Abstract
Under the uncertain statistical framework by Liu [19], there is still a lack of an effective fitting method for uncertain linear models with Box-Cox transformation of response variables. For example, for the transformation parameter Λ , the uncertain least squares estimation will produce a severely low estimation result. In this paper, we propose uncertain Box-Cox regression analysis by utilizing the uncertainty theory to model the imprecise data and applying a generalized Box-Cox transformation indexed by its parameter to validate classic regression assumptions. We use rescaled least squares to estimate unknown parameters and provide an estimate for noises followed by residual analysis for these uncertain Box-Cox regression models. We also give the forecast values and confidence intervals and use a numerical example to demonstrate our methodology. Our work sets a uniformed framework for Box-Cox transformation on the uncertain regression, and extends such regression from linear to nonlinear cases, taking the Johnson-Schumacher growth model as an example.
Highlights
Regression analysis is a useful tool for studying how a response variable is impacted by other explanatory variables
We propose the generalized uncertain Box-Cox revised regression analysis to model the quantitative relationship between uncertain dependent variables after Box-Cox transformation and uncertain independent variable with imprecisely observations, which can relax the assumptions required by classic regression analysis
Since the BoxCox transformation is applicable on both a linear model and a nonlinear model, such as the Johnson-Schumacher growth model, which was first noticed by Fang et al [10], we show the estimation of the relevant noises and describe how to perform residual analysis
Summary
Regression analysis is a useful tool for studying how a response variable is impacted by other explanatory variables. Theorem 2: For imprecise observations (xj, yj), j = 1, 2, · · · , n, where xj, yj are independent uncertain variables with regular uncertainty distributions j, j, j = 1, 2, · · · , n, respectively, the RLSE of β0, β1, β2 in the uncertain Box-Cox Johnson-Schumacher growth model. Theorem 3: For imprecise observations (xj, xj2, · · · , xjq, yj), j = 1, 2, · · · , n, where xj, xj2, · · · , xjq, yj are independent uncertain variables following uncertainty distributions j1, j2, · · · , jq, j, j = 1, 2, · · · , n, respectively, the fitted uncertain Box-Cox linear regression model is q. The plot shows that the residuals are evenly distributed around zero without clear patterns, which suggests that our model fits the data well and the homoskedasticity assumption holds
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
