a function of the cumulative hazard, the instantaneous failure rate is actually the reciprocal of the slope of the plotted data, and the slope of the plotted data corresponds to the instantaneous mean time to failure. For the data that are plotted on one of the other hazard papers and that give a curved plot, one can determine from examining the changing slope of the plot whether the tme failure rate is increasing or decreasing relative to the failure rate of the theoretical distribution for the paper. Such information on the behavior of the failure rate cannot be obtained from probability plots. 

If estimated of distribution parameters are desired from data plotted on a hazard paper, then the straight line drawn through the data should be based primarily on a fit to the data points near the center of the distribution the sample is from and not be influenced overly by data points in the tails of the distribution. This is suggested because the smallest and largest times to failure in a sample tend to vary considerably from the true cumulative hazard function, and the middle times tend to lie close to it. Similar comments apply to the probability plotting. 

Ideally, no fewer than 20 failure times, if available, should be plotted from a set of data. Often, in engineering practice there are so few failures that all should be kept in mind so that conclusions drawn from a plot are based on a limited amount of information. Note that if only selected failures from a sample are to be plotted on hazard paper, it is necessary to use all of the failures in the sample to calculate the appropriate cumulative hazard values for the plotting positions. Wrong plotting positions will result if some failures in the data are not included in the cumulative hazard calculations. A similar comment applies to the calculation of plotting positions for probability plotting. 

Figure 6 shows a cumulative probability plot of both the maximum dally and hourly NO2 averages In cities for the 1980-84 time period. The plotted values can be directly compared to the WHO guideline values of 150/tg/m3 for the maximum 24-hour level and 400/tg/m3 for the maximum 1-hour level. In both cases, about 25% of the cities worldwide exceed the guideline values. Based on these proportions of cites with NO2 concentrations above the short-term guideline values. It Is estimated that approximately 15-20 percent of urban residents In North America and Europe are at Increased risk to short-term high NO2 exposures. 

Methods. As discussed in the previous chapter, a number of approaches have been used to assess the presence of potentially toxic trace elements in water. The approaches used in this assessment include comparative media evaluation, a human health and aquatic life guidelines assessment, a mass balance evaluation, probability plots, and toxicity bioassays. Concentrations of trace elements were determined by atomic absorption spectrometry according to standard methods (21,22) by the Oregon State Department of Environmental Quality and the U.S. Geological Survey. 

Probability Plots. To distinguish between background distributions and human activity, trace element data were probability plotted using the method of Velz (10), The plots produce two separate trend lines, the intersection of which distinguishes natural from anthropogenic concentrations. Figure 10 is an illustration of the resulting plots for zinc (38),... 

Figure 10. Normal-probability plots of zinc concentrations in 20 um sediments. A - all Willamette River Basin samples B -uncontaminated area samples. B curve is an enlargement of the lower portion of A curve. Discontinuity is interpreted as concentration limit of uncontaminated sediments.

If a precipitate is allowed to undergo Ostwald ripening, or is sintered, or is caused to enter into a solid state reaction of some kind, it will often develop into a distribution which has a size limit to its growth. That is, there is a maximum, or minimum limit (and sometimes both) which the particle distribution approaches. The distribution remains continuous as it approaches that limit. The log-probability plot then has the form shown in 5.8.2. on the next page. 

Log probability plots are particularly useful when the distribution is bimodal, that is, when two separate distributions are present. Suppose we have a distribution of very small particles, say in suspension in its mother liquor. Ey an Ostwald ripening mechanism, the small particles redissolve and reprecipitate to form a distribution of larger particles. This would give us the distribution shown in 5.8.5. on the next page. 

Since we do not know the proper values for X and t, we need a way of Judging plausible values of X and t from the data. We do this by testing the transformed background measurements for normality. Our choice of a test for normality is the probability plot correlation coefficient r (12). The coefficient r is the correlation between the ordered measurements and predicted values for an ordered set of normal random observations. We denote the ordered background measure-ments by yB(l). where yB(l) < yB(2) < yBCnn) denote the... 

Each of these data sets is skewed, yet each can be transformed to normality. With no transformation applied, the probability plot correlation coefficients for the Co, Fe, and Sc data sets are 0.855, 0.857, and 0.987, respectively. For Co and Fe, the hypothesis of normality is rejected at the 0.5 percent level (12). On the other hand, the maximum probability plot correlation coefficients are 0.993, 0.990, and 0.993 for Co, Fe, and Sc, respectively. The maxima occur at (X,t) (0,0.0048), (0,0.42), and (0.457,0), respectively. These maxima are so high that they provide no evidence that the range of transformations is inadequate. Note that the (, x) values at which the maxima occur correspond to log transformations with a shift for the Co and Fe and nearly a square-root transformation for the Sc. 

Determined from log-normal probability plots of individual rate... 

Fig. 2. Conditional probability plot for Sweet and Eisenberg s (1983) hydropathy scale. The black line is the probability ( axis) that a residue is ordered given the hydropathy score indicated on the x-axis. The dashed line is the probability of disorder. Negative values for hydropathy indicate hydrophilicity, positive values indicate hydrophobicity. The area between the two curves is divided by the total area of the graph to obtain the area ratio.

It is interesting to examine the probability of finding an electron as a function of distance when s orbitals having different n values are considered. Figure 2.6 shows the radial probability plots for the 2s and 3s orbitals. Note that the plot for the 2s orbital has one node (where the probability goes to 0)... 

FIGURE 1. Molecular structure of 1,4-pentadiene (1,4-PD) presentation with thermal probability plots... 

FIGURE 2. Calculated high symmetry conformations (C2v, C2 and Dyd. S4, respectively) and experimentally determined molecular structures of 1,1-divinylcyclopropane (DVC) and tetravinylmethane (TVM) in Ci presentation with thermal probability plots of 50%... 

Next, we insert these Zeff values into their appropriate radial functions, which are gathered in Table 9.1, and use the results from these calculations to construct radial probability plots for an electron in the 35, 3p and 3d orbitals of H and Na. The six plots that result are collected in the two figures presented below ... 

Quantile probability plots (QQ-plots) are useful data structure analysis tools originally proposed by Wilk and Gnanadesikan (1968). By means of probability plots they provide a clear summarization and palatable description of data. A variety of application instances have been shown by Gnanadesikan (1977). Durovic and Kovacevic (1995) have successfully implemented QQ-plots, combining them with some ideas from robust statistics (e.g., Huber, 1981) to make a robust Kalman filter. 

Wilk, M. B., and Gnanadesikan, R. (1968). Probability plotting methods for the analysis of data. Biometrika 55, 1-17. 

Probability plot Q-Q plot P-P Plot Hanging histogram Rootagram Poissonness plot Average versus standard deviation Component-plus-residual plot Partial-residual plot Residual plots Control chart Cusum chart Half-normal plot Ridge trace Youden plot... 

FIGURE 6 (a) Normal probability plot and (b) half-normal probability or Birnbaun plot, for I I... 

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