Density Level Sets: Asymptotics, Inference, and Visualization

“…In this section, we introduce the approaches by Mammen and Polonik [14] and Chen et al [15]. These construct confidence regions for estimated level sets.…”

Section: Confidence Regions For Multivariate Quantilesmentioning

confidence: 99%

“…First, we extend two recently developed approaches for construction of level set confidence regions by Mammen and Polonik [14] and Chen et al [15] to the estimation problem at hand. Note that the multivariate quantiles considered here are level sets at specific levels of the copula.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Confidence Regions for Multivariate Quantiles

Coblenz

Dyckerhoff

Grothe

2018

Water

View full text Add to dashboard Cite

Multivariate quantiles are of increasing importance in applications of hydrology. This calls for reliable methods to evaluate the precision of the estimated quantile sets. Therefore, we focus on two recently developed approaches to estimate confidence regions for level sets and extend them to provide confidence regions for multivariate quantiles based on copulas. In a simulation study, we check coverage probabilities of the employed approaches. In particular, we focus on small sample sizes. One approach shows reasonable coverage probabilities and the second one obtains mixed results. Not only the bounded copula domain but also the additional estimation of the quantile level pose some problems. A small sample application gives further insight into the employed techniques.

show abstract

“…This approach assumes the data points to be drawn from some unknown probability distribution and defines the clusters as the basins of attraction of the maxima of the density, requiring a preliminary density estimation phase [7,5,10,11,13,15]. The theoretical analysis of this clustering framework has drawn increasing attention recently, see [6,3,9,8,2]. However, this (hard) clustering method provides a fairly limited knowledge on the structure of the data: while the partition into clusters is well understood, the interplay between clusters (respective locations, proximity relations, interactions) remains unknown.…”

Section: Introductionmentioning

confidence: 99%

A fuzzy clustering algorithm for the mode-seeking framework

Bonis

Oudot

2018

Pattern Recognition Letters

View full text Add to dashboard Cite

In this paper, we propose a new fuzzy clustering algorithm based on the modeseeking framework. Given a dataset in R d , we define regions of high density that we call cluster cores. We then consider a random walk on a neighborhood graph built on top of our data points which is designed to be attracted by high density regions. The strength of this attraction is controlled by a temperature parameter β > 0. The membership of a point to a given cluster is then the probability for the random walk to hit the corresponding cluster core before any other. While many properties of random walks (such as hitting times, commute distances, etc. . . ) have been shown to enventually encode purely local information when the number of data points grows, we show that the regularization introduced by the use of cluster cores solves this issue. Empirically, we show how the choice of β influences the behavior of our algorithm: for small values of β the result is close to hard modeseeking whereas when β is close to 1 the result is similar to the output of a (fuzzy) spectral clustering. Finally, we demonstrate the scalability of our approach by providing the fuzzy clustering of a protein configuration dataset containing a million data points in 30 dimensions.

show abstract

Density Level Sets: Asymptotics, Inference, and Visualization

Cited by 74 publications

References 66 publications

Topological Data Analysis

Topological Data Analysis

Confidence Regions for Multivariate Quantiles

A fuzzy clustering algorithm for the mode-seeking framework

Contact Info

Product

Resources

About