In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

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A tutorial on spectral clustering

Organizing data into sensible groupings is one of the most fundamental modes of understanding and learning. As an example, a common scheme of scientific classification puts organisms into a system of ranked taxa: domain, kingdom, phylum, class, etc. Cluster analysis is the formal study of methods and algorithms for grouping, or clustering, objects according to measured or perceived intrinsic characteristics or similarity. Cluster analysis does not use category labels that tag objects with prior identifiers, i.e., class labels. The absence of category information distinguishes data clustering (unsupervised learning) from classification or discriminant analysis (supervised learning). The aim of clustering is to find structure in data and is therefore exploratory in nature. Clustering has a long and rich history in a variety of scientific fields. One of the most popular and simple clustering algorithms, K-means, was first published in 1955. In spite of the fact that K-means was proposed over 50 years ago and thousands of clustering algorithms have been published since then, K-means is still widely used. This speaks to the difficulty in designing a general purpose clustering algorithm and the ill-posed problem of clustering. We provide a brief overview of clustering, summarize well known clustering methods, discuss the major challenges and key issues in designing clustering algorithms, and point out some of the emerging and useful research directions, including semi-supervised clustering, ensemble clustering, simultaneous feature selection during data clustering, and large scale data clustering.

Data clustering: 50 years beyond K-means

Gut microbial communities represent one source of human genetic and metabolic diversity. To examine how gut microbiomes differ among human populations, here we characterize bacterial species in fecal samples from 531 individuals, plus the gene content of 110 of them. The cohort encompassed healthy children and adults from the Amazonas of Venezuela, rural Malawi and US metropolitan areas and included mono- and dizygotic twins. Shared features of the functional maturation of the gut microbiome were identified during the first three years of life in all three populations, including age-associated changes in the genes involved in vitamin biosynthesis and metabolism. Pronounced differences in bacterial assemblages and functional gene repertoires were noted between US residents and those in the other two countries. These distinctive features are evident in early infancy as well as adulthood. Our findings underscore the need to consider the microbiome when evaluating human development, nutritional needs, physiological variations and the impact of westernization.

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Human gut microbiome viewed across age and geography

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Modern Applied Statistics With S

The power of feedback.

We propose a method (the ‘gap statistic’) for estimating the number of clusters (groups) in a set of data. The technique uses the output of any clustering algorithm (e.g. K-means or hierarchical), comparing the change in within-cluster dispersion with that expected under an appropriate reference null distribution. Some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature.

Estimating the number of clusters in a data set via the gap statistic

This article proposes a new quantity for assessing the number of groups or clusters in a dataset. The key idea is to view clustering as a supervised classification problem, in which we must also estimate the “true” class labels. The resulting “prediction strength” measure assesses how many groups can be predicted from the data, and how well. In the process, we develop novel notions of bias and variance for unlabeled data. Prediction strength performs well in simulation studies, and we apply it to clusters of breast cancer samples from a DNA microarray study. Finally, some consistency properties of the method are established.

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Cluster Validation by Prediction Strength

We consider the least angle regression and forward stagewise algorithms for solving penalized least squares regression problems. In Efron, Hastie, Johnstone & Tibshirani (2004) it is proved that the least angle regression algorithm, with a small modification, solves the lasso regression problem. Here we give an analogous result for incremental forward stagewise regression, showing that it solves a version of the lasso problem that enforces monotonicity. One consequence of this is as follows: while lasso makes optimal progress in terms of reducing the residual sum-of-squares per unit increase in $L_1$-norm of the coefficient $\beta$, forward stage-wise is optimal per unit $L_1$ arc-length traveled along the coefficient path. We also study a condition under which the coefficient paths of the lasso are monotone, and hence the different algorithms coincide. Finally, we compare the lasso and forward stagewise procedures in a simulation study involving a large number of correlated predictors.

Forward stagewise regression and the monotone lasso

We consider the least angle regression and forward stagewise algorithms for solving penalized least squares regression problems. In Efron et al. (2004) it is proven that the least angle regression algorithm, with a small modication, solves the lasso (L1 constrained) regression problem. Here we give an analogous result for incremental forward stagewise regression, showing that it ts a monotone version of the lasso. We also study a condition under which the coecien t paths of the lasso are monotone, and hence the dieren t algorithms all coincide. Finally, we compare the lasso and forward stagewise procedures in a simulation study involving a large number of correlated predictors.

/pdf/forward-stagewise-regression-and-the-monotone-lasso-3t7ag9dwaa.pdf

Forward Stagewise Regression and the Monotone Lasso

We study the convergence of the predictive surface of regression trees and forests. To support our analysis we introduce a notion of adaptive concentration for regression trees. This approach breaks tree training into a model selection phase in which we pick the tree splits, followed by a model fitting phase where we find the best regression model consistent with these splits. We then show that the fitted regression tree concentrates around the optimal predictor with the same splits: as d and n get large, the discrepancy is with high probability bounded on the order of sqrt(log(d) log(n)/k) uniformly over the whole regression surface, where d is the dimension of the feature space, n is the number of training examples, and k is the minimum leaf size for each tree. We also provide rate-matching lower bounds for this adaptive concentration statement. From a practical perspective, our result enables us to prove consistency results for adaptively grown forests in high dimensions, and to carry out valid post-selection inference in the sense of Berk et al. [2013] for subgroups defined by tree leaves.

Guenther Walther

Papers

Estimating the number of clusters in a data set via the gap statistic

Cluster Validation by Prediction Strength

Forward stagewise regression and the monotone lasso

Forward Stagewise Regression and the Monotone Lasso

Adaptive Concentration of Regression Trees, with Application to Random Forests