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.
Highlights
Isotonic regression is a popular statistical method based on partial order structures, which has a long history in statistics (Ayer et al 1955, Brunk 1955, van Eeden 1956)
We study the problem of estimating piecewise monotone vectors, which can be regarded as a generalization of isotonic regression that allows orderviolating changepoints
Our goal in this paper is to show that the nearly-isotonic regression can adapt to piecewise monotone vectors
Summary
Isotonic regression is a popular statistical method based on partial order structures, which has a long history in statistics (Ayer et al 1955, Brunk 1955, van Eeden 1956). Suppose that θ∗ ∈ Rn is a monotone vector satisfying θ1∗ ≤ θ2∗ ≤ · · · ≤ θn∗ , and y is a noisy observation of θ∗. The goal of the isotonic regression is to find a least-square fit under the monotone constraint: minimize y − θ 2 subject to θ1 ≤ θ2 ≤ · · · ≤ θn. The isotonic regression is the least squares estimator θ = θKn↑ over a closed convex cone Kn↑ := {θ ∈ Rn : θ1 ≤ θ2 ≤ · · · ≤ θn}. We study the problem of estimating piecewise monotone vectors, which can be regarded as a generalization of isotonic regression that allows orderviolating changepoints.
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