# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 4 Conclusions and Future Research  
We have developed the three methods of transforming the WJ distribution to the form of the Gaussian function by means of the proper Taylor expansion, showing an accurate and self-consistent approach by expanding the double exponential part around \(x=0\) (Method I) or around \(x=x_{m}\) (Method III). The relationship between the parameters \(\mu_{GS}\) and \(\sigma\) of the Gaussian function and \(\alpha\), \(\beta\) and \(\mu_{WJ}\) (or \(\
u\)) of the WJ distribution are systematically examined. Overall, the WJ distribution can represent the Gaussian function quite well, but the latter only fits the former satisfactorily under the condition of special sets of parameters. In addition, the parameter scaling relation of the WJ distribution to express the general Gaussian function is given. For assessing practical usefulness, we have performed the evaluation of typical cases on advanced magnetism and topical medical image processing by the WJ distribution and the Gaussian function separately. In consequence, the WJ distribution may be regarded as an extended Gaussian function, and our work initiates building up a bridge for the gap between relevant extreme value theories and Gaussian processes. It may be appropriate to emphasize that the WJ distribution is of interest for more study owing to its suitability for circumstances subject to the Gaussian function as well as its unique potential to describe the scenery of extreme events and critical phenomena, namely,suggesting practically a unifying platform to the pertinent data processing of those quite distinct fields. In this way, future research may proceed in three aspects. (1) It is clear that Methods I and III developed in the work are accurate and self-consistent in representing the Gaussian function by the WJ distribution, but Method II shows a limited capability in doing so. As Method II has simple expressions, however, it is easy to carry out proper normalization (direct normalization) though the solution found is far from satisfactory as outlined in the section of Method II. Nevertheless, both Methods I and III possess more complicated expressions for solving and promising possibilities of direct normalization but with great challenges in finding optimized solutions if existent. Undoubtedly, it is motivating to conduct more research along the line of direct normalization in the future. (2) We have directly compared in the text the applicability of both the WJ distribution and the Gaussian function to advanced magnetism and current medical image analyses, showing a unified framework provided by the WJ distribution in preference. It is useful to broaden and deepen such initial investigations of germane experimental databases to give more insights into relevant phenomena for mechanistic understanding and advanced applications. (3) A comparison with other distributions is constructive and should put the research in a far-reaching perspective, e.g., to enrich studies in connecting pertinent extreme value theories and Gaussian processes. Conceptualization, [PERSON] and S.G.; formal analysis, [PERSON] and [PERSON]; methodology, [PERSON] and [PERSON]; investigation, [PERSON] and [PERSON]; software, [PERSON]; computation, [PERSON]; validation, [PERSON] and [PERSON]; data curation, [PERSON] and [PERSON]; visualization, [PERSON] and [PERSON]; funding acquisition, [PERSON]; project administration, [PERSON]; resources, [PERSON] and [PERSON]; supervision, [PERSON]; writing--original draft, [PERSON] and [PERSON]; writing--review and editing, [PERSON] and [PERSON]. All authors gave final approval for publication and agreed to be held accountable for the work performed therein. All authors have read and agreed to the published version of the manuscript. The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. We are grateful to [PERSON] from Hohai University for valuable discussions and [PERSON] from Shandong University for kind support.