Robust Regression with Data-Dependent Regularization Parameters and Autoregressive Temporal Correlations

We introduce robust procedures for analyzing water quality data collected over time. One challenging task in analyzing such data is how to achieve robustness in presence of outliers while maintaining high estimation efficiency so that we can draw valid conclusions and provide useful advices in water management. The robust approach requires specification of a loss function such as the Huber, Tukey’s bisquare and the exponential loss function, and an associated tuning parameter determining the extent of robustness needed. High robustness is at the cost of efficiency loss in parameter loss. To this end, we propose a data-driven method which leads to more efficient parameter estimation. This data-dependent approach allows us to choose a regularization (tuning) parameter that depends on the proportion of “outliers” in the data so that estimation efficiency is maximized. We illustrate the proposed methods using a study on ammonium nitrogen concentrations from two sites in the Huaihe River in China, where the interest is in quantifying the trend in the most recent years while accounting for possible temporal correlations and “irregular” observations in earlier years.