An Exponential Autoregressive Time Series Model for Complex Data

Abstract

In this paper, an exponential autoregressive model for complex time series data is presented. As for estimating the parameters of this nonlinear model, a three-step procedure based on quantile methods is proposed. This quantile-based estimation technique has the benefit of being more robust compared to least/absolute squares. The performance of the introduced exponential autoregressive model is evaluated by means of four established goodness-of-fit criteria. The practical utility of the novel time series model is showcased through a comparative analysis involving simulation studies and real-world data illustrations.

1. Introduction

Time series analysis is a statistical method used to analyze and interpret data points that are collected over a sequence of time intervals. It involves studying the patterns, trends, and underlying structures within the data to make predictions or gain insights about future values. Time series data can be found in various domains, such as business, economics, finance, industry, and medicine [1,2,3,4].

Fuzzy time series analysis is an extension of traditional time series analysis that incorporates fuzzy logic, a mathematical concept that deals with uncertainty and vagueness. Fuzzy logic allows for the representation of imprecise or uncertain information by assigning degrees of membership to different categories. In the context of time series analysis, fuzzy logic can be used to model relationships and patterns in complex data when crisp boundaries are not well-defined. Thus, fuzzy time series analysis provides a flexible framework that is helpful in many ways, for instance, when handling uncertainty, embedding subjective knowledge, modeling complex relationships, considering granularity levels, and forecasting with limited data.

As common stochastic time series models cannot handle prediction problems based on data affected by uncertainties, fuzzy time series models can be consulted to handle fuzzy time series observations. Most fuzzy time series models in the literature rely on reporting crisp data (first stage), converting the predictions into fuzzy values by identifying the fuzzy logical relations (second stage), and defuzzifying them to obtain crisp predictions (third stage). Within this three-stage framework, Stage 2 assumes a pivotal role in enhancing the predictive capability of the time series model, necessitating careful attention and handling. The most prominent approaches in Stage 2 are fuzzy logical relation groups [5,6,7,8,9,10,11,12,13,14,15,16,17], machine learning techniques [18,19,20,21,22,23,24,25,26,27], and statistical methods combined with fuzzy logic [28,29,30,31].

In the scholarly literature, a handful of time series models have been devised specifically to manage fuzzy time series data [32,33,34,35,36]:

• A semiparametric model with fuzzy smooth functions and crisp parameters [32]
• A semiparametric model for observations reported by fuzzy numbers [33]
• A nonparametric additive model for fuzzy time series data [34]
• A quantile-based model for time series data reported by triangular fuzzy numbers [35]
• A nonlinear model for time series data reported by $LR$ fuzzy numbers [36]

Moreover, there are several proposals that combine time series analysis with tools of fuzzy regression analysis [37]. Fuzzy regression analysis stands out as a prominent field within fuzzy statistics, with numerous distinct approaches having been put forth to address diverse forms of fuzzy environments.

In many practical applications, nonlinear models are required in order to obtain high predictive accuracy. Nonlinear time series models offer more flexible frameworks for capturing intricacy and nonlinear dynamics [38,39,40,41,42]. They also facilitate improved forecasting, uncover hidden patterns, and provide valuable insights into the underlying processes. A general nonlinear time series model, to map the connection between the most recent time series observation ($x_t$) and the previous observations $x_{t-1},x_{t-2},\dots ,x_{t-p}$, is given by

$$x_t=f(x_{t-1},x_{t-2},\dots ,x_{t-p})+{\epsilon }_t,$$

where f is a known nonlinear function and ${\epsilon }_t$ is an error term. One powerful nonlinear time series model that can be employed to determine f is the Exponential Autoregressive (EXPAR(p)) model. This model is of autoregressive form with amplitude-dependent coefficients, so parameter estimation is a nonlinear optimization problem [43,44,45,46,47,48]. In general, the connection between $x_t$ and $x_{t-1},x_{t-2},\dots ,x_{t-p}$ in EXPAR(p) is determined by the mathematical model

$$\begin{array}{c}\hfill f(x_{t-1},x_{t-2},\dots ,x_{t-p})=\sum _{i=1}^p(a_i+(b_i+c_i{x}_{t-1})e^{-{\theta }_i{x}_{t-1}^2})x_{t-i},\end{array}$$

where $a_i,b_i,c_i\in \mathbb{R}$ and ${\theta }_i>0$ for $i=1,2,\dots ,p$. Such a model is able to generate time series data with different types of marginal distributions by restricting the parameter space in specific regions. Additionally, this model accommodates amplitude-dependent frequency, jump phenomena, and limit cycles. Notably, a distinctive attribute of this model is its ability to capture non-Gaussian characteristics inherent in the underlying time series.

In this paper, we introduce the Fuzzy Exponential Autoregressive Time Series Model (FEATSM(p)) for handling time series observations reported by $LR$ fuzzy numbers. Taking inspiration from fuzzy regression analysis [49,50,51,52], we propose a three-step estimation procedure of the nonlinear relationship in fuzzy time series data through a unified approach using quantile-based estimation errors. Quantile regression methods not only offer a means to estimate quantiles within traditional regression analysis but also significantly broaden the modeling possibilities for regression by enabling consideration of local, quantile-specific dynamics. In comparison with common least absolute methods, quantile methods have various advantages [53]. Although least squares or absolute squares techniques might demonstrate inefficiency when dealing with errors that are highly non-normal, quantile methods exhibit higher robustness. By conducting extensive comparative analysis, we evaluate the performance of the FEATSM(p) in comparison with the above time series models in the literature in the context of simulated and real data sets.

This paper is organized as follows. Section 2 gives some necessary essentials on fuzzy numbers. In Section 3, we introduce the FEATSM(p). Section 4 is then devoted to comparative analysis. The paper concludes in Section 5.

2. Basics of Fuzzy Numbers

Let $\mathbb{X}$ be a universal set on the real-line ($\mathbb{R}$). A fuzzy set is a mapping $\tilde{A}$ on $\mathbb{X}$ that assigns a degree of membership $0\le \tilde{A}\left(x\right)\le 1$ to each $x\in \mathbb{X}$. A fuzzy number (FN) $\tilde{A}$ is a convex and normalized fuzzy set on the real line $\mathbb{R}$ with an upper semi-continuous membership function of bounded support. The most commonly employed type of FNs is represented by what are known as $LR$-FNs. The membership function of an $LR$-FN$\tilde{A}={(a;l_a,r_a)}_{LR}$ is defined by

$$\begin{array}{c}\hfill {\mu }_{\tilde{A}}\left(x\right)=\left\{\begin{array}{cc}L\left({\displaystyle \frac{a-x}{l_a}}\right)& \mathrm{for}x\le a,\\ R\left({\displaystyle \frac{x-a}{r_a}}\right)& \mathrm{for}x>a,\end{array}\right.\end{array}$$

where $a\in \mathbb{R}$, $l_a>0$ and $r_a>0$ are referred to as the center value and the left and right spreads of $\tilde{A}$, respectively. The shape functions L and R are continuous and strictly decreasing functions from $[0,1]$ to $[0,1]$ satisfying $L\left(0\right)=R\left(0\right)=1$ and $L\left(1\right)=R\left(1\right)=0$. The set of all $LR$-FNs is represented by ${\mathbb{F}}_{LR}\left(\mathbb{R}\right)$. When it holds $L=R$, $\tilde{A}={(a;l_a,r_a)}_{LR}$ is called a symmetric $LR$-fuzzy number and denoted by $\tilde{A}={(a;l_a)}_L$. In this paper, we employ the most commonly used special case of $LR$-fuzzy numbers with $L\left(x\right)=R\left(x\right)=max\{0,1-x\}$, which is known as triangular fuzzy numbers (TFNs), to take into account the imprecision of the underlying data set in numerical evaluations. The membership function of a TFN $\tilde{A}={(a;l_a,r_a)}_T$ is given by:

$${\mu }_{\tilde{A}}\left(x\right)=\left\{\begin{array}{cc}\frac{x-(a-l_a)}{l_a}\hfill & \mathrm{for}a-l_a\le x\le a\hfill \\ \frac{a+r_a-x}{r_a}\hfill & \mathrm{for}a<x\le a+r_a\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

The addition operation between two $LR$-fuzzy numbers $\tilde{A}={(a;l_a,r_a)}_{LR}$ and $\tilde{B}={(b;l_b,r_b)}_{LR}$ is defined as follows:

$$\tilde{A}\oplus \tilde{B}={(a+b;l_a+l_b,r_a+r_b)}_{LR}.$$

A squared error distance [54] between two $LR$-fuzzy numbers $\tilde{A}={(a;l_a,r_a)}_{LR}$ and $\tilde{B}={(b;l_b,r_b)}_{LR}$ can be defined by

$$D^2(\tilde{A},\tilde{B})={(a-b)}^2+c_1{(l_a-l_b)}^2+c_2{(r_a-r_b)}^2,$$

where $c_1={\int }_0^1{L}^{-1}\left(\alpha \right)d\alpha$ and $c_2={\int }_0^1{R}^{-1}\left(\alpha \right)d\alpha$.

3. The Exponential Autoregressive Time Series Model for Fuzzy Data

3.1. The Model

In this subsection, the (nonlinear) exponential autoregressive time series model of order p for $LR$ fuzzy data, i.e., the FEATSM(p), is developed.

Definition 1.

Let ${\tilde{\mathbf{x}}}_{\mathbf{T}}=\{{\tilde{x}}_1,{\tilde{x}}_2,\dots ,{\tilde{x}}_T\}$ be a set of  FNs of size T. Then ${\tilde{\mathbf{x}}}_{\mathbf{T}}$ is regarded as fuzzy time series data if $\{{\tilde{x}}_1,{\tilde{x}}_2,\dots ,{\tilde{x}}_T\}$ is the vague concept of crisp time series data $\{x_1,x_2,\dots ,x_T\}$ [34,55].

Remark 1.

As discussed in the Introduction, there are many situations where it is preferable to report exact data x by a FN $\tilde{x}$ as “about x”. In such cases, $\tilde{x}$ is referred to as the vague concept of x. For instance, consider the prediction of the monthly water level of a lake (i.e., $n=30$). In common statistical methods, one is often interested in reporting the average value of the data, that is, $\overline{x}$ with $\overline{x}=\frac{1}{30}{\sum }_{i=1}^{30}{x}_i$. However, the monthly water level is not an exact quantity, so it is more promising to report it as a FN. For this purpose, one could apply the method of Buckley [56] to convert $\overline{x}$ into “about $\overline{x}$” by $\tilde{x}={(\overline{x};t_{{\alpha }_1,29}\frac{s_i}{30},t_{{\alpha }_2,29}\frac{s_i}{30})}_{LR}$, where ${\alpha }_1,{\alpha }_2\in (0,1]$, $s_i$ is the standard deviation, and $t_{{\alpha }_2,29}$ denotes the inverse t-distribution with degrees of freedom 29. Another simple approach to convert $\overline{x}$ by a FN   $\tilde{\overline{x}}$ as “about x” is based on the concept of location-scale fuzzy observations $\tilde{x}={(\overline{x};u\overline{x},(1+u)\overline{x})}_{LR}$ with $u\in (0,1]$ [57]. Throughout this paper, we employ the second method to report fuzzy data.

Based on the connection between ${\tilde{x}}_t$ and ${\tilde{x}}_{t-1},{\tilde{x}}_{t-2},\dots ,{\tilde{x}}_{t-p}$, the fuzzy nonlinear time series model is defined by

$${\tilde{x}}_t=f({\tilde{x}}_{t-1},{\tilde{x}}_{t-2},\dots ,{\tilde{x}}_{t-p})\oplus {\tilde{\epsilon }}_t,t=p+1,\dots ,T,$$

(1)

where

(1)

$\tilde{f}({\tilde{x}}_{t-1},{\tilde{x}}_{t-2},\dots ,{\tilde{x}}_{t-p})=(f(x_{t-1},x_{t-2},\dots ,x_{t-p});l_f(l_{x_{t-1}},l_{x_{t-2}},\dots ,l_{x_{t-p}}),r_f(r_{x_{t-1}},r_{x_{t-2}},$$\dots ,r_{x_{t-p}}{\left)\right)}_{LR}$ with non-negative values $l_f$ and $r_f$, and

(2)

${\tilde{\epsilon }}_t={({\epsilon }_t;l_{{\epsilon }_t},r_{{\epsilon }_t})}_{LR}$ is a fuzzy error term.

In the following, we propose a novel approach to estimate the nonlinear function f. When extending (1) by means of the exponential autoregressive model, we can assume that the center as well as the left and right spreads of the fuzzy smooth function $\tilde{f}({\tilde{x}}_{t-1},{\tilde{x}}_{t-2},\dots ,{\tilde{x}}_{t-p})$ can be expressed as:

$$\begin{array}{cc}\hfill f(x_{t-1},x_{t-2},\dots ,x_{t-p})& =\sum _{i=1}^p\left(a_i+(b_i+c_i{x}_{t-1})e^{-{\theta }_i{x}_{t-1}^2}\right)x_{t-i}\mathrm{with}{a}_i,b_i,c_i\in \mathbb{R}\mathrm{and}{\theta }_i>0,\hfill \\ \hfill l_f(l_{x_{t-1}},l_{x_{t-2}},\dots ,l_{x_{t-p}})& =\sum _{i=1}^p\left(l_{a_i}+(l_{b_i}+l_{c_i}{l}_{x_{t-1}})e^{-l_{{\theta }_i}{\left(l_{x_{t-1}}\right)}^2}\right)l_{x_{t-i}}\mathrm{with}{l}_{a_i},l_{b_i},l_{c_i},l_{{\theta }_i}>0,\hfill \\ \hfill r_f(r_{x_{t-1}},r_{x_{t-2}},\dots ,r_{x_{t-p}})& =\sum _{i=1}^p\left(r_{a_i}+(r_{b_i}+r_{c_i}{r}_{x_{t-1}})e^{-r_{{\theta }_i}{\left(r_{x_{t-1}}\right)}^2}\right)r_{x_{t-i}}\mathrm{with}{r}_{a_i},r_{b_i},r_{c_i},r_{{\theta }_i}>0.\hfill \end{array}$$

Following (1) three linear regression models can be formulated:

(1)

$f(x_{t-1},x_{t-2},\dots ,x_{t-p})={\sum }_{i=1}^p(a_i+(b_i+c_i{x}_{t-1})e^{-{\theta }_i{x}_{t-1}^2})x_{t-i}+{\epsilon }_i$, $t=p+1,p+2,$$\cdots ,T$,

(2)

$l_f(l_{x_{t-1}},l_{x_{t-2}},\dots ,l_{x_{t-p}})={\sum }_{i=1}^p(l_{a_i}+(l_{b_i}+l_{c_i}{l}_{x_{t-1}})e^{-l_{{\theta }_i}{\left(l_{x_{t-1}}\right)}^2})l_{x_{t-i}}+l_{{\epsilon }_i}$, $t=p+1,p+2,\cdots ,T$,

(3)

$r_f(r_{x_{t-1}},r_{x_{t-2}},\dots ,r_{x_{t-p}})={\sum }_{i=1}^p(r_{a_i}+(r_{b_i}+r_{c_i}{r}_{x_{t-1}})e^{-r_{{\theta }_i}{\left(r_{x_{t-1}}\right)}^2})r_{x_{t-i}}+r_{{\epsilon }_i}$, $t=p+1,p+2,\cdots ,T$.

The unknown parameters are denoted by $\Delta =\{{\Delta }_1,{\Delta }_2,{\Delta }_3\}$ with

(1)

${\Delta }_1=\{\mathit{a},\mathit{b},\mathit{c},\theta \}$, where $\mathit{a}={(a_1,a_2,\dots ,a_p)}^{\top }$, $\mathit{b}={(b_1,b_2,\dots ,b_p)}^{\top }$, $\mathit{c}={(c_1,c_2,\dots ,c_p)}^{\top }$ and $\theta ={({\theta }_1,{\theta }_2,\dots ,{\theta }_p)}^{\top }$,

(2)

${\Delta }_2=\{l_{\mathit{a}},l_{\mathit{b}},l_{\mathit{c}},l_{\theta }\}$, where $l_{\mathit{a}}={(l_{a_1},l_{a_2},\dots ,l_{a_p})}^{\top }$, $l_{\mathit{b}}={(l_{b_1},l_{b_2},\dots ,l_{b_p})}^{\top }$, $l_{\mathit{c}}=(l_{c_1},l_{c_2},$$\dots ,l_{c_p}{)}^{\top }$ and $l_{\theta }={(l_{{\theta }_1},l_{{\theta }_2},\dots ,l_{{\theta }_p})}^{\top }$,

(3)

${\Delta }_3=\{r_{\mathit{a}},r_{\mathit{b}},r_{\mathit{c}},r_{\theta }\}$, where $r_{\mathit{a}}={(r_{a_1},r_{a_2},\dots ,r_{a_p})}^{\top }$, $r_{\mathit{b}}={(r_{b_1},r_{b_2},\dots ,r_{b_p})}^{\top }$, $r_{\mathit{c}}={(r_{c_1},r_{c_2},\dots ,r_{c_p})}^{\top }$ and $r_{\theta }={(r_{{\theta }_1},r_{{\theta }_2},\dots ,r_{{\theta }_p})}^{\top }$.

Then, $\Delta$ can be estimated using three independent steps. In this regard, we implement a least quantile loss function for each step as follows:

$$\begin{array}{cc}\hfill {\widehat{\Delta }}_1& =arg\underset{{\Delta }_1}{min}\left\{\sum _{k=1}^{199}\sum _{t=p+1}^T{\rho }_{0.005k}(x_t-\sum _{i=1}^p(a_i+(b_i+c_i{x}_{t-1})e^{-{\theta }_i{x}_{t-1}^2})x_{t-i})\right\},\hfill \\ \hfill {\widehat{\Delta }}_2& =arg\underset{{\Delta }_2}{min}\left\{\sum _{k=1}^{199}\sum _{t=p+1}^T{\rho }_{0.005k}(l_{x_t}-\sum _{i=1}^p(l_{a_i}+(l_{b_i}+l_{c_i}{l}_{x_{t-1}})e^{-l_{{\theta }_i}{\left(l_{x_{t-1}}\right)}^2})l_{x_{t-i}})\right\},\hfill \\ \hfill {\widehat{\Delta }}_3& =arg\underset{{\Delta }_3}{min}\left\{\sum _{k=1}^{199}\sum _{t=p+1}^T{\rho }_{0.005k}(r_{x_t}-\sum _{i=1}^p(r_{a_i}+(r_{b_i}+r_{c_i}{r}_{x_{t-1}})e^{-r_{{\theta }_i}{\left(r_{x_{t-1}}\right)}^2})r_{x_{t-i}})\right\}.\hfill \end{array}$$

Here, ${\rho }_{\tau }\left(u\right)=u(\tau -I(u\le 0))$ with $\tau \in (0,1)$ and $I(.)$ represents the indicator function. Note that we utilize a partitioned subset $\{0.005k:k=1,2,\dots ,199\}=\{0.005,0.01,\dots ,0.995\}$ for quantile levels on $[0,1]$ in this paper. To solve the resulting optimization problem, the least local approximation approach is utilized to optimize the target functions. According to the above three steps, the fuzzy predicted response variable of the proposed fuzzy regression model can be unified as $LR$-FN

$${\tilde{\widehat{x}}}_t={({\widehat{x}}_t;l_{{\widehat{x}}_t},r_{{\widehat{x}}_t})}_{LR},$$

where

$$\begin{array}{cc}\hfill {\widehat{x}}_t& =\sum _{i=1}^p\left({\widehat{a}}_i+({\widehat{b}}_i+{\widehat{c}}_i{x}_{t-1})e^{-{\widehat{\theta }}_i{x}_{t-1}^2}\right)x_{t-i},\hfill \\ \hfill l_{{\widehat{x}}_t}& =\sum _{i=1}^p\left(l_{{\widehat{a}}_i}+(l_{{\widehat{b}}_i}+l_{{\widehat{c}}_i}{l}_{x_{t-1}})e^{-l_{{\widehat{\theta }}_i}{\left(l_{x_{t-1}}\right)}^2}\right)l_{x_{t-i}},\hfill \\ \hfill r_{{\widehat{x}}_t}& =\sum _{i=1}^p\left(r_{{\widehat{a}}_i}+(r_{{\widehat{b}}_i}+r_{{\widehat{c}}_i}{r}_{x_{t-1}})e^{-r_{{\widehat{\theta }}_i}{\left(r_{x_{t-1}}\right)}^2}\right)r_{x_{t-i}}.\hfill \end{array}$$

3.2. Performance Measures

In this subsection, we briefly introduce goodness-of-fit measures that are appropriate to compare the predictive performance of time series models using fuzzy data. Note that, initially, the model parameters are estimated using an in-sample data set of size $T^{*}$. Subsequently, the model’s performance is evaluated using the remaining data that comprise a size of $T-T^{*}$. In particular, we employ the following four established performance measures for comparative analysis in the next section. In the below explanations, A and B are two fuzzy time series models to be compared.

Mean Forecast Error (MFE):

$$\mathrm{MFE}=\frac{{\sum }_{i=T^{*}+1}^T{D}^2({\tilde{\widehat{x}}}_i,{\tilde{x}}_i)}{T-T^{*}}$$

It holds $\mathrm{MFE}\ge 0$, and it should be as close to zero as possible. Moreover, ${\mathrm{MFE}}_A<{\mathrm{MFE}}_B$ implies that model A has a better predictive accuracy than model B.

Mean Absolute Scaled Error (MASE):

$$\mathrm{MASE}=\frac{{\sum }_{i=T^{*}+1}^T{q}_i}{T-T^{*}}\mathrm{with}{q}_i=\frac{D({\tilde{\widehat{x}}}_i,{\tilde{x}}_i)}{\frac{1}{T-T^{*}}{\sum }_{i=T^{*}+1}^T{D}^2({\tilde{x}}_i,{\tilde{x}}_{i-1})}$$

The interpretation of MASE is the same as that of MFE.

Mean Absolute Relative Error (MARE):

$$\mathrm{MARE}=\frac{1}{T-T^{*}}\sum _{i=T^{*}+1}^T\frac{{\int }_0^1|{\tilde{x}}_i\left(x\right)-{\tilde{\widehat{x}}}_i\left(x\right)|dx}{{\int }_0^1{\tilde{x}}_i\left(x\right)dx}$$

It holds $\mathrm{MARE}:{\mathcal{F}}_{LR}\left(\mathbb{R}\right)\times {\mathcal{F}}_{LR}\left(\mathbb{R}\right)\to [0,\infty )$, and values close to zero indicate a high degree of similarity between the fuzzy outputs and their corresponding fuzzy predicted values.

Mean Similarity Measure (MSM):

$$\mathrm{MSM}=\frac{1}{T-T^{*}}\sum _{i=T^{*}+1}^T\frac{{\int }_0^{1}min\{{\tilde{\widehat{x}}}_i\left(x\right),{\tilde{x}}_i\left(x\right)\}dx}{{\int }_0^{1}max\{{\tilde{\widehat{x}}}_i\left(x\right),{\tilde{x}}_i\left(x\right)\}dx}$$

In contrast to the above measures, $\mathrm{MSM}:{\mathcal{F}}_{LR}\left(\mathbb{R}\right)\times {\mathcal{F}}_{LR}\left(\mathbb{R}\right)\to [0,1]$ is a similarity measure. That is, values close to 0.5 show a good degree of similarity between the fuzzy responses and their fuzzy prediction. Further, when we have ${\mathrm{MSM}}_B<{\mathrm{MSM}}_A$, then model A outperforms model B.

Remark 2.

To assess the relationship between the $\tilde{y}$’s and $\tilde{\widehat{y}}$’s, we employ the Center of Gravity (CoG) [58] to transform the $\tilde{y}$’s and $\tilde{\widehat{y}}$’s into crisp values $M_{\tilde{y}}$’s and $M_{\tilde{\widehat{y}}}$’s. The relation between the defuzzified values $M_{\tilde{y}}$’s ($M_{\tilde{\widehat{y}}}$’s) is then evaluated using graphical representations. The CoG of an $LR$-FN   $\tilde{A}={(a;l_a,r_a)}_{LR}$ can be computed by:

$$M_{\tilde{A}}=\frac{\int x{\mu }_{\tilde{A}}\left(x\right)dx}{\int {\mu }_{\tilde{A}}\left(x\right)dx}$$

4. Comparative Analysis

The performance of the FEATSM(p) is compared in the framework of a comprehensive comparative analysis, including a simulation study and real-data applications. The competitors are established time series models for fuzzy data: FSPTSM$\left(p\right)$ [32], FATSM$\left(p\right)$ [34], FQTSM$\left(p\right)$[35], and FNPTSM$\left(p\right)$ [36]. As the time series model discussed in [33] is just a slight variation of the model by [32], we exclude it from the comparisons.

Example 1.

Consider 10 sample fuzzy data sets, each of size 200, generated by the following model:

$${\tilde{x}}_t=(y_t+{\epsilon }_t;0.0005|sin(\pi y_t/4)|,(0.5+0.1|y_t|+0.05y_t^2{)/5)}_{LR},t=1,2,\dots ,200,$$

where

(1)

$y_t=1.5cos(\pi t/6)exp(-1+0.1\sqrt{t})$,

(2)

${\epsilon }_t\sim N(0,0.01)$, and

(3)

$L\left(x\right)=1-x^2$ and $R\left(x\right)=1-x$.

In each considered time series model, 75% of the data are utilized as in-sample data for parameter estimation. We fit a FEATSM$\left(3\right)$ to the data, i.e.,

$$\tilde{f}({\tilde{x}}_{t-1},{\tilde{x}}_{t-2},{\tilde{x}}_{t-3})={(f(x_{t-1},x_{t-2},x_{t-3});l_f(l_{x_{t-1}},l_{x_{t-2}},l_{x_{t-3}}),r_f(r_{x_{t-1}},r_{x_{t-2}},r_{x_{t-3}}))}_{LR},$$

where

$$\begin{array}{cc}\hfill f(x_{t-1},x_{t-2},x_{t-3})& =\sum _{i=1}^3\left(a_i+(b_i+c_i{x}_{t-1})e^{-{\theta }_i{x}_{t-1}^2}\right)x_{t-i},\hfill \\ \hfill l_f(l_{x_{t-1}},l_{x_{t-2}},l_{x_{t-3}})& =\sum _{i=1}^3\left(l_{a_i}+(l_{b_i}+l_{c_i}{l}_{x_{t-1}})e^{-l_{{\theta }_i}{\left(l_{x_{t-1}}\right)}^2}\right)l_{x_{t-i}},\hfill \\ \hfill r_f(r_{x_{t-1}},r_{x_{t-2}},r_{x_{t-3}})& =\sum _{i=1}^3\left(r_{a_i}+(r_{b_i}+r_{c_i}{r}_{x_{t-1}})e^{-r_{{\theta }_i}{\left(r_{x_{t-1}}\right)}^2}\right)r_{x_{t-i}}.\hfill \end{array}$$

The average values of the goodness-of-fit measures considering 10 sample data sets for the implemented fuzzy time series models can be found in

Table 1. Average values of the performance measures in Example 1 for $p=1$.

Time Series Model	$\overline{\mathbf{MFE}}$	$\overline{\mathbf{MASE}}$	$\overline{\mathbf{MARE}}$	$\overline{\mathbf{MSM}}$
FSPTSM$\left(1\right)$ [32] (triweight kernel)	7.08239	8.72642	2.80697	0.05886
FATSM$\left(1\right)$ [34] (Gaussian kernel)	0.02293	0.96164	1.37226	0.24014
FQTSM$\left(1\right)$ [35]	1.67571	14.08892	24.20360	0.02736
FNPTSM$\left(1\right)$ [36] (Gaussian kernel)	0.08389	1.77124	1.97235	0.17779
FEATSM$\left(1\right)$	0.10642	1.73485	1.86657	0.19975

,

Table 2. Average values of the performance measures in Example 1 for $p=2$.

Time Series Model	$\overline{\mathbf{MFE}}$	$\overline{\mathbf{MASE}}$	$\overline{\mathbf{MARE}}$	$\overline{\mathbf{MSM}}$
FSPTSM$\left(2\right)$ [32] (triweight kernel)	0.00036	0.18721	0.67350	0.56740
FATSM$\left(2\right)$ [34] (Gaussian kernel)	0.29135	3.28817	1.42827	0.09638
FQTSM$\left(2\right)$ [35]	0.40631	6.93962	17.38250	0.05400
FNPTSM$\left(2\right)$ [36] (Gaussian kernel)	0.07912	1.78195	1.81080	0.07517
FEATSM$\left(2\right)$	0.00013	0.08110	0.50606	0.61894

,

Table 3. Average values of the performance measures in Example 1 for $p=3$.

Time Series Model	$\overline{\mathbf{MFE}}$	$\overline{\mathbf{MASE}}$	$\overline{\mathbf{MARE}}$	$\overline{\mathbf{MSM}}$
FSPTSM$\left(3\right)$ [32] (triweight kernel)	$0.00154$	$0.21685$	$0.82348$	$0.48112$
FATSM$\left(3\right)$ [34] (Gaussian kernel)	$0.22023$	$4.68367$	$1.78406$	$0.10470$
FQTSM$\left(3\right)$ [35]	$0.40632$	$6.93968$	$17.38250$	$0.05400$
FNPTSM$\left(3\right)$ [36] (Gaussian kernel)	$0.01645$	$0.79601$	$1.18812$	$0.34810$
FEATSM$\left(3\right)$	0.00022	0.13849	0.53851	0.63698

and

Table 4. Average values of the performance measures in Example 1 for $p=4$.

Time Series Model	$\overline{\mathbf{MFE}}$	$\overline{\mathbf{MASE}}$	$\overline{\mathbf{MARE}}$	$\overline{\mathbf{MSM}}$
FSPTSM$\left(4\right)$ [32] (triweight kernel)	0.01247	0.60654	1.13174	0.63892
FATSM$\left(4\right)$ [34] (Gaussian kernel)	0.41809	4.11455	2.24831	0.06721
FQTSM$\left(4\right)$ [35]	1.47109	8.35295	11.07223	0.11357
FNPTSM$\left(4\right)$ [36] (Gaussian kernel)	0.36449	3.79034	1.91428	0.04388
FEATSM$\left(4\right)$	0.01036	0.06748	0.39301	0.69968

for $p=1,2,3,4$. Consulting the results in these tables, we can state that the FEATSM$\left(p\right)$ for $p=1,2,3,4$ gives more accurate predictions compared to the other models, as shown by its better performance across the considered goodness-of-fit metrics.

Example 2.

In this example, we consider a real data set on the annual global land-ocean temperature from 1990 to 2020 [59] for comparing the predictive performance. As the data set consists of average values for every month, we utilize these averages as “means of the months”. Hence, we model the data for each month t with help of TFNs in the way ${\tilde{x}}_t={(x_t;0.5x_t,1.5x_t)}_T$. The in-sample data set consists of 200 data points, and the remaining 52 data points are used for model fitting. The performance results of the best models for each of the considered time series approaches are presented in

Table 5. Performance measures of the models in Example 2.

Time Series Model	MFE	MASE	MARE	MSM	Estimated Model Parameters
FSPTSM$\left(2\right)$ [32] (triweight kernel)	$0.108$	$5.583$	$0.633$	$0.473$	$\widehat{h}=0.5$, ${\widehat{\theta }}_1=0.072$$,{\widehat{\theta }}_2=-0.002$
FATSM$\left(3\right)$ [34] (Gaussian kernel)	$0.084$	$4.475$	$0.493$	$0.581$	${\widehat{h}}_1=0.7$, ${\widehat{h}}_2=0.4$, ${\widehat{h}}_3=0.7$
FQTSM$\left(2\right)$ [35]	$0.129$	$6.564$	$0.581$	$0.633$	$\widehat{\tau }=0.95$, ${\tilde{\widehat{\theta }}}_0={(0.487;0.673,0.137)}_T$,
					${\widehat{\theta }}_1=0.240$, ${\widehat{\theta }}_2=0.364$
FNPTSM$\left(3\right)$ [36] (triweight kernel)	$0.036$	$2.975$	$0.342$	$0.696$	${\widehat{h}}_1=0.63$, ${\widehat{h}}_2=0.31$, ${\widehat{h}}_3=0.84$
					Step 1: $\widehat{\mathit{a}}={(0.444,0.060,0.495,-0.186)}^{\top }$,
					$\widehat{\mathit{b}}={(0.185,0.260,-0.312,0.366)}^{\top }$,
					$\widehat{\mathit{c}}={(-0.050,-0.024,-0.221,0.530)}^{\top }$,
					$\widehat{\theta }={(1.432,0.821,0.915,1.108)}^{\top }$,
					Step 2: $l_{\widehat{\mathit{a}}}={(0.027,0.001,0.282,0)}^{\top }$,
					$l_{\widehat{\mathit{b}}}={(0.572,0.168,0,0.171)}^{\top }$,
FEATSM$\left(4\right)$	0.027	2.462	0.282	0.758	$l_{\widehat{\mathit{c}}}={(0.048,0.169,2.118\times {10}^{-6},0)}^{\top }$,
					$l_{\widehat{\theta }}={(1.117,0.599,3.274,2.747)}^{\top }$,
					Step 3: $r_{\widehat{\mathit{a}}}={(0.476,0,0.204,0)}^{\top }$,
					$r_{\widehat{\mathit{b}}}={(1.581,2.164\times {10}^{-8},0.742,0)}^{\top },$
					$r_{\widehat{\mathit{c}}}={(0.138,0.191,0.095,0.513)}^{\top },$
					$r_{\widehat{\theta }}={(4.387,0.178,6.645,0.601)}^{\top }$.

. Following these results, the FEATSM$\left(4\right)$ outperforms the competitors in terms of all considered performance measures for the annual global land-ocean temperature data set. In addition, a graphical comparison of the performance of the FEATSM$\left(4\right)$ in Figure 1 shows that it has higher predictive accuracy in comparison with the other models since the values of $M_{{\tilde{\widehat{x}}}_t}$ and $M_{{\tilde{x}}_t}$ are close to each other.

Example 3.

This example is based on a real data set containing data on the sea surface temperature over the years 1934–2022 collected near Chumbe Island in Unguja. The data are represented by TFNs in the way $x_t={(x_t;0.2x_t,0.1x_t)}_T$ (see Figure 2). There are, in total, 88 observations, and we use 75 of them as in the sample data, while the rest is applied for model fitting. The realizations of the four considered performance measures for the time series models under investigation can be found in

Table 6. Performance measures of the models in Example 3.

Time Series Model	MFE	MASE	MARE	MSM	Estimated Model Parameters
FSPTSM$\left(3\right)$ [32] (triweight kernel)	$3.089$	$24.401$	$0.770$	$0.563$	$\widehat{h}=0.28$, ${\widehat{\theta }}_1=1.706,{\widehat{\theta }}_2=-1.118$, ${\widehat{\theta }}_3=0.256$
FATSM$\left(3\right)$ [34] (Epanechnikov kernel)	$0.127$	$4.395$	$0.236$	$0.797$	${\widehat{h}}_1=0.7$, ${\widehat{h}}_2=0.5$, ${\widehat{h}}_3=0.3$
FQTSM$\left(1\right)$ [35]	$1.131$	$15.734$	$0.498$	$0.659$	$\widehat{\tau }=0.05$, ${\tilde{\widehat{\theta }}}_0={(0.199,1.946,1.674)}_T$,
					${\widehat{\theta }}_1=0.975$
FNPTSM$\left(3\right)$ [36] (Gaussian kernel)	$0.043$	$2.407$	$0.136$	$0.868$	${\widehat{h}}_1=0.23$, ${\widehat{h}}_2=0.05$, ${\widehat{h}}_3=0.02$
					Step 1: $\widehat{\mathit{a}}={(1.668,-0.668)}^{\top },\widehat{\mathit{b}}={(1.052,0.649)}^{\top }$,
					$\widehat{\mathit{c}}={(-1.371,-0.890)}^{\top },\widehat{\theta }={(2.053,0.283)}^{\top }$,
FEATSM$\left(2\right)$	0.028	1.958	0.108	0.908	Step 2: $l_{\widehat{\mathit{a}}}={(1.638,-0.638)}^{\top },l_{\widehat{\mathit{b}}}={(0.439,0.455)}^{\top }$,
					$l_{\widehat{\mathit{c}}}={(-1.024,1.823)}^{\top },l_{\widehat{\theta }}={(1.283,1.861)}^{\top }$,
					Step 3: $r_{\widehat{\mathit{a}}}={(1.631,-0.631)}^{\top },r_{\widehat{\mathit{b}}}={(2.291,0.611)}^{\top }$,
					$r_{\widehat{\mathit{c}}}={(-0.501,4.776)}^{\top },r_{\widehat{\theta }}={(2.444,3.820)}^{\top }$.

. Following the results in Table 6, we can conclude that FEATSM$\left(2\right)$ outperforms the competitors for the sea surface temperature data set. In addition, the CoG are shown in Figure 2, also demonstrating the higher predictive accuracy of the FEATSM$\left(2\right)$ compared to the competitors.

5. Conclusions

Exponential autoregressive time series models have demonstrated their effectiveness in fitting numerous stochastic processes in real-life applications. However, numerous real-world scenarios involve time series data that are not crisp but rather fuzzy. Given the absence of a method capable of applying exponential autoregressive time series models to forecast fuzzy time series data, we have developed an exponential autoregressive time series model designed for $LR$ fuzzy numbers in this study. The coefficients of this nonlinear model were estimated using a three-step least quantile-based approach, and the model performance was assessed with the help of established goodness-of-fit metrics. Through a comprehensive comparative analysis involving simulation and real-world applications, we compared the new model to various other fuzzy data-based time series models, revealing its better performance over established alternatives. Future research could explore additional nonlinear time series models for fuzzy data, such as those based on wavelets or Generalized Autoregressive Conditional Heteroskedasticity (GARCH).