Histogram normalization

Fig. 6. Time-of-flight fusion spectrum (error bars) and simulation spectrum (histogram), normalized the number of incident muons iVM. Also plotted are simulated contributions from different resonance peaks given by the time-energy correlated events...

Keywords— Contrast normalization, endoscopy, histogram normalization, image recognition, structural similarity... 

Figure 6 shows the histogram of localized AE events vs axial position for the same time period as in fig.5. The location of the AE source corresponds, within source location errors (< 10-15 cm), to one of the welds under surveillance. The weld was known by ultrasonic examination to be affected by internal discontinuities. However, the position of the source could also correspond to one of the hangers. The steps observed in EA event accumulation have taken place during steady load operation, which normally corresponds to very low background noise conditions. This type of event, however, has not been observed afterwards. 

Fig. 5.17 Histogram of the normal modes calculated for a polyalanine polypeptide in an a-helical conformation. The height of each bar indicates the number of normal modes in each 50cm section.

The distribution curves may be regarded as histograms in which the class intervals (see p. 26) are indefinitely narrow and in which the size distribution follows the normal or log-normal law exactly. The distribution curves constructed from experimental data will deviate more or less widely from the ideal form, partly because the number of particles in the sample is necessarily severely limited, and partly because the postulated distribution... 

The data in Table 4.12 are best displayed as a histogram, in which the frequency of occurrence for equal intervals of data is plotted versus the midpoint of each interval. Table 4.13 and figure 4.8 show a frequency table and histogram for the data in Table 4.12. Note that the histogram was constructed such that the mean value for the data set is centered within its interval. In addition, a normal distribution curve using X and to estimate p, and is superimposed on the histogram. 

The volumes of water in two burets are read, and the difference between the volumes are calculated. Students analyze the data by drawing histograms for each of the three volumes, comparing results with those predicted for a normal distribution. 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

Each bin is connected to a memory location in a computer so that each event can be stored additively over a period of time. All the totaled events are used to produce a histogram, which records ion event times versus the number of times any one event occurs (Figure 31.5).With a sufficiently large number of events, these histograms can be rounded to give peaks, representing ion m/z values (from the arrival times) and ion abundances (from the number of events). As noted above, for TOP instruments, ion arrival times translate into m/z values, and, therefore, the time and abundance chart becomes mathematically an m/z and abundance chart viz., a normal mass spectrum is produced. 

The BMS deviation is a measure of the spread of values for c around the mean. A large value of O indicates that wide variations in c occur. The probability that the controlled variable hes between the values of Cl and C9 is given by the area under the distribution between Ci and Cg (histogram). If the histogram follows a normal probabihty distribution, then 99.7 percent of aU observations should lie with 3o of the mean (between the lower and upper control limits). These Emits are used to determine the quality of control. 

FIG. 8-38 Histogram plotting frequency of occurrence, c = mean, <3 = rms deviation. Also shown is fit by normal probability distribution. 

Step 1. From a histogram of the data, partition the data into N components, each roughly corresponding to a mode of the data distribution. This defines the Cj. Set the parameters for prior distributions on the 6 parameters that are conjugate to the likelihoods. For the normal distribution the priors are defined in Eq. (15), so the full prior for the n components is... 

Table 4.2 Analysis of histogram data for SAE 1018 to obtain the Normal distribution plotting positions...

Once the mean and standard deviation have been determined, the frequency distribution determined from the PDF can be compared to the original histogram, if one was constructed, by using a scaling factor in the PDF equation. For example, the expected frequency for the Normal distribution is given by ... 

We ean now plot the Normal frequeney distribution superimposed over the histogram bars for eomparison. The eurve is generated using equation 15, where the variables of interest, x, are values in steps of 10 on the x-axis from, say, 380 to 540. The Normal frequeney equation is given below, and Figure 6 shows the histogram and the Normal... 

Figure 6 Histogram for yield strength data with superimposed Normal distribution...

The eomponent shown in Figure 4 is a spaeer from a transmission system. The eomponent is manufaetured by turning/boring at the rate of 25 000 per annum and the eomponent eharaeteristie to be eontrolled, X, is an internal diameter. From the statistieal data in the form of a histogram for 40 eomponents manufaetured, shown in Figure 5, we ean ealeulate the proeess eapability indiees, Cp and Cp. It is assumed that a Normal distribution adequately models the sample data. 

Experience gained in the ZAF analysis of major and minor constituents in multielement standards analyzed against pure element standards has produced detailed error distribution histograms for quantitative EPMA. The error distribution is a normal distribution centered about 0%, with a standard deviation of approximately 2% relative. Errors as high as 10% relative are rarely encountered. There are several important caveats that must be observed to achieve errors that can be expected to lie within this distribution ... 

Rt >n for lure data Monte Carlo simulation Normal, log-nonmal, uniform, any distribution in the form of a histogram, truncated normal, beta Can conelate input parametjefs no sorting n ry to obtain the t( nthislogram IBM From .ue (... 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

In order to illustrate the type of questions which can be addressed within such an idealized lattice model, we show a histogram of the normalized order parameters M = Si/N and Q = being the system... 

The distribution function of the vectors normal to the surfaces,/(x), for the direction of the magnetic field B, in accord with the directions of the crystallographic axis (100) for the P, D, G surfaces, is presented in Fig. 6. The histograms for the P, D, G are practically the same, as they should be the differences between the histograms are of the order of a line width. The accuracy of the numerical results can be judged by comparing the histograms obtained in our calculation with the analytically calculated distribution function for the P, D, G surfaces [29]. The sohd line in Fig. 6(a) represents the result of analytical calculations [35]. 

FIGURE 9.1 Histograms show the number of new drugs (normalized for the cost of drug discovery and development in the years they were developed) as a function of the years they were discovered and developed. Adapted from [1]. 

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

Figure 1.10. The figure demonstrates what is obtained when the Monte Carlo method (cf. Section 3.5.5) is used to simulate normally distributed values each histogram (cf. Section 1.8.1) summarizes 100 measurements obviously, many do not even come close to what one expects under the label bell curve. ...

Figure 1.30. A histogram of raw weights from Figure 1.29 and the distribution of residuals that resulted after subtraction of a shifted box-car average are superimposed. The CP-curve, plotted with the (NPS) option in HISTO, is for the raw weights the corresponding curve for the residuals would be about twice as steep. The asymmetry of the raw-weight distribution is evident both in the histogram and the lack of linearity of the CP-curve it is due to many subpopulations of product being lumped into one batch. Every time a mechanic makes an adjustment on a knife, a new subpopulation is created. The residuals appear to be normally distributed, however.

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