Unimodal Regression via Prefix Isotonic Regression

Quentin F. Stout
Computer Science and Engineering
University of Michigan


Abstract: This paper gives algorithms for determining univariate unimodal regressions, that is, for determining the optimal regression which is increasing and then decreasing (more technically, nondecreasing and then nonincreasing). Such regressions arise in a wide variety of applications. They are shape-constrained nonparametric regressions, closely related to isotonic regression (i.e., regression where the regression function is nondecreasing). Unimodal orderings are occasionally called umbrella orderings.

For unimodal regression on n weighted points given in increasing abscissa order, the time required is

For weighted points, L isotonic and unimodal regression can be accomplished in O(n) time using the significantly more complex approach in L Isotonic Regression for Linear, Multidimensional, and Tree Orders. A less complicated algorithm taking O(n log n) time appeared earlier in Weighted L Isotonic Regression, along with several extensions.

Note that the optimal isotonic or unimodal regressions are not unique for L1 and L. E.g., for unweighted data 4, 0, 5, an optimal L1 isotonic regression is of the form x, x, 5, where x can be any value in [0,4], while an optimal L regression is of the form 2, 2, y, where y can be any value in [3,7]. Isotonic regression for L2 is unique, but unimodal in general isn't, as is shown by the unweighted data 2, 0, 2, which has optimal unimodal regressions of 1, 1, 2 or 2, 1, 1.

Previous algorithms were solely for the L2 metric and required Ω(n2) time. All previous algorithms used multiple calls to isotonic regression, and our major contribution is to organize these into a prefix isotonic regression, determining the regression on all initial segments. The prefix approach reduces the total time required by utilizing the solution for one initial segment to solve the next. The prefix algorithm is also useful when one is performing isotonic regression for an on-going time series, since it allows one to add new data with only a constant amortized amount of time per new data point (for the L2 metric).

Algorithms are also provided for pointwise evaluations throughout the history of the data. That is, after the prefix isotonic regressions have been constructed, then for the isotonic regression on the first m values, 0 < m ≤ n, one can determine

in O(log m) time.

Note that the prefix approach can be applied to several other approximations when only 2 pieces are desired. For example, piecewise monotonic (i.e., each piece is either monotonic increasing or monotonic decreasing), piecewise constant (also known as step functions or histogramming), piecewise linear, and piecewise quadratic.

Keywords: unimodal regression, isotonic regression, median regression, umbrella ordering, minimax, least squares, piecewise monotonic, prefix scan, pool adjacent violators (PAV)

Complete paper. This paper appears in Computational Statistics and Data Analysis 53 (2008), p. 289-297. doi:10.1016/j.csda.2008.08.005 It was submitted in 2003, and simultaneously put on the web, but a series of absurd delays occured (one of them due to me) so that the journal version appeared years later. A preliminary version appeared in ``Optimal algorithms for unimodal regression'', Computing Science and Statistics 32 (2000). Here is a link to it, but the journal version is much better and is the one that should be cited.


Related work: My interest in this problem arose from its use in adaptive sampling designs for dose-response optimization in phase I/II clinical trial.

Here is information on the fastest isotonic regression algorithms known to date. Here are papers on L1 isotonic regression (also known as isotonic median regression), L isotonic regression for arbitrary dags (also known as minimax, uniform, or Chebyshev isotonic regression), isotonic regression for points in multidimensional space, L isotonic regression for linear, multidimensional, and tree orders, and isotonic regression with a fixed number of level sets. The paper on L1 isotonic regression also includes results for Lp, 1 < p < ∞.

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