A Three-Stage Nonparametric Kernel-Based Time Series Model Based on Fuzzy Data

Citation: Hesamian, G.; Johannssen, A.; Chukhrova, N. A Three-Stage Nonparametric Kernel-Based Time Series Model Based on Fuzzy Data. Mathematics 2023,11, 2800. https:// doi.org/10.3390/math11132800 Academic Editor: Salvatore Sessa Received: 27 May 2023 Revised: 13 June 2023 Accepted: 19 June 2023 Published: 21 June 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article A Three-Stage Nonparametric Kernel-Based Time Series Model Based on Fuzzy Data Gholamreza Hesamian 1, Arne Johannssen 2,* and Nataliya Chukhrova 3 1Department of Statistics, Payame Noor University, Tehran 19395-3697, Iran; [email protected] 2Faculty of Business Administration, University of Hamburg, 20146 Hamburg, Germany 3HafenCity University of Hamburg, 20457 Hamburg, Germany; nataliya.chukhr[email protected] *Correspondence: [email protected] Abstract: In this paper, a nonlinear time series model is developed for the case when the underlying time series data are reported by LR fuzzy numbers. To this end, we present a three-stage nonparametric kernel-based estimation procedure for the center as well as the left and right spreads of the unknown nonlinear fuzzy smooth function. In each stage, the nonparametric Nadaraya–Watson estimator is used to evaluate the center and the spreads of the fuzzy smooth function. A hybrid algorithm is proposed to estimate the unknown optimal bandwidths and autoregressive order simultaneously. Various goodness-of-fit measures are utilized for performance assessment of the fuzzy nonlinear kernel-based time series model and for comparative analysis. The practical applicability and superiority of the novel approach in comparison with further fuzzy time series models are demonstrated via a simulation study and some real-life applications. Keywords: fuzzy regression; fuzzy time series model; nonparametric time series analysis; time series analysis MSC: 03E72; 37M10; 62A86 1. Introduction The field of time series analysis comprises methods used to analyze the characteristics of a response variable with respect to time. It takes into consideration the fact that observations made over time may have an internal structure (such as autocorrelations, trends, seasonal and/or cyclic variations) that should be accounted for. The main aims of time series analysis are as follows: • Trend analysis: to identify the underlying pattern or trend in the data over time, such as an upward or downward trend. • Seasonality analysis: to identify if the data exhibit a repeating pattern over a set period, such as daily, weekly, or yearly. •Forecasting: to forecast future values using historical data. • Anomaly detection: to identify any unusual or unexpected observations in the data that deviate from the normal pattern. • Model selection: to choose an appropriate model to represent the underlying relationships between variables in the data. • Noise reduction: to remove any unwanted variability or random fluctuations from the data to improve the accuracy of predictions and make the underlying patterns more clear. These aims can inform decision makers, provide insight into the underlying patterns and relationships in the data, and support the development of data-driven strategies in various fields such as economics, engineering, finance, and more (see, e.g., [1–8]). Common time series models rely on exact observations and ensure crisp predictions. However, due to various uncertainty factors, it is sometimes preferable to make predictions Mathematics 2023,11, 2800. https://doi.org/10.3390/math11132800 https://www.mdpi.com/journal/mathematics Mathematics 2023,11, 2800 2 of 17 using imprecise values. For instance, we usually observe imprecise observations in carbon emissions, social benefits and oil reserves, among others [ 9 ]. Traditional statistical time series models fail to address prediction problems based on ambiguous or vague information represented by fuzzy data. This shortcoming can be overcome by time series models that use techniques of fuzzy statistics. In general, fuzzy statistics is a branch of statistics that deals with uncertainty and imprecision, e.g., in the data. It includes, for instance, the fields of fuzzy estimation, fuzzy regression, fuzzy clustering, and fuzzy hypothesis testing [10–12]. Fuzzy time series models were originally introduced in 1993 [ 13 ], and since then they have replaced conventional (crisp) time series approaches when observations are uncertain. When considering fuzzy time series models, the prediction of future values requires three principal steps. In step 1, the exact data are reported. In step 2, through the identification of fuzzy logical relations [ 14 , 15 ], the predictions are transformed into fuzzy quantities. Finally, step 3 provides a defuzzification approach [ 16 – 22 ] to transform the fuzzy values into crisp ones. The techniques used to identify fuzzy logical relations in step 2 primarily involve fuzzy logical relation groups and matrices [ 13 , 23 – 34 ], soft computing methods [ 35 – 44 ], and statistical approaches in interaction with fuzzy logic [ 21 , 24 , 45 – 47 ]. Step 2 is an essential part of the predictive power of the presented model. Fuzzy time series models that rely on imprecise observations have attracted substantial attention in recent years, mainly due to their high applicability to real-life problems. In fact, a lot of researchers have focused on time series models using imprecise observations. The soft computing techniques employed in this framework are mostly combinations of artificial neural networks, evolutionary algorithms, fuzzy and rough sets. These approaches are widely used for crisp or fuzzy forecasts based on crisp past observations such as electricity load, stock index prices and temperature (for a review of these techniques, we refer to [ 48 – 56 ]). In addition, various methods combine techniques of time series and fuzzy regression analysis [ 57 ]. For some recent advances in fuzzy regression analysis, see [ 58 – 63 ]. The reliability of forecasting methods generally requires exact observations in the sample. But there is often only vague information that is given in terms of imprecise quantities. Moreover, there are various real-world problems related to biological, economic, environmental, medical and sociological data where we face inaccurate instead of accurate data. In many real-life applications, e.g., monthly Co 2 emission, annual sea surface temperature or the water level of a lake, conventional observations are often reported as mean values. In such cases, the data obtained are not sufficient informative since some information contained in the range of the data is neglected. To overcome this shortcoming, one alternative would be to report such kind of data as interval valued (comparable to conventional confidence intervals). However, a potential shortcoming of interval-valued data is the fact that all values within the interval have the same importance. To avoid this issue of interval-valued data, the reported data can alternatively be represented with help of fuzzy numbers [ 64 ]. These fuzzy quantities can be modeled via experts opinion, or as simple alternative, they can be constructed via a method proposed by Buckley [ 65 ]. In this approach, conventional confidence intervals are employed to construct fuzzy numbers around the conventional mean values. In addition to the abovementioned methods, there are also fuzzy time series models that rely on fuzzy data, but comparatively few overall. In this regard, Hesamian and Akbari [ 66 ] first suggested a fuzzy semi-parametric time series model ( FSPTSM ) based on fuzzy data, non-fuzzy coefficients, and fuzzy smooth functions. Secondly, Zarei et al. [ 67 ] used a specific variant of the FSPTSM [ 66 ] for triangular fuzzy data and different distance measures for fuzzy data. And thirdly, Hesamian et al. [ 68 ] introduced a forward additive time series model (FATSM) for fuzzy observations. In this paper we develop a fuzzy nonparametric time series model ( FNPTSM ) for fuzzy observations that is inspired by nonparametric regression models and kernel smoothing methods [ 57 ]. As an initial idea, note that in nonparametric regression analysis, the Nadaraya-Watson estimator [ 69 , 70 ] is fairly common. Now, let us consider the issue of pa- Mathematics 2023,11, 2800 3 of 17 rameter estimation in the nonlinear regression model xt=f(xt−1 , xt−2 , . . . , xt−p) + et with f:Rp→R . Based on this general model, a simple nonparametric way of estimating the function fis to employ the kernel-based Nadaraya–Watson estimator b f(xt) = T∗ ∑ j=p+1 wh(t,j)xj(1) with wh(t,j) = ∑p i=1Kxt−i−xj−i h ∑T∗ j=p+1∑p i=1Kxt−i−xj−i h, where K is a kernel function and h> 0 the bandwidth parameter. Note that the estimator (1) is a weighted average of x1 , x2 , . . . , xT using the weights wh(t , j) . As for determining the optimal bandwidth h, the Generalized Cross Validation (GCV) criterion bh=arg min h>0GCV(h) = arg min h>0 1 T∗−p T∗ ∑ t=p+1 xt−∑T∗ j=p+1wh(t,j)xj 1−tr(Wh) T∗−p  2 can be utilized, where tr(Wh) is the trace of the matrix Wh= [wh(t , j)] . It is a matter of fact that the estimated values of f are ensured to be within the range of the response variable. This beneficial property is one of the reasons why we apply the Nadaraya–Watson kernel-based estimator for our fuzzy time series model. By utilizing this idea, the proposed FNPTSM provides an estimation procedure of the unknown (nonlinear) relationship between the fuzzy observations in three stages. The advantage of this methodology is that it considerably decreases the complexity in the estimation procedure. While the other fuzzy time series models [ 66 – 68 ] are based on estimating the unknown components of the model by unifying the centers and spreads of fuzzy data and their corresponding predicted values, our proposed method provides a smooth estimation procedure according to three separate stages. In the framework of a simulation study and two real-data examples, the efficiency and appropriateness of the FNPTSM is assessed in comparison with previous time series models for fuzzy data by utilizing four approved goodness-of-fit criteria. The paper is organized as follows. First, we recall some necessary concepts related to fuzzy numbers in Section 2. In Section 3, the three-stage nonparametric kernel-based time series model using fuzzy data is presented. In Section 4, various application examples are given. Concluding remarks are provided in Section 5. 2. Fuzzy Numbers In this section, we introduce basic definitions of fuzzy numbers that are needed to develop our proposed method. Afuzzy set e A is a mapping on X that assigns a specific degree of membership 0 ≤µe A(x)≤ 1 to each x∈X . In addition, a fuzzy number ( FN ) e A is a convex normalized fuzzy set on the real line R with an upper semi-continuous membership function of bounded support [ 71 ]. In many real applications, vague data a can be reported as e A : “about a ”. Such fuzzy data can often be represented via a special case of FN s, so called LR - FN s, which split µe A into two curves: a part on the left and a part on the right of the modal value. So, when considering real-life applications in fuzzy environments, LR - FN s play an important role. The membership function of an LR - FN µe A(x) = (a ; la , ra)LR can be defined by: Mathematics 2023,11, 2800 4 of 17 µe A(x) =        La−x laif x≤a Rx−a raif x>a (2) In (2) , L and R are continuous and strictly decreasing functions from [ 0, 1 ] to [ 0, 1 ] satisfying L( 0 ) = R( 0 ) = 1 and L( 1 ) = R( 1 ) = 0. In addition, a∈R represents the modal value, while la> 0 and ra> 0 are the left spreads and right spreads of e A , respectively. The set of all LR - FN s is represented by FLR(R) . A special case of an LR - FN is the so-called triangular fuzzy number (TFN), whose membership function has the following form: µe A(x) =        x−(a−la) laa−la≤x≤a a+ra−x raa<x≤a+ra 0 otherwise There are various operations that can be defined between two LR - FN s, i.e., between e A= (a ; la , ra)LR and e B= (b ; lb , rb)LR . For instance, as we need both operations in this paper, we define Addition and Scalar multiplication of e Aand e Bin the following [72]: • Addition: e A⊕e B= (a+b;la+lb,ra+rb)LR • Scalar multiplication: λ⊗e A=(λa;λla,λra)LR if λ>0 (λa;−λra,−λla)RL if λ<0 Moreover, there are numerous concepts used to define distances between two LR - FN s e A= (a ; la , ra)LR and e B= (b ; lb , rb)LR [ 71 ]. Here, we utilize the squared error distance measure D for performance evaluation of the FNPTSM in comparison with other models. It is defined as D(e A,e B) = (((a−b)2+c1(la−lb)2+c2(ra−rb)2)/3)0.5 with c1=R1 0L−1(α)dαand c2=R1 0R−1(α)dα[73]. 3. Nonparametric Kernel-Based Time Series Model for Fuzzy Data In this section, the FNPTSM is developed along with the suggested parameter estimation method. 3.1. The Model First, we recall the definition of fuzzy time series data. Definition 1. Let exT={ex1 , ex2 , . . . , exT be a set of FN s of size T . Then, exT is called fuzzy time series data if {ex1,ex2, . . . , exTis the vague concept of ordinary time series data {x1,x2, . . . , xT}[68,74]. As discussed in the Introduction, there are many situations where it is preferable to report exact data xby an FN e xas “about x”. Then, e xis the respective vague concept of x. Definition 2. Let exT={e x1 , e x2 , . . . , e xT be fuzzy time series data. The FNPTSM for fuzzy time series data exTis then defined by e xt=e f(e xt−1,e xt−2, . . . , e xt−p)⊕eet, (3) where 1. e xt= (xt;lxt,rxt)LR, 2. e f(e xt−1,e xt−2, . . . , e xt−p) = (f(xt−1,xt−2, . . . , xt−p); lf(lxt−1,lxt−2,...,lxt−p),rf(rxt−1,rxt−2,...,rxt−p))LR, Mathematics 2023,11, 2800 5 of 17 3. eet= (et;let,ret)LR’s are fuzzy errors, where et∈Rand let,ret∈R+. Remark 1. Note that (3) provides an FN in the form e x∗ t= (x∗ t ; lx∗ t , rx∗ t)LR with x∗ t=f(xt−1 , xt−2 , . . . , xt−p) + et , lx∗ t=lf(lxt−1,...,lxt−p)+let and rx∗ t=rf(rxt−1,...,rxt−p)+ret with t= 1, 2, . . . , T . According to Definition 1, as {x1 , x2 , . . . , xT} is ordinary time series data, ex∗ T={e x∗ 1 , e x∗ 2 , . . . , e x∗ T is also a vague concept of ordinary time series data {x∗ 1 , x∗ 2 , . . . , x∗ T . Thus, the proposed fuzzy time series model (3)generates new fuzzy time series data. 3.2. Three-Stage Estimation Method for the Nonlinear Fuzzy Smooth Function Below, we suggest a three-stage method to estimate the unknown fuzzy smooth function e f in (3) . For this purpose, the fuzzy predictions are obtained based on a within-sample forecast xT∗=x1 , x2 , . . . , xT∗> with T∗<T . From (3) , one can get three ordinary nonlinear time series models as (1) xt=f(xt−1 , xt−2 , . . . , xt−p) + et , (2) lxt=lf(lxt−1,...,lxt−p)+let , and (3) rxt=rf(rxt−1,...,rxt−p)+ret for t= 1, 2, . . . , T∗ . Therefore, to estimate the fuzzy smooth function at ex= (x ; lx , rx)T with x= (x1 , x2 , . . . , xp)> , lx= (lx1 , lx2 , . . . , lxp)> and rx= (rx1 , rx2, . . . , rxp)>, we follow the three-stage procedure below: •Stage (1): Consider the nonlinear regression model lxt=lf(xt−1,xt−2,...,xt−p)+let . Based on the time series data lxt= (lxt−1 , . . . , lxt−p)> , we employ the weighted Nadaraya–Watson estimator to estimate lf for a within-sample forecast T∗≤T at lx= (lx1, . . . , lxp)>)as lb f(lxt)= T∗ ∑ j=p+1 whl(t,j)lxj, where whl(t,j) = ∑p i=1Klxt−i−lxj−i hl ∑T∗ j=p+1∑p i=1Klxt−i−lxj−i hl(4) with kernel function K( . ) and bandwidth parameter hl> 0. The optimal value of hl can be estimated by implementing the GCV criterion, bhl=arg min hl>0GCV(h) = arg min hl>0 1 T∗−p T∗ ∑ t=p+1  lxt−∑T∗ j=p+1whl(t,j)lxj 1−tr(Whl) T∗−p    2 , (5) where tr(Whl) is the trace of the matrix Whl= [whl(t , j)] with whl(t , j) as defined in (4) . •Stage (2): Consider the nonlinear regression model rxt=rf(xt−1,xt−2,...,xt−p)+ret . Based on the within-sample time series forecast data rxt= (rxt−1 , . . . , rxt−p)> , t= 1, 2, . . . , T∗ , the weighted Nadaraya–Watson estimation of rf at rx= (rx1 , . . . , rxp)>) can be established via rb f(rxt)= T∗ ∑ j=p+1 whr(t,j)rxj, where whr(t,j) = ∑p i=1Krxt−i−rxj−i hr ∑T∗ j=p+1∑p i=1Krxt−i−rxj−i hr(6) Mathematics 2023,11, 2800 6 of 17 and hr> 0 is a bandwidth parameter. The optimal value of hr can be estimated using the GCV criterion, bhr=arg min hr>0GCV(h) = arg min hr>0 1 T∗−p T∗ ∑ t=p+1 rxt−∑T∗ j=p+1whr(t,j)rxj 1−tr(Whr) T∗−p  2 , (7) where tr(Whr)is the trace of the matrix Whr= [whr(t,j)] with whr(t,j)as defined in (6). •Stage (3): Consider the nonlinear regression model xt=f(xt−1 , xt−2 , . . . , xt−p) + et . Based on the within-sample time series forecast data (xt= (xt−1 , xt−2 , . . . , xt−p)>) , t=1, 2, . . . , T∗, a nonparametric estimator fcan be achieved as b f(xt) = T∗ ∑ j=p+1 wh(t,j)xj, where wh(t,j) = ∑p i=1Kxt−i−xj−i h ∑T∗ j=p+1∑p i=1Kxt−i−xj−i h(8) and bandwidth parameter h> 0. Similar to the previous stages, the optimal value of his estimated with the help of the GCV criterion, bh=arg min h>0GCV(h) = arg min h>0 1 T∗−p T∗ ∑ t=p+1 xt−∑T∗ j=p+1wh(t,j)xj 1−tr(Wh) T∗−p  2 , (9) where tr(Wh)is the trace of the matrix Wh= [wh(t,j)] with wh(t,j), as defined in (8). Therefore, the forecast e xT∗+k with time lag k∈N can be achieved by an LR - FN via eb xT∗+k= (b xT∗+k;lb xT∗+k,rb xT∗+k)LR with b xT∗+k= T∗+k−1 ∑ j=p+1 ∑p i=1Kxt−i−xj−i bh ∑T∗+k−1 j=p+1∑p i=1Kxt−i−xj−i bh·xj, lb xT∗+k= T∗+k−1 ∑ j=p+1 ∑p i=1Klxt−i−lxj−i bhl ∑T∗+k−1 j=p+1∑p i=1Klxt−i−lxj−i bhl·lxj, rb xT∗+k= T∗+k−1 ∑ j=p+1 ∑p i=1Krxt−i−rxj−i bhr ∑T∗+k j=p+1∑p i=1Krxt−i−rxj−i bhr·rxj. According to Stages (2) and (3), it can be seen that the spreads of the fuzzy prediction e xT∗+kare always non-negative. Remark 2. Since the proposed time series model relies on fuzzy data, let us recall the previous time series models based on fuzzy data [ 66 – 68 ]. First, Hesamian and Akbari [ 66 ] proposed a fuzzy semi-parametric autoregressive integrated moving average (ARIMA) model as follows: e xi= p M l=1 (θl⊗e xi−l⊕e f(ti)⊕eei),i=p+1, . . . , T. Mathematics 2023,11, 2800 7 of 17 The parameters of the model are estimated by employing a hybrid method including a nonparametric kernel-based method and least absolute deviations. For a second time series model based on fuzzy data, Zarei et al. [ 67 ] applied the method [ 66 ] to estimate the model parameters and the fuzzy smooth function based on a specific distance, kernel and triangular fuzzy numbers. Finally, Hesamian et al. [68] proposed the fuzzy nonlinear time series model e xt=e f(e xt−1,e xt−2, . . . , e xt−p)⊕eet,t=1, 2, . . . , T, where e f(e xt−1,e xt−2, . . . , e xt−p) = p M l=1 fl(e xt−l). As for the estimation of the unknown fuzzy smooth functions e fl , they applied a forward additive nonparametric technique. Remark 3. We have extended some common performance measures used to compare the predictive accuracy of different time series models that we implement in Section 4. For this purpose, a time series model is first estimated based on a within-sample fuzzy time series dataset of size T∗<T and then the performance of the model is evaluated via the remaining fuzzy time series dataset of size T −T∗. 1. Mean Forecast Error: MFE =∑T t=T∗+1D2(eb xt,e xt) T−T∗ 2. Mean Absolute Scaled Error: MASE =∑T t=T∗+1qt T−T∗ with qt=D(eb xt,e xt) 1 T−T∗∑T t=T∗+1D2(e xt,e xt−1) 3. Basis of the Index of Agreement: BIA =1−∑T t=T∗+1D2(e xt,eb xt) ∑T t=T∗+1(D(e xt,ex) + D(ex,eb xt))2 with ex=∑T t=T∗+1e xt T−T∗ 4. Mean Similarity Measure: MSM =1 T−T∗ T ∑ t=T∗+1Rmin{eb xt(x),e xt(x)}dx Rmax{eb xt(x),e xt(x)}dx Let A and B be two fuzzy time series models. As MSM :FLR(R)×FLR(R)→[ 0, 1 ] is a similarity measure, values of MSM above 0.5 show a good degree of similarity between the fuzzy responses and their fuzzy predictions. If we observe MSMB<MSMA , then model A outperforms model B . Further, if MFEA<MFEB , MASEA<MASEB or BIAA<BIAB , then model A acts better in terms of prediction accuracy compared to model B. Remark 4. While the proposed estimation procedure does not depend on the shape functions L and R corresponding to fuzzy data, the performance measures MFE , MASE and MSM depend on these shape functions. Therefore, the selected type of the shape functions L and R may affect the prediction criteria. For instance, assume that the data have reported by e xt= (xt , lxt , rxt)LR with L(x) = 1 −x and R(x) = √1−x . That is, c1=1 2 and c2=2 3 . Therefore, the distance between Mathematics 2023,11, 2800 8 of 17 e xt and its prediction is D2(e xt , eb xt) = (xt−b xt)2+1 2(lxt−lb xt)2+2 3(rxt−rb xt)2 . This implies that the MFE criterion is more sensitive to right spreads than to left spreads in this case. Considering L(x) = R(x) = 1 −x , it can be seen that D2(e xt , eb xt) would be equally dependent from the left and right spreads. However, when we compare the performance of fuzzy time series models, it is reasonable that the shape functions L and R are assumed to be the same for all the considered models. Thus, following this approach, the performance criteria are not sensitive to the selection of L and R since c1and c2remain fixed for each model. 3.3. Selection of Autoregressive Order and Optimal Bandwidths When implementing the FNPTSM (3) , it is necessary to select the optimal bandwidths h , hl and hr , to choose the kernel function and to determine the autoregressive order p . The procedure used to select the autoregressive order and the optimal bandwidths is proposed as follows: (1) Let p=1. (2) (2.1) Compute bhp lbased on (5). (2.2) Compute bhp rbased on (7). (2.3) Compute bhpbased on (9). (3) Let p=p+1 and return to (2) until b p=arg min pRMSEp, where RMSEp=v u u t∑T∗ i=p+1D2(eb xi,e xi) T∗−p. Then, b p,bhp,bhp land bhp rare the optimal values. 4. Numerical Examples In this section, the effectiveness of the FNPTSM is investigated considering a simulation study and application examples that rely on fuzzy data. Recall that there are three other time series models that are based on fuzzy data (see Remark 2), i.e., the models introduced by Hesamian and Akbari [ 66 ], Zarei et al. [ 67 ] and Hesamian et al. [ 68 ]. However, as the method of Zarei et al. [ 67 ] is based on Hesamian and Akbari’s method [ 66 ] (with a different distance measure), we omit this technique in the comparisons below. Thus, we compare our proposed method with the models suggested by Hesamian and Akbari ( FSPTSM ) [ 66 ] and Hesamian et al. ( FATSM ) [ 68 ] via three different kernel functions (Gaussian, Epanechnikov, and triweight). Example 1. In this example, 10 fuzzy datasets, each of size 300, are generated by the following FNPTSM: e xt=f(e xt−1,e xt−2,e xt−3)⊕eet,t=4, 5, . . . , 300, where 1. f(e x1,e x2,e x3) = x1−cos(x2)−expx3 1+|x3|; cos2 0.9 3 ∏ j=1 lxj!, exp 0.002 3 ∏ j=1 rxj!!LR 2. e xj= (xj ; lxj , rxj)LR , j= 1, 2, 3 are the initial values with xj∼N( 0, 1 ) , and lxj and rxj are random variables following U(0, 0.2)and U(0, 0.9), respectively, 3. eet= (et ; let , ret)LR with et∼N( 0, 4 ) , let and ret are random variables following U( 0, 0.4 ) and U(0, 0.5), respectively, and 4. L(x) = 1−x2and R(x) = 1−x. Mathematics 2023,11, 2800 9 of 17 The kernels Gaussian, Epanechnikov, and triweight are applied to predict e xt . Based on the 10 sample fuzzy datasets (each of size 300), the mean values of the goodness-of-fit measures and their corresponding bandwidth mean values are summarized in Table 1. Consulting the results for the FNPTSM , it is evident that the best results among various kernels are obtained via the Gaussian kernel (lowest values of MFE , MASE and largest values of BIA , MSM ). In addition, the results of the FSPTSM and FATSM can also be found in Table 1. Comparing these results with the results of the FNPTSM , it is obvious that the FNPTSM provides more accurate predictions compared to both other methods for all three kernels, as all the considered goodness-of-fit measures show better results for the FNPTSM . That is, we observe the lowest values of MFE , MASE and the largest values of BIA, MSM for the FNPTSM. Table 1. The mean performance measures of the FNPTSM , FSPTSM and FATSM corresponding to some specific kernels in Example 1. Method Kernel Results Goodness-of-Fit Criteria FNPTSM Gaussian MFE =1.0452 bh=0.45 MASE =1.6089 b hl=0.04 BIA =0.9996 b hr=0.22 MSM =0.4167 Epanechnikov MFE =1.1728 bh=0.66 MASE =1.6478 b hl=0.05 BIA =0.9992 b hr=0.39 MSM =0.3953 triweight MFE =1.2482 bh=1.89 MASE =1.6339 b hl=0.08 BIA =0.9991 b hr=0.54 MSM =0.3721 FSPTSM Gaussian hopt =0.07 MFE =8.9728 b θ1=0.5575 MASE =4.2652 b θ2=−0.0956 BIA =0.9536 b θ3=−0.1247 MSM =0.2207 Epanechnikov hopt =0.02 MFE =5.2530 b θ1=−0.6424 MASE =4.5383 b θ2=−0.5168 BIA =0.9603 b θ3=−0.4258 MSM =0.2920 triweight hopt =0.13 MFE =11.5231 b θ1=0.3757 MASE =3.8643 b θ2=−0.1576 BIA =0.9738 b θ3=−0.2568 MSM =0.2133 Mathematics 2023,11, 2800 16 of 17 28. Cheng, S.H.; Chen, S.M.; Jian, W.S. Fuzzy time series forecasting based on fuzzy logical relationships and similarity measures. Inf. Sci. 2016,327, 272–287. [CrossRef] 29. Sadaei, H.J.; Enayatifar, R.; Abdullah, A.H.; Gani, A. 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