Isotonic Regression via Partitioning

Algorithms are given for determining weighted isotonic regressions satisfying order constraints specified via a directed acyclic graph (DAG). For the L1 metric a partitioning approach is used which exploits the fact that L1 regression values can always be chosen to be data values. Extending this approach, algorithms for binary-valued L1 isotonic regression are used to find Lp isotonic regressions for 1<p<∞. Algorithms are given for trees, 2-dimensional and multidimensional orderings, and arbitrary DAGs. Algorithms are also given for Lp isotonic regression with constrained data and weight values, L1 regression with unweighted data, and L1 regression for DAGs where there are multiple data values at the vertices.