Piecewise Linear Regression for Massive Data through Dataspheres

We propose enhancing the tting of linear regression models to massive multidimensional data by partitioning the data using a DataSphere (proposed in our previous work) and tting piecewise linear regression models to each class in the representation. Nonlinear models typically involve several iterations through the data and require the knowledge of every data point. Linear regression models, on the other hand, can be computed just from second order summaries. Hence piecewise linear models are often used to capture nonlinear relationships using rectangular binning schemes. A disadvantage of rectangular grids is that the number of classes in the partition grows exponentially with the number of dimensions. We propose using the DataSphere partition to divide the data space for tting piecewise linear regression models, where the number of classes grows linearly in the number of dimensions. The partition is based on distance from a center (distance layers) and direction of maximumvariance (directional pyramids). The method exploits the fact that linear regression coeecients can be computed from second order moments that are generated during the creation of the DataSphere representation. An additional advantage is exibility, since the regression models for a coarser DataSphere representation (fewer distance layers for instance) can be recomputed from the summaries without going back to the raw data.