RDELA—a Delaunay-triangulation-based, location and covariance estimator with high breakdown point

Abstract

We propose an approach that utilizes the Delaunay triangulation to identify a robust/outlier-free subsample. Given that the data structure of the non-outlying points is convex (e.g. of elliptical shape), this subsample can then be used to give a robust estimation of location and scatter (by applying the classical mean and covariance). The estimators derived from our approach are shown to have a high breakdown point. In addition, we provide a diagnostic plot to expand the initial subset in a data-driven way, further increasing the estimators’ efficiency.

Appendix: Breakdown points of RDELA based estimators

Theorem 1

Let X be a collection of n≥d+1 points x ₁,…,x _n in dimension d, x _i ∈ℝ^d, i=1,…,n, in general position, and let T(X)={t _k , k=1,…,l} be the corresponding Delaunay triangulation (where t _k denotes a d-simplex with $\mathbf{t}_{k}=\{\mathbf{x}_{i_{1}},\ldots,\mathbf{x}_{i_{d+1}}\}\subset \mathbf{X}$), and let μ _n and Σ _n be the RDELA estimators of location and covariance (as described in Sects. 2.2 and 2.3). Then the breakdown point ε ^∗ (defined as the smallest fraction of outliers that can take the estimate over all bounds) of these estimators is ε ^∗(μ _n ,X)=⌊(n+1)/2⌋/n and ε ^∗(Σ _n ,X)≥⌊(n−d+1)/2⌋/n respectively.

Proof

We first show that ε ^∗(μ _n ,X) is at least ⌊(n+1)/2⌋/n.

To let the location estimate break down at least one point in the chosen subset has to grow to infinity. If for some x∈X we have ∥x∥→∞, then ∥x−y∥→∞ for all y∈X. For a subset of X consisting of d+1 points and its corresponding triangulation object t _k with x,y∈t _k it follows (due to the circumcircle property) that the radius of t _k grows also to infinity r(t _k )→∞.

Suppose that X=X′∪X″, where |X′|>|X″|=m=⌊(n−1)/2⌋, and X″ will be altered in an arbitrary way to cause the procedure’s breakdown. Denote Y⊂X the subset chosen by the RDELA procedure with |Y|=|X′|=n−m. Furthermore, let $\widetilde{\mathbf {T}}_{\mathbf{X}^{\prime}}= \{\mathbf {t}'_{k}=\{\mathbf{x}_{i_{1}},\ldots,\mathbf{x}_{i_{d+1}}\},\mathbf{x}_{i_{j}}\in \mathbf{X}^{\prime} \}$ as well as $\widetilde{\mathbf {T}}_{\mathbf {Y}}= \{\mathbf {t}^{y}_{k}=\{\mathbf{x}_{i_{1}},\ldots,\mathbf{x}_{i_{d+1}}\},\mathbf{x}_{i_{j}}\in \mathbf {Y}\}$ the sets of the corresponding triangulation objects with $\widetilde{\mathbf {T}}_{\mathbf{X}^{\prime}},\widetilde{\mathbf {T}}_{\mathbf {Y}}\subseteq \mathbf {T}(\mathbf{X})$. Then the radii of all triangulation objects $\mathbf {t}'_{k}$ are bounded, i.e. ∃α∈ℝ with $r(\mathbf {t}'_{k})<\alpha$.

Suppose now ∃Y′⊂Y in such a way that ∥x−y∥≥2⋅α where x∈Y′ and y∈Y∖Y′. Then it follows from the circumcircle property that for the triangulation objects $\mathbf {t}^{y'}_{k} \in\widetilde{\mathbf {T}}_{\mathbf {Y}}$ with $\mathbf{x},\mathbf {y}\in \mathbf {t}^{y'}_{k}$:

$$r\bigl(\mathbf {t}^{y'}_k\bigr) \geq\alpha. $$

But on the other hand if Y=X′, for all triangulation objects $\mathbf {t}^{y}_{k} \in\widetilde{\mathbf {T}}_{\mathbf {Y}}$ holds $r(\mathbf {t}^{y}_{k}) < \alpha$ as stated above. Hence, the algorithm described in Sect. 2.3 always terminates in X′. This proves that the breakdown point of the RDELA location estimator is at least ⌊(n+1)/2⌋/n.

The Delaunay triangulation is invariant against orthogonal transformations because its calculation is purely based on Euclidean distances which are invariant against this type of transformation. Consequently, this is also true for the estimators derived from the Delaunay triangulation.

As for orthogonal equivariant location estimators the maximum breakdown point is proved to be ⌊(n+1)/2⌋/n (Lopuhaä and Rousseeuw, 1991), this also ends the proof.

Now, we show that ε ^∗(Σ _n ,X)≥⌊(n−d+1)/2⌋/n.

Denote 0<λ ₁≤⋯≤λ _d <∞ the eigenvalues of Σ _n (Y). The covariance estimator may break down by explosion (meaning that λ _d →∞) or by implosion (if λ ₁→0). For Σ _n to explode, at least one observation y from the chosen subset Y has to grow arbitrarily, ∥y∥→∞. This case is covered by the location estimator’s proof.

For the case of implosion note that each chosen subset Y which is of size ⌊(n+d+1)/2⌋ by construction contains at least d+1 unaltered points x _i ∈X′. Because X′ is in general position each subset of X′ consisting of d+1 points is also in general position. Due to this Σ _n is positive semidefinite (and λ _i >0 ∀i). This proves that ε ^∗(Σ _n ,X) is at least ⌊(n−d+1)/2⌋/n. □

It can be assumed that the breakdown point of the covariance estimator is at most ⌊(n−d+1)/2⌋/n. However, this has not been proved yet. This is primarily due to the fact that a triangulation is not defined for degenerated point configurations, which in turn are required to let the covariance estimate implode.