Representing data using histograms

We describe two simple ways in which discrete data may be treated to obtain the required areas (1) use of the data in histogram form, and (2) use of the trapezoid rule. These are illustrated in Figures 19.6 and. 7, respectively, in which 10 data points are plotted to represent a response curve. The curve drawn in each case is unnecessary for the calculations, but is included to indicate features of the approximations used. 

It is extremely useful to move beyond a subjective and qualitative analysis of the spatial distribution of sample components, and to begin to explore the quantitative information contained within chemical imaging data sets. One of the most powerful statistical representations of an image does not even maintain spatial information. A chemical image can be represented as a histogram, with intensity along the x-axis and the number of pixels with that intensity along the y-axis. This is a statistical and quantitative... 

These ten results represent a sample from a much larger population of data as, in theory, the analyst could have made measurements on many more samples taken from the tub of low-fat spread. Owing to the presence of random errors (see Section 6.3.3), there will always be differences between the results from replicate measurements. To get a clearer picture of how the results from replicate measurements are distributed, it is useful to plot the data. Figure 6.1 shows a frequency plot or histogram of the data. The horizontal axis is divided into bins , each representing a range of results, while the vertical axis shows the frequency with which results occur in each of the ranges (bins). 

Figure 3.17. Interatomic distances in CsCI. The distances are given for the CsCI compound (cubic, cP2-CsCl type, a = 411.3 pm) with Cs and Cl in the representative positions 0, 0, 0, and A, A, A respectively, white and black atoms in Fig. 3.8. In the tables the first two groups of distances (in pm) are given as positions of each atom around the reference atom. Notice that not only atoms in the reference cell but also those in the adjacent cells must be considered (see Figs. 3.8 (d)-(f)). At the right side, the corresponding histograms using the reduced distances d/dmm are shown the first two bars summarize the data contained in the table.

The samples of unknown composition—21-23 and samples 1-20, 24-34 (Appendix I) were those of Aroclors of variable composition. Variables 5-73 are isomer concentrations (Variable 74, the total PCB concentration in ppm was not included in the analysis). Variables 5-73 represent the fractional composition or isomer proportional concentration values. Representative concentration histograms of the data set are presented in Figure 13. Four PLS components were extracted and then used to estimate the Aroclor content of the unknowns and of a standard sample (No. 24). The Aroclor standard is a mixture of three Aroclors in the ratio of 033 0.33 0 0.33. Chromatograms of the samples for which the PLS estimates were made (Table VI) were similar when compared to a chromatogram of a similar mixture of standards. 

Another way in which these kinds of data are sometimes represented is as a cumulative curve in which the total number (or fraction) of particles nT>, having diameters less (sometimes more) than and including a particular d, are plotted versus dr Figure 1.18b shows the cumulative plot for the same data shown in Figure 1.18a as a histogram. The cumulative curve is equivalent to the integral of the frequency distribution up to the specified class mark. Cumulative distribution curves are used in Chapter 2 in connection with sedimentation. 

Fig. 3. Phosphorylation level of Akt and GSK-3/3 following focal cerebral ischemia. (A) Western blot analysis of phospho-Akt (Ser473) (p-Akt) and total Akt performed on brain cortical homogenates from rats sacrificed 24 h after MCAo shows a trend toward a decrease of p-Akt immunoreactivity in the ipsilateral (I), ischemic, cortex as compared to contralateral (C), nonischemic, side. This trend toward a reduction also occurred for total Akt expression, so that phosphorylation level (expressed as the ratio p-Akt/Akt) is not significantly affected by ischemia. The result is representative of three independent experiments. Histograms in (C) show the results of the densitometric analysis of the autoradioraphic bands corresponding to p-Akt, total Akt, and /3-actin. p-Akt and Akt levels were normalized to the values yielded by /3-actin and Akt phosphorylation was expressed by the ratio of p-Akt/total Akt data are reported as mean S.E.M. ( = 3 per group). The same samples were used for the subsequent western blot analysis of phospho-GSK-3/3 (Ser9) (p-GSK-3/3) and total GSK-3/3 and a representative result of three independent experiments is shown in (B). The results of the densitometric analysis of the bands corresponding to p-GSK-3/3, total GSK-3/3, and a-tubulin are reported in (C). p-GSK-3/3 and total GSK-3/3 were normalized to the values yielded by a-tubulin whereas GSK-3/3 phosphorylation was calculated from the ratio of p-GSK-3/3/total GSK-3/3 data are reported as mean S.E.M. (n = 3 per group).

Fig. 2. Representative HUVEC transfection efficacy of C32 and method for FACS gating. The one-dimensional histogram of C32/DNA transfected cells (56.5% positive) (left) includes some falsely positive cells that are excluded during the two-dimensional analysis of the same data set as shown by the FACS density plot gating for the negative control (0% positive) (middle) and the C32/DNA transfection (44.7% positive) (right). Ratio of GFP fluorescence (x-axis) to background fluorescence (y-axis) is used to accurately gate positive cells.

The stem and leaf diagram of Fig. 5.3 and the dot plot of Fig. 5.4 carry shape information about the distribution of aluminum contents in a manner very similar to the histogram of Fig. 5.2. But the stem and leaf and dot diagrams do so without losing the exact identities of the individual data points. The box plot of Fig. 5.5 represents the middle half of the data with a box divided at the 50th percentile (or in statistical jargon, the median) of the data, and then uses so-called whiskers to indicate how far the most extreme data points are from the middle half of the data. 

Usually, 10 to 15 pages of text with additional tables as needed should suffice for the integrated summary. It should represent a perspective on the completed animal studies at the time the sponsor decided that human trials were appropriate. Use of visual data displays (e.g., box plots, histograms, or distribu-... 

Fig. 17 Energy as a function of a T-T distance and b T-T-T angle used in the simulation procedure (calculated as smoothing spline fits to Boltzmann equilibrium interpretations of the histogrammed data taken from 32 representative zeolite crystal structures). Only the central portions are shown, c The contribution to the energy sum for the merging of two symmetry-related atoms merging is only permitted when the two atoms are at less than a defined minimum distance [84], Reproduced with the kind permission of the Nature Publishing Group (http //www.nature.com/)...

It is useful to distinguish two forms of probability distribution, discontinuous and continuous. As an example of a discontinuous distribution consider the outcome of throwing a die. The chance outcome of a series of throws can be represented as a discontinuous probability distribution. There is only a limited number of possible outcomes and the results can be shown in the form of a block histogram. The second kind of distribution would arise in, e.g., replicate measurements of the fluorescence intensity of a sample. These observations will differ as a result of statistical variation, as discussed in the previous section, but instead of being a single chance event as in the case of the die, many chance factors will contribute to the observed variation in the fluorescence data. The variation observed in this case is an example of a continuous probability distribution. Although it is true in principle... 

While a plotted curve assumes a continuous relationship between the variables by interpolating between individual data points, a histogram involves no such assumptions and is the most appropriate representation if the number of data points is too few to allow a trend line to be drawn. Histograms are also used to represent frequency distributions (p. 265), where the y-axis shows the number of times a particular value of x was obtained (e.g. Fig. 37.3). As in a plotted curve, the x-axis represents a continuous variable which can take any value within a given range, so the scale must be broken down into discrete classes and the scale marks on the x-axis should show either the mid-points (mid-values) of each class (Fig. 37.3), or the boundaries between the classes. 

From the total sample set (48 samples), 45 samples were used as calibration samples. The three samples excluded from the calibration set were selected on the basis of a representative variation of their active ingredient concentrations, and finally used as unknown test samples to predict the content of their active ingredients. Partial least squares (PLS) models for each active ingredient were developed with the Unscrambler Software (version 9.6 CAMO Software AS, Oslo, Norway) from the MSC-pretreated median spectra of all pixels of each of the 45 calibration sample images. Based on these calibration models, the predictions of the active ingredient content for each pixel of the imaging data of the three test samples and their evaluation as histograms, contour plots and RGB plots was performed with Matlab v. 7.0.4 software (see below). 

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