Variable selection in uncertain regression analysis with imprecise observations

Abstract

Variable selection is crucial in order to better investigate relationships between variables in regression analysis. However, sometimes data are collected in an imprecise way and can not be described by random variables. As a result, classical variable selection methods are invalid. Characterizing these imprecise observations as uncertain variables, this paper presents the uncertain lasso estimate and the de-biased uncertain lasso estimate to select variables and estimate unknown parameters, respectively. Moreover, a way to choose the tuning parameter using cross-validation is suggested. Finally, numerical examples are documented to show our methods in detail.

Appendix

Proof of Corollary 1

According to Definitions 1 and 2, the uncertain lasso estimate and de-biased uncertain lasso estimate for linear regression model (11) solve minimization problems (12) and (13), respectively. Since

$$\begin{aligned} \beta _{0}+\sum _{j=1}^{p} \beta _{j}\tilde{x}_{ji} \end{aligned}$$

is increasing with respect to $\tilde{x}_{ji}$ when $\beta _{j} > 0$ and decreasing with respect to $\tilde{x}_{ji}$ when $\beta _{j} \le 0$ for each i $(i=1,2,\ldots , n)$, the corollary follows from Theorems 1 and 2 immediately. $\square$

Proof of Corollary 2

According to Definitions 1 and 2, the uncertain lasso estimate and de-biased uncertain lasso estimate for the regression model (14) solve minimization problems (15) and (16), respectively. It follows from the operational law of uncertain variable (p. 55 of Liu 2015) that inverse uncertainty distributions of uncertain variables $\ln \tilde{y}_{i}$ are $\ln F_{i}^{-1}(\alpha )$, $i=1,2,\ldots ,n$, respectively. Since

$$\begin{aligned} \beta _{0}+ \sum _{j=1}^{p} \beta _{j} \tilde{x}_{ji} \end{aligned}$$

is increasing with respect to $\tilde{x}_{ji}$ when $\beta _{j} > 0$ and decreasing with respect to $\tilde{x}_{ji}$ when $\beta _{j} \le 0$ for each i $(i=1,2,\ldots , n)$, the corollary follows from Theorems 1 and 2 immediately.

$\square$