A new instance-based learning method is presented for regression problems with high-dimensional data. As an instance-based approach, the conventional method, KNN, is very popular for classification. Although KNN performs well on classification tasks, it does not perform as well on regression problems. We have developed a new instance-based method, called Regression by Partitioning Feature Projections (RPFP) which is designed to meet the requirement for a lazy method that achieves high levels of accuracy on regression problems. RPFP gives better performance than well-known eager approaches found in machine learning and statistics such as MARS, rule-based regression, and regression tree induction systems. The most important property of RPFP is that it is a projection-based approach that can handle interactions. We show that it outperforms existing eager or lazy approaches on many domains when there are many missing values in the training data.