#
Applying a Flexible Fuzzy Adaptive Regression to Runoff Estimation^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Simplified Reasoning System by Using Crisp Regression

_{1}and β

_{2}.

_{1}and β

_{2}, the following membership functions are modulated (Figure 1). From the picture, it is obvious that the following property holds:

IF P (annual precipitation) is low, THEN y (annual Runoff) is $\left(y={a}_{10}+{a}_{11}P\right)$ | $\}$ (MODEL 1) |

IF P (annual precipitation) is high, THEN y (annual Runoff) is $\left(y={a}_{20}+{a}_{21}P\right)$ |

_{1}and β

_{2}are known, then the next linear system of algebraic equations is produced [3,4] in order to determine the vector

**θ**:

**b**is the matrix that contains the values of the measured runoff (dependent variables)

**θ***can be found:

_{1}and β

_{2}are known, the coefficients of the crisp linear equations can be determined with respect to Equations (3) and (5).

## 3. Proposed Simplified Architecture of the Fuzzy Rule Based System by Using Fuzzy Regression

IF P is low, THEN Annual Runoff, y is $\left(\tilde{y}={\tilde{a}}_{10}+{\tilde{a}}_{11}P\right)$ | $\}$ (MODEL 2) |

IF P is high, THEN Annual Runoff, y is $\left(\tilde{y}={\tilde{a}}_{20}+{\tilde{a}}_{21}P\right)$ |

## 4. The Proposed Learning Process with the Use of PSO

_{1}, β

_{2}). More details about the method can be found in [8,9,10,11].

_{1}, β

_{2}, which are the aforementioned parameters of the membership functions. Two cases can be distinguished. The first one is the use of crisp regression, and hence, the least square method is activated, whilst the second choice is the use of the fuzzy model of Tanaka. In the case of the Tanaka fuzzy regression being used, for each candidate solution, a linear programming problem is used.

## 5. Case Study

^{2}, running from north to south along 40 km in the Huelva province. The mean annual precipitation is 574 mm/year, and mean annual runoff is 106 mm/year. It is regulated by the Piedras and Los Machos reservoirs, which are operated for water supply and irrigation. The Aguas river is a short coastal river in the south of Spain, running along 65 km through the east of the province of Almería. The contributing basin is 547 km

^{2}. The climate is semiarid, with mean annual precipitation of 334 mm/year and mean annual runoff of 41 mm/year.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The proposed method and the conventional regression applied in order to assess a relationship between annual precipitation and runoff in the case of Rio Piedras.

**Figure 3.**The evolution of the parameter (β

_{1}, β

_{2}) after a sufficient number of iterations in the case of Rio Piedra.

**Figure 4.**The proposed method and the conventional regression applied in order to assess a relationship between annual precipitation and runoff in the case of Rio de Aguas.

**Figure 5.**The proposed method with fuzzy regression curves applied in order to assess a fuzzy relationship between annual precipitation and annual runoff in the case of Rio Piedras.

**Figure 6.**Fuzzy estimation of the runoff in the case that the annual runoff is y = 98.8538 mm in the case of Rio Piedras.

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## Share and Cite

**MDPI and ACS Style**

Spiliotis, M.; Garrote, L.
Applying a Flexible Fuzzy Adaptive Regression to Runoff Estimation. *Environ. Sci. Proc.* **2023**, *25*, 85.
https://doi.org/10.3390/ECWS-7-14308

**AMA Style**

Spiliotis M, Garrote L.
Applying a Flexible Fuzzy Adaptive Regression to Runoff Estimation. *Environmental Sciences Proceedings*. 2023; 25(1):85.
https://doi.org/10.3390/ECWS-7-14308

**Chicago/Turabian Style**

Spiliotis, Mike, and Luis Garrote.
2023. "Applying a Flexible Fuzzy Adaptive Regression to Runoff Estimation" *Environmental Sciences Proceedings* 25, no. 1: 85.
https://doi.org/10.3390/ECWS-7-14308