Evaluation of linear regression techniques for atmospheric applications: the importance of appropriate weighting

Linear regression techniques are widely used in atmospheric science, but they are often improperly applied due to lack of consideration or inappropriate handling of measurement uncertainty. In this work, numerical experiments are performed to evaluate the performance of five linear regression techniques, significantly extending previous works by Chu and Saylor. The five techniques are ordinary least squares (OLS), Deming regression (DR), orthogonal distance regression (ODR), weighted ODR (WODR), and York regression (YR). We first introduce a new data generation scheme that employs the Mersenne twister (MT) pseudorandom number generator. The numerical simulations are also improved by (a) refining the parameterization of nonlinear measurement uncertainties, (b) inclusion of a linear measurement uncertainty, and (c) inclusion of WODR for comparison. Results show that DR, WODR and YR produce an accurate slope, but the intercept by WODR and YR is overestimated and the degree of bias is more pronounced with a low <i>R</i>² <i>X</i><i>Y</i> dataset. The importance of a properly weighting parameter <i>λ</i> in DR is investigated by sensitivity tests, and it is found that an improper <i>λ</i> in DR can lead to a bias in both the slope and intercept estimation. Because the <i>λ</i> calculation depends on the actual form of the measurement error, it is essential to determine the exact form of measurement error in the <i>X</i><i>Y</i> data during the measurement stage. If a priori error in one of the variables is unknown, or the measurement error described cannot be trusted, DR, WODR and YR can provide the least biases in slope and intercept among all tested regression techniques. For these reasons, DR, WODR and YR are recommended for atmospheric studies when both <i>X</i> and <i>Y</i> data have measurement errors. An Igor Pro-based program (Scatter Plot) was developed to facilitate the implementation of error-in-variables regressions