A conventional time series is composed by real numbers whereas a fuzzy time series is composed by fuzzy sets [27] , where a fuzzy set is a class with fuzzy boundaries. In order to define a fuzzy time series let $U$ be the universe of discourse as defined in Equation 1 and a fuzzy set $A$ of $U$ is defined by Equation (2).

$U=\left\{u_1,u_2,\cdots ,u_n\right\}$ (1)

$A=\left\{\frac{f_A\left(u_1\right)}{u_1}+\frac{f_A\left(u_2\right)}{u_2}+\cdots +\frac{f_A\left(u_n\right)}{u_n}\right\}$ (2)

Based on [3] , $f_A$ is the membership function of $A$, $f_A:U\to \left[0,1\right]$. $f_A\left(u_i\right)$ indicates the degree of membership of $u_i$ in $A$, where $f_A\left(u_i\right)\in \left[0,1\right]$ and $1\le i\le n$.

The Equations (1) and (2) presented define key concepts in fuzzy time series. Equation (1) establishes the universe of discourse $U$, which represents the set of possible values in the time series. Equation (2) then defines a fuzzy set $A$ over $U$, where each element $u_i$ is assigned a degree of membership $f_A\left(u_i\right)$ within the interval $\left[0,1\right]$, as determined by the membership function $f_A$. These fuzzy sets serve as the foundation for defining a fuzzy time series, which, unlike conventional time series composed of precise numerical values, consists of fuzzy sets representing linguistic variables. The definitions provided further formalize fuzzy time series by describing their structure and classification, distinguishing between time-invariant and time-variant cases based on how values evolve over time.

The background definitions of fuzzy time series can be reviewed as follows.

Definition 2.1 [14] : Let $Y\left(t\right)$ $\left(t=0,1,2,\cdots \right)$, a subset of $ℝ$, be the universe of discourse on which fuzzy sets $f_i\left(t\right)$ $\left(i=1,2,\cdots \right)$ are defined, and let $F\left(t\right)$ be a collection of $f_i\left(t\right)$ $\left(i=1,2,\cdots \right)$. Then $F\left(t\right)$ is called a fuzzy time series on $Y\left(t\right)$ $\left(t=0,1,2,\cdots \right)$.

$F\left(t\right)$ can be regarded as a linguistic variable [28] , and $f_i\left(t\right)$ $\left(i=1,2,\cdots \right)$ can be viewed as possible linguistic values of $F\left(t\right)$ where $f_i\left(t\right)$ $\left(i=1,2,\cdots \right)$ are represented by fuzzy sets. Note that $F\left(t\right)$ is a function of time since its values can be different at different times. According to [27] , if $F\left(t\right)$ is caused by $F\left(t-1\right)$ only, then this relationship is represented by $F\left(t-1\right)\to F\left(t\right)$.

Definition 2.2 [3] : Let $F\left(t\right)$ be a fuzzy time series. If for any time $t$, a displacement at $t$ causes a displacement at $F\left(t\right)$ and $F\left(t\right)$ only has finite elements, then $F\left(t\right)$ is called a time-invariant fuzzy time series. Otherwise, it is called a time-variant fuzzy time series.

The above definitions will be used throughout the paper.

A Red-Black tree is a self-balancing binary search tree that ensures balance by performing rotations and recoloring of nodes, following specific rules [13] . The relevance of this data structure can be observed in the work developed by [29] , where a concurrent RBT version is presented. In this paper, we focus only on the insertion function, which proceeds as follows:

1) If the tree is empty, create a new node as the root and color it black.

2) If the tree is not empty, insert the new node as a red leaf.

3) If the parent of the new node is black, no further action is needed, and the insertion is complete.

4) If the parent of the new node is red, check the color of the parent’s sibling (the new node’s “uncle”):

*If the uncle is black or absent, perform a suitable rotation and recolor the nodes to restore balance.

*If the uncle is red, recolor both the parent and the uncle to black, and check whether the grandparent (parent of the new node’s parent) is not the root. If the grandparent is not the root, recolor it and recheck for any red-red conflicts, treating the grandparent as the new node.

In this step, stock index data is collected. The dataset should consist of a time series representing the closing prices of an index or stock. For the examples in this paper, the data was sourced from historical records on the http://www.investing.com website, which is widely trusted by traders for its accurate stock market information.

The time series data is input into the model in array format, as shown in Equation (3):

$x\left(t\right)=\left\{x_1,x_2,\cdots ,x_n\right\}$ (3)

In Equation (3), $t$ is the time index, where $x_1$ is the first value and $x_n$ is the last.

In this step, a new array is created from the time series, where duplicate elements are removed, resulting in $x_u\left(t\right)$. This step is crucial for the subsequent red-black tree construction.

Here, the red-black tree is built from the array $x_u\left(t\right)$, ensuring that no duplicate nodes are included, as required by the model. A detailed example of constructing a red-black tree is provided in [30] .

For example, given the following time series:

$x\left(t\right)=\left\{6,7,15,7,10,2,8,20,5\right\}$ (4)

After constructing the red-black tree from the data in Equation (4), the resulting structure is shown in Figure 3 .

In this step, clusters are generated to define fuzzy sets, with each cluster corresponding to a distinct fuzzy set.

The clustering procedure is based on the parent-child difference in node values. Its advantages stem from ensuring that similar values are grouped within the same cluster. This way, the clusters maintain elements with values that preserve closer statistical measurements. Additionally, since the method relies on differences rather than absolute values, it is resistant to global shifts or biases in the data. Finally, it is also advantageous because it reduces noise in the data, provided there is a trend in the related values.

The clustering algorithm follows these steps:

For the tree shown in Figure 3 , the clusters formed are as follows:

Fuzzification refers to the process of converting the time series, which consists of real numbers ( $x_1,x_2,\cdots ,x_n$), into a fuzzy time series composed of fuzzy sets ( $A_1,A_2,\cdots ,A_c$). Here, $n$ is the number of time series elements, and $c$ is the number of fuzzy sets (which is equal to the number of clusters from Step 4). In this example, $n=9$ and $c=4$.

The fuzzification process is as follows:

The result of this process is a fuzzy time series, as shown in Table 1 .

In this step, fuzzy rules are created based on the fuzzy time series. A typical fuzzy dependency rule is represented as:

$A\left(t\right)\to A\left(t+1\right)$ (5)

here, $A\left(t\right)$ is the antecedent (precedent) and $A\left(t+1\right)$ is the consequent.

For each fuzzy time series value, a rule is generated to predict the next fuzzy time series value. For example, if the first fuzzy time series value is A1 (corresponding to the original time series value of 6), this implies the next fuzzy time series value will also be A1 (corresponding to 7 in the original time series).

The fuzzy rules corresponding to the fuzzy time series from Equation (4) are shown in Table 2 .

Next, fuzzy rules are grouped. If multiple rules share the same antecedent, they are combined into a single group rule with the common antecedent and all their consequents. For instance, the following rules

$\begin{array}{l}A1\to A1\\ A1\to A2\\ A1\to A7\end{array}$ (6)

can be grouped as:

$A1\to A1,A2,A7$ (7)

Table 3 shows the fuzzy group rules (FGRs) derived from Table 2 . Note that the fuzzy group rules include all consequents, even when repeated.

In this step, linguistic forecasting is performed using the fuzzy group rules (FGRs) from Step 6. Given the fuzzy time series $A\left(t\right)$, the forecast for the next time step is obtained by applying the appropriate fuzzy rule.

For example, consider the fuzzy time series from Equation (8):

$A\left(t\right)=A1,A1,A4,A1,A2,A3,A2,A4,A1$ (8)

To forecast the second sample, we first identify the fuzzy value for $t=1$, which is $A1$. Referring to the FGRs in Table 3 , the corresponding rule is $A1\to A1,A4,A2$. Thus, the forecast for $t=2$ is $A1,A4,A2$.

In the final step, the linguistic forecast is converted back to a numerical value using defuzzification. To explain this, consider the fuzzy forecast for the second sample, as shown in Equation (9):

$\text{fuzzy\_forecast}\left(2\right)=\left\{A1,A4,A2\right\}$ (9)

Each forecast set is assigned a weight, as shown in Table 4 .

Next, the average of each cluster center is computed, as shown in Table 5 .

Finally, the numerical forecast is computed as the weighted average of the cluster centers. For example, for the second sample, the forecast is:

$\text{forecast}\left(2\right)=\frac{6\times 1+17.5\times 2+9\times 3}{1+2+3}=11.333$ (10)

This completes the process for forecasting subsequent samples. Note that Steps 1 through 6 represent the training phase, and Steps 7 and 8 are used repeatedly to make forecasts for future samples.

The in-sample test was conducted using the closing prices of the Ibovespa (Brazilian stock index) from 03/29/2018 to 10/11/2018, obtained from http://investing.com/ . A sample of the data used for the model forecasts is presented in Table 6 .

The performance of the proposed model relative to existing models is summarized in Table 7 , based on four different performance metrics.

The next section discusses the out-of-sample test to further validate the effectiveness of the proposed model.

The out-of-sample test was conducted as follows:

1) Train the model using the Ibovespa dataset from 03/29/2018 to 10/11/2018.

2) Predict the price for the next sample (10/15/2018). (Only business days are considered).

3) On 10/12/2018, Brazil observed a holiday, and the stock market was closed. Since 10/13/2018 and 10/14/2018 were non-business days, there were no trades on these dates.

4) The forecast for 10/15/2018 was recorded as forecast (1).

For subsequent forecasts, the training dataset was updated by shifting it one day ahead, using data from 04/02/2018 to 10/15/2018 (real prices, not forecasts). This process was repeated for four steps, yielding forecasts forecast (1) to forecast (4). The results for all methods are shown in Table 8 .

The performance comparison among all methods in the out-of-sample test is shown in Table 9 .

As observed, the forecasts generated by the proposed model are closer to the actual Ibovespa data compared to those of the other models, demonstrating superior predictive performance.

The RMSE value serves as a measure of model performance, where lower RMSE values indicate higher forecasting accuracy. The proposed model exhibits the lowest RMSE among the models analyzed, signifying its superior performance.

The RMSE relative to the Naïve model provides an additional benchmark. A value greater than one indicates that the model performs worse than the Naïve model, while a value below one signifies better performance. The Naïve model assumes that the forecast for the next period is equal to the actual value of the current period. The proposed model not only achieves an RMSE relative to the Naïve model below one but also outperforms all other models in this regard.

Finally, the R-squared value reaches its highest level in the proposed model, indicating its superior ability to explain the variability in the original data.