# Estimating Piecewise Monotone Signals

## TL;DR

This paper analyzes the nearly-isotonic regression for estimating piecewise monotone signals, providing risk bounds and an algorithm for general graphs, showing it performs nearly as well as an oracle estimator.

## Contribution

It derives adaptive risk bounds for nearly-isotonic regression and introduces a versatile algorithm applicable to weighted graphs.

## Key findings

- Risk bounds are adaptive to piecewise monotone signals.
- Nearly-isotonic regression performs close to an oracle estimator.
- The proposed algorithm works on general weighted graphs.

## Abstract

We study the problem of estimating piecewise monotone vectors. This problem can be seen as a generalization of the isotonic regression that allows a small number of order-violating changepoints. We focus mainly on the performance of the nearly-isotonic regression proposed by Tibshirani et al. (2011). We derive risk bounds for the nearly-isotonic regression estimators that are adaptive to piecewise monotone signals. The estimator achieves a near minimax convergence rate over certain classes of piecewise monotone signals under a weak assumption. Furthermore, we present an algorithm that can be applied to the nearly-isotonic type estimators on general weighted graphs. The simulation results suggest that the nearly-isotonic regression performs as well as the ideal estimator that knows the true positions of changepoints.