Isotonic Regression via Partitioning

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