## 2 Material and Methods
### ARCH-Type Models  
By applying nonlinear models to hydrologic processes, especially streamflow modeling, it is worth mentioning that the conventional linear models primarily focus on the average of data (first-order moment) because they do not consider the second moment of data (variance), thus, their application is not enough for modeling stochastic data. By using linear models, experts cannot seize the nonlinear characteristics of hydrological data [48]. Besides, methods for working with changes to variance over time are necessary for water resources management developments [50]. Defining nonlinear methods for modeling variance variation is essential in modeling and forecasting. For this purpose, [PERSON] (1982) introduced the ARCH model. Equations (1) and (2) illustrate the ARCH model [51].  
\[\varepsilon_{t}\ =\ \sigma_{t}.z_{t} \tag{1}\]  
\[\sigma_{t}^{2}\ =\ \alpha_{0}+\sum_{i\ =\ 1}^{q}\alpha_{i}\varepsilon_{t-i}^{2} \tag{2}\]  
where \(\sigma_{t}^{2}\) is conditional variance, \(\varepsilon_{t}\) is a discrete-time stochastic process, and \(\alpha_{0}\) is the ARCH model's parameters, \(q\) is the model's order, and \(z_{t}\) is the normal and standard series.  
This study presents 12 steps for modeling the stochastic parts of time-series using ARCH-type methods:  
Step 1: Data collection. All stream data were first collected on a monthly scale.  
Step 2: Data Preprocessing. It included checking data lengths, investigating statistical parameters, and data stationery.  
Step 3: Data normalization. This used the Delleur and Karamouz method [52] and defined the average and standard deviation of streamflow data.  
\[Q_{nt}\ =\ Ln\ (Q_{t}\ +\ c) \tag{3}\]  
Step 4: Defining \(Z_{t}\).  
\[Z_{t}\ =\ \frac{x_{t}-\overline{X}}{S_{d}} \tag{4}\]  
Step 5: Fitting the best AR models to stream time-series data.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Streams**} & \multirow{2}{*}{**Stations**} & \multicolumn{4}{c}{**Geographic Properties**} \\ \cline{3-5}  & & **Longitude** & **Latitude** & **Elevation (m)** \\ \hline
**Nazluchai** & Tapik & 00-44-54 & 00-37-40 & 1410 \\
**Baranduzchai** & Dizaj & 00-45-04 & 00-37-23 & 1320 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The geographical properties of monthly streamflow of both streams.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rivers** & **Stations** & \(X_{min}\) & \(X_{max}\) & \(X_{mean}\) **(m3/s)** & \(S_{x}\) & \(C_{xx}\) \\ \hline
**Nazluchai** & Tapik & 0.017 & 104.378 & 11.101 & 15.502 & 2.822 \\
**Baranduzchai** & Dizaj & 0.031 & 46.125 & 7.269 & 7.996 & 2.055 \\ \hline \hline \end{tabular} \(X_{min}\), \(X_{max}\), \(X_{mean}\), \(S_{x}\) and \(C_{xx}\) denote the minimum, maximum, mean, standard deviation, and skewness coefficient, respectively.  
\end{table}
Table 2: Statistical properties of monthly streamflow data.  
Step 6: Defining \(\varepsilon_{t}\).  
Step 7: Comparison of different ARCH-type models. At this point, the best-performing method was selected.  
Step 8: Defining different scenarios for estimating the deterministic part of the modeling equations.  
Step 9: Dividing data into two sections (calibration and validation). At this point, the best input combination and selection method can be defined.  
Step 10: Running DDMs for defining the deterministic part of the streamflow modeling formula.  
Steps 11 and 12: Selecting the best-performing hybrid model for estimating streamflow.  
As a conclusion, autoregressive conditional heteroscedastic residuals occur when the ARCH model considers the assumption that all data are normal, but the conditional variance of the residuals fluctuates linearly with squared residuals. The main difference of ARCH models with the other time-series models like AR, MA, and ARMA is that they can be applied on squared residuals of data.