Uncertain Regression Model Research Articles

Estimating Unknown Parameters and Disturbance Term in Uncertain Regression Models by the Principle of Least Squares

In the field of statistics, uncertain regression analysis occupies an important position. It can thoroughly analyze data sets contained in complex uncertainties, aiming to quantify and reveal the intricate relationships between variables. It is worth noting that the traditional least squares method only takes into account the reduction in the deviations between predictions and observations, and fails to fully consider the inherent characteristics of the correlation uncertainty distributions under the uncertain regression framework. In light of this, this paper constructs a statistical invariant with symmetric uncertainty distribution based on the observations and the disturbance term. It also proposes the least squares estimation of unknown parameters and disturbance term in the uncertain regression model based on the least squares principle and, combined with the mathematical properties of the normal uncertainty distribution, gives a numerical algorithm for solving specific estimates. Finally, in order to verify the effectiveness of the least squares estimation method proposed in this paper, we also design two numerical examples and an empirical study of forecasting of electrical power output.

Ridge estimation for uncertain regression model with imprecise observations

Ridge estimation for uncertain regression model with imprecise observations

Estimation of uncertainty distribution function by the principle of least squares

In order to estimate the unknown parameters in an uncertainty distribution function, this article uses the principle of least squares that minimizes the sum of the squared deviations between the uncertainty distribution and the empirical distribution of the observed data. After that, the principle of least squares is applied to determining the uncertain disturbance term of uncertain regression model and uncertain time series model, and estimating the unknown parameters in uncertain differential equation. Finally, in order to illustrate the proposed method, some real-world examples are provided, including PetroChina stock price, electricity price, grain yield, China’s population, and beef price.

RETRACTED: Uncertain regression model based on Huber loss function

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219433.

The parameter estimations for uncertain regression model with autoregressive time series errors

Uncertain regression model with autoregressive time series errors studies the relations between response variables and explanatory variables when the errors are dependence. The estimations of the unknown parameters are worth exploring in further research. The least square (LS) estimations were proposed to estimate the unknown parameters. However, when the outliers appear, we find that the LS is not effective. So, in this article, the least absolute deviations (LAD) and the maximum likelihood estimations (MLE) are proposed to estimate the unknown parameters. Moreover, some examples are given to verify the applicability and practicability of these two methods. The comparative analysis is given to verify that the proposed methods are more advantageous and effective than the least square method when the observed data are affected by outliers. Finally, the example of exploring the relationships between GDP and related influencing factors for China, from 2004 to 2020, is given to testify the practicability of the methods.

Moment estimation for uncertain regression model with application to factors analysis of grain yield

Uncertain regression analysis is a powerful analytical tool to model the relationships between explanatory variables and the response variable by uncertainty theory. One of the core problems in uncertain regression analysis is to estimate the unknown parameters of an uncertain regression model and the uncertain disturbance term. In this paper, the moment estimation of uncertain regression model is proposed, which can determine both the uncertain regression model and the disturbance term at one time. After that, the uncertain hypothesis test is used to test whether the estimated uncertain regression model is appropriate. Furthermore, a real-world example of factors analysis of grain yield is provided to illustrate the moment estimation. Finally, as a byproduct, this paper also indicates that the stochastic regression model cannot model the agriculture data.

Uncertain hypothesis testing of multivariate uncertain regression model

The multivariate uncertain regression model reveals the relationship between the explanatory and response variables to us very effectively. In this paper, firstly, the uncertain maximum likelihood estimation method for the parameters of the one-dimensional uncertain regression model is extended to the multivariate uncertain regression model to obtain estimates of the parameters. Secondly, in order to determine the reasonableness of the estimated values that are obtained by the various parameter estimation methods, uncertain hypothesis testing is applied to the multivariate uncertain regression model. Finally, some numerical examples are given to verify the feasibility of the method.

Uncertain regression model with moving average time series errors

As a basic model, an uncertain regression model with autoregressive time series errors has been investigated. This paper proposes another fundamental model—uncertain regression model with moving average time series errors—by assuming that the errors of regression model have a moving average structure. Then the principle of least squares is used to estimate the unknown parameters in the model. Based on the fitted model, the forecast value and confidence interval of the future data are derived. Finally, an example is presented to verify the feasibility of this approach.

Uncertain significance test for regression coefficients with application to regional economic analysis

This paper presents a statistical tool of uncertain significance test that uses uncertainty theory to test whether certain prespecified regression coefficients can be regarded as zero. A numerical example is given to illustrate how to test the significance of regression coefficients in an uncertain regression model. In order to compare uncertain significance test with stochastic significance test, both of these significance testing approaches are applied in studying the relationship between GDP and four indicators, including urban population scale, volume of foreign trade, fiscal expenditure, and water resource. The results show that uncertain significance test is more appropriate than stochastic significance test.

Uncertain regression model with autoregressive time series errors.

Uncertain regression model is a powerful analytical tool for exploring the relationship between explanatory variables and response variables. It is assumed that the errors of regression equations are independent. However, in many cases, the error terms are highly positively autocorrelated. Assuming that the errors have an autoregressive structure, this paper first proposes an uncertain regression model with autoregressive time series errors. Then, the principle of least squares is used to estimate the unknown parameters in the model. Besides, this new methodology is used to analyze and predict the cumulative number of confirmed COVID-19 cases in China. Finally, this paper gives a comparative analysis of uncertain regression model, difference plus uncertain autoregressive model, and uncertain regression model with autoregressive time series errors. From the comparison, it is concluded that the uncertain regression model with autoregressive time series errors can improve the accuracy of predictions compared with the uncertain regression model.

A new uncertain linear regression model based on equation deformation

When the observed data are imprecise, the uncertain regression model is more suitable for the linear regression analysis. Least squares estimation can fully consider the given data and minimize the sum of squares of residual error and can effectively solve the linear regression equation of imprecisely observed data. On the basis of uncertainty theory, this paper presents an equation deformation method for solving unknown parameters in uncertain linear regression equations. We first establish the equation deformation method of one-dimensional linear regression model and then extend it to the case of multiple linear regression model. We also combine the equation deformation method with Cramer’s rule and matrix and propose the Cramer’s rule and matrix elementary transformation method to solve the unknown parameters of the uncertain linear regression equation. Numerical example show that the equation deformation method can effectively solve the unknown parameters of the uncertain linear regression equation.

Tukey’s biweight estimation for uncertain regression model with imprecise observations

The purpose of regression analysis is to study how a response variable has a relation to a vector of explanatory variables. Traditionally, statisticians assume that the observation data are precise, and we can get some exact values. However, in many cases, the imprecise observation data are available. We assume that these data are uncertain variables in the sense of uncertainty theory. In this paper, the Tukey biweight or bisquare family of loss functions is applied to estimate unknown parameters satisfying the uncertain regression model. First, the Tukey biweight estimations of three types of regression models are given, namely linear, asymptotic and Michaelis–Menten. Then an empirical study is presented to verify the feasibility of this approach. Finally, the effectiveness of this method in weakening the outliers influence is shown by the comparative analysis.

Multivariate uncertain regression model with imprecise observations

The multivariate regression model is a mathematical tool for estimating the relationships among some explanatory variables and some response variables. In some cases, observed data are imprecise. In order to model those imprecise data, we can employ uncertainty theory to design the uncertain regression model by regarding those data as uncertain variables. Parameters estimation is an important topic in the uncertain regression model. In this paper, we explore a method of parameters estimation by the principle of least squares in the multivariate uncertain regression model containing more than one response variables and assuming both explanatory variables and response variables as uncertain variables. Besides, when the new explanatory variables are given, we propose an approach to obtain the forecast value and the confidence interval of the response variables. At last, a numerical example of the multivariate uncertain regression model is showed.

A new uncertain regression model and its application

Uncertain linear regression (ULR) model based on symmetric triangular uncertain set has been studied early. This paper extends the symmetric triangular uncertain coefficients to asymmetric triangular uncertain coefficients and builds two methods for estimating the parameters of ULR model. Our aim is to minimize the differences of the uncertain membership functions between the observed and estimated values. Firstly, we propose a linear programming method, whose principle is to minimize the sum of the absolute values of the differences between left width and right width of two triangular uncertain sets for each index i. Secondly, we develop a new nonlinear programming method by maximizing the overlaps of acreage of the estimated and real triangular uncertain sets in a particular \(h_i\)-cut. Then, a criterion is established to evaluate the performance of the proposed approaches. Finally, we use an example based on industrial water demand data of China to illustrate our proposed approaches which are reasonable and compare the explanatory power of the ULR model and traditional linear regression (TLR) model using the presented evaluation criteria, which shows that the performance of the ULR model is obviously better than the TLR model.

Residual and confidence interval for uncertain regression model with imprecise observations

Residual and confidence interval for uncertain regression model with imprecise observations

An uncertain regression model

PurposeThe purpose of this paper is to propose an uncertain regression model with an intrinsic error structure facilitated by an uncertain canonical process.Design/methodology/approachThis model is suitable for dealing with expert's knowledge ranging from small to medium size data of impreciseness. In order to have a rigorous mathematical treatment on the new regression model, this paper establishes a series of new uncertainty concepts sequentially, such as uncertainty joint multivariate distribution, the uncertainty distribution of uncertainty product variables and uncertain covariance and correlation based on the axiomatic uncertainty theoretical foundation. Two examples are given for illustrating a small data regression analysis.FindingsThe uncertain regression model is formulated and the estimation of the model coefficients is developed.Practical implicationsThe paper is devoted to a regression model to handle a small amount of data with mathematical rigor.Originality/valueThe theory and the methodology of the uncertain canonical process regression is proposed for the first time. It addresses the practical challenges of small data size modelling.