Uncertain least squares estimation model based on relative error

Abstract

Uncertain least squares estimation is one of the important methods to deal with imprecise data, which can fully consider the influence of given data on regression equation and minimize the absolute error. In fact, some scientific studies or observational data are often evaluated in terms of relative error, which to some extent allows the error of the forecasting value to vary with the size of the observed value. Based on the least squares estimation and the uncertainty theory, this paper proposed the uncertain relative error least squares estimation model of the linear regression. The uncertain relative error least squares estimation minimizes the relative error, which can not only solve the fitting regression equation of the imprecise observation data, but also fully consider the variation of the error with the given data, so the regression equation is more reasonable and reliable. Two numerical examples verified the feasibility of the uncertain relative error least squares estimation, and compared it with the existing method. The data analysis shows that the uncertain relative error least squares estimation has a good fitting effect.

Keywords

Relative error least squares estimation relative error least squares estimation uncertainty theory

1 Introduction

Regression analysis is a widely used statistical analysis method, which can better determine the interdependence between variables. Regression analysis can be divided into unary regression analysis and multiple regression analysis according to the number of independent variables. If the regression analysis contains only two variables, and the relationship between them can be approximately expressed by a straight line. This kind of regression analysis is called unary linear regression analysis. A regression analysis is called multiple linear regression analysis if it contains more than one variable and the relationship between the variables is linear. The least squares method is one of the most commonly used methods to solve the unknown parameters of linear regression equation. It makes the sum of the distances from each point to the fitted line the shortest, that is, the least squares method can minimize the absolute error, so as to obtain a better fitting line. Zhang and Fu [1] studied several problems of least squares method in regression analysis. Wang [2] applied the least squares method to establish the regression model of the air temperature curve. Many expert scholars have carried out in-depth studies on the least square method and applied to many fields, which will not be listed one by one here.

The absolute error reflects the deviation of the forecasting result from the true value, but cannot reflect the relative degree of the error. Sometimes the absolute error is relatively large, but the deviation from the actual data may be small. Sometimes the absolute error is relatively small, but the deviation from the actual data may be large. Relative error is the ratio of absolute error to true value, so it is a percentage. Generally speaking, the relative error can better reflect the credibility of the forecasting. For example, the surveyor uses the same ruler to measure objects with a length of 1cm and 10cm respectively. Assuming that the absolute errors of their measured values are the same, but the relative errors of the former are an order of magnitude larger than the latter, it indicates that the measured values of the latter are more reliable. So it is more meaningful to study the relative error of data. In fact, some scientific studies and observations are often evaluated on the basis of relative error. Therefore, many experts and scholars have studied and applied the relative error. Li [3] proposed least squares method based on relative error. Qi [4] put forward relative error analysis of least squares estimation for unary linear regression. Jin, Huang and Shan [5] proposed a new linear regression model from electric load forecasting of shandong province. Yun and Cao [6] proposed data fitting in the sense of relative error based on optimization. Gai and Zhang [7] proposed single-index relative error regression models. The relative error least squares method can minimize the sum of relative error squares, and the regression equation obtained is more effective and practical.

In the traditional mathematical statistics, the least squares method of relative error can better solve the regression equation of given precision data. But in fact, the observation data we get are often imprecise, or there is no historical data, so the relative error least squares method is powerless. Liu [10] proposed uncertainty regression analysis based on uncertainty theory can help to solve this problem. In order to describe the relationship between imprecision data, Liu [8] founded the uncertainty theory in 2007 and gradually improved it [9 –13]. Uncertainty theory can deal with the uncertainty in reality very well, and many experts and scholars have conducted in-depth research on it. Uncertainty theory has been widely used [14 –17]. In 2010, Liu [10] started his research on uncertain statistics. In order to estimate the unknown parameters in the uncertain distribution, Liu put forward the principle of least squares. Since then, many experts and scholars have systematically studied uncertain statistics [18 –20]. In 2018, Yao and Liu [21] proposed the least squares estimation and Song and Fu [22] presented the least squares method of uncertain multiple linear regression. Wang, Li and Guo [23] put forward a new uncertain regression model in 2020. Wang et al. [24, 25] proposed two new uncertain linear regression models in 2021. Shi et al. [26] proposed total least squares estimation model based on uncertainty theory in 2022. Ye and Liu [27] proposed uncertain hypothesis test with application to uncertain regression analysis in 2022.

Uncertain linear regression model is an important part of uncertain regression analysis and an effective tool to deal with imprecise data. At present, least squares estimation is one of the best methods to solve the unknown parameters of uncertain linear regression equation. It minimizes the sum of squares of residuals, and can also forecast data effectively and get confidence interval. However, although the least squares estimation is not the absolute error in the traditional sense, it still solves the degree of deviation of the estimated value from the true value. The least squares estimation can only get the magnitude of the error, but can not reflect the credibility of the estimated value. Other methods for solving the unknown parameters of uncertain linear regression equations are similar without considering the relative degree of errors. In fact, some scientific studies need to evaluate the relative error of the data. On the condition that the sum of squares of relative fitting errors is minimum, References [3 –7] put forward the relative error least squares method and discusses some related problems. The significance of the relative error least squares method is that the larger the observed quantity is, the larger the actual measurement error is allowed, so it has more extensive use and practical significance. The relative error least squares method can only deal with precise data, but can not solve the regression equation of imprecise data. On the basis of previous studies [3], this paper proposed an uncertain relative error least squares estimation model. The uncertain relative error least squares estimation model minimizes the sum of squares of relative errors and obtains a more realistic regression equation. Compared with the absolute error least squares estimation model, the uncertain relative error least squares estimation model proposed in this paper minimizes the sum of the squares of the relative errors, which not only makes the regression equation more realistic, but also has a better fitting effect than the absolute error model. At the same time, the relative error least squares estimation model can be used to solve the unary linear regression equation, and can also be extended to multiple regression, which can be used to deal with both imprecise data and precise data.

The main organizational structure of this paper is as follows: In section 2, we introduce the uncertain least squares estimation, which can effectively solve the regression equation. In section 3, we propose uncertain relative error least squares estimation model. This paper first deduces the formula of unknown parameters of unitary linear regression equation and extends it to the case of multiple linear regression equation. In section 4, we verify the feasibility of the model with two numerical examples, and compare the calculated results with the existing methods. The results shows that the relative error least square estimation model has a good fitting effect. Finally, we compare the absolute error with the relative error, analyze the superiority of the uncertain relative error least squares estimation model, and summarize the whole paper.

2 Uncertain regression model

In order to describe the relationship between imprecise data, Liu [8] founded uncertainty theory in 2007. Readers interested in uncertainty theory can read Reference [13]. This section mainly introduces the uncertain regression equation.

Let (x₁, x₂, ⋯, x_n) be a vector of explanatory variables, and let y be a response variable. If the functional relationship between (x₁, x₂, ⋯, x_n) and y can be expressed by a regression model

$\begin{matrix} y = f (x_{1}, x_{2}, \dots, x_{n} ∣ β) + ɛ, \end{matrix}$ (1)

where β is an unknown vector of parameters, and ɛ is a disturbance term [13].

In particular, Liu [13] called $\begin{matrix} y = β_{0} + β_{1} x_{1} + β_{2} x_{2} + \dots + β_{n} x_{n} + ɛ \end{matrix}$ (2) a linear regression model.

Now supposed that we have a set of imprecise data $({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, \dots, {\tilde{x}}_{in}, {\tilde{y}}_{i}), i = 1, 2, \dots, m$ where ${\tilde{x}}_{i 1}$ , ${\tilde{x}}_{i 2}$ , ⋯, ${\tilde{x}}_{in}$ , ${\tilde{y}}_{i}$ are independent uncertain variables with regular uncertainty distributions Φ_i1, Φ_i2, ⋯, Φ_in, Ψ_i, i = 1,2,⋯,m, respectively.

Yao-Liu [21] proposed the least squares estimate of unknown parameter β of linear regression model. The parameter β is the solution of the following minimization problem $min_{β} \sum_{i = 1}^{n} E [({\tilde{y}}_{i} - f ({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, \dots, {\tilde{x}}_{in} ∣ β))^{2}] .$ (3)

If the minimization solution is β^*, then the fitting regression equation is determined by y = f (x₁, x₂, ⋯, x_n ∣ β^*). Then for each index i(i = 1,2,⋯,m), the term

${\tilde{ɛ}}_{i} = {\tilde{y}}_{i} - f ({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, \dots, {\tilde{x}}_{in} ∣ β^{*})$ (4) is called the i-th residual [28].

We call

$y = β_{0}^{*} + β_{1}^{*} x_{1} + β_{2}^{*} x_{2} + \dots + β_{p}^{*} x_{p}$ (5) the multiple linear regression equation.

If the disturbance term ɛ is uncertain variable, its expected value and variance can be estimated as

$\begin{matrix} \hat{e} = \frac{1}{n} \sum_{i = 1}^{m} E [{\tilde{ɛ}}_{i}], \end{matrix}$ (6) and

${\hat{σ}}^{2} = \frac{1}{n} \sum_{i = 1}^{m} E [({\tilde{ɛ}}_{i} - \hat{e})^{2}],$ (7) where ${\tilde{ɛ}}_{i}$ are the i-th residual, i = 1,2,⋯,m, respectively [28].

Lio-Liu [28] suggested that the forecast uncertain variable of response variable y with respect to $({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ is determined by $\hat{y} = f ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n} ∣ β^{*}) + \hat{ɛ}, \hat{ɛ} \sim N (\hat{e}, \hat{σ}),$ (8) and the forecast value is defined as the expected value of the forecast uncertain variable $\tilde{y}$ , i.e., $μ = E [f ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n} ∣ β^{*})] + \hat{e} .$ (9) Take α (e.g., 95%) as the confidence level, and find the minimum values b such that $\hat{Ψ_{i}} (μ + b) - \hat{Ψ_{i}} (μ - b) \geq α$ , the confidence interval of response variable y is $\begin{matrix} μ \pm b . \end{matrix}$ (10)

3 Uncertain relative error least squares estimation model of linear regression

The relative error least squares estimation model studies the relative error of observation data, which minimizes the sum of relative errors and reflects the relative accuracy of estimates. For imprecise data, the model first calculates the expected value of each relative error, and then minimizes the sum of squares of the expected values.

3.1 Unary relative error least squares estimation

We always assume that there is a linear relationship between ${\tilde{x}}_{i}$ and ${\tilde{y}}_{i}$ (i = 1, 2, ⋯ , m) . The main steps of the relative error least squares estimation model are as follows.

Step 1. Construct the relative error least squares estimation model.

According to the imprecise data ${\tilde{x}}_{i}$ and ${\tilde{y}}_{i}$ (i = 1, 2, ⋯, m). we obtain the least squares estimation model as y = β₀ + β₁x₁ + ɛ, where β₀, β₁ are unknown parameters, and ɛ is an uncertain disturbance term. The relative error least squares estimation model is to solve the ratio of the difference between each observation value and the estimated value to the observed value, and then find the minimum of the sum of the relative errors according to the uncertain expectation. The relative error least squares estimation model is

$\begin{matrix} M (β_{0}, β_{1}) = min_{β_{0}, β_{1}} \sum_{i = 1}^{m} [\frac{{\tilde{y}}_{i} - (β_{0} + β_{1} {\tilde{x}}_{i})}{{\tilde{y}}_{i}}]^{2} . \end{matrix}$ (11)

Step 2. Solve the regression parameters β₀ and β₁ of the model.

Equation (11) is a binary elementary function of β₀ and β₁. In order to solve β₀ and β₁, equation (11) takes the first partial derivative of β₀ and β₁, respectively, and makes its first partial derivative equal to 0. The following equations are obtained

$\begin{matrix} \frac{\partial M}{\partial β_{0}} & = - 2 \sum_{i = 1}^{m} [\frac{{\tilde{y}}_{i} - (β_{0} + β_{1} {\tilde{x}}_{i})}{{\tilde{y}}_{i}}] \frac{1}{{\tilde{y}}_{i}} = 0, \\ \frac{\partial M}{\partial β_{1}} & = - 2 \sum_{i = 1}^{m} [\frac{{\tilde{y}}_{i} - (β_{0} + β_{1} {\tilde{x}}_{i})}{{\tilde{y}}_{i}}] \frac{\tilde{x_{i}}}{{\tilde{y}}_{i}} = 0 . \end{matrix}$ (12)

By deforming equation (12), we obtain the following equation $\begin{matrix} β_{0} \sum_{i = 1}^{m} \frac{1}{{\tilde{y}}_{i}^{2}} + β_{1} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}^{2}} & = \sum_{i = 1}^{m} \frac{1}{{\tilde{y}}_{i}}, \\ β_{0} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}^{2}} + β_{1} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}^{2}}{{\tilde{y}}_{i}^{2}} & = \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}} . \end{matrix}$ (13)

In equation (13), $({\tilde{x}}_{i}, {\tilde{y}}_{i})$ , i = 1, 2, ⋯ , m, are imprecise data. In uncertainty theory, $({\tilde{x}}_{i}, {\tilde{y}}_{i})$ , i = 1, 2, ⋯ , m, are uncertain variables and the expected value is required before the equation can be solved. Taking the expected value of both sides of equation (13), we get

$\begin{matrix} β_{0} \sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}^{2}}] + β_{1} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}^{2}}] & = \sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}}], \\ β_{0} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}^{2}}] + β_{1} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}^{2}}{{\tilde{y}}_{i}^{2}}] & = \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}}] . \end{matrix}$ (14)

Denoted

$\begin{matrix} \bar{\tilde{Y}} = \frac{1}{m} \sum_{i = 1}^{m} E [{\tilde{y}}_{i}], \bar{\tilde{X}} = \frac{1}{m} \sum_{i = 1}^{m} E [{\tilde{x}}_{i}] . \end{matrix}$ (15)

By solving equation (14), we obtain estimates of β₀ and β₁

$\begin{matrix} β_{1}^{*} = & \frac{\sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}}] \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}^{2}}]}{(\sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}^{2}}])^{2} - (\sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}^{2}}{{\tilde{y}}_{i}^{2}}]) (\sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}^{2}}])} \\ - & \frac{(\sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}}]) (\sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}^{2}}])}{(\sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}}{{\tilde{y}}_{i}^{2}}])^{2} - (\sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i}^{2}}{{\tilde{y}}_{i}^{2}}]) (\sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}^{2}}])}, \end{matrix}$ (16) and $β_{0}^{*} = \bar{\tilde{Y}} - β_{1}^{*} \bar{\tilde{X}} .$ (17)

According to the expected value calculation formula in reference [13], the specific calculation formula of equations (17) are $\begin{matrix} β_{1}^{*} = & \frac{\sum_{i = 1}^{m} \int_{0}^{1} \frac{1}{Ψ_{i}^{- 1} (1 - α)} d α \sum_{i = 1}^{m} \int_{0}^{1} \frac{Φ_{i}^{- 1} (α)}{[Ψ_{i}^{- 1} (1 - α)]^{2}} d α}{(\sum_{i = 1}^{m} \int_{0}^{1} \frac{Φ_{i}^{- 1} (α)}{[Ψ_{i}^{- 1} (1 - α)]^{2}} d α)^{2} - (\sum_{i = 1}^{m} \int_{0}^{1} \frac{[Φ_{i}^{- 1} (α)]^{2}}{[Ψ_{i}^{- 1} (1 - α)]^{2}} d α) (\sum_{i = 1}^{m} \int_{0}^{1} \frac{1}{[Ψ_{i}^{- 1} (1 - α)]^{2}} d α)} \\ - & \frac{(\sum_{i = 1}^{m} \int_{0}^{1} \frac{Φ_{i}^{- 1} (α)}{Ψ_{i}^{- 1} (1 - α)} d α) (\sum_{i = 1}^{m} \int_{0}^{1} \frac{1}{[Ψ_{i}^{- 1} (1 - α)]^{2}} d α)}{(\sum_{i = 1}^{m} \int_{0}^{1} \frac{Φ_{i}^{- 1} (α)}{[Ψ_{i}^{- 1} (1 - α)]^{2}} d α)^{2} - (\sum_{i = 1}^{m} \int_{0}^{1} \frac{[Φ_{i}^{- 1} (α)]^{2}}{[Ψ_{i}^{- 1} (1 - α)]^{2}} d α) (\sum_{i = 1}^{m} \int_{0}^{1} \frac{1}{[Ψ_{i}^{- 1} (1 - α)]^{2}} d α)} \end{matrix}$ (18) and

$\begin{matrix} β_{0}^{*} = & \frac{1}{m} \sum_{i = 1}^{m} \int_{0}^{1} Ψ_{i}^{- 1} (α) d α \\ - β_{1}^{*} \frac{1}{m} \sum_{i = 1}^{m} \int_{0}^{1} Φ_{i}^{- 1} (α) d α . \end{matrix}$ (19)

Equations (19) are specific calculation formulas for calculating unknown parameters. By substituting known data, the estimated value of unknown parameters can be obtained, thus obtaining the fitting equation.

Step 3. Data analysis and forecasting.

After we get the estimated values $β_{0}^{*}$ and $β_{1}^{*}$ , the equation $y = β_{0}^{*} + β_{1}^{*} x$ is the unary linear regression equation of the imprecise data. As for the estimated value of $β_{0}^{*}$ and $β_{1}^{*}$ , we need to conduct data analysis to further judge the fitting effect. According to the actual need, we can also forecasting, and get forecasting intervals according to the accuracy requirement.

Relative error least squares estimation model is actually to solve the minimum problem of relative error by partial derivative. The method has more practical significance and the overall fitting effect is good.

3.2 Multivariate relative error least squares estimation

The basic idea and derivation process of the relative error least squares estimation model of multiple linear regression are similar to the Unary relative error least-squares estimation model. However, due to the large number of variables, the calculation process is more complicated. In order to simplify the calculation of the relative error least squares estimation model of multiple linear regression, we derive it by means of matrix equation.

We assumed that there is a linear functional relationship between uncertain variables vector $({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}$ , $\dots, {\tilde{x}}_{in})$ and ${\tilde{y}}_{i}$ , i = 1, 2, ⋯ , m. Supposed that the multiple linear regression equation satisfies the following equations

$\begin{matrix} y_{1} & = β_{0} + β_{1} {\tilde{x}}_{11} + β_{2} {\tilde{x}}_{12} + \dots + β_{n} {\tilde{x}}_{1 n}, \\ y_{2} & = β_{0} + β_{1} {\tilde{x}}_{21} + β_{2} {\tilde{x}}_{22} + \dots + β_{n} {\tilde{x}}_{2 n}, \\ \dots \\ y_{m} & = β_{0} + β_{1} {\tilde{x}}_{m 1} + β_{2} {\tilde{x}}_{m 2} + \dots + β_{n} {\tilde{x}}_{mn} . \end{matrix}$ (20) where β₀, β₁, β₂, ⋯ , β_n are unknown parameters in the linear regression equation.

The main steps of the relative error least squares estimation model are as follows.

Step 1. Construct the relative error least squares estimation model.

Similar to the unary relative error least squares estimation model, the multivariate relative error least squares estimation model also calculates the ratio of the difference between each observed value and the estimated value and the observed value, and then minimizes the sum of the relative errors according to the uncertain expectation. Multivariate relative error least squares estimation model is

$\begin{matrix} M (β_{0}, β_{1}, β_{2}, \dots, β_{n}) \\ = min_{β_{0}, β_{1}, β_{2}, \dots, β_{n}} \sum_{i = 1}^{m} [\frac{{\tilde{y}}_{i} - (β_{0} + \sum_{j = 1}^{n} {\tilde{x}}_{ij} β_{j})}{{\tilde{y}}_{i}}]^{2} . \end{matrix}$ (21)

Step 2. Solve the regression parameters of the model.

Equation (21) can be regarded as a basic elementary function of β₀, β₁, β₂, ⋯ , β_n. According to the extreme value theory of functions, if a differentiable function has an extreme value, its partial derivative equation must have a solution. We take the partial derivatives of equation (21) for β₀, β₁, β₂, ⋯ , β_n respectively, and we obtain the following equation.

$\begin{matrix} \frac{\partial M}{\partial β_{0}} & = - 2 \sum_{i = 1}^{m} (\frac{{\tilde{y}}_{i} - β_{0} - \sum_{j = 1}^{n} β_{j} {\tilde{x}}_{ij}}{{\tilde{y}}_{i}}) \frac{1}{{\tilde{y}}_{i}} = 0, \\ \frac{\partial M}{\partial β_{1}} & = - 2 \sum_{i = 1}^{m} (\frac{{\tilde{y}}_{i} - β_{0} - \sum_{j = 1}^{n} β_{j} {\tilde{x}}_{ij}}{{\tilde{y}}_{i}}) \frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}} = 0, \\ ⋮ \\ \frac{\partial M}{\partial β_{n}} & = - 2 \sum_{i = 1}^{m} (\frac{{\tilde{y}}_{i} - β_{0} - \sum_{j = 1}^{n} β_{j} {\tilde{x}}_{ij}}{{\tilde{y}}_{i}}) \frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}} = 0 . \end{matrix}$ (22)

After collating and calculating equation (22), the following equation is obtained.

$\begin{matrix} β_{0} \sum_{i = 1}^{m} \frac{1}{{\tilde{y}}_{i}^{2}} + & β_{1} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}^{2}} + β_{2} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 2}}{{\tilde{y}}_{i}^{2}} + \dots \\ + β_{n} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}} = \sum_{i = 1}^{m} \frac{1}{{\tilde{y}}_{i}}, \\ β_{0} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}^{2}} + & β_{1} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 1}^{2}}{{\tilde{y}}_{i}^{2}} + β_{2} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 1} {\tilde{x}}_{i 2}}{{\tilde{y}}_{i}^{2}} + \dots \\ + β_{n} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 1} {\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}} = \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}}, \\ ⋮ \\ β_{0} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}} + & β_{1} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 1} {\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}} + β_{2} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i 2} {\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}} + \dots \\ + β_{n} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{in}^{2}}{{\tilde{y}}_{i}^{2}} = \sum_{i = 1}^{m} \frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}} . \end{matrix}$ (23)

In Equation (23), ${\tilde{x}}_{ij}, {\tilde{y}}_{i}$ , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n are imprecise data. According to the expected value calculation [13], we take the expected values of both sides of equation (23) and arrange them to get the following equation. $\begin{matrix} β_{0} \sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}^{2}}] + β_{1} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}^{2}}] + β_{2} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 2}}{{\tilde{y}}_{i}^{2}}] \\ + \dots + β_{n} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] = \sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}}], \\ β_{0} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}^{2}}] + β_{1} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1}^{2}}{{\tilde{y}}_{i}^{2}}] + β_{2} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1} {\tilde{x}}_{i 2}}{{\tilde{y}}_{i}^{2}}] \\ + \dots + β_{n} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1} {\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] = \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}}], \\ ⋮ \\ β_{0} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] + β_{1} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1} {\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] + β_{2} \sum_{i = 1}^{m} \\ E [\frac{{\tilde{x}}_{i 2} {\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] + \dots + β_{n} \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{in}^{2}}{{\tilde{y}}_{i}^{2}}] = \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}}] . \end{matrix}$ (24)

We introduce the following matrices $\bar{\tilde{X}}, \bar{\tilde{Y}}$ and β respectively, and the equation (24) is expressed as a matrix equation.

$\bar{\tilde{X}} = [\begin{matrix} \sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}^{2}}] & \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}^{2}}] & \dots & \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] \\ \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}^{2}}] & \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1}^{2}}{{\tilde{y}}_{i}^{2}}] & \dots & \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1} {\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] \\ \dots & \dots & \dots & \dots & \dots \\ \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] & \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1} {\tilde{x}}_{in}}{{\tilde{y}}_{i}^{2}}] & \dots & \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{in}^{2}}{{\tilde{y}}_{i}^{2}}] \end{matrix}],$ (25)

$\begin{matrix} \bar{\tilde{Y}} = [\sum_{i = 1}^{m} E [\frac{1}{{\tilde{y}}_{i}}], \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{i 1}}{{\tilde{y}}_{i}}], \dots, \sum_{i = 1}^{m} E [\frac{{\tilde{x}}_{in}}{{\tilde{y}}_{i}}]]^{T}, \end{matrix}$ (26)

$β = {[\begin{matrix} β_{0}, β_{1}, \dots, β_{n} \end{matrix}]}^{T} .$ (27)

Therefore, the matrix equation form of Equation (24) is

$\begin{matrix} \bar{\tilde{X}} β = \bar{\tilde{Y}} . \end{matrix}$ (28)

Here, $\bar{\tilde{X}}$ is a real symmetric matrix, and if $\bar{\tilde{X}}$ is an invertible square matrix, multiply both sides of equation (28) by the inverse of $\bar{\tilde{X}}$ to obtain the solution of equation (21). The solution of equation (21) is as follows

$\begin{matrix} β = (\bar{\tilde{X}})^{- 1} \bar{\tilde{Y}} . \end{matrix}$ (29)

If $\bar{\tilde{X}}$ is irreversible, we get the augmented matrix $(\bar{\tilde{X}} ⋮ \bar{\tilde{Y}})$ of equation (28). We can obtain the solution of equation (28) through the method of uncertain linear regression model of equation deformation. See Reference [25] for details.

Step 3. Data analysis and forecasting.

After obtaining the estimate of β^* in step 2, we have the multiple linear regression equation $\bar{\tilde{Y}} = \bar{\tilde{X}} \cdot β^{*}$ for the imprecise data. As for the estimated value of β^*, we need to conduct data analysis to further judge the fitting effect. According to the actual need, we can also forecasting, and get forecasting intervals according to the accuracy requirement.

The derivation process of multivariate relative error least squares estimation method is similar to that of unary relative error least squares estimation method, but due to many variables, the solution of the model is more complicated. In the process of solving the model, we use the inverse matrix and matrix elementary transformation and other related content, requires readers to have a certain matrix foundation.

4 Numerical examples

In the previous section, we derived the uncertain relative error least squares estimation model in detail. In this section, we verify the feasibility of the relative error model through two numerical examples and compare the results with the least squares estimation and the uncertain linear regression model based on the slope mean, respectively. The results of numerical examples show that the uncertain relative error least squares estimation model has the better effect.

4.1 Example of imprecise data

In this part, we verify the feasibility of the uncertain relative error least squares estimation model through an example of imprecise data.

It is assumed that ( ${\tilde{x}}_{i}$ , ${\tilde{y}}_{i}$ )(i = 1, 2, ⋯ , 10) are imprecise data provided in Table 1, where ${\tilde{x}}_{i}$ , ${\tilde{y}}_{i}$ (i = 1,2,⋯,10) are independent linear uncertain variables with regular uncertainty distributions Φ_i and Ψ_i(i = 1,2,⋯,10) respectively.

Table 1
Imprecise data (Linear uncertainty distribution)

i 1 2 3 4 5

y _i ℒ(4,6) ℒ(5,7) ℒ (6, 8) ℒ (5, 9) ℒ (8, 10)

x _i ℒ (1, 3) ℒ(2,4) ℒ(3,5) ℒ(4,6) ℒ(5,7)

i 6 7 8 9 10

y _i ℒ (9, 11) ℒ (10, 12) ℒ (11, 15) ℒ (17, 19) ℒ (13, 15)

x _i ℒ(6,8) ℒ(7,9) ℒ(8,10) ℒ(9,11) ℒ(10,12)

i	1	2	3	4	5
y _i	ℒ(4,6)	ℒ(5,7)	ℒ (6, 8)	ℒ (5, 9)	ℒ (8, 10)
x _i	ℒ (1, 3)	ℒ(2,4)	ℒ(3,5)	ℒ(4,6)	ℒ(5,7)
i	6	7	8	9	10
y _i	ℒ (9, 11)	ℒ (10, 12)	ℒ (11, 15)	ℒ (17, 19)	ℒ (13, 15)
x _i	ℒ(6,8)	ℒ(7,9)	ℒ(8,10)	ℒ(9,11)	ℒ(10,12)

The linear regression equations obtained by the relative error least squares estimation, least squares estimation and uncertain slope mean model are shown in Table 2. From the numerical point of view, there is little difference between the three equations, indicating that the fitting effect of the three methods should be similar.

Table 2

The regression equation

Model	Equation
least squares estimation	y = 1.2606x + 1.8061
relative error least squares estimation	y = 1.1800x + 2.2018
uncertain slope mean model	y = 1.1973x + 2.0526

The expected value and variance of the disturbance terms of the three models are shown in Tables 4. The results show that the expected value and variance of the disturbance term of the relative error least squares estimation model are the smallest and the fitting effect is the better.

Table 3

Expected value of the disturbance term

Model	Expected value
least squares estimation	-0.0505
relative error least squares estimation	0.0000
uncertain slope mean model	0.0034

Table 4

Variance of the disturbance term

Model	Variance
least squares estimation	0.1146
relative error least squares estimation	0.0976
uncertain slope mean model	0.0993

The relative error least squares estimation can solve the forecast value of imprecise data and find the confidence interval. We assumed that $\tilde{x} \sim L (11, 13)$ is a new imprecise data. The forecast value of y is

$μ = 15.3619$

If we take the confidence level α = 95%, and the disturbance term is subject to an uncertain normal distribution $N (\hat{e}, {\hat{σ}}^{2})$ . According to the method of solving the confidence interval of References [28] and [29], the confidence interval of y is

$15.3619 \pm 0.6310$

In this numerical example, the uncertain relative error least squares estimation model considers the relative error of the given observation data, which makes the regression equation more practical. Compared with least squares estimation and the uncertain linear regression model based on the slope mean, uncertain relative error least squares method has better fitting effect. However, the numerical example is unary linear equation, and its solving process and analysis degree are not too complicated. The calculation of the multivariate relative error least squares estimation model is very complicated due to the large number of variables.

4.2 Example of precise data

In this part, we verify the feasibility of the relative error least squares estimation model with an example of precise data. In terms of absolute error and residual sum of squares, the model has achieved good results.

Assuming that (x_i,y_i), i = 1, 2, ⋯ , 10 are precise data provided in Table 5.

Table 5
Precise data

i 1 2 3 4 5 6 7 8 9 10

y _i 2 4 5 6 10 15 14 16 21 20

x _i 1 2 3 4 5 6 7 8 9 10

i	1	2	3	4	5	6	7	8	9	10
y _i	2	4	5	6	10	15	14	16	21	20
x _i	1	2	3	4	5	6	7	8	9	10

The linear regression equations are obtained by relative error least squares estimation, least squares estimation and the uncertain linear regression model based on the slope mean are shown in the Table 6. From the coefficient of the equation, the fitting effect of relative error least squares estimation and least squares estimation are similar, and the fitting effect of the uncertain linear regression model based on the slope mean is the worst.

Table 6

The linear regression equations

Model	Linear regression equations
least squares estimation	y = 2.2121x - 0.8667
relative error least squares	y = 2.2088x - 0.8478
estimation
uncertain slope mean model	y = 1.9508x - 0.0888

Tables 8 are statistical tables of sum of absolute error and sum of squared residuals of the three models respectively. The results shows that the relative error least squares estimation model has the best fitting effect.

Table 7

The absolute error

Model	The absolute error
least squares estimation	11.2970
relative error least squares estimation	11.2716
uncertain slope mean model	11.3984

Table 8

The residual sum of squares

Model	Residual sum of squares
least squares estimation	18.3879
relative error least squares estimation	18.3888
uncertain slope mean model	18.4025

In general, this is a numerical example of precise data, and the relative error least squares estimation model is also feasible. The data of the fitting results are analyzed and compared with the existing methods, the fitting effect is the better.

5 Conclusion

Absolute error is the difference between the estimated value and the true value, which can reflect the magnitude of the deviation of the estimated value from the true value. Relative error is the ratio of absolute error to the estimated value or the average value of multiple estimates. It is a dimensionless value, which can compare the reliability of different estimation results and reflect the impact of error on data more truly. The relative error can better reflect the reliability of estimated value, so it is of great practical significance to study the relative error of linear regression equation.

In the numerical analysis of practical problems, in order to study the impact of error on data, we often need to solve the relative error regression equation of data. In practice, however, the data we obtain may not be precise enough, or there may be no historical data at all. The traditional relative error cannot solve these problems. The relative error least squares estimation model proposed in this paper can solve this problem well, and can solve the fitting equations of precise data and imprecise data. The relative error least squares estimation is not only related to the magnitude of the estimated value, but also can reflect the influence of the estimated value on the actual value, making it more reliable. In this paper, we first introduce the least squares estimation of uncertain regression models, which can deal with the regression fitting problem of given data well. Then combined with the relative error, we proposed the uncertain relative error least squares estimation. On the basis of the unary relative error least squares estimation model, we extend the uncertain relative error least squares estimation to the multiple linear regression model. In the case of few variables, the relative error least squares estimation model is not complicated to solve. When there are many variables or a large amount of data, we should use computer programming or software to perform calculations and solve problems, which is also the direction we strive towards.

Footnotes

Acknowledgments

This research was supported by the Natural Science Foundation of Shandong Province (No.ZR2019BG015).

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Uncertain least squares estimation model based on relative error

Abstract

Keywords

1 Introduction

2 Uncertain regression model

3.1 Unary relative error least squares estimation

4.1 Example of imprecise data

Table 5 Precise data i 1 2 3 4 5 6 7 8 9 10 y i 2 4 5 6 10 15 14 16 21 20 x i 1 2 3 4 5 6 7 8 9 10

Footnotes

Acknowledgments

References

Table 5
Precise data

i 1 2 3 4 5 6 7 8 9 10

y _i 2 4 5 6 10 15 14 16 21 20

x _i 1 2 3 4 5 6 7 8 9 10