## Abstract

Solar still productivity (SSP) is of vital importance in solar desalination project planning and management. In this investigation, the applicability of adaptive neuro-fuzzy inference systems (ANFIS) and multiple nonlinear regression (MNLR) in modeling SSP is investigated. Eight different membership functions (MFs) were used with ANFIS approach. Solar radiation, relative humidity, feed flow rate, total dissolved solids of feed, and brine are used as inputs to the models. The outcomes of the ANFIS are compared with those of the MNLR with respect to correlation coefficient (CC), root mean square error (RMSE), overall index of model performance (OI), and mean absolute error (MAE). Comparison results illustrate the generalized bell MF with ANFIS model has better accuracy than the other seven MFs in modeling SSP. Performance evaluation criteria show the predictive abilities of ANFIS and MNLR models were very similar and can be suggested to predict SSP effectively. Using the ANFIS model, the average value of CC, RMSE, OI, and MAE was 0.96, 0.05 L/m^{2}/h, 0.91, and 0.04 L/m^{2}/h, respectively. The corresponding values for the MNLR model were CC = 0.97, RMSE = 0.06 L/m^{2}/h, OI = 0.93, and MAE = 0.05 L/m^{2}/h. One of the advantages of MNLR model is using explicit equations.

## INTRODUCTION

Water scarcity is a major danger to humanity in the future. Consequently, converting saline water into pure water is one of the feasible solutions to meet water demand. A solar still is the simplest solar desalination device, but its productivity is very low. Hence, numerous experimental investigations and studies in recent years, such as those by Selvaraj & Natarajan (2018), Panchal & Mohan (2017), Lal *et al.* (2017), and Kaviti *et al.* (2016), have been done to evaluate, enhance, and raise the productivity of solar stills. The main disadvantage of experimental work is its cost and the fact that it is time-consuming. Thus, mathematical modeling for productivity is considered suitable to evaluate the performance of solar stills and facilitate assessment on the basis of efficiency.

In this study, an adaptive neuro-fuzzy inference system (ANFIS) as a mathematical modeling technique was developed and used for the performance analysis of an inclined solar passive solar still. The usage of ANFIS has been applied broadly in most areas of science and technology, especially in the field of solar engineering by several investigators (Amirkhani *et al.* 2015; Jović *et al.* 2016; Mohammadi *et al.* 2016; Yaïci & Entchev 2016; Belhachat & Larbes 2017; Halabi *et al.* 2018; Mashaly & Alazba 2018; Shabaan *et al.* 2018). ANFIS is the combination of artificial neural networks (ANNs) and fuzzy logic (FL) (Jang 1993). Compared with ANNs and FL, ANFIS needs less data learning and performs better in forecasting with its rule-based inference mechanism.

This study describes the application of ANFIS to the problem of predicting solar still productivity (SSP) and discusses the precision of eight different membership functions, namely, triangle, trapezoidal, generalized bell, Gaussian, two-sided Gaussian, Pi curve, product of two sigmoidal functions, and difference between two sigmoidal functions (DSIGMF), in SSP estimation. The second objective of this investigation aimed at comparing ANFIS as an intelligence technique and the multiple nonlinear regression (MNLR) as a classical technique for SSP forecasting. The input parameters to the MNLR model were the same as the ones used in the ANFIS models. The predictive capabilities of the models were assessed using standard statistical performance evaluation criteria.

## MATERIALS AND METHODS

### Experimental setup

The experiments were conducted at the Agricultural Research and Experiment Station, Department of Agricultural Engineering, King Saud University, Riyadh, Saudi Arabia (24°44′10.90″N, 46°37′13.77″E), during the period of February–April 2013, where the weather data were obtained from a weather station (model: Vantage Pro2, manufacturer: Davis, USA) located close to the experimental site (24°44′12.15″N, 46°37′14.97″E). The solar still system utilized in the experiments consists of one stage of C6000 panel (Carocell Solar Panel, F CUBED Ltd, Australia). The area of the used panel was 6 m^{2}. The solar still is manufactured as a panel using modern cost-effective materials, such as coated polycarbonate plastic. The panel heats and distills a film of water flowing over the absorber mat of the panel. The panel was fixed at an angle of 29° to horizontal. The basic construction materials were galvanized steel legs, aluminum frame, and polycarbonate covers. The transparent polycarbonate was coated from inside by special coating material to prevent fogging (patent for F CUBED, Australia). A cross-sectional view of the solar still is presented in Figure 1. The working operation of the available system is summarized in the following paragraphs.

The water was fed to the panel using a centrifugal pump (model: PKm 60, 0.5 HP, Pedrollo, Italy) with a constant flow rate of 10.74 L/hr. Eight drippers/nozzles drip the feed causing a film to flow over the absorbent mat. Under the absorbent mat there is an aluminum screen which helps to distribute the dropped water over the absorber mat. Beneath the aluminum screen, there is a plate, also made from aluminum. The aluminum was selected for the manufacturing process because aluminum is a hydrophilic material that assists even distribution of the sprayed water. The water flows through and over the absorbent mat, and as the solar energy is absorbed and partially collected inside the panel, water is heated and hot air is naturally circulated within the panel. The hot air flows in the upper part towards the top, and then reverses the direction towards the bottom. By this circulation, the humid air touches the cooled surfaces of the transparent polycarbonate cover and the bottom polycarbonate layer, thus water condenses and flows down the panel and is collected as a distilled stream. Seawater was used as feed water input to the system. The solar still system was run during the period from 23/02/2013 to 23/04/2013. Raw seawater was obtained from the Gulf, Dammam, East Saudi Arabia (26°26′24.19″N, 50°10′20.38″E). The initial concentration of the total dissolved solids (TDS), pH, density (, and electrical conductivity (EC) of the raw seawater were 41.4 ppt, 8.02, 1.04 g.cm^{−3}, and 66.34 mS cm^{−1}, respectively. The productivity or the amount of distilled water produced (SSP) during a time period by the system was obtained by collecting the cumulative amount of water produced over time. The temperatures of the feed (T_{F}) and brine (T_{B}) were measured by using thermocouples (T-type, UK). Temperature data for feed brine water were recorded on a data logger (model: 177-T4, Testo, Inc., UK) at 1 min intervals. The amount of feed water (M_{F}) was measured by calibrated digital flow meter mounted on the feed water line (micro-flo, Blue-White, USA). The amount of brine water and distilled water were measured by graduated cylinder. TDS and EC were checked using calibrated (TDS) meter (Cole-Parmer Instrument, Vernon Hills, USA). A pH meter (model: 3510 pH meter, Jenway, UK) was utilized to determine the pH. *ρ* was measured by a digital density meter (model: DMA 35_{N}, Anton Paar, USA). The seawater was fed separately to the panel using the pump mentioned above. The residence time for the water to pass through the panel was about 20 minutes. Consequently, the flow rate for feed water, distilled water, and brine water were measured at 20 min. Also, the total dissolved solids of feed water (TDSF) and of brine water (TDSB) were measured every 20 minutes. The weather data, such as ambient temperature (To), relative humidity (RH), wind speed (WS), and solar radiation (SR) were obtained from the weather station mentioned above. Here, there is one dependent variable which is the SSP and nine independent variables which are To, RH, WS, SR, TF, TB, TDSB, TDSF, and MF.

### Input parameters

In this study, field data obtained from the experimental work were used for the training, testing, and validation of the MNLR and ANFIS models. One of the most important steps in the modeling process for satisfactorily predicting results is the selection of the input parameters, since these parameters determine model structures and affect the weighted coefficient and the overall performance of the model. For this purpose, a correlation matrix is performed to assess relationships between the dependent parameter (SSP) and the independent parameters (To, RH, WS, SR, TF, TB, MF, TDSF, and TDSB) as given in Table 1. This matrix allows us to recognize how each parameter affects SSP and eventually which parameter(s) should be used as input in MNLR and ANFIS models. Furthermore, this matrix displays the findings of correlation analysis conducted between each pair of parameters. The strongest correlation is observed between SSP and SR with Pearson's correlation coefficient (CC) of +0.734. Furthermore, SSP is found to be well correlated with TDSF with CC = −0.402. The + and − signs refer to positive correlation and negative correlation, respectively. This agrees with the findings of Mashaly *et al.* (2016). Also, there is a significant correlation between SSP and MF and TDSB with CC = 0.25, and −0.172, respectively. On the other hand, a very weak correlation is found between SSP and To and TF with CC = −0.072, and −0.061, respectively; consequently, we do not consider them as input parameters. The correlation analysis also led to the exclusion of the WS and TB due to their high collinearity with other parameters, though there are significant correlations with the SSP. Although some of the parameters also appear correlated to others, these were included in the modeling process since their inclusion was found to improve its prediction performance, primarily by enhancing the CC. The same argument was also invoked to consider RH as an input parameter with low CC. The descriptive statistics of the input parameters used in the training, testing, and validation are shown in Table 2.

To | RH | WS | SR | TF | TB | MF | TDSF | TDSB | SSP | |
---|---|---|---|---|---|---|---|---|---|---|

To | 1.00 | −0.66 | −0.14 | −0.15 | 0.91 | 0.06 | 0.44 | −0.01 | −0.15 | −0.07 |

RH | −0.66 | 1.00 | −0.08 | 0.15 | −0.80 | 0.05 | −0.72 | 0.23 | 0.45 | 0.01 |

WS | −0.14 | −0.08 | 1.00 | 0.22 | −0.01 | 0.33 | −0.34 | 0.64 | 0.49 | −0.31 |

SR | −0.15 | 0.15 | 0.22 | 1.00 | −0.09 | 0.82 | −0.27 | 0.22 | 0.39 | 0.73 |

TF | 0.91 | −0.80 | −0.01 | −0.09 | 1.00 | 0.13 | 0.48 | 0.06 | −0.11 | −0.06 |

TB | 0.06 | 0.05 | 0.33 | 0.82 | 0.13 | 1.00 | −0.40 | 0.49 | 0.57 | 0.40 |

MF | 0.44 | −0.72 | −0.34 | −0.27 | 0.48 | −0.40 | 1.00 | −0.75 | −0.84 | 0.25 |

TDSF | −0.01 | 0.23 | 0.64 | 0.22 | 0.06 | 0.49 | −0.75 | 1.00 | 0.94 | −0.40 |

TDSB | −0.15 | 0.45 | 0.49 | 0.39 | −0.11 | 0.57 | −0.84 | 0.94 | 1.00 | −0.17 |

SSP | −0.07 | 0.01 | −0.31 | 0.73 | −0.06 | 0.40 | 0.25 | −0.40 | −0.17 | 1.00 |

To | RH | WS | SR | TF | TB | MF | TDSF | TDSB | SSP | |
---|---|---|---|---|---|---|---|---|---|---|

To | 1.00 | −0.66 | −0.14 | −0.15 | 0.91 | 0.06 | 0.44 | −0.01 | −0.15 | −0.07 |

RH | −0.66 | 1.00 | −0.08 | 0.15 | −0.80 | 0.05 | −0.72 | 0.23 | 0.45 | 0.01 |

WS | −0.14 | −0.08 | 1.00 | 0.22 | −0.01 | 0.33 | −0.34 | 0.64 | 0.49 | −0.31 |

SR | −0.15 | 0.15 | 0.22 | 1.00 | −0.09 | 0.82 | −0.27 | 0.22 | 0.39 | 0.73 |

TF | 0.91 | −0.80 | −0.01 | −0.09 | 1.00 | 0.13 | 0.48 | 0.06 | −0.11 | −0.06 |

TB | 0.06 | 0.05 | 0.33 | 0.82 | 0.13 | 1.00 | −0.40 | 0.49 | 0.57 | 0.40 |

MF | 0.44 | −0.72 | −0.34 | −0.27 | 0.48 | −0.40 | 1.00 | −0.75 | −0.84 | 0.25 |

TDSF | −0.01 | 0.23 | 0.64 | 0.22 | 0.06 | 0.49 | −0.75 | 1.00 | 0.94 | −0.40 |

TDSB | −0.15 | 0.45 | 0.49 | 0.39 | −0.11 | 0.57 | −0.84 | 0.94 | 1.00 | −0.17 |

SSP | −0.07 | 0.01 | −0.31 | 0.73 | −0.06 | 0.40 | 0.25 | −0.40 | −0.17 | 1.00 |

To, ambient temperature; RH, relative humidity; WS, wind speed; SR, solar radiation; TF, temperature of feed water; TB, temperature of brine water; MF, feed flow rate; TDSF, total dissolved solids of feed; TDSB, total dissolved solids of brine; SSP, solar still productivity.

Statistical parameters | Input parameters | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Training | Testing | Validation | |||||||||||||

RH (%) | SR (W/m^{2}) | M_{F} (L/min) | TDS_{F} (PPT) | TDS_{B} (PPT) | RH (%) | SR (W/m^{2}) | M_{F} (L/min) | TDS_{F} (PPT) | TDS_{B} (PPT) | RH (%) | SR (W/m^{2}) | M_{F} (L/min) | TDS_{F} (PPT) | TDS_{B} (PPT) | |

AVG | 23.36 | 590.03 | 0.21 | 81.36 | 96.69 | 23.29 | 566.11 | 0.21 | 77.08 | 91.72 | 23.51 | 613.12 | 0.21 | 78.59 | 95.09 |

SE | 1.24 | 17.10 | 0.00 | 2.79 | 2.77 | 2.43 | 33.56 | 0.01 | 5.17 | 5.54 | 2.57 | 44.87 | 0.01 | 7.66 | 7.27 |

MED | 18.88 | 635.60 | 0.24 | 75.60 | 95.80 | 18.52 | 585.88 | 0.24 | 68.95 | 86.05 | 20.40 | 654.88 | 0.24 | 74.95 | 89.45 |

SD | 13.08 | 180.99 | 0.04 | 29.49 | 29.34 | 13.77 | 189.86 | 0.04 | 29.24 | 31.32 | 10.29 | 179.46 | 0.04 | 30.66 | 29.09 |

MIN | 12.90 | 75.10 | 0.13 | 41.40 | 46.20 | 13.00 | 188.29 | 0.13 | 41.60 | 49.00 | 13.00 | 157.62 | 0.13 | 41.70 | 51.80 |

MAX | 66.95 | 920.69 | 0.25 | 130.00 | 132.80 | 70.00 | 890.50 | 0.25 | 129.30 | 130.50 | 50.19 | 880.60 | 0.25 | 128.90 | 130.40 |

Statistical parameters | Input parameters | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Training | Testing | Validation | |||||||||||||

RH (%) | SR (W/m^{2}) | M_{F} (L/min) | TDS_{F} (PPT) | TDS_{B} (PPT) | RH (%) | SR (W/m^{2}) | M_{F} (L/min) | TDS_{F} (PPT) | TDS_{B} (PPT) | RH (%) | SR (W/m^{2}) | M_{F} (L/min) | TDS_{F} (PPT) | TDS_{B} (PPT) | |

AVG | 23.36 | 590.03 | 0.21 | 81.36 | 96.69 | 23.29 | 566.11 | 0.21 | 77.08 | 91.72 | 23.51 | 613.12 | 0.21 | 78.59 | 95.09 |

SE | 1.24 | 17.10 | 0.00 | 2.79 | 2.77 | 2.43 | 33.56 | 0.01 | 5.17 | 5.54 | 2.57 | 44.87 | 0.01 | 7.66 | 7.27 |

MED | 18.88 | 635.60 | 0.24 | 75.60 | 95.80 | 18.52 | 585.88 | 0.24 | 68.95 | 86.05 | 20.40 | 654.88 | 0.24 | 74.95 | 89.45 |

SD | 13.08 | 180.99 | 0.04 | 29.49 | 29.34 | 13.77 | 189.86 | 0.04 | 29.24 | 31.32 | 10.29 | 179.46 | 0.04 | 30.66 | 29.09 |

MIN | 12.90 | 75.10 | 0.13 | 41.40 | 46.20 | 13.00 | 188.29 | 0.13 | 41.60 | 49.00 | 13.00 | 157.62 | 0.13 | 41.70 | 51.80 |

MAX | 66.95 | 920.69 | 0.25 | 130.00 | 132.80 | 70.00 | 890.50 | 0.25 | 129.30 | 130.50 | 50.19 | 880.60 | 0.25 | 128.90 | 130.40 |

AVG, average value; SE, standard error; MED, median; SD, standard deviation; MIN, minimum value; MAX, maximum value; RH, relative humidity; SR, solar radiation; MF, feed flow rate; TDSF, total dissolved solids of feed; TDSB, total dissolved solids of brine.

### Adaptive neuro-fuzzy inference system (ANFIS)

*x*and

*y*are the inputs and

*A*

_{1},

*B*

_{1},

*A*

_{2},

*B*

_{2}are fuzzy sets p

_{1}, p

_{2}, q

_{1}, q

_{2}, r

_{1}, and r

_{2}are the coefficients of the output function that are determined during the training.

**Layer 3**is the normalization layer (layer of average nodes). In this layer, the i

^{th}node is a circle labeled N, which computes the normalized firing strength as follows:

**Layer 4**is the defuzzification layer (layer of consequent nodes). In this layer, every node i marked by a rectangle is an adaptive node with a node function. The output of this layer is calculated by: where {p

_{i}, q

_{i}, r

_{i}} is the parameter set of this node.

*X*is normalized value,

_{n}*X*is measured value of the variable,

_{i}*X*is maximum measured value, and

_{max}*X*is minimum measured value.

_{min}MATLAB software (MATLAB 8.1.0.604, R2013a, MathWorks Inc., USA) was used to develop the ANFIS model from the experimental data to forecast SSP. The Sugeno-type fuzzy inference system was used in the modeling of SSP. The grid partition method is employed to classify the input data and in making the rules (Jang & Sun 1995). In the modeling process, we employ eight different types of input MFs, including triangle (TRIMF), trapezoidal (TRAPMF), generalized bell (GBELLMF), Gaussian (GAUSSMF), two-sided Gaussian (GASUSS2MF), Pi curve (PIMF), the product of two sigmoidal functions (PSIGMF), and DSIGMF. The output MF was selected as a linear function. Moreover, a hybrid learning algorithm that combines the least-squares estimator and the gradient descent method is utilized to estimate the optimum values of the FIS parameters of the Sugeno-type (Jang & Sun 1995). The number of epochs was chosen as 50 owing to their small error.

### Multiple nonlinear regression (MNLR)

*X*

_{1}is RH (%),

*X*

_{2}is SR (W/m

^{2}),

*X*

_{3}is MF (L/min),

*X*

_{4}is TDSF (PPT),

*X*

_{5}is TDSB (PPT), are the second-orders of these parameters,

*X*

_{1}

*X*

_{2}

*,*, etc. are the interaction between each two parameters,

*Y*is the dependent parameter (i.e., SSP, L/m

^{2}/h) and

*A*

_{0}

*A*

_{50}are regression coefficients.

As in the ANFIS modeling, the MNLR modeling process includes three stages, namely, training, testing, and validation. The data division is the same used in the ANFIS modeling. Therefore, the training, testing, and validation sets have 112, 32, and 16 data points, respectively. The Statistical Package for Social Science (IBM SPSS Statistics 23) program (SPSS Inc., Chicago, IL, USA) was used to develop the nonlinear regression analysis by stepwise method. Stepwise method is a technique to build a model by adding or removing predictor parameters, typically through a series of F-tests or T-tests. The parameters to be added or removed are selected based on the test statistics of the estimated coefficients.

### Statistical evaluation of ANFIS and MNLR models' performance

*SSP*denotes the measured value,

_{o,i}*SSP*is the predicted value, is the mean of measured values, is the mean of predicted values, is maximum measured value, is minimum measured value, and n is the whole number of observations.

_{p,i}## RESULTS AND DISCUSSION

### Performances of ANFIS models

We developed eight ANFIS models with eight different types of input membership functions (MFs). The used MFs were TRIMF, TRAPMF, GBELLMF, GAUSSMF, GASUSS2MF, PIMF, DSIGMF, and PSIGMF. As we discussed, the eight developed ANFIS models during the training process with additional details oriented to the best ANFIS model. The ANFIS models developed have five inputs (RH, SR, MF, TDSF, and TDSB) and one output (SSP). Table 3 shows the outcomes of the statistical parameters, CC, RMSE, OI, and MAE, which are numerical indicators used to assess the agreement between observed and predicted SSP values during the training stage. For all the ANFIS models, in the input layer, five neurons were incorporated. For each neuron, the same three MFs were considered with three linguistic terms {low, medium, high} and accordingly, 243 (3 × 3 × 3 × 3 × 3) rules were developed for implementation of ANFIS models. The model properties of the ANFIS model structures are listed in Table 3. However, the ANFIS models' CC values ranged from 0.9998 to 0.9999, RMSE values from 0.0007 to 0.0048 L/m^{2}/h, OI values from 0.9972 to 0.9996, and MAE values from 0.0005 to 0.0027 L/m^{2}/h. The CC and OI values are very close to 1 while RMSE and MAE values are close to 0, indicating excellent agreement between the measured results and predicted results from the ANFIS models during the training stage. These findings emphasize the accuracy and efficiency of ANFIS models for estimating SSP by using the eight MFs.

MFs | Model properties | Statistical parameters | |||||||
---|---|---|---|---|---|---|---|---|---|

NN | NLP | NNP | TNP | NFR | CC | RMSE | OI | MAE | |

TRIMF | 524 | 1,458 | 45 | 1,503 | 243 | 0.9999 | 0.0040 | 0.9977 | 0.0027 |

TRAPMF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9998 | 0.0048 | 0.9972 | 0.0026 |

GBELLMF | 524 | 1,458 | 45 | 1,503 | 243 | 0.9999 | 0.0013 | 0.9993 | 0.0007 |

GAUSSMF | 524 | 1,458 | 30 | 1,488 | 243 | 0.9999 | 0.0014 | 0.9992 | 0.0007 |

GAUSS2MF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9999 | 0.0012 | 0.9994 | 0.0007 |

PIMF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9999 | 0.0041 | 0.9976 | 0.0022 |

PSIGMF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9999 | 0.0007 | 0.9996 | 0.0005 |

DSIGMF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9999 | 0.0007 | 0.9996 | 0.0005 |

MFs | Model properties | Statistical parameters | |||||||
---|---|---|---|---|---|---|---|---|---|

NN | NLP | NNP | TNP | NFR | CC | RMSE | OI | MAE | |

TRIMF | 524 | 1,458 | 45 | 1,503 | 243 | 0.9999 | 0.0040 | 0.9977 | 0.0027 |

TRAPMF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9998 | 0.0048 | 0.9972 | 0.0026 |

GBELLMF | 524 | 1,458 | 45 | 1,503 | 243 | 0.9999 | 0.0013 | 0.9993 | 0.0007 |

GAUSSMF | 524 | 1,458 | 30 | 1,488 | 243 | 0.9999 | 0.0014 | 0.9992 | 0.0007 |

GAUSS2MF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9999 | 0.0012 | 0.9994 | 0.0007 |

PIMF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9999 | 0.0041 | 0.9976 | 0.0022 |

PSIGMF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9999 | 0.0007 | 0.9996 | 0.0005 |

DSIGMF | 524 | 1,458 | 60 | 1,518 | 243 | 0.9999 | 0.0007 | 0.9996 | 0.0005 |

NN, number of nodes; NLP, number of linear parameters; NNP, number of nonlinear parameters; TNP, total number of parameters; NFR, number of fuzzy rules; CC, coefficient of correlation; RMSE, root mean square error; OI, overall index of model performance; MAE, mean absolute error.

The performance, in the training stage, for all MFs is approximately the same, but in relative terms, the highest performance is obtained with GBELLMF in the training process. The CC, RMSE, OI, and MAE for GBELLMF were 0.9999, 0.0013 L/m^{2}/h, 0.9993, and 0.0007 L/m^{2}/h, respectively. This agrees with the results of Taghavifar & Mardani (2015). However, the best/selected ANFIS structure for SSP prediction was obtained by using GBELLMF and consisted of five layers. The 1st layer of the developed model included the input parameter (RH, SR, M_{F}, TDS_{F}, and TDS_{B}) membership functions (MFs). This layer provides the input parameter values to the following layer. The 2nd layer was a MF layer, which determined and checked the weights for each MF. For the best ANFIS model, the fuzzification layer (2nd layer) contained 15 nodes and 45 nonlinear parameters. The rule layer (3rd layer) with 524 nodes achieved a pre-condition matching process for fuzzy rules. The 4th layer (the defuzzification layer) with 524 nodes and 1,458 linear parameters took the inference of the rules and generated output values. The 5th layer summed up and combined the inputs and transformed the fuzzy classification into a binary outcome. Overall, the total number of parameters and fuzzy rules were 1,503 and 243, respectively, for the ANFIS model using GBELLMF.

### Performances of MNLR models

The final equations obtained from MNLR are presented in Table 4. The SE, t-stat, and *p*-value for independent variables in the equations are illustrated in Table 5. MNLR models are developed by iteratively adding and removing the terms from a multi-nonlinear model based on their significance within a regression by using SPSS software. All five independent variables (*RH*, *SR*, *MF*, *TDSF,* and *TDSB*), their second-orders and interaction terms were used to create predicting models for SSP. MNLR produced 11 models with 1–8 predictor variables, where *SR · MF* (interaction between *SR* and *MF*) was involved in each set of predictor variables, as demonstrated in Table 4. Overall, the interaction between *SR* and *MF* has the most effect on SSP modeling.

Model | Mathematical expression | CC | SEE |
---|---|---|---|

MNLR-1 | 0.859 | 0.1256 | |

MNLR-2 | 0.926 | 0.0931 | |

MNLR-3 | 0.955 | 0.0734 | |

MNLR-4 | 0.972 | 0.0589 | |

MNLR-5 | 0.977 | 0.0533 | |

MNLR-6 | 0.978 | 0.0518 | |

MNLR-7 | 0.978 | 0.0516 | |

MNLR-8 | 0.979 | 0.0498 | |

MNLR-9 | 0.979 | 0.0480 | |

MNLR-10 | 0.980 | 0.0464 | |

MNLR-11 | 0.981 | 0.0442 |

Model | Mathematical expression | CC | SEE |
---|---|---|---|

MNLR-1 | 0.859 | 0.1256 | |

MNLR-2 | 0.926 | 0.0931 | |

MNLR-3 | 0.955 | 0.0734 | |

MNLR-4 | 0.972 | 0.0589 | |

MNLR-5 | 0.977 | 0.0533 | |

MNLR-6 | 0.978 | 0.0518 | |

MNLR-7 | 0.978 | 0.0516 | |

MNLR-8 | 0.979 | 0.0498 | |

MNLR-9 | 0.979 | 0.0480 | |

MNLR-10 | 0.980 | 0.0464 | |

MNLR-11 | 0.981 | 0.0442 |

CC: coefficient of correlation; SEE: standard error of the estimate.

Model | Intercept | Independent parameters | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

SR.MF | MF.RH^{2} | MF^{2}.TDSF^{2} | MF.TDSB | RH^{2}.SR^{2} | RH^{2}.TDSF^{2} | MF.TDSF^{2} | SR^{2}.MF^{2} | SR.RH^{2} | |||

MNLR-1 | SE | 0.037 | 2.88 × 10^{−04} | ||||||||

t-stat | −3.154 | 17.589 | |||||||||

p-value | 0.002 | 9.52 × 10^{−34} | |||||||||

MNLR-2 | SE | 0.032 | 2.22 × 10^{−04} | 7.50 × 10^{−05} | |||||||

t-stat | −8.559 | 25.435 | 9.555 | ||||||||

p-value | 8.08 × 10^{−14} | 1.19 × 10^{−47} | 4.43 × 10^{−16} | ||||||||

MNLR-3 | SE | 0.033 | 1.78 × 10^{−04} | 6.30 × 10^{−05} | 5.00 × 10^{−05} | ||||||

t-stat | −3.181 | 30.360 | 8.224 | −8.217 | |||||||

p-value | 0.002 | 1.06 × 10^{−54} | 4.80 × 10^{−13} | 4.98 × 10^{−13} | |||||||

MNLR-4 | SE | 0.039 | 1.85 × 10^{−04} | 5.70 × 10^{−05} | 1.04 × 10^{−04} | 0.004 | |||||

t-stat | −8.409 | 24.311 | 5.618 | −11.119 | 7.766 | ||||||

p-value | 1.97 × 10^{−13} | 2.22 × 10^{−45} | 1.54 × 10^{−07} | 1.50 × 10^{−19} | 5.18 × 10^{−12} | ||||||

MNLR-5 | SE | 0.037 | 1.79 × 10^{−04} | 1.20 × 10^{−04} | 9.60 × 10^{−05} | 0.003 | 2.62 × 10^{−11} | ||||

t-stat | −7.627 | 23.277 | −1.834 | −13.063 | 9.185 | 4.986 | |||||

p-value | 1.09 × 10^{−11} | 1.82 × 10^{−43} | 0.069 | 8.14 × 10^{−24} | 3.84 × 10^{−15} | 2.00 × 10^{−06} | |||||

MNLR-6 | SE | 0.036 | 1.82 × 10^{−04} | 1.36 × 10^{−04} | 9.70 × 10^{−05} | 0.003 | 2.91 × 10^{−11} | 1.84 × 10^{−09} | |||

t-stat | −7.328 | 22.117 | −0.266 | −12.204 | 9.026 | 5.776 | −2.663 | ||||

p-value | 5.02 × 10^{−11} | 2.59 × 10^{−41} | 0.791 | 7.37 × 10^{−22} | 9.33 × 10^{−15} | 7.88 × 10^{−08} | 0.009 | ||||

MNLR-7 | SE | 0.036 | 1.74 × 10^{−04} | 8.50 × 10^{−05} | 0.003 | 2.60 × 10^{−11} | 1.58 × 10^{−09} | ||||

t-stat | −7.464 | 23.253 | −13.852 | 9.523 | 6.316 | −3.266 | |||||

p-value | 2.45 × 10^{−11} | 1.99 × 10^{−43} | 1.58 × 10^{−25} | 6.64 × 10^{−16} | 6.41 × 10^{−09} | 0.001 | |||||

MNLR-8 | SE | 0.034 | 1.72 × 10^{−04} | 2.43 × 10^{−04} | 0.004 | 2.56 × 10^{−11} | 2.02 × 10^{−09} | 3.40 × 10^{−05} | |||

t-stat | −8.784 | 21.961 | −8.947 | 11.013 | 7.881 | −5.545 | 4.335 | ||||

p-value | 3.24 × 10^{−14} | 4.80 × 10^{−41} | 1.41 × 10^{−14} | 3.28 × 10^{−19} | 3.19 × 10^{−12} | 2.21 × 10^{−07} | 3.40 × 10^{−05} | ||||

MNLR-9 | SE | 0.050 | 0.001 | 2.38 × 10^{−04} | 0.004 | 2.62 × 10^{−11} | 1.96 × 10^{−09} | 3.30 × 10^{−05} | 3.00 × 10^{−06} | ||

t-stat | −3.729 | 2.896 | −8.538 | 10.258 | 8.705 | −6.080 | 4.249 | 2.993 | |||

p-value | 3.13 × 10^{−04} | 0.005 | 1.21 × 10^{−13} | 1.78 × 10^{−17} | 5.19 × 10^{−14} | 2.01 × 10^{−08} | 4.70 × 10^{−05} | 0.003 | |||

MNLR-10 | SE | 0.049 | 0.001 | 2.33 × 10^{−04} | 0.003 | 7.92 × 10^{−11} | 3.00 × 10^{−09} | 3.50 × 10^{−05} | 3.00 × 10^{−06} | 8.23 × 10^{−08} | |

t-stat | −3.284 | 2.016 | −9.258 | 10.190 | 0.032 | −6.309 | 5.249 | 3.863 | 3.004 | ||

p-value | 0.001 | 0.046 | 3.31 × 10^{−15} | 2.79 × 10^{−17} | 0.975 | 7.13 × 10^{−09} | 8.25 × 10^{−07} | 1.96 × 10^{−04} | 0.003 | ||

MNLR-11 | SE | 0.048 | 0.001 | 2.31 × 10^{−04} | 0.003 | 2.45 × 10^{−09} | 3.30 × 10^{−05} | 3.00 × 10^{−06} | 2.61 × 10^{−08} | ||

t-stat | −3.303 | 2.042 | −9.373 | 10.307 | −7.725 | 5.540 | 3.953 | 9.566 | |||

p-value | 0.001 | 0.044 | 1.70 × 10^{−15} | 1.39 × 10^{−17} | 7.31 × 10^{−12} | 2.30 × 10^{−07} | 1.41 × 10^{−04} | 6.27 × 10^{−16} |

Model | Intercept | Independent parameters | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

SR.MF | MF.RH^{2} | MF^{2}.TDSF^{2} | MF.TDSB | RH^{2}.SR^{2} | RH^{2}.TDSF^{2} | MF.TDSF^{2} | SR^{2}.MF^{2} | SR.RH^{2} | |||

MNLR-1 | SE | 0.037 | 2.88 × 10^{−04} | ||||||||

t-stat | −3.154 | 17.589 | |||||||||

p-value | 0.002 | 9.52 × 10^{−34} | |||||||||

MNLR-2 | SE | 0.032 | 2.22 × 10^{−04} | 7.50 × 10^{−05} | |||||||

t-stat | −8.559 | 25.435 | 9.555 | ||||||||

p-value | 8.08 × 10^{−14} | 1.19 × 10^{−47} | 4.43 × 10^{−16} | ||||||||

MNLR-3 | SE | 0.033 | 1.78 × 10^{−04} | 6.30 × 10^{−05} | 5.00 × 10^{−05} | ||||||

t-stat | −3.181 | 30.360 | 8.224 | −8.217 | |||||||

p-value | 0.002 | 1.06 × 10^{−54} | 4.80 × 10^{−13} | 4.98 × 10^{−13} | |||||||

MNLR-4 | SE | 0.039 | 1.85 × 10^{−04} | 5.70 × 10^{−05} | 1.04 × 10^{−04} | 0.004 | |||||

t-stat | −8.409 | 24.311 | 5.618 | −11.119 | 7.766 | ||||||

p-value | 1.97 × 10^{−13} | 2.22 × 10^{−45} | 1.54 × 10^{−07} | 1.50 × 10^{−19} | 5.18 × 10^{−12} | ||||||

MNLR-5 | SE | 0.037 | 1.79 × 10^{−04} | 1.20 × 10^{−04} | 9.60 × 10^{−05} | 0.003 | 2.62 × 10^{−11} | ||||

t-stat | −7.627 | 23.277 | −1.834 | −13.063 | 9.185 | 4.986 | |||||

p-value | 1.09 × 10^{−11} | 1.82 × 10^{−43} | 0.069 | 8.14 × 10^{−24} | 3.84 × 10^{−15} | 2.00 × 10^{−06} | |||||

MNLR-6 | SE | 0.036 | 1.82 × 10^{−04} | 1.36 × 10^{−04} | 9.70 × 10^{−05} | 0.003 | 2.91 × 10^{−11} | 1.84 × 10^{−09} | |||

t-stat | −7.328 | 22.117 | −0.266 | −12.204 | 9.026 | 5.776 | −2.663 | ||||

p-value | 5.02 × 10^{−11} | 2.59 × 10^{−41} | 0.791 | 7.37 × 10^{−22} | 9.33 × 10^{−15} | 7.88 × 10^{−08} | 0.009 | ||||

MNLR-7 | SE | 0.036 | 1.74 × 10^{−04} | 8.50 × 10^{−05} | 0.003 | 2.60 × 10^{−11} | 1.58 × 10^{−09} | ||||

t-stat | −7.464 | 23.253 | −13.852 | 9.523 | 6.316 | −3.266 | |||||

p-value | 2.45 × 10^{−11} | 1.99 × 10^{−43} | 1.58 × 10^{−25} | 6.64 × 10^{−16} | 6.41 × 10^{−09} | 0.001 | |||||

MNLR-8 | SE | 0.034 | 1.72 × 10^{−04} | 2.43 × 10^{−04} | 0.004 | 2.56 × 10^{−11} | 2.02 × 10^{−09} | 3.40 × 10^{−05} | |||

t-stat | −8.784 | 21.961 | −8.947 | 11.013 | 7.881 | −5.545 | 4.335 | ||||

p-value | 3.24 × 10^{−14} | 4.80 × 10^{−41} | 1.41 × 10^{−14} | 3.28 × 10^{−19} | 3.19 × 10^{−12} | 2.21 × 10^{−07} | 3.40 × 10^{−05} | ||||

MNLR-9 | SE | 0.050 | 0.001 | 2.38 × 10^{−04} | 0.004 | 2.62 × 10^{−11} | 1.96 × 10^{−09} | 3.30 × 10^{−05} | 3.00 × 10^{−06} | ||

t-stat | −3.729 | 2.896 | −8.538 | 10.258 | 8.705 | −6.080 | 4.249 | 2.993 | |||

p-value | 3.13 × 10^{−04} | 0.005 | 1.21 × 10^{−13} | 1.78 × 10^{−17} | 5.19 × 10^{−14} | 2.01 × 10^{−08} | 4.70 × 10^{−05} | 0.003 | |||

MNLR-10 | SE | 0.049 | 0.001 | 2.33 × 10^{−04} | 0.003 | 7.92 × 10^{−11} | 3.00 × 10^{−09} | 3.50 × 10^{−05} | 3.00 × 10^{−06} | 8.23 × 10^{−08} | |

t-stat | −3.284 | 2.016 | −9.258 | 10.190 | 0.032 | −6.309 | 5.249 | 3.863 | 3.004 | ||

p-value | 0.001 | 0.046 | 3.31 × 10^{−15} | 2.79 × 10^{−17} | 0.975 | 7.13 × 10^{−09} | 8.25 × 10^{−07} | 1.96 × 10^{−04} | 0.003 | ||

MNLR-11 | SE | 0.048 | 0.001 | 2.31 × 10^{−04} | 0.003 | 2.45 × 10^{−09} | 3.30 × 10^{−05} | 3.00 × 10^{−06} | 2.61 × 10^{−08} | ||

t-stat | −3.303 | 2.042 | −9.373 | 10.307 | −7.725 | 5.540 | 3.953 | 9.566 | |||

p-value | 0.001 | 0.044 | 1.70 × 10^{−15} | 1.39 × 10^{−17} | 7.31 × 10^{−12} | 2.30 × 10^{−07} | 1.41 × 10^{−04} | 6.27 × 10^{−16} |

The CC values associated with each of the 11 models ranged from 0.859 to 0.981. Corresponding standard errors of the estimate (SEE) ranged from 0.0442 to 0.1256 L/m^{2}/h. It can be noted from Table 4 that the absence or presence of some of the input variables in the MNLR models significantly affects the performances of these models. MNLR-1 with just the *SR · MF* performed relatively worst, with CC = 0.859 and SSE = 0.1256 L/m^{2}/h. MNLR-2 performed better than MNLR-1, owing to the presence of *MF.RH*^{2}. The CC value of MNLR-2 was increased by 7.80% than that for MNLR-1. Furthermore, the SEE value of MNLR-2 was decreased by 25.88% than that for MNLR-1. In MNLR-3, adding the *MF*^{2}.*TDSF*^{2} caused an increase of 3.13% in CC's value, decrease of 21.16% in SEE's value compared to MNLR-2. The same trend was followed with MNLR-4 by the inclusion of both *MF*^{2}.*TDSF*^{2} and *MF.TDSB* (CC = 0.972, SEE = 0.0589 L/m^{2}/h). Following MNLR-4, the accuracy was dramatically unchanged. The value of CC was between 0.977 and 0.981, and SEE ranged from 0.0533 L/m^{2}/h to 0.0442 L/m^{2}/h. However, MNLR-11 showed the best model for estimating SSP (CC = 0.981, SEE = 0.0442 L/m^{2}/h) which involved *SR.MF, MF.TDSF, MF.TDSB, RH*^{2}.*TDSF*^{2}, *MF.TDSF*^{2}, *SR*^{2}.*MF*^{2}, and *SR.RH*^{2} as predictor variables. Furthermore, the MNLR-11 model provides better performance in estimating SSP, as is reflected in the values of the statistical parameters (Table 5). The selected model has shown the significance of all independent variables at the significance level of 5% (*p*-value <0.05). The standard error (SE) determines the accuracy of the estimate of the coefficient. The smaller the SE, the more accurate the estimate. The significance of each coefficient of the obtained MNLR model was determined by t-stat and *p*-value, which are displayed in Table 5. The larger the t-stat and the smaller the *p*-value, the more significant is the corresponding coefficient. The absolute value of the t-stat should be greater than the critical t-value.

### Comparison between ANFIS and MNLR models

In this section, performance comparison is made between the ANFIS and MNLR models. The performance of these models was assessed according to statistical criteria such as CC, RMSE, OI, and MAE. The findings of applying these models are compared in Table 6. During the training process, it is clear from Table 6 that the values predicted by the ANFIS fit perfectly with the observed values during the training process as reflected in the values of the statistical indicators. For the ANFIS model, the CC (0.999) value is very close to 1, and the RMSE value (0.001 L/m^{2}/h) is close to 0. The corresponding OI value was 0.999, which is very close to 1. The MAE value, as well, was 0.001 L/m^{2}/h, which was very close to 0. These results indicate that the ANFIS model is better than the performance of the MNLR model (CC = 0.981, RMSE = 0.053 L/m^{2}/h, OI = 0.948, and MAE = 0.041 L/m^{2}/h) in the training process, as indicated in Table 6.

Statistical criteria | Training | Testing | Validation | |||
---|---|---|---|---|---|---|

ANFIS | MNLR | ANFIS | MNLR | ANFIS | MNLR | |

CC | 0.999 | 0.981 | 0.959 | 0.962 | 0.915 | 0.972 |

RMSE (L/m^{2}/h) | 0.001 | 0.053 | 0.070 | 0.066 | 0.085 | 0.055 |

OI | 0.999 | 0.948 | 0.910 | 0.921 | 0.830 | 0.935 |

MAE (L/m^{2}/h) | 0.001 | 0.041 | 0.045 | 0.051 | 0.063 | 0.044 |

Statistical criteria | Training | Testing | Validation | |||
---|---|---|---|---|---|---|

ANFIS | MNLR | ANFIS | MNLR | ANFIS | MNLR | |

CC | 0.999 | 0.981 | 0.959 | 0.962 | 0.915 | 0.972 |

RMSE (L/m^{2}/h) | 0.001 | 0.053 | 0.070 | 0.066 | 0.085 | 0.055 |

OI | 0.999 | 0.948 | 0.910 | 0.921 | 0.830 | 0.935 |

MAE (L/m^{2}/h) | 0.001 | 0.041 | 0.045 | 0.051 | 0.063 | 0.044 |

CC, coefficient of correlation; RMSE, root mean square error; OI, overall index of model performance; MAE, mean absolute error.

In the testing stage, ANFIS statistics were CC = 0.959, RMSE = 0.070 L/m^{2}/h, OI = 0.910, and MAE = 0.045 L/m^{2}/h. However, the CC values in the MNLR model were about 0.31% more accurate than for the ANFIS model, as shown in Table 6. The RMSE value of the ANFIS model was almost 1.06 times that of the value of the MNLR model (0.066 L/m^{2}/h). The OI value for the ANFIS model was less than its value for the MNLR model (0.921). The value of MAE for the MNLR model (0.051 L/m^{2}/h) was almost 1.13 times that of the ANFIS model. It is quite clear that the results for both models are very close at this stage.

In the validation stage, the MNLR model had a CC value of about 6.23% more accurate than that of the ANFIS model, as shown in Table 6. The value of RMSE for the ANFIS model (0.085 L/m^{2}/h) was almost 1.55 times that of the MNLR model (0.055 L/m^{2}/h). The OI value for the MNLR model was 12.65% more accurate than that of the ANFIS model. The MAE value of 0.063 L/m^{2}/h for the ANFIS model was almost 1.43 times that of the MNLR (0.044 L/m^{2}/h). The CC, RMSE, OI, and MAE values confirm that ANFIS model performed relatively poorly during the validation phase. However, generally, Table 6 clearly shows that the predictive performance of ANFIS and MNLR models is somewhat similar. These outcomes indicate that both models can be used successfully for SSP modeling.

Another representation of the results generated using the ANFIS and MNLR models, is demonstrated in Figure 3, where the observed and predicted SSP values from the models during the training, testing, and validation stages are compared. These are in the form of a scatter (1:1) plot (left panel) and a relative error plot (right panel). The data were mostly evenly and tightly distributed around the 1:1 line, as illustrated in Figure 3, and there was a very close agreement between the observed and predicted values by the ANFIS model during the training process, while most of the values were inaccurate with the MNLR model during the same process. In the testing stage, Figure 3 illustrates that the most of the values predicted by the ANFIS model were acceptable, while the majority of predicted values were somewhat imprecise when using the MNLR model. In the validation stage, there is a similarity in performance between the two models. However, the average relative errors were 8.11%, 6.71%, and 5.05% for the MNLR model during the training, testing, and validation phases, respectively. The corresponding values for the ANFIS model were generally lower at 0.01%, −1.12%, and 6.23% during the training, testing, and validation phases, respectively. In general, both models showed efficiency and accuracy in the SSP prediction process as in Table 6 and Figure 3.

## CONCLUSIONS

The main objective of this study was to assess the performance of ANFIS and MNLR models for estimating SSP of an inclined passive solar still. Five parameters, relative humidity, SR, feed flow rate, total dissolved solids of feed, and total dissolved solids of brine, were used as inputs for the ANFIS and MNLR models. The SSP was used as a target variable. The performance of the models was assessed by comparing the prediction results for both approaches with the experimental results by using a variety of standard statistical parameters, namely, CC, RMSE, OI, and MAE. These comparisons revealed that the ANFIS and MNLR models can be suggested to forecast SSP effectively because of fast, accurate, and reliable results. There was agreement between the predicted and observed data for the ANFIS and MNLR models. Generally, the trend of the MNLR results is the same as that of the ANFIS model. The ANFIS model gave accurate estimates, but the MNLR models are easier to use, as they compute the SSP using explicit algebraic equations.

## ACKNOWLEDGEMENTS

The project was financially supported by King Saud University, Vice Deanship of Research Chairs.