Global Climate Models (GCMs) are numerical models that simulate complex physical processes within the Earth's climate system and are essential for understanding and predicting climate change. However, GCMs suffer from systemic biases due to simplifications made to the underlying physical processes. GCM output therefore needs to be bias corrected before it can be used for future climate projections. Most common bias correction methods, however, cannot preserve spatial, temporal, or inter-variable dependencies. We propose a new semi-parametric conditional density estimation (SPCDE) for density correction of the joint distribution of daily precipitation and maximum temperature data obtained from gridded GCM spatial fields. The Vecchia approximation is employed to preserve dependencies in the observed field during the density correction process, which is carried out using semi-parametric quantile regression. The ability to calibrate joint distributions of GCM projections has potential advantages not only in estimating extremes, but also in better estimating compound hazards, like heat waves and drought, under potential climate change. Illustration on historical data from 1951-2014 over two 5x5 spatial grids in the US indicate that SPCDE can preserve key marginal and joint distribution properties of precipitation and maximum temperature, and predictions obtained using SPCDE are better calibrated compared to predictions using asynchronous quantile mapping and canonical correlation analysis, two commonly used bias correction approaches.

Traditional methods which the features form cluster structures, as can be seen in often rely on R.I.P or its relaxed variants. However, many machine learning [Lehiste, 1976] and computer vision in real applications, features are often correlated problems [Lan et al., 2013; Lowe, 2004]. Due to the fact that to each other, which makes these assumptions many features extractors are similar to each others and they too strong to be useful. In this paper, we reflect the characteristics of the same image, vision features study the sparse recovery problem in which the feature are often correlated and have cluster structures. This correlation matrix is strictly non-R.I.P.. We prove that is even stronger in those systems that have thousands when features exhibit cluster structures, which often to millions of features [Lan et al., 2013; Gan et al., 2015a; happens in real applications, we are able to recover 2015b].