Morse-Smale Regression.

Area of Science:

• Computational topology
• Statistical modeling
• Data analysis

Background:

• Partition-based regression typically uses quality-of-fit for domain decomposition.
• Existing methods often lack explicit consideration of the underlying topological structure.
• Topological features can reveal important functional characteristics of data.

Purpose of the Study:

• Introduce a novel partition-based regression approach integrating topological information.
• Develop a method for topologically meaningful data segmentation.
• Provide a new metric for assessing the topological accuracy of regression estimates.

Main Methods:

• Employ a discrete approximation of the Morse-Smale complex for domain segmentation.
• Generate regression models based on partitions with single minima and maxima.
• Introduce a topological accuracy criterion to complement geometrical error measures.

Main Results:

• The Morse-Smale regression yields interpretable models with good predictive capacity.
• The topological accuracy measure is sensitive to over-fitting.
• Comparisons show Morse-Smale regression achieves comparable or improved fits over state-of-the-art methods.

Conclusions:

• The proposed Morse-Smale regression effectively integrates topological insights into data analysis.
• This approach offers a valuable tool for understanding complex datasets, particularly in scientific simulations.
• The method provides enhanced interpretability and predictive performance, with a new metric for topological accuracy.