### 1. Introduction

### 1.1. Study Area

*N*and 33Â°32â€²

*N*and 49Â°30â€²

*E*and 49Â°52â€²

*E*. The area includes four cities including Shahrekord, Frieden, Lenjan and Isfahan and covers part of the Chahar Mahal Bakhtiari province (1.7% of the total area). Fig. 1 indicates the study area [1]. Average annual rainfall varies from 1,600 mm in the Zard Kuh Mountains to less than 40 mm in the eastern regions of Isfahan [9].

### 2. Materials and Methods

### 2.1. Factor Analysis

##### (1)

$$KMO=\frac{\xe2\u02c6\u2018\xe2\u02c6\u2018{r}_{ij}^{2}}{\xe2\u02c6\u2018\xe2\u02c6\u2018{r}_{ij}^{2}+\xe2\u02c6\u2018\xe2\u02c6\u2018{a}_{ij}^{2}}$$_{ij}is the correlation coefficient of indicator

*x*

*and indicator of*

_{i}*x*

*;*

_{j}*aij*is the offset correlation coefficient of index

*x*

*and indicator*

_{i}*x*

*.*

_{j}*KMO*values of close to one indicate that the correlation between pairs of variables can be explained by other variables. Therefore, justifying the application of variable factor analysis is provable. The following steps should be carried out for factor analysis.

Creating a matrix of correlations between the water quality parameters which is a square matrix of correlation coefficients.

Determining

*KMO*to demonstrate the suitability of factor analysis.Factors should be partially rotated around the origin, to obtain a new position.

Finally, the number of factors equal of the correlation matrix that is considered to be greater than one [11â€“15].

### 2.2. ANN Modeling

### 2.3. Architecture of the Network

### 2.4. The TDNN Network

### 2.5. RBF Neural Network

##### (3)

$${h}_{j}=\frac{\xe2\u02c6\dots \mathrm{\xe2\u20ac\u2030}(X-{c}_{j})}{{\mathrm{\xce\xb4}}_{j}}$$*c*

*is a neuron center and*

_{j}*Î´*

*is the neuronâ€™s central span. Nonlinear function is due to Î¦ functions. Neurons have a linear function in the output layer and the output of*

_{j}*y*

*in neuron*

_{k}*k*in the output layer is obtained from Eq. (4) [20]:

*W*

*is a synaptic weight connecting of j-th of hidden layer and neuron*

_{kj}*k*of output layer and m is the number of hidden layer neurons.

### 2.6. Model Efficiency

*N*is the number of data;

*A*

*is the actual data and*

_{t}*F*

*is the predicted data.*

_{t}^{2}) and the Index of Agreement (IA) indicate the reliability of the model [22]. R

^{2}and IA can be illustrated as follows:

##### (7)

$${R}^{2}={\left(\frac{{\xe2\u02c6\u2018}_{i=1}^{i}({A}_{t}-\stackrel{\xc2\xaf}{A})\mathrm{\xe2\u20ac\u2039}({F}_{t}-\stackrel{\xc2\xaf}{F})}{{\xe2\u02c6\u2018}_{t=1}^{n}Z{({A}_{t}-\stackrel{\xc2\xaf}{A})}^{2}){\xe2\u02c6\u2018}_{i=1}^{i}{(\mathrm{\xe2\u20ac\u2039}{({F}_{t}-\stackrel{\xc2\xaf}{F})}^{2})}^{0.5}}\right)}^{2}$$##### (8)

$$IA=d=1.0-\frac{{\xe2\u02c6\u2018}_{t=1}^{N}{({A}_{t}-{F}_{t})}^{2}}{{\xe2\u02c6\u2018}_{t=1}^{N}(\xe2\u02c6\pounds {F}_{t}-\stackrel{\xc2\xaf}{A})\mathrm{\xe2\u20ac\u2030}+{\xe2\u02c6\pounds {A}_{t}-\stackrel{\xc2\xaf}{A})\xe2\u02c6\pounds}^{2}}$$*Ä€*and

*FÌ„*are the means of the actual data and the predicted data, respectively.

### 3. Results and Discussion

*p*value less than 0.05, a null hypothesis is confirmed, and a significant correlation exists between the variables.

_{3}

^{âˆ’1}), chloride (Cl

^{âˆ’}) and calcium (Ca

^{2+}), which are the most important parameters in water quality of the river Zayanderud. We selected the mentioned parameters as input parameters to the ANN.

### 3.1. The ANN Results

^{+}, Mg

^{2+}, carbonate parameters and the results of factor analysis, which were HCO

_{3}

^{âˆ’1}, Clâˆ’, Ca

^{2+}and TH as input data to the ANN. In the sensitivity analysis the accuracy of the factor analysis will be examined. To avoid overfitting, in this study, we used 6-fold cross validation to compute the true error estimation. We applied six sets of data, in which the testing data were changed. The results of each five subsamples validating for both TDNN and RBF networks are listed in the Table 6.

### 3.2. The Time Delay Neural Network (TDNN) Results

^{2}) and the IA between the predicted TDS and the observed data were 0.957 and 0.986 which means the accuracy of the model in predicting TDS parameters was acceptable.

### 3.3. The RBF Neural Network Results

^{2}between the predicted data and the observed data for TDS in the Mosian station was 0.997 and the IA was 0.999.

^{2}of TDNNs with RBF neural network, RBF predicting of TDS indicates more accurate than TDNNs (Fig. 5, Fig. 7).

^{2}between the observed and the predicted TDS concentrations was 0.859 and 0.949, respectively. Their input data to the model was only the amount of flow. However, in our study, eight parameters including pH, Na

^{+}, Mg

^{2+}, Carbonat(CO

_{3}

^{âˆ’2}), HCO

_{3}

^{âˆ’1}, Clâˆ’, Ca

^{2+}and TH were used as input parameters to the ANN. We also carried out a sensitivity analysis to determine the roles of each input parameter in predicting of TDS in the river. Asadollahfardi et al. [2] predicted the TDS of the Talkherud River, Iran, using an MLP and ELMAN methods. They studied two stations and the R

^{2}between the observed and the predicted TDS concentrations was 0.964 and 0.96, respectively. Their input data was rate of flow. The R

^{2}in our study is larger than their study and the input parameter for their work was only the amount of flow in the river. Nemati et al. [8] applied an ANN to predict TDS of the Siminehrud River, Iran. Its R

^{2}was 0.841. The R

^{2}of our study was 0.999 and for selecting of suitable input parameters, we applied factor analysis.

### 3.4. Sensitivity Analysis

^{2+}and SO

_{4}

^{2âˆ’}had the greatest impact on the TDS prediction. After that, Clâˆ’, HCO

_{3}

^{âˆ’1}and TH were effective, respectively. Except the SO

_{4}

^{2âˆ’}, the results are the same as factor analysis results for selection of input parameters.

### 4. Conclusions

For TDS prediction in TDNN, R

^{2}and IA between the predicted data and the observed data were 0.957 and 0.986, respectively, which mean that our two neural network results are acceptable.The R

^{2}and the IA between the predicted and the observed data for predicting TDS in the RBF was 0.997 and 0.999. The TDNN contained 2 hidden layers with 15 neurons in each layer and the RBF with one hidden layer containing 100 neurons.The MSE, RMSE and MBE for the TDNN were 0.0006, 0.0603 and 0.843, respectively. For the RBF neural network the mentioned errors were 0.0001, 0.43 and 0.516, respectively.

The result of the RBF is more accurate than the TDNN in the prediction of TDS in Zayanderud River.

The results of sensitivity analysis indicated that Ca

^{2+}and SO_{4}^{2âˆ’}had the highest effect on the TDS prediction. According to its results, all the parameters from factor analysis had an important role in changes of TDS. The SO_{4}^{2âˆ’}was not mentioned in the results of factor analysis.