Morse–Smale Regression

Journal Article · Thu Jan 19 04:00:00 EST 2012 · Journal of Computational and Graphical Statistics

DOI:https://doi.org/10.1080/10618600.2012.657132· OSTI ID:1226951

Gerber, Samuel ^[1]; Rubel, Oliver ^[2]; Bremer, Peer -Timo ^[3]; Pascucci, Valerio ^[1]; Whitaker, Ross T. ^[1]

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1. Univ. of Utah, Salt Lake City, UT (United States)
2. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
3. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

This paper introduces a novel partition-based regression approach that incorporates topological information. Partition-based regression typically introduces a quality-of-fit-driven decomposition of the domain. The emphasis in this work is on a topologically meaningful segmentation. Thus, the proposed regression approach is based on a segmentation induced by a discrete approximation of the Morse–Smale complex. This yields a segmentation with partitions corresponding to regions of the function with a single minimum and maximum that are often well approximated by a linear model. This approach yields regression models that are amenable to interpretation and have good predictive capacity. Typically, regression estimates are quantified by their geometrical accuracy. For the proposed regression, an important aspect is the quality of the segmentation itself. Thus, this article introduces a new criterion that measures the topological accuracy of the estimate. The topological accuracy provides a complementary measure to the classical geometrical error measures and is very sensitive to overfitting. The Morse–Smale regression is compared to state-of-the-art approaches in terms of geometry and topology and yields comparable or improved fits in many cases. Finally, a detailed study on climate-simulation data demonstrates the application of the Morse–Smale regression. Supplementary Materials are available online and contain an implementation of the proposed approach in the R package msr, an analysis and simulations on the stability of the Morse–Smale complex approximation, and additional tables for the climate-simulation study.

• Research Organization: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
• Sponsoring Organization: USDOE
• DOE Contract Number: AC52-07NA27344
• OSTI ID: 1226951
• Report Number(s): LLNL-JRNL--519551
• Journal Information: Journal of Computational and Graphical Statistics, Journal Name: Journal of Computational and Graphical Statistics Journal Issue: 1 Vol. 22; ISSN 1061-8600
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Morse–Smale complex
adaptive
nonparametric
partition
segmentation