*5.2. Multiple Regression Analyses*

5.2.1. Multiple Regression Analysis for the Reconstruction of the Unit Stream Power Formula

As a natural sequence, and to further test the successfulness of the results, the unit stream power formula Equation (4) was rebuilt. Basically, knowing both the dependent and independent variables, the coefficients of the equation were recalculated.

In the case that the conventional least square-based regression is used, the following relation is achieved:

$$\log \mathcal{C}\_{\mathcal{F}} = -0.2215 \mathcal{X}\_1 - 0.4369 \mathcal{X}\_2 + 1.7105 \mathcal{X}\_3 - 0.4271 \mathcal{X}\_4 - 0.4742 \mathcal{X}\_5 + 4.9998 \tag{24}$$

The coefficient of determination, *R*2, is equal to:

$$R^2 = 1 - \frac{\sum\_{j=1}^{m} \left(\log \mathcal{C}\_{t\_j} - \log \mathcal{C}\_{F\_j}\right)^2}{\sum\_{j=1}^{m} \left(\log \mathcal{C}\_{t\_j} - \overline{\log \mathcal{C}\_t}\right)^2} = 0.857\tag{25}$$

where log*Ctj* , log*CFj* , log*Ct* are the measured *j*th value of the concentration, the calculated based on Equation (24) (crisp regression) and the mean value, respectively.

#### 5.2.2. Fuzzy Regression Analysis

In the case that the aforementioned fuzzy regression is used, the following equation is produced:

$$\begin{aligned} \log \mathcal{C}\_{\mathcal{F}} &= 0.1602 X\_1 + (0.2842, 0.1193) X\_2 + (1.2643, 0.1394) X\_3 - 0.1694 X\_4 \\ &+ (-0.2768, 0.1191) X\_5 + (4.1880, 0.4014) \end{aligned} \tag{26}$$

The first term in the parentheses expresses the central value and the second term the semi-width of the produced fuzzy coefficient.

The total amount of uncertainty, namely the sum of the semi-widths regarding the available data, is:

$$J = \left\{ 432c\_0 + \sum\_{j=1}^{432} \sum\_{i=1}^{5} c\_i |X\_{ij}| \right\} = 289.069 \tag{27}$$

The projection of the produced logarithmic concentration Equation (26), with respect to several input variables, is presented in Figure 2.

**Figure 2.** Projection of the achieved fuzzy relation regarding the log of the total sediment concentration with respect to (**a**) *X*3, (**b**) *X*2, (**c**) *X*4.

As can be seen, all the available data are included within the produced fuzzy band. Furthermore, the use of only one input variable—in this case, variable *X*3—is separately investigated (Figure 3). In case that the crisp linear regression is used with the *X*<sup>3</sup> as the only independent variable, the results are similar, and the squared correlation coefficient, *r*2, is equal to 0.801. Hence, the linear dependence can be suggested. In this case, Equation (26) becomes:

> 432 *j*=1 *c*3|*X*3| ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭

$$
\log \mathcal{C}\_{\mathcal{F}} = (1.1872, \, 0.2243)X\_3 + (4.5954, \, 0.4948) \tag{28}
$$

= 343.305 (29)

and the corresponding objective function obtains the following value:

*J* = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 432*c*<sup>0</sup> +

**Figure 3.** Fuzzy linear regression to achieve the log of the total sediment concentration with respect to *X*3.

Based on the value of the objective function, *J*, it is obvious that this simplification (i.e., considering *X*<sup>3</sup> as the only independent variable) increases the uncertainty. However, the usefulness is that the emphasis is put on the subtraction of the critical unit stream power from the exerted unit stream power, as a main independent variable. Going in the opposite direction, if only the variable *X*<sup>3</sup> is removed, then the uncertainty of the produced fuzzy band is greater than the above value (*J* = 396.75).

An interesting perspective is that by adopting a polynomial form, in the above simplification, a small reduction of the uncertainty is achieved and hence, Equation (28) cannot be further improved. Indeed, a small reduction of the fuzzy band is achieved, if the fourth-degree polynomial regression is used.

In Figure 3, the observations against the results of the conventional linear regression, as well as the results of fuzzy linear and polynomial regressions, by using only *X*<sup>3</sup> as input variable, are depicted. As it can be observed from the figure, the data of the fuzzy fourth-degree polynomial regression and the data of the fuzzy linear regression almost overlap for the most part. However, the fourth-degree polynomial regression presents an "irrational behavior" in the area of low *X*<sup>3</sup> values, from a physical meaning point of view. To better explain this, as the difference "exerted unit stream power minus critical unit stream power" (here, represented by *X*3) grows larger, a higher sediment transport, and therefore a higher sediment concentration is expected. Simply put, *logCF* and *X*<sup>3</sup> are similar amounts and the increase of one by the decrease of the other is not justified. The negligible reduction of the uncertainty, as well as the "irrational behavior" of the fuzzy fourth-degree polynomial regression, indicates the improperness of the polynomial models for these data. From the above it is concluded that the auxiliary variable *X*<sup>3</sup> is the most significant parameter parameter. The use of high polynomial extension to Equation (28) did not improve the results. Equation (26) results in significantly less uncertainty and should be preferred.

However, a fuzzy band with high spread will include all the data, but this will be a non-useful approach. Therefore, another suitability measure, *JJ*, is proposed, which is equal to the mean ratio of the total spread - log*CFj* <sup>+</sup> <sup>−</sup> log*CFj* − to the central value (μ = 1, with the index *j*), log*CFj* , and when it is applied for Equation (26), it leads to the following result [37]:

$$\|f\| = \frac{1}{N} \sum\_{j=1}^{N} \frac{\left(\log \mathcal{C}\_{\mathcal{F}\_j} + \log \mathcal{C}\_{\mathcal{F}\_j}\right)}{\overline{\log \mathcal{C}\_{\mathcal{F}\_j}}} = 0.5644\tag{30}$$

where *N* is the number of data; in brief, the measure *JJ* expresses the mean uncertainty of the produced fuzzy band as a percentage of the central value. It is desirable to get low values for *JJ* [37]. At this point, it must be clarified that from a sediment transport point of view, the results can be characterized as sufficiently good. It should be noted that the measure *JJ* takes a better value compared with the corresponding *JJ* measure achieved by Kaffas et al. [79]. However, that study was based only on the experimental data of Guy et al. [65].
