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.