Ridge Regression With Intuitionistic Fuzzy Input and Output: A Parametric Approach
Subject Areas : Fuzzy Optimization and Modeling JournalZahra Behdani 1 * , Majid Darehmiraki 2
1 - Department of Mathematics, Behbahan Khatam Alanbia University of Technology,
Khouzestan, Iran
2 - Department of Mathematics, Behbahan Khatam Alanbia University of Technology,
Khouzestan, Iran
Keywords: Intuitionistic Fuzzy Number, Regression Model, Ridge regression, Distance,
Abstract :
Ridge regression is a model that is frequently used and has numerous effective applications, particularly in the management of correlated factors in a multiple regression model. Additionally, multicollinearity poses a significant risk in fuzzy regression models when it comes to predictions. In order to solve this problem, we bring together the fuzzy regression model with the ridge regression technique. Regarding the evaluation of the coefficients of the ridge fuzzy regression model, the algorithm that we have suggested makes use of the parametric estimation approach. In this article, we examine the ridge regression in the intuitionistic fuzzy environment. We assume that the input and output data are intuitionistic fuzzy numbers. Since in the regression analysis we need to calculate the distance between the variables, we define a new fuzzy parametric distance. Also, the goodness of fit of the model with the indicators of the mean square of the prediction error has been investigated in simulation examples and real data.
1. Akram, M., Sarwar, M., & Borzooei, R. A. (2018). A novel decision-making approach based on hypergraphs in intuitionistic fuzzy environment. Journal of Intelligent & Fuzzy Systems, 35(2), 1905-1922.
2. Atanassov, K. T. (2012). On intuitionistic fuzzy sets theory (Vol. 283). Springer.
3. Choi, S. H., Jung, H. Y., & Kim, H. (2019). Ridge fuzzy regression model. International Journal of Fuzzy Systems, 21(7), 2077-2090.
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7. Flores-Sosa, M., Aviles-Ochoa, E., Merigo , J. M., & Kacprzyk, J. (2022). The OWA operator in multiple linear regression. Applied Soft Computing, 124, 108985.
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17. Shakouri, B., Abbasi Shureshjani, R., Daneshian, B., & Hosseinzadeh Lotfi, F. (2020). A parametric method for ranking intuitionistic fuzzy numbers and its application to solve intuitionistic fuzzy network data envelopment analysis models. Complexity, 1-25.
18. Szmidt, E. (2014). Distances and similarities in intuitionistic fuzzy sets (Vol. 307). Switzerland: Springer International Publishing.
19. Tanaka, H., Lee, H. (1998). Interval regression analysis by quadratic programming approach. IEEE Trans. Fuzzy Systems. 6(4),437-481.
20. Torra, V., & Narukawa, Y. (2009, August). On hesitant fuzzy sets and decision. In 2009 IEEE international conference on fuzzy systems (pp. 1378-1382). IEEE.
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E-ISNN: 2676-7007 | Fuzzy Optimization and Modelling Journal 5(2) (2024) 19-31 |
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Contents lists available at FOMJ
Fuzzy Optimization and Modelling Journal
Journal homepage: https://sanad.iau.ir/journal/fomj/ | ||
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Paper Type: Research Paper
Ridge Regression With Intuitionistic Fuzzy Input and Output: A Parametric Approach
Zahra Behdania,*, Majid Darehmirakia
a Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Khouzestan, Iran
A R T I C L E I N F O |
| A B S T R A C T Ridge regression is a model that is frequently used and has numerous effective applications, particularly in the management of correlated factors in a multiple regression model. Additionally, multicollinearity poses a significant risk in fuzzy regression models when it comes to predictions. In order to solve this problem, we bring together the fuzzy regression model with the ridge regression technique. Regarding the evaluation of the coefficients of the ridge fuzzy regression model, the algorithm that we have suggested makes use of the parametric estimation approach. In this article, we examine the ridge regression in the intuitionistic fuzzy environment. We assume that the input and output data are intuitionistic fuzzy numbers. Since in the regression analysis we need to calculate the distance between the variables, we define a new fuzzy parametric distance. Also, the goodness of fit of the model with the indicators of the mean square of the prediction error has been investigated in simulation examples and real data.
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Article history: Received 29 December 2023 Revised 27 April 2024 Accepted 18 June 2024 Available online 26 June 2024 | ||
Keywords: Intuitionistic fuzzy number Regression model Ridge regression Distance
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1. Introduction
In statistics, linear regression is a linear model approach between the response variable and one or more explanatory variables. Regression is often used to discover the model of linear relationship between variables. In this case, it is assumed that one or more descriptive variables whose value is independent of the other variables or under the control of the researcher, can be effective in predicting the response variable whose value is not dependent on the descriptive variables and under the control of the researcher. The purpose of regression analysis is to identify the linear model of this relationship. Since in the real world we often face imprecise data, it is better to use fuzzy logic to model the inherent uncertainty in these data. In a general division, the types of fuzzy regression can be divided into the following three models:
· Fuzzy regression in the case where the relationship between the variables is assumed to be fuzzy. In other words, the regression equation coefficients are considered fuzzy.
· Fuzzy regression in the case that the variables (either prediction or response) are imprecise and fuzzy.
· Fuzzy regression in the case that both variables and coefficients of the model are considered fuzzy.
It should be mentioned that the variation in fuzzy regression is not limited to the above modes. Rather, the methods that have been proposed for each mode have created a lot of diversity in fuzzy regression methods. Like classical regression, which is based on the principle or principles on which model parameters are estimated, fuzzy regression can be divided into possible regression methods, regression of the least squares of the fuzzy error, and regression of the least absolute magnitude of the fuzzy error.
Hassamian et al. [10] investigated fuzzy non-parametric regression model with fuzzy responses and exact predictors. Li et al. [14] investigated fuzzy multiple linear regression with LR numbers. In it, they presented a calculation formula for regression parameters and introduced two new distances between fuzzy numbers. Authors of Flores-Sosa et al. [7] applied ordinary least squares and ordered weighted average to solve multiple linear regression. In Durso and Chachi [5], the authors proposed Ordered Weighted Averaging to solve regression models with exact/fuzzy inputs and fuzzy output. Hasanpour et al. [9] estimated the fuzzy using diamond meter. They tried to make the estimates as close to the real value as possible with the ideal planning method and by comparing the error of their method with the error of Diamond [4] method, they showed the superiority of their method. Rabiei et al. [16] using the fuzzy regression method for a set of data with input and fuzzy output.
Intuitionistic fuzzy sets (IFSs) were first introduced by Atanassov [2] as an extension of Zadeh's fuzzy sets [23]. These IFSs serve as a mathematical framework for representing sets that are non-crisp and characterized by uncertainty. In the context of these sets, we establish functions that determine both membership and non-membership. In this particular scenario, it is possible to establish a hesitation function, which can be defined as the disparity between the "membership function" and the "one minus non-membership function." Through the use of Interval Type-2 Fuzzy Sets (IFSs), we are able to effectively represent and analyze incomplete information. Numerous scholars have made significant contributions to the field of Interval Type-2 Fuzzy Systems (IFSS) in terms of both theoretical advancements and practical implementations (see Szmidt [18] for more details). Akram et al. [1] introduced a unique decision-making approach using hypergraphs within the context of intuitionistic fuzzy environments. This technique was subsequently used in practical scenarios [17]. A hybrid technique that is based on recurrent neural networks was published in Karbasi et al., [11] for the purpose of approximating the coefficients (parameters) of a ridge fuzzy regression model that has LR-fuzzy output and crisp inputs. This problem was solved in Choi et al. [3] by the use of alpha-level estimation approach. The authors of reference Kim and Jung [13] developed a fuzzy ridge estimator that is independent of the distance between fuzzy numbers.
The use of distance and similarity measurements is crucial for identifying the dissimilarities between two entities. Decision making, pattern recognition, image processing, machine learning, market prediction, and so on are just some of the numerous possible uses for distance and similarity measures. Wang [21] first presented a computational formula for the similarity measure of fuzzy collections. Many scholars have taken an interest in this issue ever then and have gone into further depth. Different fuzzy set, intuitionistic fuzzy set, and fuzzy multiset distance and similarity metrics have been presented. There are a number of distance measures in common usage, but three of the most well-known are the Hamming distance, the Euclidean distance, and the Housdorff distance. The authors of Li et al. [15] examine the fuzziness of fuzzy sets, similarity measures, and connection measures. It was investigated by Ejegwa and Adamu [6] how far apart two intuitionistic fuzzy sets of second type may be. Weighted distance measure for intuitionistic fuzzy sets using the Choquet integral with regard to the non-monotonic fuzzy measure was presented by Torra and Narukawa [20]. Distance and similarity measures based on hesitant fuzzy sets were expanded by Xia and Xu [22].
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also known as Tikhonov regularization, named for Andrey Tikhonov, it is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters Kennedy [12]. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias [8].
When it comes to fuzzy multiple regression models, multicollinearity is a major concern, just as it is in conventional statistical theory when it comes to traditional multiple regression models. We offer the ridge fuzzy regression model, which is a combination of the ridge regression model and the fuzzy regression model. The purpose of this model is to lessen the impact of multicollinearity when it is present. We offer a parametric estimation approach that is based on the parametric distance measure in order to generate the ridge fuzzy regression model. Among the prominent features of the proposed meter is that decision makers can determine the distance between two fuzzy numbers based on their decision level by choosing the appropriate alpha values. And as a result, the regression model can be determined by changing the parameters of this distance according to the level of decision-making desired by the decision-makers. The remaining parts of this study are as follows:
The basics of IFSs, IFNs and related arithmetic operators are presented in Section 2. In this section, a new parametric approach for the distance of IFNs is proposed in general using the concepts of decision level and degree of uncertainty. The advantages of the proposed method are shown through a suitable example. In section 3, we will review the concepts of regression and estimation of regression parameters using the meter defined in the previous section. In the next section, we introduce ridge regression to estimate model parameters with an example.
2. Fuzzy preliminaries
We summarize below the basic concepts of intuitionistic fuzzy set theory, such as intuitionistic fuzzy numbers, intuitionistic fuzzy arithmetic, and the ranking of intuitionistic fuzzy numbers.
Definition 1. Suppose 
is the universal set, then 
is a fuzzy set by the following representation:
(1)
where 
correspond to membership value of each element of universal set respect to and is defined according to relation (2):
(2)
Definition 2. The intuitionistic fuzzy set has the following representation
(3)
where and 
respectively correspond to membership and non membership values and are defined according to equations (4) and (5):
(4)
(5)
A function 
is called hesitancy function for each 
can be represented by relation:
(6)
It is clear that the value of 
is a number between zero and one.
Definition 3. For two intuitionistic fuzzy sets 
and 
in universal set , the following propositions are valid:
Definition 4. An arbitrary intuitionistic fuzzy number (IFN) such as defines an intuitionistic fuzzy set on the axis of real numbers that membership and non-membership functions are introduced corresponding to relations (7) and (8):
(7)
=
(8)
where 
and 
are two continuous and strictly decreasing function from 
to 
and 
. Also 
and 
are called respectively left and right spreads and 
. A LR type IFN is denoted by 
When 
, we obtain a special type of IFNs called triangular intuitionistic fuzzy numbers (TriIFN), which are known as an important class of intuitionistic fuzzy numbers.
Definition 5. Let 
and 
are two TriIFNs and 
is an arbitrary positive number. Then,
1. 
Definition 6. Let 
be an TriIFN. 
cut, 
cut and 
cut for are respectively:
In the following, we express the parametric form of a fuzzy number.
Definition 7. Parametric form of a fuzzy number including two functions 
which apply in the following conditions:
1. 
is a bounded left continuous non-decreasing function over 
2. 
is a bounded left continuous non-increasing function over
3. 
.
Similar to Definition 7, we can also define parametric IFNs. The parametric form of an IFN is as 
where 
which apply in the conditions of Definition 7.
Now we can define a parametric distance measure between two numbers according to what was said about the representation of fuzzy numbers. Also, according to the mentioned materials, a TriIFN can be displayed as follows: 
where 
and 
.
Definition 8. Let 
and 
be two IFNs. The parametric distance between two numbers is defined as follows:
(9)
Theorem 1. The defined function (9) is a meter, which means it applies to the following properties:
Proof. Items 1 and 2 are clear. For item 3, we have:
£
Assume 
and 
be two LR IFNs. Then
In a special case where two numbers are triangular, we have:
With simplify the above equation we have:
Example 1. Let 
and 
be two TriIFNs. In this case, we have:
Table 1 shows the value of distance 
and 
for different 
and . Also, in Figure 1, you can see the membership function of two IFNs and . that alpha and beta represent different levels of decision making. As alpha increases, a higher decision level is selected, and conversely, if alpha is a lower value, it means that this problem is solved at a lower level of decision making. It is the opposite for beta, i.e. a low beta value indicates problem solving at a higher level of decision making. The data in the Table 1 shows that the distance between these two fuzzy numbers for members of the set that have a membership function of 0.8 and a non-membership function of 0.2 is equal to 0.023 (second row of the table). The members of the set whose membership is less than 0.4 and their non-membership is 0.8 is estimated at 0.21.
Figure 1. Membership functions of 
and 
in Example 1.
Table 1. and distance values for different 
and 
in Example1
Num. |
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1 | 0.2 | 0.8 | 0.22 | ||
2 | 0.8 | 0.2 | 0.023 | ||
3 | 0.5 | 0.5 | 0.11 | ||
4 | 0.4 | 0.8 | 0.21 | ||
5 | 0.6 | 0.6 | 0.13 | ||
6 | 0.3 | 0.4 | 0.12 | ||
7 | 0 | 1 | 0.31 | ||
8 | 1 | 0 | 0 |
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7 |
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8 |
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9 |
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11 |
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Num. |
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| MSE |
1 | 0.2 | 0.8 | (147.8,18.87,14.54,-167.08) | (232.9662,72,51,-42.72) | 32504.59 |
2 | 0.8 | 0.2 | (248.99,17.98,13.46,-165.19) | (272.11,66.94,-47.74) | 507.86 |
3 | 0.5 | 0.5 | (174.88,18.48,14.08,-163.56) | (252.1946,70.51,-42.24) | 7935.62 |
4 | 0.4 | 0.8 | (152.58,18.77,14.42,-165.75) | (238.42,72.04,-42.21) | 23108.69 |
5 | 0.6 | 0.6 | (170.60,18.51,14.12,-163.51) | (250.91,70.72,-41.95) | 8887.91 |
6 | 0.3 | 0.4 | (161.56,18.61,14.23,-164.04) | (246.33,71.28,-41.73) | 12919.25 |
7 | 0 | 1 | (138.94,19.11,14.81,-170.34) | (220.47,73.52,-44.16) | 63485.7 |
8 | 0.99 | 0.01 | (285.89,17.68,13.09,-164.94) | (284.96,64.97,-49.84) | 0.064 |
9 | 0.4 | 0.3 | (171.41,18.49,14.09,-163.13) | (252.22,70.62,-41.78) | 7713.43 |
Num. |
|
| MSE |
1 | 0 | (285.89,17.67,13.09,-164.95) | 0.01835 |
2 | 0.001 | (263.65,17.72,13.14,-162.21) | 0.018352 |
3 | 0.003 | (227.79,17.78,13.21,-157.73) | 0.018355 |
4 | 0.007 | (178.15,17.86,13.29,-151.28) | 0.01836 |
5 | 0.009 | (160.27,17.89,13.31,-148.85) | 0.018361 |
6 | 0.015 | (122.25,17.93,13.33,-143.36) | 0.018365 |
7 | 0.035 | (65.11,17.91,13.2,-133.12) | 0.018369 |
8 | 0.07 | (32.20,17.74,12.81,-123.34) | 0.018366 |
9 | 0.127 | (14.55,17.43,12.164,-112.64) | 0.018359 |
10 | 0.225 | (4.81,16,96,11.19,-99.20) | 0.01835 |
11 | 0.35 | (0.72,16.48,10.21,-86.47) | 0.018346 |
12 | 0.65 | (-1.74,15.69,8.61,-66.28) | 0.018353 |
13 | 0.85 | (-2.06,15.33,7.90,-57.36) | 0.018362 |
14 | 0.94 | (-2.11,15.20,7.63,-54.08) | 0.018367 |
15 | 1 | (-2.12,15.12,7.47,-52.09) | 0.018369 |
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