Segmentimin

Segmentimin is a class of methods used to partition a one-dimensional sequence into contiguous segments while minimizing a cost function. The approach underpins piecewise-constant and piecewise-linear approximation of signals, time series, or genomic data. The term reflects the combination of segmentation and minimization common to change-point analysis and signal processing.

Formally, given a sequence y1,...,yn, segmentimin seeks a partition 0 = t0 < t1 < ... < tm = n and a model for each segment. The objective is to minimize C = sum_{i=1}^m cost(y_{t_{i-1}+1 ... t_i}; model_i) + Penalty(m). Segment cost can be the sum of squared errors to a constant or a line; the penalty discourages excessive segmentation.

Computations typically rely on dynamic programming to find the exact optimum, with practical variants employing pruning, fast data structures, or approximate routines to scale to large datasets. Online versions exist to detect segment boundaries as data arrives.

Segmentimin has applications in finance for regime detection, climatology for climate-structure analysis, genomics for copy-number segmentation, and signal processing for denoising and feature extraction. In imaging, segmentimin ideas extend to two-dimensional data for piecewise-smooth image approximation.

Variants include robust costs that resist outliers, multivariate extensions for several synchronized sequences, and Bayesian formulations that place priors on the number and location of segments. The framework adapts to different model assumptions, data types, and application goals.