Normalized histogram

The name seems to reflect no more than the fact that the resulting estimates of PDFs are as always based on (normalized) histograms accumulated in the sampling process. 

If we are working with normalized histograms where the buckets contain the probability that a particular color is found in the image, then the normalization can be omitted because we have IIm (c) = 1. This particular method of comparing histograms was defined... 

Fig. 8.2. Compared with the narrow peak in the normal histogram at the upper left, it can be seen that a single peak with a wide coefficient of variation (CV) or skewed profile may mask a near-diploid malignant cell line. In addition, an extra small peak at the 4C position may result from clumping of nuclei, cycling cells, or a true tetra-ploid abnormality. Data courtesy of Colm Hennessy.

The normalized histogram of the fractional successes was fitted by a continuous function. It was the Bayesian distribution for required to find the Bayesian confidence intervals for the true value. The analysis showed that, indeed, the three-second half-life falls within the 95 percent confidence interval. In a similar later work [29], the measurement time was 1.5 seconds and as many as 50 counts were registered. The results, as such, clearly indicated that more decays had taken place beyond 1.5 seconds, so that the half-life is of the order of several seconds. 

The normalized histogram of the frequency of successes was again fitted by a smooth curve to evaluate the confidential intervals. The other two possible combinations of the data points, which are shown in the bottom of Fig. 6.5, were simulated in a similar way. 

Fig. 7 Angular distribution of nanofiber substrates and the cells cultured on them are shown in normalized histograms, (a) Substrate was collected on a solid plate, and (b) substrate was collected on a rotating mandrel [114]...

Fig. 2. The normalized histogram distribution functions of the diffusion coefficients for samples 1 --------------, 2 and...

Figure 6 Prior PDFs for the breaking ductility ratio plotted against normalized histograms of posterior samples for updating model class M2 with (a) data set T>cc and (b) data...

The difference in the information gained for the two data sets is illustrated by Figure 6, which shows normalized histograms of the posterior samples for the breaking ductility ratio fx in model class Mi, plotted with the prior PDF. Updating with data set VcCi which features little deterioration, results in a broad posterior PDF that is fairly similar to the prior PDF. However, updating with T>ov results in a very peaked posterior PDF for /x that indicates more information has been extracted from the data. 

Figure 11 Normalized histogram of calculated vesicle radius (open bars) overlaid with the known radius histogram (shaded bars) from Schubert et al. (1980) (14 bins, 10 nm/bin each). (Reproduced from/. Neurosci. Meth. with permission [31].)...

This conversion gives us training data in a format suitable for analysis by standard supervised learning algorithms. Let us denote the normalized histogram for the j input image by hj and denote... 

The purpose of overlapping the histogram for each block is to enhance the lesion for comparison with normal. This will produce a new histogram as shown in Fig. 8. The optimal threshold is determined by the ROl indicator, which the intensity level of normal histogram is reached zero pixels. 

The exact density of states is given by g E)=pcm E T)Z T)e and since the normalized histogram hi E)/Mj obtained in the rth simulation thread is an estimator for the canonical distribution functionpcan(E Tt), the density of states in this thread is estimated by... 

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. 

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