Comparative Assessment of Multiple Linear Regression and Fuzzy Linear Regression Models

Abstract

Prognosticating crop yield still remains as one of the challenging tasks in agriculture. Even though multiple linear regression methodology has dominated the area of predictive modelling, it is constrained to the assumption that the underlying relationship is assumed to be crisp or precise. Consequently, it often fails to provide satisfactory results when this assumption is violated in realistic situations. Fuzzy linear regression methodology is one of the promising and potential techniques to overcome this lacuna. Moreover, this fuzzy methodology can efficiently handle the problem of multicollinearity. In this paper, an attempt has been made to comparatively assess the efficiency of conventional regression models with their fuzzy counterparts using data on sweet corn yield (t/ha), total weed dry matter (g/m2) at 30 DAS and total weed density (no./m2) at 30 DAS. Model efficiency is computed in terms of average width of the prediction intervals. Efficiency of the models is also assessed in the presence of correlated explanatory variables. Outcomes emanated from the study clearly show the higher relative efficiency of fuzzy linear regression technique in comparison with the widely used simple and multiple linear regression techniques. This study also reveals that the fuzzy methodology has clear advantages over the conventional regression methodology in dealing with correlated explanatory variables.

Introduction

Regression analysis is a very powerful and widely used technique in numerous fields of research [26]. It is basically carried out to explain the variation of the response variable in terms of the variation of the explanatory variable(s), assuming there exists a linear relationship between the response and explanatory variable(s) [20, 39]. One major drawback of this methodology is that the underlying relationship is assumed to be crisp or precise, as it gives a precise value of response for a set of values of explanatory variables. However, in realistic situations, the underlying relationship is not a crisp function of a given form; it contains some vagueness or impreciseness. Therefore, a forceful assumption of a crisp relationship may lead to the loss of some vital information. To address this vagueness, a very promising technique of fuzzy linear regression (FLR) has been developed [51]. It has stemmed from the idea of considering fuzzy uncertainty as ambiguity and vagueness [1, 10]. Tanaka et al. [11, 45] was the first to introduce FLR model by utilising the concept of linear programming to determine the regression coefficients as fuzzy numbers. Model proposed by Tanaka et al., even though quite common and useful, is constrained to only symmetrical triangular fuzzy numbers [42]. Buckley and Eslami [9] developed a fuzzy least square regression model to overcome this limitation. The derived model, however, is based on non-linear programming concepts and requires a lot of computations. Pan et al. [38] suggested a matrix-driven multivariate FLR approach with the capability of handling both the functions of asymmetric and symmetric triangular membership to improve computational efficiency.

It is now a well-recognized fact that agriculture is a ‘soft science’ rather than a ‘hard science’ as there always exists some amount of ‘vagueness’ or ‘impreciseness’ or ‘fuzziness’ in the underlying phenomenon, and/or explanatory variable(s), and/or response variable. Consequently, incorporation of this aspect in the conventional linear regression model is necessitated for the purpose of more realistic modelling [21, 29, 30, 49]. Kandala and Prajneshu [23] demonstrated the superiority of fuzzy methodology over the usual multiple regression approach, when variables (both explanatory and response) are all crisp, but the underlying phenomenon is assumed to be fuzzy. Kandala and Prajneshu [25] also employed FLR methodology to determine age–length relationship in pearl oyster by fitting fuzzy von Bertalanffy growth model. Despite the popularity and the sheer power of this methodology, the empirical performance of fuzzy linear regressions especially in presence of correlated explanatory variables has rather been inconclusive [8, 15, 44].

Hence, in this paper, an effort has been made to evaluate the suitability of FLR models as yield prediction models in agriculture in comparison with the conventional multiple linear regression (MLR) models using weed dry matter and weed density data. Rest of the paper is organised as follows. Related works are discussed in “Related works”. Regression methodologies to be investigated are described in detail in “Methodologies”. Empirical results obtained from the field data are presented in “Empirical results”. “Conclusion”, finally, concludes the paper.

Related Works

Misfits between observed and estimated values in fuzzy regression are assumed to be due to system fuzziness or fuzziness of regression coefficients [43]. Modarres et al. [35] emphasised the relative simplicity in programming and computation in case of FLR modelling. Keeping this in mind, several researchers have successfully applied FLR models in divergent domains. Song et al. [50] carried out short-term load forecasting for the holidays by employing FLR method. Abdullah and Zamri [2] employed fuzzy regression model to predict road accidents in Malaysia. Shafi [47], based on their findings, suggested to use fuzzy regression model instead of multiple regression model to determine tumour size in colorectal cancer. Azadeh et al. [7] also reported the superiority of fuzzy algorithm over the conventional regression approach in predicting gas consumption in Iran.

As a consequence of upsurging research interest in fuzzy regression modelling, various extensions as well as improvement of the already existing methods have been observed [6]. Yang and Liu [59] proposed a robust estimation procedure for FLR models to deal with outliers. Hassanpour et al. [18] evaluated FLR models by comparing membership functions. Lu and Wang [32] proposed an enhanced FLR model with more flexible spreads. Razzaghnia and Pasha [41] introduced a new mathematical programming approach to estimate the parameters of FLR with crisp/fuzzy input and fuzzy output. Taghizadeh et al. [53] designed a multi-level FLR model to predict energy demand. Liu and Chen [31] proposed an approach to optimise h value for FLR analysis using minimum fuzziness criteria with symmetric triangular fuzzy numbers. Khan and Valeo [28] demonstrated that use of non-linear membership functions is more appropriate to deal uncertain environmental data as compared to the typical triangular representations. Zhang [65] introduced a FLR model based on centroid method. Darwish et al. [14] presented a regression model for a special case of interval type-2 fuzzy sets based on the least squares estimation technique. Shafi et al. [46] proposed a hybrid approach by augmenting FLR with symmetric parameter model and fuzzy c-means method. Pérez-Cañedo et al. [40] proposed two new fuzzy goal programming methods based on linear and Chebyshev scalarisations with a view to solve the problem of fully fuzzy multi-objective linear programming.

Methodologies

Simple and Multiple Linear Regression Model

In a simple linear regression (SLR) model, there is only one explanatory variable, whereas in a multiple linear regression model, multiple explanatory variables are utilised to predict the outcome of a response variable assuming the relationship between the explanatory variables and the response variable as linear. Multiple linear regression is basically the extension of simple linear or ordinary least squares (OLS) regression by allowing more than one explanatory variable to rely on the mean function E(Y) [16]. A usual representation of multiple linear regression is:

$$Y = \beta_{0} + \beta_{1} X_{1} + \beta_{2} X_{2} + \cdots + \beta_{k} X_{k} + \varepsilon ,$$
(1)

where Y and \(X_{i}\) (i = 1, 2, …, k) represent response and explanatory variables, respectively. \(\beta_{i}\) (i = 0, 1, …, k) are model parameters and ε represents the random error term.

Fuzzy Linear Regression Model

Zadeh [61, 62] introduced the concept of fuzziness for the first time. Fuzzy theory, according to Zadeh, emerged after evidencing the failure of traditional techniques of systems analysis, when the relationships between variables are too vague or complex [63, 64]. In such situations of uncertainty, the fuzzy number is defined and applied to real-world problems. Emergence of various concepts connecting fuzzy theories with most of the mathematical field has led to the introduction of fuzzy real line [13], fuzzy topology [19], etc. Anand and Bharatraj [5] identified the resemblances in behaviour and properties between triangular fuzzy numbers and Pythagorean triples.

Deviations between observed and estimated values, in conventional regression analysis approach, are assumed to be due to random errors. However, in reality these may be due to indefiniteness of structure of a system or imprecise observations. In such cases, the associated uncertainty turns into ‘fuzziness’, not into ‘randomness’. Fuzzy linear regression model, proposed by Tanaka et al. [55], is represented by the following equation:

$$Y = A_{0} + A_{1} X_{1} + A_{2} X_{2} + \cdots + A_{k} X_{k} .$$
(2)

The explanatory variables \({X_i}{\text{'s}}\) are assumed to be precise. The response variable (Y), however, is assumed to be fuzzy, which consequently implies the fuzzy nature of the parameters. An assumption that \({A_i}{\text{'s}}\) are symmetric fuzzy numbers (i.e., vagueness can be expressed as equidistant from centre) is also made so that these can be represented by intervals. For instance, \(A_{i}\) can be expressed as:

$$A_{i} = \left\langle {a_{{i{\text{c}}}} ,a_{{i{\text{w}}}} } \right\rangle ,$$
(3)

where \(a_{{i{\text{c}}}}\) and \(a_{{i{\text{w}}}}\) represent the centre and radius (i.e., associated vagueness) respectively. The belief of regression coefficient around \(a_{{i{\text{c}}}}\) in terms of symmetric triangular membership function is described by this fuzzy set. It should be also noted that this methodology is applicable, when the underlying phenomenon is fuzzy, i.e., both the response variable and the relationship is fuzzy in nature. In alternate notations, \(A_{i}\) can also be expressed as [22]:

$$A_{i} = [a_{{i{\text{L}}}} ,a_{{i{\text{R}}}} ],$$
(4)

where \(a_{{i{\text{L}}}} = a_{{i{\text{c}}}} - a_{{i{\text{w}}}}\) and \(a_{{i{\text{R}}}} = a_{{i{\text{c}}}} + a_{{i{\text{w}}}}\). Parameters in fuzzy regression methodology are estimated by minimising total vagueness in the model, i.e., sum of radii of prediction intervals. We can rewrite the Eq. (2) as:

$$Y_{j} = A_{0} + A_{1} X_{1j} + A_{2} X_{2j} + \cdots + A_{k} X_{kj} .$$
(5)

Utilising Eq. (3), we can further obtain:

$$y_{j} = \left\langle {a_{0c} ,a_{0w} } \right\rangle + \left\langle {a_{1c} ,a_{1w} } \right\rangle x_{1j} + \left\langle {a_{2c} ,a_{2w} } \right\rangle x_{2j} + \cdots + \left\langle {a_{kc} ,a_{kw} } \right\rangle x_{kj} = \left\langle {y_{jc} ,y_{jw} } \right\rangle .$$
(6)

Thus,

$$y_{jc} = a_{0c} + a_{1c} x_{1j} + a_{2c} x_{2j} + \cdots + a_{kc} x_{kj} ,$$
(7)

And

$$y_{jw} = a_{0w} + a_{1w} \left| {x_{1j} } \right| + a_{2w} \left| {x_{2j} } \right| + \cdots + a_{kw} \left| {x_{kj} } \right|.$$
(8)

As \(y_{jw}\), being radius, cannot be negative, absolute values of \(x_{ij}\) are considered on the right hand side of Eq. (8). Suppose there are n data points; each comprising of a(k + 1)-row vector. Then the parameters are estimated by minimising the total vagueness of the model-data set combination, subject to the constraint that within the estimated value of response variable, each data point must fall. This can be envisaged as the following linear programming problem [56], which can be solved by Simplex procedure [54]:

$${\text{Minimize}}\;\sum\limits_{j = 1}^{n} {(a_{0w} + a_{1w} \left| {x_{1j} } \right| + a_{2w} \left| {x_{2j} } \right| + ... + a_{kw} \left| {x_{kj} } \right|)} ,$$
(9)
$${\text{subject }}\;{\text{to}}\;\left\{ {\left( {a_{{0{\text{c}}}} + \sum\limits_{i = 1}^{k} {a_{{i{\text{c}}}} x_{ij} } } \right) - \left( {a_{{0{\text{w}}}} + \sum\limits_{i = 1}^{k} {a_{{i{\text{w}}}} x_{ij} } } \right)} \right\} \le Y_{j} ,$$
(10)
$$\left\{ {\left( {a_{{0{\text{c}}}} + \sum\limits_{i = 1}^{k} {a_{{i{\text{c}}}} x_{ij} } } \right) + \left( {a_{{0{\text{w}}}} + \sum\limits_{i = 1}^{k} {a_{{i{\text{w}}}} x_{ij} } } \right)} \right\} \ge Y_{j} ,$$
(11)

and

$$a_{{i{\text{w}}}} \ge 0.$$
(12)

Figure 1 schematically represents the methodology of comparing SLR and MLR models with FLR models.

Fig. 1
figure1

Schematic representation of the comparative assessment methodology

Empirical Results

Data and Implementation

The study utilises the data pertaining to a field experiment conducted during the spring season of 2017 in N. E. Borlaug Crop Research Centre of G. B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, India. The experiment was laid out in a randomized block design with three replications. Sweet corn variety ‘Sugar 75’ was utilised for the experimental purpose. Data on sweet corn yield (t/ha), total weed dry matter (g/m2) at 30 DAS (Days After Sowing) and total weed density (no./m2) at 30 DAS collected from 48 samples are utilised in evaluating the efficiency of two approaches.

Total weed dry matter \((X_{1} )\) and total weed density \((X_{2} )\), even though associated, have well-known effect on yield [4, 12, 33, 34, 36, 37, 52, 58]. Therefore, in both approaches, individually and combinedly, these are used as explanatory variable(s) to assess the influence(s) on sweet corn yield.

Experimental Findings

Table 1 shows the results of SLR and MLR models. Total weed dry matter and total weed density have shown significant negative impact on yield in the respective SLR model. However, in case of the MLR model considering both total weed dry matter and total weed density as regressors, we notice the change of sign in case of the coefficient of total weed density, even though the model as well as all the parameters are significant at 1 percent level. This observed alteration in sign may be the pronounced effect of multicollinearity [3, 17, 48, 57, 60] (as \(r_{{X_{1} X_{2} }}\) = 0.978).

Table 1 Summary of simple and multiple linear regression models for assessing sweet corn yield

Table 2 briefs the FLR models. We observe that vagueness associated with the centre is lesser when both the explanatory variables, instead of only one of these, are included in the model. Average width of prediction intervals of SLR/MLR and FLR models are reported in Table 3. Figures 2a–c, respectively, show the prediction intervals of both SLR/MLR model and FLR model for all the three cases under study.

Table 2 Summary of fuzzy linear regression models for assessing sweet corn yield
Table 3 Average width of prediction intervals of the linear regression models under study
Fig. 2
figure2figure2

Prediction intervals of both simple/multiple linear regression model and fuzzy linear regression model by considering a total weed dry matter as single explanatory variable, b total weed density as single explanatory variable and c both total weed dry matter and total weed density as explanatory variables

It is obvious from Table 3 that the average width is much larger in case of SLR and MLR models as compared to their fuzzy counterparts. This clearly suggests the higher relative efficiency of FLR technique in comparison with SLR and MLR techniques [8, 27]. At this juncture, it is worth mentioning that in case of FLR, when both total weed dry matter and total weed density are considered for the model, average width is observed to be smaller than when only one of these are considered. However, inclusion of both the explanatory variables in the model enlarges the average width in MLR approach. Earlier Kandala and Prajneshu [24] also obtained similar results in crop yield forecasting using remotely sensed data, when the two highly correlated explanatory variables, viz., normalised difference vegetation index (NDVI) and ratio vegetation index (RVI) were utilised in the model. Thus, FLR approach, unlike MLR approach, is very capable of handling the situations, when the explanatory variables are highly correlated.

Conclusion

Appropriate methodology to obtain the relationship between response and explanatory variables while dealing with fuzzy phenomenon is FLR rather than SLR or MLR. This study has compared the efficiency of SLR and MLR methodology with FLR methodology using data on sweet corn yield, total weed dry matter and total weed density. Efficiency of the models are computed in terms of average width of the prediction intervals. As in reality, the underlying phenomenon is fuzzy, fuzzy methodology has provided smaller average width than the conventional SLR and MLR methodology. Thus the present study validates the theoretical results obtained in the literature based on a real experimental set up in the field of agriculture. An attempt has also been made to compare the models in presence of correlated explanatory variables. Outcomes clearly show the superiority of FLR methodology over the conventional linear regression methodology in presence of multicollinearity. This capability of handling multicollinearity would be particularly useful, when large number of interrelated variables are expected to predict an outcome variable.

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Acknowledgements

The data presented in the current paper are obtained from the experiment conducted for the thesis work of Mr. Prithwiraj Dey at G. B. Pant University of Agriculture and Technology, Pantnagar, India. The authors deeply acknowledge the resources and financial aids received from GBPUA&T, Pantnagar, India and ICAR, India during the entire duration of the experiment.

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Correspondence to Pramit Pandit.

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This article is part of the topical collection “Computational Statistics” guest edited by Anish Gupta, Mike Hinchey, Vincenzo Puri, Zeev Zalevsky and Wan Abdul Rahim.

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Pandit, P., Dey, P. & Krishnamurthy, K.N. Comparative Assessment of Multiple Linear Regression and Fuzzy Linear Regression Models. SN COMPUT. SCI. 2, 76 (2021). https://doi.org/10.1007/s42979-021-00473-3

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Keywords

  • Average width
  • Fuzzy linear regression
  • Model efficiency
  • Multicollinearity
  • Multiple linear regression