# Ckmeans.1d.dp v4.3.0

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## Optimal, Fast, and Reproducible Univariate Clustering

Fast, optimal, and reproducible weighted univariate clustering by dynamic programming. Four types of problem including univariate k-means, k-median, k-segments, and multi-channel weighted k-means are solved with guaranteed optimality and reproducibility. The core algorithm minimizes the sum of (weighted) within-cluster distances using respective metrics. Its advantage over heuristic clustering in efficiency and accuracy is pronounced at a large number of clusters k. Weighted k-means can also process time series to perform peak calling. Multi-channel weighted k-means groups multiple univariate signals into k clusters. An auxiliary function generates histograms that are adaptive to patterns in data. This package provides a powerful set of tools for univariate data analysis with guaranteed optimality, efficiency, and reproducibility.

The Ckmeans.1d.dp package provides fast, optimal, and reproducible univariate clustering by dynamic programming. It is practical to cluster millions of sample points with optional weights in seconds on a typical desktop computer using a single processor core.

Four types of problem including univariate $k$-means, $k$-median, $k$-segments, and multi-channel weighted $k$-means are solved with guaranteed optimality and reproducibility. The core algorithm minimizes the (weighted) sum of within-cluster distances using respective metrics. Its advantage over heuristic clustering in efficiency and accuracy is increasingly pronounced as the number of clusters $k$ increases. Weighted $k$-means can also optimally segment time series to perform peak calling. An auxiliary function generates histograms that are adaptive to patterns in data. This package provides a powerful set of tools for univariate data analysis with guaranteed optimality, efficiency, and reproducibility.

The Ckmeans.1d.dp algorithm clusters (weighted) univariate data given by a numeric vector $x$ into $k$ groups by dynamic programming (Wang and Song, 2011). It guarantees the optimality of clustering---the total of within-cluster sums of squares is always the minimum given the number of clusters $k$. In contrast, heuristic univariate clustering algorithms may be non-optimal or inconsistent from run to run. As unequal non-negative weights are supported for each point, the algorithm can also segment a time course using the time points as input and the values at each time point as weight. Utilizing the optimal clusters, a function can generate histograms adaptive to patterns in data.

Apart from the time for sorting $x$, the default weighted univariate clustering algorithm takes a runtime of $O(kn)$, linear in both sample size $n$ and the number of clusters $k$, using a new divide-and-conquer strategy integrating a previous theoretical result on matrix search (Aggarwal et al., 1987) and a novel in-place search space reduction method. The space complexity is $O(kn)$. This method is numerically stable.

This package provides a powerful alternative to heuristic clustering and also new functionality for weighted clustering, segmentation, and peak calling with guaranteed optimality.

## Functions in Ckmeans.1d.dp

 Name Description Univariate Clustering Optimal (Weighted) Univariate Clustering plot.Ckmeans.1d.dp Plot Optimal Univariate Clustering Results MultiChannel.WUC Optimal Multi-channel Weighted Univariate Clustering plotBIC Plot Bayesian Information Criterion as a Function of Number of Clusters Univariate Segmentation Optimal Univariate Segmentation Ckmeans.1d.dp-package Optimal, Fast, and Reproducible Univariate Clustering plot.Cksegs.1d.dp Plot Optimal Univariate Segmentation Results ahist Adaptive Histograms print.Ckmeans.1d.dp Print Optimal Univariate Clustering Results print.Cksegs.1d.dp Print Optimal Univariate Segmentation Results No Results!