**6. Conclusions**

The objective of this research is to transform the arithmetic coefficients of the total sediment transport rate formula of Yang, into fuzzy numbers, and thus create a fuzzy relationship that will provide a fuzzy band of in-stream sediment concentration. A very large set of experimental data, in flumes, was used for the fuzzy regression analysis. The reason for selecting the fuzzy regression is that it provides a fuzzy band not only for the coefficients of the independent variables, but for the final result, as well, which is the total sediment concentration. This means that the resulting sediment concentration is not a crisp value, but a range of values, which stretch to a value equal to the semi-width on both sides of the central value. It is proved well, by the results, that this range of values deals efficiently with the uncertainties and the ambiguous nature of sediment transport processes. Apart from the measurement errors, the computational part, and specifically the physical simplifications, i.e., one-dimensional, uniform and steady flow, grain size distribution, etc., increase the uncertainty. An interesting perspective is that even if the validation data are observations from natural rivers, where significant uncertainty and simplifications take place, these are successfully captured by the proposed fuzzy band. The minimum advantage of the fuzzy band produced is that all the data must be included. However, a main criterion is the produced width of the fuzzy band. Based on this criterion, the authors concluded that the simplification of using only one variable should be avoided, and furthermore that a determinant variable is the subtraction of the critical unit stream power from the exerted unit stream power (*X*3). Nevertheless, a simplification based on the *X*<sup>3</sup> (i.e., only the variable *X*<sup>3</sup> is taken into account) leads to a fuzzy linear curve that can be used to interpret the phenomenon. The produced fuzzy band compared with the central values indicates the good performance of the proposed fuzzy curve. In terms of elaboration of the original data utilized by Yang for the establishment of the unit stream power theory, this research goes the closest possible to what could be called "fuzzy twin" of Yang's stream sediment transport formula.

**Author Contributions:** Conceptualization, M.S. (Mike Spiliotis); Methodology, M.S. (Mike Spiliotis), K.K. and V.H.; Validation, M.S. (Mike Spiliotis), K.K. and M.S. (Matthaios Saridakis); Investigation, K.K.; Data curation, K.K.; Writing—original draft preparation, K.K. and M.S. (Mike Spiliotis); Writing—review and editing, V.H., K.K., M.S. (Mike Spiliotis) and M.R.; Visualization, M.S. (Mike Spiliotis), M.S. (Matthaios Saridakis) and K.K.; Supervision, V.H. and M.S. (Mike Spiliotis). All authors have read and agreed to the published version of the manuscript.

**Funding:** K.K. and M.R. were supported by the project Sediplan–r (FESR1002) and financed by the European Regional Development Fund (ERDF) Investment for Growth and Jobs Programme 2014–2020.

**Acknowledgments:** The authors would like to express their thankfulness to the reviewers for their comments and suggestions, which helped to improve the quality of this study.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

The Nash-Sutcliffe Efficiency (NSE), proposed by Nash and Sutcliffe in 1970, is defined as one minus the sum of the squared differences between the observed and predicted data normalized by the variance of the observed values [88]:

$$NSE = 1 - \frac{\sum\_{j=1}^{m} \left(\boldsymbol{y}\_{j} - \boldsymbol{\hat{y}}\_{j}\right)^{2}}{\sum\_{j=1}^{m} \left(\boldsymbol{y}\_{j} - \overline{\boldsymbol{y}}\right)^{2}}\tag{A1}$$

where *yj* are the observed values; *y*ˆ*<sup>j</sup>* the predicted values; *y* the mean observed value; and *j* = 1, ... , *m*.

The use of NSE is not restricted solely in regression models, but extends for any application of hydrological modeling and, therefore, in its general use, it takes values between −∞ and 1, i.e., *NSE* ∈ (−∞, +1]. An efficiency lower than zero indicates that the mean value of the observed time series would have been a better predictor than the model [89]. In such cases, the model should be rejected.

At this point, the relation between the coefficient of determination, *R*2, and the Nash-Sutcliffe Efficiency, NSE, should be clarified. In case of a multiple regression, *R*<sup>2</sup> and NSE are simply identical. However, the coefficient of determination has a value bounded between 0 and 1 [90], that is *<sup>R</sup>*<sup>2</sup> <sup>∈</sup> [0, 1] [82].

An interesting perspective is the comparison of the correlation coefficient, *r*, with the Nash-Sutcliffe Efficiency. In case of a conventional linear regression model with only one independent variable, the squared value of the correlation coefficient, *r*2, is equal to *R*<sup>2</sup> = *NSE*. The correlation coefficient, *r*, indicates the strength and the direction of a linear relationship with respect to the data, whilst it cannot imply causation.

In the present model and with regard to the training set, namely the laboratory data, in the case of the crisp multiple regression, the NSE is identical to the *R*2, and obviously non-negative, i.e., *NSE* = *R*<sup>2</sup> = 0.857. However, when the crisp multiple regression equation is used upon the other validation measurements, from natural streams, then it takes negative values, i.e., *NSE* = −1.207.

Finally, if only one independent variable is used (in this case, *X*3), again for the training data, then the NSE is identical to the *R*<sup>2</sup> and with the squared value of the correlation coefficient, *r*, *NSE* = *R*<sup>2</sup> = *r*<sup>2</sup> = 0.801.
