The membership values of elements belonging to a fuzzy set can be represented by a straight line, known as a linear function. This function is characterized by three main parameters (boundaries): a, b, and c. It can be defined according to the Equation 3.

Where a<b<c ϵ R .In Figure 1a illustrates the graph of the triangular membership function (see Hasan and Mohammad [15]).

It is also a linear function, and it is distinguished from other membership functions by having four parameters (boundaries). a, b, c, d This function can be defined according to the Equation 4,

Where < b<c<d∈R . In Figure 1b illustrates the graph of the trapezoidal membership function (see Alavala [20]).

It is one of the decision-making algorithms extensively studied by many researchers in the literature for building linear regression models as an important tool in the decision-making process [26]. This algorithm was designed to eliminate the bias in selecting the most significant variables in modeling important phenomena in regression analysis, thereby providing a good fit to the relevant experimental data [24]. Its operation is based on the chi-square test applied to residuals. The algorithm is constructed within the framework of piecewise constant linear regression models with univariate splits [27].

At each terminal node, the sample mean is calculated to serve as the estimate, followed by the computation of residuals. The node is then split into two groups: the first group contains positive residuals, while the second group contains non-positive residuals. The idea behind this division is to detect random patterns within these two groups using a sign test on the residuals at each node. A chi-square test can be employed to examine the association between the signed residuals at each node (categorized as belonging or not belonging) represented as rows in the test, and the total frequencies of the predicted values for c splits represented as columns [28]. If the predictive variable y is categorical with ccc categories, a 2 × c contingency table is constructed, where the two rows correspond to the two residual groups and the c columns correspond to the predicted categories [29]. In the case where y is a quantitative variable, these values can be divided into columns according to a specific scheme, commonly using four groups representing the quartiles of the sample. This results in a 2 × 4 contingency table [30].

Binary decision trees are used to estimate the parameters of non-parametric regression models in the context of fuzzy set theory. Suppose that the function μ_A(x)represents the membership function of an element x to the fuzzy set A, and we attempt to find a solution by designing a decision tree with outcomes{either x∈A or x∋A} where the value 1 represents membership of the element x to the set A, and the value 0 represents non-membership of the element x to the set A [23].

The degree of membership of the element x to the fuzzy set A allows for any real value between 0 and 1, which means the following [31]:

μ_A(x) ∈ [ 0, 1].

By using partial membership in a fuzzy set, the strict logical constraints of set membership can be relaxed, thereby improving the performance of decision trees when dealing with fuzzy sets. This results in enhanced performance of decision trees in regression models in terms of flexibility and robustness [3].

Based on the literature of crisp decision trees, the GUIDE method can be used to estimate the regression model. Suppose there are P explanatory (independent) variables.

Let x_i = (x₁, x₂, ..., x_P) i = 1, ..., p

Categorical variables do not pose a significant challenge, except in the case of fuzzy partitions of categorical variables. The values of the response variable represent the prediction target and can be either categorical (for classification) or real-valued (for regression) [32].

According to the definition of a fuzzy decision tree, all nodes of the tree are connected to the root node t₀, where the decision tree is constructed using a hierarchical splitting strategy. The feature space is divided through a hierarchical sequence of logical tests into a set of non-overlapping regions, making the decision-making process straightforward. Each internal node in the decision tree corresponds to a test in the hierarchical structure used to construct the decision tree of interest [2].

Suppose the decision tree has been constructed up to a certain level, and the binary (terminal) nodes t_i are characterized by the membership function μ_i(x). Each node is split into two branches: one representing the value 1, which satisfies the logical test, and the other representing the value 0, which does not satisfy the logical test, as defined by Equation 5 [1].

The absolute membership degree and the number of training examples in the terminal node t₁ are expressed as Equation 6.

In a regression problem, the node t_i provides a prediction for the value of the response variable, which is equal to the mean of the response variable y for the training samples associated with t_i and it is represented as Equation 7 [33].

In the regression model, the binary (terminal) node t_i represents the predicted value of the response variable. This value corresponds to the average of the response variable y calculated from the training samples assigned to node t_i. The mathematical formula for computing this average is given by Equation 8 [2].

where

N_train Represents the number of training samples for each n = 1, ..., N_train.

μ_i(.) Represents the membership function for each i = 0, 1, 2, ..., where i denotes the number of nodes in the tree.

y_n ∀ n = 1, ..., N_trainAll elements of the response variable in the training set N_train.

x_n ∀ n = 1, ..., N_train All elements of the dependent variable in the training set N_train.

Equation 2 can be expressed through building the tree as given in Equations 9, 10:

The terminal (leaf) nodes for all branches of the tree are calculated and denoted byT~, such that the total number of nodes in the tree can be computed using the following formula [28].

Similarly, the number of internal (branch) nodes of the tree can be determined according to the following formula [26]:

The terminal nodes are used as predictive variables (y¯1,y¯2,...), and the average value of the dependent variable within each terminal node is calculated based on the number of observations that satisfy the test at that terminal node. Referring back to the definition in the formula above [31]. When applying the decision tree to a regression problem, it is essential to understand the membership relationship between the terminal (branch) nodes and the root node through the absolute degree of this membership, based on the above Equation 5 can be rewritten in a more detailed form as follows.

Where R: right node, L: left node, and μ_i(x). The absolute degree of membership for the original node t_i can be calculated by repeatedly applying Equation 11 up to the root node. At this point, all points belong to the root node, Therefore, Equation 12 is holds true [24]:

All successful splits originating from the root node, repeated recursively, represent the sequence of connected logical tests from the root node to the terminal node t_i. This implies that the estimate (mean) at each terminal node corresponds to the outcome of the test at that terminal node. The sum of these estimates across all terminal nodes provides the overall estimate (global mean) of all decision nodes in the regression tree, which corresponds to the overall mean in a general regression model. Therefore, the following Equation 13 must hold at every internal (splitting) node to ensure the consistency and accuracy of the model [33].

In the regression problem, the predicted value y given by the tree for a specific input feature vector x_test is expressed by the Equation 14:

Where l denotes the test set size (i.e., the number of observations in the test set).

y¯l is given in Equation 6 such that, by construction, only one of the membership values {μl(xtest);tl∈T~} equals 1, while the rest are zero. The error rate of the tree based on the training set is given by Equation 15 [1].

Unless a stopping criterion is specified, the decision tree continues to grow until R_train(T) = 0:

However, there exists an optimal-sized tree beyond which, despite the fact that R_train(T) continues to decrease monotonically, the predictive performance deteriorates (i.e., the true error rate or the unbiased estimation error increases). To avoid the problem of overfitting in the decision tree built from the training set and to obtain a tree of optimal size, several strategies have been proposed.

One of the most important methods to address excessive branching in tree growth is pruning, which involves removing branches that contain insignificant splits [5].

For the implementation of the work, the GUIDE algorithm processes four types of data (see Loh [30, p. 12]) as follows

In this paper, we presented a new statistical method for measuring and analyzing climate change in a specific geographical area and in a specific period of time, relying on data on two important variables (temperature and wind speed) for the purpose of identifying the risks of this phenomenon and taking appropriate precautions in the near and distant future to deal with this natural emergency that is increasing over time. The tool of this study the decision tree regression method, was employed as a new extension of its application in the phenomenon of climate change, relying on two variables and using fuzzy logic to divide the data into groups as an accurate description of the climate situation.

Here we must study and analyze the emerging phenomena according to the reference literature for those interested in studying the field of knowledge from all its aspects and relationships. Here, we employed the decision tree approach as a primary tool, relying on it to construct a regression model for the fuzzy data of both study variables. In the previous section, we presented the quantitative and fuzzy general description of the two variables (Tables 1, 2).

We utilized fuzzy logic to transform the quantitative and real-valued data of the two variables temperature, and wind speed, which are characterized by ambiguity and uncertainty, into categorical variables represented by clearer attributes and partitions. These transformed variables may simultaneously belong to two groups at the same time, as shown in Table 2. Figure 4 illustrates the fuzzy membership functions (triangular) for each partition (attribute) of the temperature variable, as described in Table 2.

In this section, in line with the objectives of the research, the two variables are examined separately and in a detailed manner. Figures 4, 5 present the partitions of each variable based on their respective root nodes adopted in this study. Furthermore, Figures 6a–e, 7a–e provide a comprehensive illustration of the hierarchical branching structure at each decision node of the decision tree model constructed using the real data corresponding to the two variables, respectively. In light of this approach, the binary-division sets (binary-division decision trees type) for the purpose of employing the chi-square test, the data set is divided into two types: first, the negative-signal set and second, the positive-signal set. Then, each node of the tree is constructed to represent a specific fuzzy partition based on fuzzy (linguistic) rules. This process transforms the crisp input data into fuzzy sets relying on membership functions, which in turn act as splitting nodes in the tree branching into five child nodes. According to the hierarchical structure of the decision tree partitions, these form partial groups based on quartiles. Subsequently, these groups are further divided into two terminal (leaf) nodes corresponding to positive and negative difference signals. Based on the calculated test statistic, decisions are made using the contingency tables method for the chi-square test. The related tables and the following section explain in detail.

To carry out the analysis using the steps of the GUIDE algorithm discussed in Section 8.1 on the data of the climate change variables (temperature and wind speed), we will construct 2 × 4 contingency tables for the chi-square test to measure the independence of the error sign between the predicted and actual values. This will be done for both the observed and expected frequencies across the four fuzzy group partitions (columns). Then, we calculate the chi-square test statistic and compare it with the tabulated critical value (x(3,0.05)2=7.8147)according to the following hypothesis testing: the null hypothesis, which represents the levels of sign error that are independent of the levels for the fuzzy partition groups, against the alternative hypothesis, which represents the levels of sign error that are dependent on the levels for the fuzzy partition groups. Figures 8a, b explained the descriptive for fuzzy partition groups.

Figure 8a illustrates the triangular membership functions for the temperature variable. And Figure 8b illustrates the triangular membership functions for the wind speed variable, which are measuring some climate change factors during the study period in Dhi Qar Governorate. and the characteristics that represent the fuzzy sets used to classify the temperature variable into five fuzzy groups (very cold, cold, mild, hot, and very hot) and the wind speed variable into five fuzzy groups (very slow, slow, mild, strong, and very strong). In order to give a clear and accurate perception of the weather condition and remove the ambiguity from the general description of the condition to a more accurate description closer to measuring.

Figure 4 showed the definition of the root node of the temperature variable, and Figure 5 showed the definition of the root node of the wind speed variable. Figures 6a–e shows the branches of the regression tree of the temperature variable for the fuzzy partitions that we explained in Figure 8a and which relied in its calculations on the steps of the GUIDE algorithm to clarify the work using this approach and relying on the analysis in Table 3. For the purpose of clarifying the work in Table 3, (r) represents the number of repetitions in each cell that corresponds to the error signal for each of (the observed repetition and the expected repetition) and the corresponding subset of the total sets that were divided through the three quartile values (q₁ = 22.3, q₂ = 29.5, q₃ = 43.7). Figures 7a–e shows the branches of the regression tree for the wind speed variable for the fuzzy partitions that we explained in Figure 8b and which relied in its calculations on the steps of the GUIDE algorithm to clarify the work using this approach and relying on the analysis in Table 4. For the purpose of clarifying the work in Table 4, (r) represents the number of repetitions in each cell that corresponds to the error signal for each of (the observed repetition and the expected repetition) and the corresponding subset of the total sets that were divided through the three quartile values (q₁ = 5.8, q₂ = 7.4, q₃ = 9.15) in fuzzy logic.

Tables 3, 4 display the classification outcomes derived from the analysis of a fuzzy dataset. These results were validated using two independent tree-based models. Figures 6a–e, 7a–e demonstrate the underlying decision-making process, which leverages the inherent tree structure within fuzzy regression. This process involves the division of the sample space into quartiles to provide a comprehensive description of the variable states. In the case of analyzing the temperature variable, the output results are in Table 3, where the second and third columns display the fuzzy group divisions and the error signal for each division. The rest of the columns represent the branching results for the variable according to the fuzzy categorical division of the error signal. Thus, the descriptive results of the repetition of cases can be analyzed in the cells corresponding to the chi-square table as follows: As for the first division, very cold weather has a main influence and is the focus of our attention, where 339 observations with a rate of 80% corresponded to descriptions of cold weather (the error sign is positive), which compares with rate 20% for very cold weather. Analyzing the first quartile, 22 observations with a rate of 20% were classified as cold weather, while 86 observations with rate 80% were categorized as very cold weather. As for the remaining quarters of this category division, they were 106 and 105 observations, with rates of 100% for cold and rates of 0% for very cold, respectively. As for the rest of the category classifications, they can take the same analysis, and they are moderate with mild climate category divisions and are considered not influential in the main way (cold, intermediate, and hot). As for the fifth division, very hot, it has a main influence and is the focus of our attention. We recorded 328 observations, with 77% of the total observations, compared to 23% who described the weather as very hot. As for the fourth quarter, which described the weather as hot, there were 9 observations, with a rate of 8.5%, while 97 observations described the weather as very hot, representing 91.5%. For Table 3, we found that the calculated chi-square value for the very cold of the temperature variable was 316.4632, and compared with the corresponding tabular chi-square value, which was [x(3,0.05)2=7.8147], we note that the calculated value is greater than the tabular value, and thus the null hypothesis is rejected and the hypothesis that states that the levels of the residuals signal are not independent of the partial sums is accepted, and this means that the first fuzzy set for the temperature variable affects climate change. And, we found that the calculated chi-square value for the very hot of the temperature variable was 378.2436, and compared with the corresponding tabular chi-square value, which was [x(3,0.05)2=7.8147], we note that the calculated value is greater than the tabular value, and thus the null hypothesis is rejected and the hypothesis that states that the levels of the residuals signal are not independent of the partial sums is accepted, and this means that the Fifth fuzzy set for the temperature variable affects climate change.

We conclude from Table 3 that the fuzzy classifications were more accurate in describing temperatures when using the fuzzy regression tree structure based on the selected categorical classifications provided by fuzzy logic. This improves the accuracy of the results shown in Table 3 and provides a clear understanding for both the general public in Dhi-Qar Governorate who are interested in weather conditions and researchers studying climate change, enabling decision-makers to take immediate and future action to mitigate and prevent temperature increases and to utilize them in electricity generation by developing alternative plans and environmentally friendly policies.

When analyzing the results for the wind speed variable in Table 4, the second and third columns display the fuzzy group classifications and the error signal for each classification. The remaining columns present the results of the variable's branches according to the fuzzy categorical division of the error signal. Accordingly, the descriptive results of case repetitions can be analyzed in the cells corresponding to the chi-square table as follows: The number of observed frequencies in the sample describing the weather as strong winds was 256 observations, accounting for 60%, compared to 40% who described the weather as stormy. In the first and second quarters, 46 observations, a rate of 21% described the weather as having strong winds, while 169 observations, a rate of 79%, described it as stormy. The number of observations in the sample describing the weather as stormy was 337, accounting for 79%, compared to 21% who described it as very stormy. In the fourth quarter, 18 observations, a rate of 17% described the weather as stormy, while 88 observations, a rate of 83% described it as very stormy. For Table 4, we found that the calculated chi-square value for the strong of the wind speed variable was 302.1632 and compared with the corresponding tabular chi-square value, which was [x(3,0.05)2=7.8147], we note that the calculated value is greater than the tabular value, and thus the null hypothesis is rejected and the hypothesis that states that the levels of the residuals signal are not independent of the partial sums is accepted, and this means that the fourth fuzzy set for the wind speed variable affects climate change. And, we found that the calculated chi-square value for the very strong wind speed variable was 333.9847 and compared with the corresponding tabular chi-square value, which was [x(3,0.05)2=7.8147], we note that the calculated value is greater than the tabular value, and thus the null hypothesis is rejected and the hypothesis that states that the levels of the residuals signal are not independent of the partial sums is accepted, and this means that the fifth fuzzy set for the wind speed variable affects climate change.

We conclude from Table 4 that the structure of the fuzzy regression tree, based on the selected categorical classifications provided by fuzzy logic, played a significant role in monitoring weather. Analyzing the results presented in Table 4 provides information both the general public in Dhi-Qar Governorate who are interested in weather, and for researchers focused on climate change and alternative energy. These findings can assist decision-makers in utilizing wind speed as a source of clean and renewable energy. Wind energy can be harnessed across various fields, with one of the most important being the generation of electricity in an environmentally friendly manner.