# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### Analyses of Typical Cases
#### 3.6.2 Application to Topical Medical Imaging Analyses  
It is noteworthy that the data set considered in the above deliberation has provided an example to demonstrate the representation of the WJ distribution for the Gaussian function in addition to the pertinency to extreme events and critical phenomena. Due to its skewedness; however, a concern could arise about the legitimacy of applying the Gaussian function to it as a result of possible misinterpretation in the procedure. To resolve the issue and give more examples in practical application, we are going to present the data acquired from the same experiment [7], both with and without skewedness, to directly appraise the applicability of the WJ distribution and the Gaussian function in the following. The experiment reports a multidisciplinary fitting approach to improve the localization of parathyroid glands from their Single-Photon Emission Computed Tomography (SPECT) images using generalized Gaussian functions for quantitative assessment of preoperative parathyroid SPECT/CT (CT, Computed Tomography) scintigraphy results in a large patient cohort in combination with Particle Swarm Optimization (PSO) modeling and generates the distributions of several important parameters. Two sets of experimental data are retrieved to initiate a preliminary evaluation of the effectiveness of the Gaussian function and the WJ distribution in analyzing such cases. One is the distribution of the mean scale errors for random topology, and the other is the distribution of the mean amplitude errors for random topology. Since the data are not normalized, we need to introduce a normalization factor \(k\) (\(k_{GS}\) for the Gaussian function and \(k_{WJ}\) for the WJ distribution) to modify the corresponding Equations (1a) and (2) in the way of  
\[f_{m}(x,\mu_{GS},\sigma)=\frac{k_{GS}}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu_{GS} )^{2}}{2\sigma^{2}}} \tag{30}\]  
and  
\[f_{m}(x,\alpha,\beta,\
u)=\frac{k_{WJ}\beta}{\Gamma(\alpha/\beta)}e^{-\alpha( x-\
u)-e^{-\beta(x-\
u)}} \tag{31}\]  
Figure 9 shows the fitting results and related analyses. Figure 9 presents the outcomes on the distribution of the mean scale errors for random topology (in dots) assessed by the Gaussian function (in a continuous curve) and the WJ distribution (in dashed line) separately. It is easily recognizable that both the Gaussian function and the WJ distribution can excellently match the data points, giving virtually no discernible difference between them. The fitting parameters read \(\alpha=19.06\), \(\beta=0.1444\), \(\
u=35.84\) and \(k_{WJ}=0.1027\) (\(R^{2}=0.9981\)) for the WJ distribution and \(\mu_{GS}=2.043\), \(\sigma=0.6022\) and \(k_{GS}=1026\) (\(R^{2}=0.9980\)) for the Gaussian function, correspondingly. Statistically substantiating the visual observation above, the correlation coefficients, almost one (0.9981 for the WJ distribution and 0.9980 for the Gaussian function), manifest that both of them have the same, equal goodness of fit with a slightly better performance of the WJ distribution if any. The obtained normalization factors, \(k_{GS}\) and \(k_{WJ}\), are essentially the same, in accord with the precise estimation of the experimental data by both the WJ distribution and the Gaussian function. As recorded in Figure 9, the good fits are mirrored in the calculated deviations of the fittings, with no apparent preference in spreading between the WJ distribution and the Gaussian function. Figure 9: Comparative analyses of experimental data for different parameters obtained from the topical medical studies.