Examples Gaussian curve

Finite resolution and partial volume effects. Although this can occur in other areas of imaging such as MRS, it is particularly an issue for SPECT and PET because of the finite resolution of the imaging instruments. Resolution is typically imaged as the response of the detector crystal and associated electron to the point or line source. These peak in the center and fall off from a point source, for example, in shapes that simulate Gaussian curves. These are measures of the ability to resolve two points, e.g. two structures in a brain. Because brain structures, in particular, are often smaller than the FWHM for PET or SPECT, the radioactivity measured in these areas is underestimated both by its small size (known as the partial volume effect), but also spillover from adjacent radioactivity... 

Example EH Using a Spreadsheet to Find Area Beneath a Gaussian Curve... 

The absorption curves given by coal macerals approached the horizontal (magnetic field strength) axis more slowly than a Gaussian distribution curve. Shape analysis (16) showed that over much of the curve, the form closely approximated a Lorentzian distribution curve, but both positive and negative deviations were found in the wings of the curves (that is, in various examples, the curves approached the axis either somewhat more or somewhat less rapidly... 

In our example, all the uncertainties have the same magnitude. This restriction is not necessary to derive the equation for a Gaussian curve. 

From experience with many determinations, we find that the distribution of replicate data from most quantitative analytical experiments approaches that of the Gaussian curve shown in Figure 6-2c. As an example, consider the data in the spreadsheet in Table 6-2 for the calibration of a 10-mL pipet. In this experiment a small flask and stopper were weighed. Ten milliliters of water were transferred to the flask with the pipet, and the flask was stoppered. The flask, the stopper, and the water were weighed again. The temperature of the water was also measured to determine its density. The mass of the water was then calculated by taking the difference between the two masses. The mass of water divided by its density is the volume delivered by the pipet. The experiment was repeated 50 times. 

Figure 6-4a shows two Gaussian curves in which we plot the relative frequency y of various deviations from the mean versus the deviation from the mean. As shown in the margin, curves such as these can be described by an equation that contains just two parameters, the population mean p. and the population standard deviation a. The term parameter refers to quantities such as pu and a that define a population or distribution. This is in contrast to quantities such as the data values x that are variables. The term statistic refers to an estimate of a parameter that is made from a sample of data, as discussed below. The sample mean and the sample standard deviation are examples of statistics that estimate parameters p. and a, respectively. 

Because of area relationships such as these, the standard deviation of a population of data is a useful predictive tool. For example, we can say that the chances are 68.3 in 100 that the random uncertainty of any single measurement is no more than 1(T. Similarly, the chances are 95.4 in 100 that the error is less than 2cr, and so forth. The calculation of areas under the Gaussian curve is described in Feature 6-2. 

Fig 3. Example data consisting of three classes (A, B C) with two descriptors (FI F2) illustrates the Fisher linear discrimination technique. When the data is projected onto the Fisher line, between-class scatter (dashed arrows) is maximized and within-class scatter (solid arrows) is minimized. Gaussian curves for each class, defined by the mean and standard deviation upon projection onto the Fisher line, are also provided. 

Figure 1.3 Chromatographic peaks, (a) The concept of retention time. The hold-up time is the retention time of an unretained compound in the column (the time it took to make the trip through the column) (b) Anatomy of an ideal peak (c) Significance of the three basic parameters and a summary of the features of a Gaussian curve (d) An example of a real chromatogram showing that while travelling along the column, each analyte is assumed to present a Gaussian distribution of concentration.

In Fig. 2-2(a-e) is shown the distribution of X on the adsorbent bed after 4, 8, 12, 16, and 20 transfers, respectively. As X moves down the adsorbent bed, a symmetrical concentration distribution or band is formed. The center of the band after each of these transfers moves down the bed exactly one-half as far as the solvent front, as predicted by Eq. (2-1). As the band moves down the column, it widens and eventually approaches the shape of a Gaussian curve. This is illustrated in Fig. 2-3 for the preceding example after 20 transfers. Mathematically the distribution of X on the adsorbent bed (compare Table 2-1) is given by the coefficients of the binomial expansion (x - - y) , where n is the number of solvent transfers that have taken place. At large values of n the resulting distribution can be shown to converge to a Gaussian distribution. 

Often, it is found that certain data profiles are best described by non-linear bases. A typical example is curves from e.g. infrared spectroscopy [20] which are well described by a linear combination of Lorentzian peaks. Sometimes it is also possible to use Gaussian peaks. The Lorentzian peak function is written as... 

This is a good example of a Gaussian curve superposed by an underlying exponential function, which is eliminated by higher-order differentiation (Fig. 4-5). 

Another problem with this method is it tends to overestimate the diameter due to the inclusion of the background intensity when the fitting process tries to minimize the residual errors as shown in Fig. 3(b). For example, the diameter for the vessel using the half-of-the-amplitude definition is measured to be 60 pixels, whereas fitting of a twin-Gaussian curve to the simulated vessel, the diameter is 75 pixels. 

Figure 14.21 depicts the curves for five imprecise classes that assign degrees of truth to the input values. In most cases, these curves are triangular, but they can also follow the Gaussian curve or any other mathematical function. In this example, the x-axis represents the absolute input value and the y-axis stands for the degree of truth to the respective membership curve. Any element can thus belong to more than one imprecise set (membership curve). For example, a yarn fineness... 

The multiple quantum spectrum associated with this six-quantum coherence is shown in Fig. 9. The intensities of the peaks associated with higher orders of coherence drop off roughly as a Gaussian curve, and the highest order of coherence developed depends upon the time allowed for this development. In the present example, only five peaks were developed no matter how long time was allowed for higher coherences to develop, indicating that with the... 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

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