An adaptive fuzzy semi-parametric regression model using TPB and ABC-BPNN

In this paper, a new hybrid method is developed that incorporates the rationale of truncated power basis (TPB) and artificial bee colony-back propagation neural networks (ABC-BPNN) into an adaptive fuzzy semi-parametric regression model yielding more accurate results and better generalization ability. The proposed adaptive fuzzy semi-parametric regression model comprises sub-model formulation and approximates the observed fuzzy outputs from the outside employing neural networks computation, such that the proposed adaptive fuzzy regression model better explains the inherent dependence and vagueness that exist in a given dataset. Using the cross-validation criterion and absolute deviation-based distance measures for LR-type fuzzy numbers, a target function optimization problem of constructing the adaptive fuzzy semi-parametric regression is performed by solving the smooth function, bandwidth of kernel function, and regression coefficients. This strategy significantly increases the goodness of fit for the proposed algorithm and offers a dependence framework among magnitude and uncertainty for the fuzzy regression model. We also use three formula measures to evaluate the fit quality of the regression results within each membership function as well as in the center or spread tendency property, respectively. The proposed algorithm is numerically evaluated on three experimental examples including a simulation study and two practical cases to prove its practicality and efficiency. Comparative analyses of our proposed method are provided to support its cogency, and the results show that our proposed model is more effective and stable than some other existing fuzzy regressions.


Introduction
The regression analysis is commonly used statistical methodology for the purpose of studying the dependence between a real phenomenon and other real phenomena (Thrane 2019).However, there exist deviations between the observed data and their corresponding estimates due to imprecise data.Moreover, we encounter many situations where necessary assumptions for statistical regression analysis cannot be met, because they are not based on random uncertainty.Fuzzy regression methods were proposed to model the relationship between variables, when the available data are fuzzy, the relationships are imprecise, or the underlying statistical assumptions are not fulfilled.They also have wide appli-B Keli Jiang 990739889@qq.com 1 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China cations for analyzing complex systems including economic systems, social systems, engineering systems, and environmental systems where the vagueness of human subjective judgment is influential (Lin et al. 2012;Chachi et al. 2014;López et al. 2016;Tzimopoulos et al. 2016;Mirzaei et al. 2018;Alshammari et al. 2020;Abu Arqub et al. 2023).Tanaka et al. (1982) was the first to put forward the fuzzy linear regression problem for crisp explanatory variables and crisp/fuzzy response variables as a linear programming problem.To estimate the parameters of the regression model, they minimized the system fuzziness subject to including the data points of each sample within a specified feasible interval.Since then, fuzzy regression model has attracted some researchers' interests and has been investigated and applied in a variety of areas.Yen et al. (1999) extended the results of a fuzzy linear regression model that used symmetric triangular parameters to one with asymmetric fuzzy triangular coefficients.Lee and Chen (2001) formulated a generalized fuzzy regression model using the generalized fuzzy linear function and proposed a non-linear programming model to identify the fuzzy parameters.After that, there are several studies making some development and improvement (Hojati et al. 2005;Liu and Chen 2013;Chen et al. 2016;de Andrés-Sánchez 2017;Spiliotis et al. 2020).Celmin , š (1987) and Diamond (1988) used certain distances between fuzzy numbers, studied the least-squares approaches of estimating the unknown parameters to the problem of fuzzy regression modeling.Chang and Lee (1996) proposed a fairly general fuzzy regression technique based on the least-squares approach to estimate the modal value and the spreads separately.Xu and Li (2001) developed a fuzzy analog by a distance defined on a fuzzy number space and dealt with the question of fuzzy multivariable linear regression by least squares.This approach was investigated and improved by some authors [see for example (Nasrabadi and Hashemi 2008;Shen et al. 2010;Salmani et al. 2017;Chachi 2018;Gao and Lu 2018)].
For many practical problems, the functional form between the inputs and the output is frequently unknown and cannot be obtained easily.Along this line of consideration, several researchers have proposed the use of non-parametric fuzzy regression methods involving neural networks (Abu Arqub 2017; Abu Arqub et al. 2021).Ishibuchi and Tanaka (1992) were the pioneers in combining fuzzy regression analysis and back propagation neural networks to fit the upper and lower bounds of interval-valued fuzzy numbers.Cheng and Lee (1999a) combined a fuzzy inference system for fuzzy regression analysis with the learning ability of neural networks.Dunyak and Wunsch (2000) generalized linear and non-linear fuzzy regression analysis using neural networks models with general fuzzy number inputs, outputs, weights and biases.Alex (2004) discussed a fuzzy regression neural network system for a fuzzy normal regression model.Zhang et al. (2005) applied a fuzzy radial basis function network to fuzzy non-linear regression analysis for multiple LR-type fuzzy data.Mosleh et al. (2013) presented a hybrid approach based on fuzzy neural networks for approximate fuzzy parameters of fuzzy linear and nonlinear regression models with fuzzy outputs and crisp inputs.He et al. (2018) used a random weight network to develop a fuzzy non-linear regression model, where inputs/outputs are trapezoidal fuzzy numbers, respectively.Karbasi et al. (2020) presented a hybrid scheme based on recurrent neural networks for approximate fuzzy coefficients of fuzzy linear and polynomial regression models with fuzzy outputs and crisp inputs.Prakaash and Sivakumar (2021) implemented a new data prediction system using an optimized machine learning algorithm based on recurrent neural networks with fuzzy classifier and fuzzy regression model.Asadollahfardi et al. (2022) investigated the performance prediction of the reactor in removing acid red 14 using radial basis function, adaptive neuro-fuzzy inference system, and fuzzy regression analysis.
In addition to the neural networks algorithms, there are other effective non-parametric techniques that have been utilized to improve fuzzy regression analysis.Cheng and Lee (1999b) have fuzzified the non-parametric regression techniques k-nearest neighbor and kernel smoothing, that are applied to fuzzy regression equations.Petit-Renaud and Denoeux ( 2004) considered a non-parametric regression approach which is based on fuzzy belief assignment.Wang et al. (2007) presented a non-parametric fuzzy regression model based on weighted least squares and a local linear smoothing technique with a cross-validation procedure.Farnoosh et al. (2012) dealt with ridge estimation of fuzzy non-parametric regression model with multivariate crisp inputs and triangular fuzzy output.Chachi et al. (2014) presented a fuzzy regression model based on the nonparametric multivariate adaptive regression splines approach.Note that being an entirely non-parametric approach, it usually causes some drawbacks such as the dimensionality issue, difficulty of interpretation, and lack of extrapolation capability.Notably, fuzzy semi-parametric regression model is more advantageous and flexible than the corresponding fuzzy non-parametric regression.Hesamian et al. (2017) suggested a hybrid procedure based on curve fitting methods and least absolute deviations to investigate a semiparametric partially linear model with fuzzy inputs, fuzzy outputs, fuzzy smooth functions, and non-fuzzy coefficients.Akbari and Hesamian (2019a) extended the conventional semi-parametric partial linear regression model with fuzzy predictors and fuzzy response when multicollinearity and outlier occur.Also, Akbari and Hesamian (2019b) extended the conventional elastic net multiple linear regression model adopted with a semi-parametric method to fuzzy predictors and responses.Hesamian et al. (2021) constructed a fuzzy univariate regression model utilizing the non-parametric estimator and developed a hybrid algorithm to estimate the bandwidth and fuzzy regression coefficients.
It is worth noting that although some previous studies have adopted various algorithms to solve the optimization problems for generating fuzzy regression models, they did not investigate the simultaneous impact of semi-parametric techniques and neural networks on the centers of response variables in an adaptive fuzzy regression model.So far as the authors know, there has not been any research on an adaptive fuzzy semi-parametric multivariate regression analysis with fuzzy predictors and fuzzy responses.The present work provides an attempt to extend a linear adaptive fuzzy regression model via semi-parametric regression technique, spline basis function, and neural networks-based method.In this regard, our proposed method consists of suggesting the least absolute error distance measures between two LRtype fuzzy numbers and approximating the observed output centers with artificial bee colony-back propagation neural networks computation.Then, the classical semi-parametric technique, truncated power basis, and adaptive fuzzy regression model are combined to achieve a realistic regression model and more stable results.In the proposed approach, an algorithm based on minimization of the cross-validation criterion is adopted to select the optimal bandwidth to estimate the crisp smooth function and an optimization model involving the objectives of minimizing the total distance errors is applied to determine the final optimal crisp regression coefficients.Effectiveness and advantages of the proposed methods are then examined by comparing our results in terms of some common goodness-of-fit criteria.To make a comparative study, a simulation experiment and two practical problems are performed between the proposed adaptive fuzzy semi-parametric regression model and some existing fuzzy regression models.The numerical and comparative results declare that the proposed adaptive fuzzy semi-parametric regression model allows us to provide sufficiently accurate results in fuzzy regression analysis and can neutralize the disruptive effects of possible fuzzy outliers in the estimation process.
The rest of this paper is organized as follows.Section 2 recalls some basic information about LR-type fuzzy number, linear adaptive fuzzy regression model, and fuzzy semiparametric partially linear model, which are essential for the proposed method in this paper.Section 3 describes the formalization of the proposed adaptive fuzzy semi-parametric regression model using spline basis and neural networks for studying the functional dependence of an LR-type fuzzy response variable on a set of LR-type fuzzy explanatory variables.Afterward, the performance criteria of different models are reported in terms of some measures to assess the goodness of fit of the proposed model.In Sect.4, using the results of the illustrative and applicative examples, we devote some comparative studies to demonstrating the effectiveness of the proposed model.Finally, we make some concluding remarks related to the present work for further research in Sect. 5.

Preliminaries
In this section, the basic concepts about LR-type fuzzy number, linear adaptive fuzzy regression model, and fuzzy semi-parametric partially linear model are explained below.

LR-type fuzzy number
An LR-type fuzzy number A is described by its membership function, such that (Zimmermann 2011) (1) (2) (3) Kelkinnama and Taheri (2012) have defined a distance based on absolute deviation between LR-type fuzzy numbers as follows: where ω = dx represent the influence of the shape of the membership function on the distance.It can be proved that D is a metric on LR-type fuzzy numbers and is the average of errors which is computed not only in the centers but in the left and right endpoints of cut sets for two fuzzy numbers.

Linear adaptive fuzzy regression model
We consider a group of n LR-type fuzzy number sample data, denoted by x i0 , x i1 , ..., x i p , y i , i = 1, 2, ..., n (specify x i0 = (1;1, 1) L R ).Let x i j = (x i j ;l x i j ,r x i j ) L R be the independent variable, y i = (y i ;l y i , r y i ) L R be the dependent variable( j = 0, 1, ..., p).Then, the linear adaptive fuzzy regression model with fuzzy inputs and fuzzy output has the following form D' Urso (2003): where in ( 5), y i , l y i and r y i present the interpolated values of the center, left and right spread for the ith output; α j , β j and δ j are the coefficients for the regression model on the center; u, v, g, and z are the coefficients for the regression models on the left and right spreads; i , λ i , and ρ i are the residuals of the models on the centers and on the spreads, respectively, and may be considered as fuzzy errors due to the fuzzy structure of the system.The basic idea consists in simultaneously modeling the centers and the spreads of the LR-type response variables by means of three sub-models.The first sub-model (referred to as core regression model) explains the centers of the fuzzy observations, while the other two sub-models (referred to as spread regression model) are built over the first one and yield their spreads.The linear adaptive fuzzy regression model is formulated in such a way as to take into account possible linear dynamic relationship between the size of the spreads and the magnitude of the estimated centers, as it is often necessary in real case studies.
The least-squares iterative estimates for the coefficients of the fuzzy regression model are obtained minimizing the squared Euclidean distance between the fuzzy values estimated by the model and ambiguous data really observed.The optimization problem can be expressed as where π c , π l , π r are positive weights.

Fuzzy semi-parametric partially linear model
Semi-parametric partially linear model as a powerful tool to incorporate statistical parametric and non-parametric regression analyses has gained attentions in many reallife applications.Given a set of n fuzzy observed data x i1 , x i2 , ..., x i p , y i , i = 1, 2, ..., n , a fuzzy semi-parametric partially linear model with LR-type fuzzy inputs and LR-type fuzzy output can be built as follows Hesamian et al. (2017): where β j is an unknown real valued to be estimated; τ i is an additional covariate in which τ i ∈ [0, 1] and i is a random error with mean zero.The coefficient β j is assumed to be the parametric part of the model, and the unknown fuzzy smooth function f (τ i ) represents the non-parametric part of the model.It is mentioned that the fuzzy semi-parametric partially linear model can be reduced to a common fuzzy linear regression when we use a constant number instead of f (τ i ) into the model.
To find the best fuzzy semi-parametric partially linear model of the form (7), a two-step procedure is suggested based on curve fitting methods and least absolute deviations.Using the metric D in (4), Hesamian et al. (2017) minimized the sum of distances between the observed outputs and estimated outputs, i.e., n i=1 D( y i , y i ), which is equivalent to minimize the following expression, to estimate the components of the fuzzy regression model:

Formulation of the proposed model
Here, we introduce a new adaptive fuzzy regression method using semi-parametric technique and neural network computation based on the fuzzy inputs and fuzzy output data.

Adaptive fuzzy semi-parametric regression model using TPB
truncated power basis is a s th -order spline with knot points at t j1 < t j2 < ... < t js and a real-valued polynomial function terming up to degree d on each of the intervals (−∞, t j1 ], [t j1 , t j2 ], ..., [t js , ∞).The truncated power basis allows for smoother estimates as linear combinations of these basis functions have d −1 continuous derivatives.The prediction errors of this spline regression approach using truncated power basis can be considered unbiased estimators over their entire domains, which often provides the analyst with additional flexibility for the formulation of exploratory analyses (Perperoglou et al. 2019).In this regard, we consider the following adaptive fuzzy semi-parametric regression model applying TPB: where α jq , β jk , u, v, g and z are unknown coefficients, i , λ i and ρ i indicate residuals and f (τ i ) is an unknown smooth function, τ i ∈ [0, 1] is an additional covariates.
It is mentioned that the fuzzy regression model in ( 9) can be reduced to an adaptive fuzzy linear regression model for crisp inputs and LR-type fuzzy output when the observations of inputs are reduced to the crisp values and we use a constant number instead of f (τ i ) into the model with d = 1 and max 1≤i≤n x i j ≤ t j1 , j = 1, 2, ..., p.
Therefore, the unknown smooth function in the core regression model can be evaluated by Nadaraya-Watson estimator (Cai 2001) and h > 0 controls the amount of smoothing which is known as bandwidth of the kernel function K (•).
By substituting expression (10) for the corresponding term in the core regression model of (9), we get Convert the above expression as follows: we may obtain an adaptive fuzzy semi-parametric regression model using TPB: Remark 1 The corresponding regression model for the symmetric case with fuzzy inputs x i j =(x i j ; η i j ) L and output y i =(y i ; η i ) L , i=1, 2, ..., n, j=1, 2, ..., p as follows: It is easy to derive from the above-mentioned procedure the following equations: Remark 2 The proposed fuzzy semi-parametric regression model is adaptive model, where the dynamic of the spreads is somehow dependent on the magnitude of the estimated centers.The non-adaptive version of the adaptive fuzzy semiparametric regression model is shown as follows: where α jq , β jk , jq , jk , γ jq and φ jk are unknown coefficients; f 1 (τ i ), f 2 (τ i ) and f 3 (τ i ) are unknown smooth functions.From the similar procedure, we could obtain the following equations: In terms of number of estimated coefficients, the adaptive regression model proposed is more parsimonious than its non-adaptive version.As the number of input variables increases, the divergence between the number of the coefficients of adaptive and non-adaptive models is increasing.

Adaptive fuzzy semi-parametric regression model using TPB and ABC-BPNN
In this section, based on the absolute error distance, we fit an adaptive fuzzy semi-parametric regression model using TPB and ABC-BPNN.We consider a three-layered BPNN with some input units, hidden units, and one output neuron that has the weights and biases.ABC-BPNN based on some concepts of ABC and BPNN fully considers the influence of the shortcomings of BPNN on the prediction trends.The ABC algorithm is an optimization method proposed by imitating bee behavior, which is used to optimize the weights and biases of the randomly initialized BPNN.The ABC algorithm has the characteristics of simple operation, less control parameters, and high search precision.Experiment results show that ABC-BPNN method fits the prediction trend more accurately, and has faster convergence and more stable prediction results (Liu and Meng 2019;Wang et al. 2019).
The ABC algorithm is aimed at finding proper solutions and optimizing them in the searching environment, which adapts the ideologies used by bees during the honey collection procedure.The employed bees search for the nectar sources in the neighborhood and the following bees mark the location of nectar sources, and then, the scout bees identify the new location to replace the corresponding nectar source.We repeat the procedure until the number of iterations reaches the maximal iteration number and utilize the optimal structure of BPNN to perform the final output prediction.The flow structure of ABC-BPNN algorithm is shown in Fig. 2.
Here, an input vector considering the different combinations of the centers or spreads of dependent or independent variables is presented to the input layer.We want to adjust the weights and biases using the probability function according to the rules relating to the degree of fitness, a specified maximum number of update failure, and the maximum number of cycle.Then, the optimal weights and biases are derived for performing the neural networks.The target values for the centers of outputs corresponding to the input vector denoted by y * m , m = 1, 2, ..., n can be calculated by BPNN.Therefore, by replacing y m in the core regression model of (13) with y * m , the adaptive fuzzy semi-parametric regression model based on TPB and ABC-BPNN is Then, the computational procedure for estimating optimal value of bandwidth should be implemented.The role of the smoothing parameter h is to adjust the degree of smoothness of the estimates.In this paper, a leave-one-out cross-validation procedure is employed, which uses a single observation from the whole dataset as the validation dataset, and the remaining observations as the training dataset.The validation process is repeated until each observation of the whole dataset is used once as a validation dataset.Based on absolute error distance, the smoothing parameter h should be where y (i) represents the estimate of y i after deleting the ith sample.
Similar to common linear regression, the fuzzy regression coefficients are estimated by minimizing (20) which is a constrained non-linear programming problem Now, combating the above optimization problems, the following algorithm is suggested in this paper to find the optimal value for bandwidth and crisp regression coefficients.Given n observations on one LR-type fuzzy dependent variable y i = (y i ; l y i , r y i ) L R and p LR-type fuzzy independent variables x i j = (x i j ; l x i j , r x i j ) L R , i = 1, 2, ..., n, j = 1, 2, ..., p.
Step 1: Select a combination subset with respect to the center and/or spread of y i and/or x i j .
Step 2: For each combination of the components, perform the shift and range transformation on the corresponding n observations.Step 3: The outputs of BPNN are the estimation measures for the centers of dependent variables based on ABC algorithm.
Step 4: Select a kernel function, and calculate the optimal bandwidth by the cross-validation procedure.
Step 5: Regression coefficients are estimated by means of minimizing (20).
Step 6: Steps 1-5 are repeated until the developed model based on every combination subset with respect to the center and/or spread of y i and/or x i j is achieved.
In this case, it would be better to select the fitted network and employ the estimated parameters asso-ciated to the values of indices for evaluating the performance of fuzzy regression models.
Remark 3 For a selected value of h, if the estimated left or right spread at some center is negative, one way to handle this problem is as suggested in D' Urso (2003), to set the negative spread to be zero at center.

Evaluation of the model
In fuzzy environment, based on the difference between the membership value of the observed fuzzy number and the estimated fuzzy number, we can use the error of the fitting of the membership functions as a measure of goodness of fit introduced by Kim and Bishu (1998).Suppose that y i and y i are the actual output and the estimated output based on the developed model, respectively.The mean of errors (ME) using Kim and Bishu distance is defined by where the integrals are only calculated over intervals containing the support of the fuzzy numbers.
The smaller the difference between the two fuzzy numbers is, the closer the value of ME is to zero, and the accuracy of the model is higher.However, when the observed value and the predicted value are not overlapped, the value of this measure of performance will be the same regardless of the amount of distance between the observed and its estimation.
To overcome this problem, the mean absolute percentage error for the center, left, and right spread of output (MAPE c , MAPE l and MAPE r ), respectively, is used for comparing and assessing the accuracy performance of regression models In addition, the mean of similarity measures (MSM) is denoted as follows: where min and max operators are used for intersection and union of two fuzzy sets, respectively.When MSM is larger, the predicted outputs of the fuzzy regression model are closer to the actual outputs.Hence, a larger MSM indicates that the corresponding fuzzy regression model is better in terms of the degree of similarity.

Illustrative examples and competitive studies
In this section, we examine the feasibility and effectiveness of the proposed adaptive fuzzy regression solution procedure via three numerical examples including a simulation study and two applied examples.The first and third example have fuzzy numbers for both the inputs and output, and the second example has fuzzy numbers only for the output.All fuzzy numbers are assumed to be from class of triangular fuzzy numbers.Among three cases, putting d = 2.The smooth function of the three examples is the triweight kernel function Example 1 On 100 samples, we have simulated a fuzzy output variable and two fuzzy input variables ( y i , x i1 , x i2 ) from the Model I as follows: where x i1 and x i2 are the observed center samples of inputs from U (6, 12) and U (−2, 4), respectively; l x i1 and l x i2 are the observed left spread samples of inputs from U (0.1, 1); r x i1 and r x i2 are the observed right spread samples of inputs from U (0.1, 1).The residuals e i ∼N (0, 1) and e l i , e r i ∼U (0.01, 0.1); ξ i and ν i are the observed random samples from U (0.01, 0.1).
.., 100.Then, we repeat the designed model 1000 times to investigate the performance of our proposed model.In each simulation, the estimations of coefficients are computed by solving the minimization problem.We compare in this example our adaptive fuzzy semiparametric regression (denoted as AFR ABC−BP ) with the models given by D' Urso and Gastaldi (2002); D' Urso (2003); Hesamian et al. (2017), respectively.And the results of non-adaptive version of our model (simply as NAFR ABC−BP ), using TPB alone in our model (termed as AFR) and using TPB combined with BPNN in our model (termed as AFR BP ) are also used for comparison.The mean values of goodness-of-fit measures are shown in the left panel of Table 1.In Fig. 3, the horizontal axes represent the center values for the two independent variables, the vertical axis represents the center value for the dependent variable, in which average values of regression coefficients over 1000 scenarios are given.Figure 3 depicts the graphical dependence of center values for output on the center values for inputs.
The left panel of Table 1 and Fig. 3 reveal that the results of AFR, AFR BP and AFR ABC−BP are expected to coincide.As it can be seen from Fig. 3, the produced deviations are evidently small in the AFR, AFR BP and AFR ABC−BP models which seem to be a bit closer to the observed dependence compared with the NAFR ABC−BP model.The D' Urso and Gastaldi (2002); D' Urso (2003) and Hesamian et al. (2017) models display a relatively poor agreement between estimates and data, where the observed dependence is significantly away from the estimated dependence.The AFR, AFR BP , and AFR ABC−BP models give better fitting to the data and produce a little bias in the dependence of output on the inputs than the other four models.
To study how our approach performs when the data are spoiled by outliers, the input-output pattern of Model I is used, but the 10% of target output is replaced by the data generated according to Model II as follows: Model II The corresponding results of optimal models are given in the right panel of Table 1.By comparing the results provided in Table 1 based on the goodness-of-fit indices, we see that coefficient estimates for our method are little impacted by the outliers in the fuzzy response variable, while the presence of outliers heavily affects the estimates for the other models and the fit performance of these models are bad.The difference between each goodness-of-fit measure for our model in the uncontaminated and contaminated cases is less  than those of the other models.The results imply that the proposed method is relatively stable to the outliers.
To further illustrate the performance of the proposed method, we contaminate the dataset by adding an increasing percentage P={5%, 10%, 15%, 20%} of outliers generated from the pattern of Model II and estimate the coefficients, respectively.Repeating this contamination mechanism 1000 times for each percentage, we propose a comparative analysis.The impact of outlier magnitude on the proposed method is visualized in Fig. 4.
It is seen from Fig. 4 that as the contamination rate of outliers increases from 0% to 20%, the reported ME, MAPE c , MAPE l , and MAPE r values also increase, and MSM values decrease progressively as expected.Although the contamination rate reaches 20%, the proposed method produces the lower errors and continues to work better than these conventional linear adaptive fuzzy regression models and fuzzy semi-parametric partially linear model.This implies that even the contamination rate is influential on the fitting indices of the proposed method, it still provides the satisfactory results, since the distortions caused by outliers are more extreme   The coefficients of the proposed fuzzy regression model with h opt = 0.023 are calculated, and then, the objective model is obtained as As expected, the fitting performance of the proposed method is better than that of the other methods.In the light of this information, the minimum values for ME, MAPE c , and MAPE l terms are obtained and the proposed fuzzy model makes MSM term maximum demonstrated in Table 2.
Figure 5 gives the graphical comparison on the trend of fitting values for output in this study.The proposed method provides appropriate intervals and the actual fuzzy output values are basically located in the estimated value intervals.The trend is apparent from the plot, revealing the fact that the coefficient estimates obtained from the proposed method yielded the smaller deviations between the observed outputs and the estimated outputs than the others and the actual outputs are nearer the target outputs.It can be concluded that there is a good performance of the fitness on this real dataset for our proposed model.
Example 3 Here, we work on the dataset consisting of daily stock price of five different airlines from January 1988 to October 1991 (see https://sci2s.ugr.es/keel/dataset.php?cod=77/) to verify the performance of the proposed model.This dataset contains 150 numerical observations with five continuous variables.Among the five variables, the fifth variable in the dataset is considered as the response variable, with the remaining four variables indexed as explanatory variables based on the sequence in the dataset.
Then, considering the accuracy of measurements in observations, all variables are transformed into triangular fuzzy numbers using the fuzzification method described by He et al. (2016).Assume that there is a real number c i and we can obtain a triangular fuzzy number (c i ;l i , r i ) T , l i and r i can be determined by the following expressions: are the mean value and variance of variable.Note that the fuzzified variable could be an asymmetric triangular fuzzy number in Fig. 6, which shows that two endpoints c i − l i and c i + r i make the corresponding squares equal, and in this study, we select a smaller value 0.05 that affects the fuzzification on c i .
The corresponding results of the derived adaptive fuzzy semiparametric regression model using TPB and ABC-BPNN for such data are shown in Table 3.
In Fig. 7, the illustration of the fitting is displayed.As shown in Fig. 7, the proposed model produced the smaller ME value and larger MSM value compared to those of the other models.Examining the MAPE indicator, the values for centers and left spreads are much lower than the others.It can be concluded that our method is significantly fitted and efficient when compared to the existing methods in the literature.

Conclusions and further research
We investigated a novel adaptive fuzzy semi-parametric regression model with fuzzy inputs and fuzzy output data by taking the advantage of the neural networks and spline basis function.The proposed regression model is based on the well-known linear adaptive fuzzy regression model because of its simplicity and flexibility, and this paper has enlarged the scope of application of the linear adaptive fuzzy regression.First, the adaptive fuzzy regression model along with the semi-parametric technique and TPB is defined and allows us to simultaneously estimate the regression coefficients and smooth function by optimizing an objective function using absolute deviation metric.Second, the integration of the ABC-BPNN into the adaptive fuzzy semi-parametric regression model is a flexible computing framework and universal approximator applied to the forecasting problems of the fuzzy regression with a stable degree of accuracy.Third, to illustrate and validate the proposed model, we have carried out a step-by-step practical application and discussed our results by comparing with those obtained by some classical and recently proposed approaches.With a general point of view, at least based on the results of three numerical examples, the proposed model dominates the other models and can be an effective and rational way of improving forecasting accuracy.
Further developments of this study should need the investigation of more variety domains for the proposed model, simulation experiments to evaluate the impact of missing data and more kinds of outliers on the global performance of the model, and additional work into the total computational time consumed in the regression model.In line with the results obtained, further works could extend the proposed approach for more complex fuzzy regression models as well as the development of a comparative study with the traditional methods.

Fig. 3
Fig. 3 Comparison in the impact of observed centers of inputs on observed/estimated centers of outputs in Example 1

Fig. 4
Fig. 4 Impacts of different proportions of outliers on the performance of the proposed method

Fig. 5 Fig. 6
Fig. 5 Observed outputs (in violet color) and estimated outputs (in blue color) from the models in Example 2 (color figure online)

Table 1
Fitting performances of the fuzzy regression models in the uncontaminated and contaminated cases The mean values of goodness-of-fit measure are obtained from 1000 times repetition for each model procedure

Table 2
The performances of various fuzzy regression models in Example 2