*3.3.1 M5 tree model*

The M5T model was first introduced by [47]. Model trees simplify the theories of regression trees and there are constant values at their leaves [48]. M5T model is established in relation to a binary decision tree where linear regression functions are placed in the terminal node (leaf) and a relationship is developed between dependent and independent variables through it [49]. Model development involves two stages; the first stage involves in creation of a decision tree by using a split criterion


*Note: DD = Degree decimal; P= Precipitation; Tmax= Maximum temperature; Tmin= Minimum temperature; RH = Relative humidity; SR = Solar radiation.*

#### **Table 2.**

*List of meteorological stations in the Hunza basin.*

whereas in the second stage overgrown tree is pruned for designing the model tree [25]. The splitting stage in the M5T model is composed of regression function at the leaves instead of class labels and continuous numerical attributes can be estimated through it [36]. The splitting criterion for the M5T model procedure is based on the standard deviation reduction (SDR) function achieved in every node. This criterion points out the error in that node and the minimum expected error is calculated by the model because of testing each attribute in that node [50, 51]. The SDR in the M5T model can be calculated by the following Equation [47]:

$$\text{SDR} = \text{sd}(\text{M}) - \sum \frac{|\text{M}\_i|}{|\text{M}|} \text{sd}(\text{M}\_i) \tag{1}$$

Where *SDR* specifies the standard deviation reduction and sd indicates standard deviation; *M* specifies a set of examples that reaches the node; whereas *Mi* signifies the subset of examples that have the *i th* outcome of the potential set.

Because of the splitting or branching process, data in child nodes (smaller nodes) have less SD than parent nodes (greater nodes). The division process often results in producing a large tree-like structure which causes overfitting and this issue can be resolved by pruning back the tree [52], for instance by substituting a subtheme with a leaf. Pruning the overgrown tree and substitution of subthemes with linear regression functions are performed in the second stage of model designing. This method of producing the model tree separates the parameter space into subspaces and builds in each of them a linear regression model.
