**2.2 Auto-correlation function (ACF) and partial auto-correlation function (PACF)**

For the selection of proper input combinations of rainfall and runoff, the autocorrelation function (ACF) [55] and Cross-correlation function (CCF) [56] were employed for runoff data and rainfall-runoff data, respectively, with a 95% confidence level. From the **Tables 1** and **2**, it can be seen that the cross-relation in the rainfall and runoff dataset is poor, which may be an issue for modeling of rainfall-runoff phenomenon [57]. So, the partial autocorrelation was used between these two input variables. It is concluded from the results shown in **Table 1** that three lag times of rainfall and runoff datasets will be efficient for the modeling process. Based on results, the following input combinations were used in this study;


**Partial Auto-Correlation Function (PACF) Input Variable Coefficients Standard Error t Stat Pvalue Lower 95% Upper 95% Lower 95.0% Upper 95.0% Qobs** 89224.45 30235.64 2.95 0.00 29953.86 148495.05 29953.86 148495.05 **P(t)** �1449.13 3534.18 �0.41 0.68 �8377.14 5478.88 �8377.14 5478.88 **P(t-1)** �840.46 3838.46 �0.22 0.83 �8364.94 6684.02 �8364.94 6684.02 **P(t-2)** �239.51 3838.46 �0.06 0.95 �7764.01 7284.99 �7764.01 7284.99 **P(t-3)** �1139.05 3534.17 �0.32 0.75 �8067.04 5788.94 �8067.04 5788.94

*Evaluating the Performance of Different Artificial Intelligence Techniques… DOI: http://dx.doi.org/10.5772/intechopen.98280*

#### **Table 1.**

*Partial auto-correlation function (PACF) between rainfall and runoff data.*


#### **Table 2.**

*Auto-correlation between runoff and rainfall data.*

7.C7 Q(t-3), Q(t-2), Q(t-1), P(t-3), P(t-2), P(t-1), Pt

Where Q is discharge (m3 /sec), P is precipitation (mm), and it is Time (sec). There we created different time lags of Q and P to test and train the models, i.e. (t), (t-1), (t-2) and (t-3). These parameters are arranged to create different input combinations C1, C2, C3, C4, C5, C6 and C7, which are used for testing and training AI techniques to get better results.

#### **2.3 Support vector machine (SVM)**

A brief description of the SVM has been mentioned in this study, whereas the theory SVM [24] was discussed by many researchers in detailed, i.e. [28–30]. According to [24] in the SVM technique, independent variable x helps estimate the dependent variable y. The relationship between x and y was determined by the given function like other regression scenarios;

$$\mathbf{f}(\mathbf{x}) = (\mathbf{f}(\mathbf{x}).\mathbf{w}\mathbf{T}.\mathcal{J}(\mathbf{x}) + \mathbf{b})\tag{1}$$

$$\mathbf{f}(\mathbf{y}) = \mathbf{f}(\mathbf{x}) + \text{noise} \tag{2}$$

where Ø is kernel function which can be defined as; it takes to input information and changes it into the desired shape. Various SVM algorithms practice diverse sorts of kernel functions. There are many kinds of these functions. i.e. sigmoid, polynomial, non-linear, linear, and RBF. b is a constant, w is the coefficient of vector, w and b are the constraints of the regression function. In contrast, noise is elaborated by error tolerance (e). During the training of the SVM model, a process of association of successive optimization of the error function in which can be achieved. There are two kinds of SVM models based on the error function, such as e-SVM (Regression I) and t-SVM (Regression II) [58]. In this study, BRF Regression, I is

engaged because for prediction like rainfall-runoff purposes. [59, 60] proposed that the training time of SVM decreased by selecting the automatic RBF kernel function because it efficiently selected the proper kernel function constraints. As compared to V-fold validation is consumed less Time and more efficient. Let consider (x<sup>i</sup> j ) j=1 … … . Ni Rd. is the dataset of i, and Ni is the number of training samples of i class. Whereas i = 1,2,3 … ...L and L is the number of classes in the dataset, then RBF is;

$$\mathbf{K}(\mathbf{x}, \mathbf{x}', \sigma) = \exp\left(-\frac{\mathbf{I}\mathbf{I}\mathbf{x} - \mathbf{x}'\mathbf{I}\mathbf{I}^1/2}{2\,\sigma^1/2}\right) \tag{3}$$

K is a kernel function, (x‵, <sup>σ</sup>) are elements of R<sup>d</sup> and <sup>σ</sup> element of R-0 which is corresponding constraints. It has two major possessions, i. the cosine value of training dataset ≥1, and it must be more than 0. ii. The norm in the dataset must be 1 [61] shown in **Figure 2**.

As in this study, RBF based kernel is used, so the following expression is used to calculate the mean of values;

$$\mathbf{b}(\boldsymbol{\sigma}) = \frac{\mathbf{1}}{\sum\_{i=1}^{L} \sum\_{l=1, l \neq 1}^{L} \mathbf{N} \mathbf{i} \mathbf{N} \mathbf{j}} \sum\_{i=1}^{L} \sum\_{l=1, l \neq 1}^{L} \sum\_{l=1}^{L} \sum\_{k=1}^{L} k(\mathbf{x}l \ (i), \mathbf{x}k(j), \boldsymbol{\sigma}) \tag{4}$$

Therefore, b(σ) is calculated in a pattern that (σ) is must be greater than 0 but not less or 0. The σ can be calculated in SVM based on RBF kernel function by solving the given steps;

1.To determine the best constraint, the given expression is optimized.

**Figure 2.**

*Working layout of RBF based kernel support vector machine.*

*Evaluating the Performance of Different Artificial Intelligence Techniques… DOI: http://dx.doi.org/10.5772/intechopen.98280*

2.Applying the RBF kernel function further utilizes the V-fold cross validation to determine the best constraint (penalty constraint).

$$\mathbf{M} \text{in}(\sigma) \,\mathbf{J}(\sigma) = \mathbf{1} - (\sigma) + \mathbf{b}(\sigma) \tag{5}$$

Based on [66] theory of RBF kernel-based SVM, the technique is employed in this research for rainfall-runoff modeling.

## **2.4 M5 model tree**

In the M5 model tree machine learning technique, the following principle converted the space into the area and made the linear regression model. The model's outcome is shown in the modular model, committee machine, with linear models specially designed on appropriate subsets of input space. This design is not innovative. Fusion of specialized technique ("local" model) is passed down in modeling. The finding can clear analogy among Model Trees (MTs), and a combo of linear models utilized in dynamic hydrology since the 1970s- evident paper on multilinear techniques is by [62]. Model tree M5, based on the information theory principle, will have divided multi-dimensional space and create the models automatically based on quality criterion. The number of models can also be varying in number. Computational intelligence techniques combined the numerous models and possibly the combination theory and data-driven outcomes are supporters in hydrology. (example [63], in the fuzzy system, combined hydrological techniques). Computational requirement for model tree raises rapidly with dimensionality [34]. Model tree tackles the task efficiently with high dimension-up to hundreds of attributes. The main advantage of tree models instead of the regression model is that they are smaller than regression trees. The strength of the decision is clear, and regression parameters do not normally involve various variables. M5 algorithm is used for inducing a model tree, which works as shown in.

Suppose collection T of example training is available. Each example is categorized by the values of non-variable set of attributes and has target value. Goal is to build a model with associated target values of training and their input attributes. The efficiency of the model will be calculated by the accuracy, which is forecasting that targets unknown cases shown in **Figure 3**.

**Figure 3.** *Working layout of M5 model tree.*

### **2.5 Model performance**

Different performance evaluation criteria were used to evaluate the reliability of AITs of the rainfall-runoff process [22, 64] 1) Co-efficient of determination (R2 ) [65]; (2) Normalized root mean square error (NRMSE) [66]; (3) Nash-Sutcliffe Coefficient of efficiency (COE) [67] (4) Mean square error (MSE) [68] were used.

$$R^2 = \frac{n(\Sigma \infty \mathfrak{y}) - (\Sigma \infty)(\Sigma \mathfrak{y})}{\sqrt{n[\Sigma \infty 2 - (\Sigma \infty 2)][\Sigma \mathfrak{y} 2 - (\Sigma \mathfrak{y} 2)]}} \tag{6}$$

$$\text{NRMSE} = \frac{\sqrt{\sum\_{i=1}^{N} (\text{Qobs} - \text{Qpre})^2}}{\sigma} \tag{7}$$

$$COE = 1 - \frac{\sum\_{i=1}^{N} (Qobs - Qpre)^2}{\sum\_{i=1}^{N} (Qobs - Qmean)^2} \tag{8}$$

$$\text{MSE} = \frac{\sum\_{i=1}^{N} \left( \text{Qobs} - \text{Qpre} \right)^{2}}{N} \tag{9}$$

Where, Qobs and Qpre are the observed and predicted flows, respectively, while Qmean is the mean of observed flows. *R*<sup>2</sup> tells us how the fit line of regression approaches the actual data in regression. Value 1 illustrates that the line efficiently fits the real data.

## **3. Results and discussions**

#### **3.1 Rainfall forecasting**

Flow Duration Curves (FDCs) were employed to evaluate the applied AITs against the percent of Time. FDCs for all input combinations (C1, C2, C3, C4, C5, C6 and C7) showed a good relationship with applied AITs in both training and testing seasons. To understand the behavior of applied AITs with the Jhelum River basin, the FDCs analysis was executed at nine rainfall stations for the modeling of the rainfall-runoff process as the runoff data was collected from the Mangla reservoir from time duration 1981–2012, the behavior of all techniques necessary to understand throughout the catchment.

The observed hydrographs of low, medium and high percentile flow extracted by the AITs (GEP, RBF-SVM and M5 Model Tree) to access the capability. [52, 69, 70] revealed that the FDCs exposed the relationship between the observed and modeled percentile flow and exceedance probability in the designated time duration. From 1 to 10%, the flow is considered high, 11–89% the flow is medium while, 90–100% the flow is referred to as low flows, which can be clearly seen from.

Furthermore, the percentile flows from 11 to 49%, and 50–89% are considered high medium and low medium flows. The outcomes of FDCs exposed that the GEP was better AIT for high flows and medium-high flows, and it better bonds with FDC of observed flow. Whereas the FDC of the M5 Model Tree better bonds with medium-low and low percentile flows. While RBF-SVM better bonded with the FDCs of low percentile flows. GEP was compared to other AITs was found more accurate DDM and found highly efficient. **RBF-SVM** these trends are shown in **Figure 4**.

*Evaluating the Performance of Different Artificial Intelligence Techniques… DOI: http://dx.doi.org/10.5772/intechopen.98280*

#### **Figure 4.**

*Flow duration curve (FDC) of observed and simulated daily streamflow in all rivers for various combination C1, C2, C3, C4, C5, C6 and C7 are labeled as A, B, C, D, E, F, and G respectively of Mangla watershed for the time periods 1981–2012.*

In RBF kernel-based SVM modeling, the functionality and importance of input combinations were achieved by adjusting the model parameters Gamma, C and P. In other words, the successful application of the RBF-SVM model dependent on accurate determinations of these model parameters. **Figure 5** and **Table 3** show the output results of different input combinations regarding model evaluation performance criterion. It can be clearly seen that RBF-SVM has potential and explicit good performances in training and testing durations of rainfall-runoff modeling. Furthermore, all input combinations employed in this research showed good performance. R<sup>2</sup> , COE, MSE and NRMSE for the training period were found 0.99, 1.00, 21245.92 and 820420.17m<sup>3</sup> /sec with input C3 and 0.99, 1.00, 21475.00 and 825413.21 m<sup>3</sup> /sec respectively with input C6. But input combinations C2 and C4 were found poor combination during training of model with results 0.16, 1.00, 16623.59, 833046.88 m<sup>3</sup> /sec and 0.11, 1.00, 980.10, 988371.24 m<sup>3</sup> /sec respectively. The behavior of RBF-SVM found poor in both cases due to which showed deprived results. By examining the model evaluation parameters in testing periods, it can be seen that the RBF-SVM model performed and obtain better prediction accuracy. R<sup>2</sup> , COE, MSE and NRMSE for the testing duration were found 1.00, 1.00, 188.52 and 1437.96 m<sup>3</sup> /sec with C1 and 1.00, 1.00, 147.81 and 1128.49 m<sup>3</sup> /sec with input C5, respectively.

#### **Figure 5.**

.

*Hydrographs of RBF-SVM model for overall training and testing rainfall and runoff datasets of Mangla watershed.*


#### **Table 3.**

*Training and testing outcomes of statistics of RBF-SVM model with different input combinations.*

*Evaluating the Performance of Different Artificial Intelligence Techniques… DOI: http://dx.doi.org/10.5772/intechopen.98280*


#### **Table 4.**

*Training and testing outcomes of statistics of M5 model tree with different input combinations.*
