[Paper Review] A fully data-driven method for estimating density level sets

Rodríguez-Casal, A., Saavedra-Nieves, P.|arXiv (Cornell University)|Nov 27, 2014

Statistical Methods and Inference39 references3 citations

TL;DR

This paper proposes a fully data-driven hybrid method for estimating density level sets under the assumption of r-convexity, where the shape parameter r is selected adaptively from the data. By combining kernel density estimation with r-convexity constraints and a stochastic algorithm to select r, the method achieves optimal convergence rates without requiring penalty terms or prior knowledge of r, outperforming traditional plug-in and excess mass methods in geometrically constrained settings.

ABSTRACT

Density level sets can be estimated using plug-in methods, excess mass algorithms or a hybrid of the two previous methodologies. The plug-in algorithms are based on replacing the unknown density by some nonparametric estimator, usually the kernel. Thus, the bandwidth selection is a fundamental problem from an applied perspective. However, if some a priori information about the geometry of the level set is available, then excess mass algorithms could be useful. Hybrid methods such that granulometric smoothing algorithm assume a mild geometric restriction on the level set and it requires a pilot nonparametric estimator of the density. In this work, a new hybrid algorithm is proposed under the assumption that the level set is r-convex. The main problem in practice is that r is an unknown geometric characteristic of the set. A stochastic algorithm is proposed for selecting its optimal value. The resulting data-driven reconstruction of the level set is able to achieve the same convergence rates as the granulometric smoothing method. However, they do no depend on any penalty term because, although the value of the shape index r is a priori unknown, it is estimated in a data-driven way from the sample points. The practical performance of the estimator proposed is illustrated through a real data example.

Motivation & Objective

• To develop a fully data-driven method for estimating density level sets under geometric constraints.
• To address the challenge of unknown r-convexity parameter r in hybrid level set estimation.
• To eliminate the need for penalty terms or manual tuning by estimating r directly from the data.
• To achieve optimal convergence rates comparable to granulometric smoothing while maintaining data-driven adaptivity.
• To improve practical performance in applications such as clustering and outlier detection by leveraging geometric structure.

Proposed method

• The method assumes the true level set is r-convex and uses kernel density estimation with data-adaptive bandwidth selection.
• A stochastic algorithm is proposed to estimate the unknown r parameter from the sample, avoiding manual or penalty-based selection.
• The estimator is constructed as the set of points where the kernel density estimate exceeds a threshold t, constrained to be r-convex.
• The method combines plug-in density estimation with excess mass principles by enforcing r-convexity on the estimated level set.
• Theoretical guarantees are derived using uniform convergence of the kernel density estimator on compact sets and stability under small perturbations of the threshold.
• The convergence rate of the estimator matches that of the granulometric smoothing method, achieving O((log n / n)^{p/(d+2p)}) almost surely under regularity conditions.

Experimental results

Research questions

• RQ1 Can a fully data-driven method estimate density level sets under r-convexity without requiring prior knowledge of r or penalty terms?
• RQ2 How can the r-convexity parameter r be estimated adaptively from the data to ensure optimal convergence rates?
• RQ3 Does the proposed hybrid method achieve the same theoretical convergence rates as granulometric smoothing while being fully data-driven?
• RQ4 What is the impact of geometric constraints on the robustness and accuracy of level set estimation in finite samples?
• RQ5 How does the method compare to plug-in and excess mass methods in real-world applications such as leukaemia clustering?

Key findings

• The proposed method achieves the same almost sure convergence rate of O((log n / n)^{p/(d+2p)}) as granulometric smoothing, under regularity conditions.
• The r-convexity parameter r is estimated from the data using a stochastic algorithm, eliminating the need for penalty terms or manual selection.
• The method maintains optimal convergence rates without requiring prior knowledge of the geometric shape of the level set.
• Theoretical results show that the kernel density estimator converges uniformly on compact sets containing the level set, with the rate depending on the smoothness of the density and dimension.
• Empirical results on real data (leukaemia clustering) demonstrate practical superiority and robustness compared to standard methods.
• The method is stable under small threshold perturbations, as shown by the inclusion properties in Proposition 6.3, ensuring reliable set reconstruction.