# 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
(**a**) Distribution of mean scale errors: Experimental data (dots), Gaussian function (continuous curve) and WJ distribution (dash); (**b**) Distribution of mean amplitude errors: Experimental data (dots), Gaussian function (continuous curve) and WJ distribution (dash); (**c**) Calculated deviations of mean scale errors: Gaussian function (dots) and WJ distribution (triangles); (**d**) Calculated deviations of mean amplitude errors: Gaussian function (dots) and WJ distribution (triangles); (**e**) Fitting Gaussian function by WJ distribution for the case of mean scale errors: Gaussian function (dots) and WJ distribution (curve); (**f**) Fitting Gaussian function by WJ distribution for the case of mean amplitude errors: Gaussian function (dots) and WJ distribution (curve). Figure 9b displays the fitting results on the distribution of the mean scale errors for random topology (in dots) appraised by the Gaussian function (in a continuous curve) and the WJ distribution (in dashed line) individually. In terms of the figure, both the Gaussian function and the WJ distribution can capture the major features of the data, but the performance becomes much worse for the Gaussian function than for the WJ distribution. The relevant parameters from the fittings are \(\alpha=0.5157\), \(\beta=0.4840\), \(\
u=3.343\) and \(k_{WJ}=0.2218\) (\(R^{2}=0.9846\)) for the WJ distribution and \(\mu_{GS}=3.700\), \(\sigma=2.152\) and \(k_{GS}=0.2032\) (\(R^{2}=0.9531\)) for the Gaussian function, correspondingly. Examining the correlation coefficients, 0.9846 for the WJ distribution and 0.9531 for the Gaussian function, statistically reveals the degrees of deviation, which show much better performance by the WJ distribution than by the Gaussian function. Nonetheless, the acquired normalization factors, \(k_{GS}\) and \(k_{WJ}\), are quite different, reflecting the fact of the dissimilar estimation capabilities in gauging the experimental data between the WJ distribution and the Gaussian function. As shown in Figure 9d, such discrepancy is straightforwardly revealed in the calculated deviations of the fittings, which indicate that the maximum deviation from the Gaussian function can be twice as large as from the WJ distribution. Subsequently, we apply the WJ distribution to refit the Gaussian function with the parameter set obtained from the fitting above. In the case of the mean scale errors, the parameter set of \(\mu_{GS}=2.043\), \(\sigma=0.6022\) and \(k_{GS}=0.1026\) is used, and the fitting outcome is given in Figure 9e with the parameter values of \(\alpha=20.2\), \(\beta=0.1358\), \(\
u=38.85\) and \(k_{WJ}=0.1026\) (\(R^{2}=0.9997\)) for the fitted WJ distribution. As anticipated, the agreement between the WJ distribution and the Gaussian function is, in virtue, seamless as the correlation coefficient of 0.9997 is essentially one, confirming once more the representation of the WJ distribution for the Gaussian function. In the case of the mean amplitude errors, the parameter set is \(\mu_{GS}=3.700\), \(\sigma=2.152\) and \(k_{GS}=0.2032\), and the fitting result is shown in Figure 9f with the parameter values of \(\alpha=5.204\), \(\beta=0.04077\), \(\
u=122.6\) and \(k_{WJ}=0.2032\) (\(R^{2}=0.9997\)) for the fitted WJ distribution. Analogous to the case of the mean scale errors, the superior agreement, as reflected in the outstanding correlation coefficient of 0.9997, between them proves once again the representation of the WJ distribution for the Gaussian function. In summary, the preceding analysis shows that the WJ distribution, overall better than the Gaussian function, can give a unified approach to treating the data of the different parameters in such topical medical studies for the well-being of the patients. The unification should offer valuable insights to deepen the understanding of the experimental results and potentially facilitate the operative recovery.