Pith. sign in

REVIEW 3 major objections 5 minor 39 references

$\beta$-integrated local depth and corresponding partitioned local depth representation

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proposes β-integrated local depth, a weighted integral of β-local depth that smooths fluctuations, keeps global-depth properties, and via its PILD decomposition improves depth classifiers and outlier detection.

desk verdict A genuinely new local depth construction and a useful PILD matrix, but the continuity proof has a gap and the empirical claims run ahead of the evidence. read the letter →

arxiv 2506.14108 v2 pith:XUSZYBCE submitted 2025-06-17 math.ST stat.MEstat.TH

Siyi Wang , Alexandre Leblanc , Paul D. McNicholas This is my paper · ORCID
classification math.STstat.MEstat.TH MSC 62G0562G2062H30
keywords β-integratedlocaldepthdatapartitionedPILDmatrixclassificationoutlierdetectiondepth-basedclassifier
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes β-integrated local depth (β-ILD), the weighted integral of β-local depth with respect to the locality parameter β, and claims that this smoothing makes local depth less sensitive to the choice of β while preserving the structural guarantees of global depth: consistency, T-invariance, continuity in x, and vanishing at infinity. A discrete partition of the integral yields the PILD matrix, whose entries record how much each point contributes to the local depth of every other point and whose row sums recover β-ILD; column sums are then an interpretable local-centrality score. The paper argues, and demonstrates on simulations and benchmark datasets, that classifiers and outlier detectors built on β-ILD and PILD outperform their unintegrated depth-based counterparts, especially for non-convex or multimodal distributions. A curious reader should care because this offers a principled way to remove (or reduce) the tuning-parameter burden of local depth and to turn depth values into pairwise influence scores usable by other algorithms.

What carries the argument

The load-bearing object is the integrated local depth $ILD(x|P_X,W)=\int_0^1 LD(\beta|x,P_X)\,dW(\beta)$ (Definition 2.4), with $W$ a probability measure on $(0,1]$, plus its discrete partition $PILD(z|x,X)=\sum_{i=1}^b \mathbf{1}_{\{z\in N_{\beta_i}^x\}} \frac{LD(\beta_i|x,X)}{\lceil n\beta_i\rceil} \int_{\beta_{i-1}}^{\beta_i} w(\beta)\,d\beta$ (Eq. 11). The integration smooths the local-depth curve; the bound of Proposition 2.1 (max adjacent fluctuation of ILD ≤ $\Delta/2$, where $\Delta$ is the max jump of LD) is the quantitative guarantee that this smoothing works. The partitioning then converts the scalar depth into a pairwise contribution matrix whose row sums equal ILD, so that column sums—how much a point is leaned on by others, weighted by their local depth—serve as the paper's new interpretable centrality measure.

What would settle it

Construct an absolutely continuous distribution and a quasi-concave depth satisfying Assumption 1 for which the central regions $R_\beta^x$ do not converge in Hausdorff distance as $x_n\to x$ (e.g., a depth with non-strictly-monotone or non-convex level sets), and check whether the corresponding β-LD is continuous in x; a discontinuity there would invalidate the proof of Property 2 and the paper's continuity claim for ILD.

Watch

Extended reading notes

Core claim

The central claim is that the map $\beta\mapsto LD(\beta|x,P_X)$, which can jump substantially between neighbouring $\beta$ levels, can be profitably replaced by $ILD(x|P_X,W)=\int_0^1 LD(\beta|x,P_X)\,dW(\beta)$. Proposition 2.1 shows that the adjacent fluctuation of this integrated version is at most half the maximum jump of the unintegrated local depth, so integration genuinely stabilizes the measure. The paper then shows that when β-ILD is computed in discrete steps and each point's share of the neighborhood is tracked, the resulting PILD matrix decomposes each point's local depth into pairwise contributions, with row sums exactly equal to ILD and column sums acting as a local-centrality score. On the theoretical side, β-ILD inherits consistency, T-invariance, continuity in x, centrality in the extreme-locality limit, and vanishing at infinity from the underlying depth; on the applied side, the paper exhibits classifiers and outlier scores based on these objects that improve over standard depth-based algorithms across several setups.

Load-bearing premise

The continuity of β-ILD in x is proved under the extra, unverified assumption that central regions of the symmetrized distributions converge in the Hausdorff sense as x varies, which needs strict monotonicity and convexity of those regions; if that convergence fails for some depth satisfying the stated assumptions, the continuity claim is not established.

Editorial extensions

If this is right

  • With a uniform weight over $(0,1]$, the parameter-free Full-ILD classifier outperforms the max-depth classifier in most of the paper's classification setups, and the cross-validated B-ILD and B-PILD variants achieve the best average ranks among all tested classifiers.
  • Column sums of the PILD matrix give an outlier score that beats global depth, β-LD, and β-ILD on the benchmark datasets; the PILD-based similarity matrix also improves LOF on WPBC, WDBC, and SpamBase.
  • The PILD matrix provides a point-to-point decomposition of local depth where each row sums to β-ILD, making it a depth-based counterpart of the PaLD matrix used in local community depth.
  • β-ILD retains T-invariance, consistency, and vanishing at infinity automatically when the underlying depth does, so the integration does not sacrifice the structural guarantees that make depth functions attractive.
  • Proposition 2.1's half-jump bound means the choice of locality level matters less for β-ILD than for β-LD, which is the theoretical basis for the method's improved stability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same integration-over-tuning-parameter recipe could stabilize other local depth families, such as kernelized spatial depth over bandwidth σ or lens depth over radius, with an analogous bound whenever the family is bounded and stepwise continuous in its parameter.
  • The PILD matrix is effectively a directed, weighted graph on the sample with row sums equal to a global centrality score; normalizing and symmetrizing it (as done for LOF) suggests it could seed spectral clustering, community detection, or low-dimensional embedding, none of which the paper explores.
  • A distinction the paper does not draw: a point's column sum in PILD need not align with its own ILD, so a point with low ILD can still be a strong contributor to others' local depth; testing whether such 'influential non-central' points behave differently in clustering or outlier detection would be a natural follow-up.
  • If the Hausdorff-convergence premise fails for some depth, the continuity claim would need repair rather than the whole construction, since the integral in β may still smooth out isolated discontinuities in x—an extension worth checking.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes β-integrated local depth (β-ILD), defined as ILD(x|P_X,W)=∫_0^1 LD(β|x,P_X)dW(β), where LD is the Paindaveine–Van Bever β-local depth. The authors argue that integrating over the locality parameter smooths the unstable sample β-LD function (Proposition 2.1), and that the resulting functional inherits consistency, T-invariance, continuity in x, vanishing at infinity, and an extreme-locality centrality property. They introduce a paired/partitioned matrix PILD whose row sums equal β-ILD, and they report simulation studies for classification and outlier detection that are claimed to show significant improvements over global and local depth methods.

Significance. If the theoretical claims are established, β-ILD would be a useful tool for removing the sensitivity of local depth to the locality parameter, and the PILD matrix provides a genuinely interpretable local-centrality summary. The paper has clear strengths: Definition 2.4 is simple and natural; Proposition 2.1 gives a clean, parameter-free smoothing bound; the strategy of transferring consistency and invariance properties from β-LD to β-ILD by dominated convergence is sound in outline; and the simulation setups are described in enough detail to be reproducible. However, the continuity-in-x proof rests on an unverified and partly false central-region convergence argument, and Property 4 has a normalization problem in its mean-value step. These gaps affect advertised inherited properties, so the theoretical core of the paper is not yet established in its current form.

major comments (3)
  1. [§3, Property 2 (Appendix proof)] The proof of continuity in x is not valid under the stated assumptions. Step 2 invokes Dyckerhoff [38] to assert Hausdorff convergence of the central regions R^β_{x_n} to R^β_x, but that result requires strict monotonicity and convexity of central regions, whereas Property 2 only assumes quasi-concavity plus Assumption 1. Quasi-concavity gives convex superlevel sets, not strict monotonicity, and no verification is supplied that the symmetrized mixture distributions satisfy the extra Dyckerhoff conditions. More seriously, the proof then uses the implication that Hausdorff convergence R^β_{x_n} → R^β_x entails eventual inclusion of every z∈R^β_x in R^β_{x_n}. This implication is false: the intervals [1/n,1+1/n] converge to [0,1] in Hausdorff distance, yet 0 belongs to none of the approximating intervals. Consequently, the claimed pointwise convergence of indicators 1_{R^β_{x_n}} → 1_{R^β_x} is not justified, and the subsequent dominated-convergence step also requires a boundary condition μ(∂R^β_x)=0 that is not established. Since continuity of ILD in x is one of the advertised inherited properties, this is a load-bearing gap; it can likely be repaired by adding strict-monotonicity/convexity and boundary-measure-zero assumptions, but as written Property 2 is not proven.
  2. [§3, Property 4] The proof of the extreme-locality centrality property appears to misuse the weighted mean value theorem. In the statement, W is fixed as a probability measure on (0,1], and ILD_B(x|P)=∫_0^B LD(β|x,P)dW(β). The proof then applies the mean value theorem as if ∫_0^B w(β)dβ=1 for every B, but for a fixed W this integral tends to 0 as B→0. The sentence "where W(0,B]=1" appears to redefine W for each B; if ILD_B is instead defined with W conditionally normalized to (0,B], this is a different object from the original integrated depth and the notation should say so explicitly. Without such normalization, the displayed mean-value identity is incorrect, and the conclusion lim_{B→0} ILD_B(x|P)=lim_{β→0} LD_β(x|P) does not follow from the argument given.
  3. [§4.1–4.2, Tables 1–2] The empirical claims of "significant improvements" are not supported by uncertainty quantification or by the stated parameter-selection protocol. In Table 1, the text states that PILD and B-PILD are "significantly better" in some setups and that certain methods "consistently outperform" others, but no standard errors, confidence intervals, or significance tests are reported across the 100 simulations. In Table 2, the authors say "we select the optimal parameters that maximize precision" for all methods; since this selection is performed on the same data used for evaluation, the reported precision values are oracle-tuned and cannot be compared directly as estimates of unsupervised out-of-sample performance. The abstract's claim of significant improvements should either be accompanied by proper error bars and non-oracle tuning, or be qualified accordingly.
minor comments (5)
  1. [Definition 2.3] The β-neighborhood is defined as "the set of ⌈nβ⌉ points with highest depth values", but ties in depth values are not addressed; a tie-breaking rule or a deterministic convention is needed for the sample β-LD and PILD to be well defined.
  2. [Eq. (7) and Section 3] Definition 2.4 allows an arbitrary probability measure W, but the sample formula in Eq. (7) and the theoretical results in Section 3 assume an absolutely continuous W with density w. Please state explicitly where absolute continuity is required and what happens for discrete W.
  3. [Table 2] Several entries in the WDBC row are run together as "0.60.70.2" and similar strings, making the table unreadable; the entries need proper spacing or column separation.
  4. [Throughout] There are several typographical errors, including "multivatiate" in the introduction and "varing" in the conclusion; a careful proofread is needed.
  5. [Property 3] The phrase "the probability measure W remains unchanged under any transformation of P" is imprecise; since W is a measure on (0,1], it does not depend on P, so the intended condition should be stated more clearly, for example that W is the same measure in both ILD(T(x)|P_{T(X)}) and ILD(x|P_X).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central derivation integrates an external β-LD definition and relies on independent prior results; the one self-citation is peripheral.

full rationale

The paper's derivation chain is not circular. Definition 2.4 defines β-ILD as an integral of the previously defined β-local depth, which is itself cited from Paindaveine and Van Bever [1]; this is a generalization of an external notion, not a redefinition of the paper's own output. Consistency (Property 1) is obtained by applying the Dominated Convergence Theorem to the external consistency result Theorem 3.1 from [1]. Continuity in x (Property 2) is attempted via a separate external result of Dyckerhoff [38]; although the proof contains a questionable implication (Hausdorff convergence of central regions does not by itself give eventual pointwise inclusion), that is a correctness gap, not a circular reduction. T-invariance (Property 3) follows from Lemma 1 and the invariance of central regions, and centrality and vanishing at infinity (Properties 4 and 5) are inherited from β-LD results in the literature. The PILD matrix in Eq. (11) is constructed from β-neighborhoods and β-LD values, and the statement that rows sum to ILD is a designed normalization rather than a fitted quantity later presented as a prediction. The only self-citation, [21] (Wang, Leblanc and McNicholas, 'Depth-based local center clustering'), appears in an illustrative sentence listing applications of depth-based similarity and is not load-bearing for any theorem or empirical claim. No equation is equivalent to another by construction, and no fitted parameter is renamed as a prediction. Hence the correct circularity finding is 'no significant circularity'.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central definition itself is self-contained, but the inherited properties rely on Paindaveine and Van Bever's consistency and continuity results, on Assumption 1, and on an unstated strict-monotonicity and convexity condition in the continuity proof. The only fitted quantities are the tuning parameters B1 and W used in applications.

free parameters (3)
  • Locality upper bound B1 in uniform weight w_B(β) = Classification: CV medians 0.5, 0.15, 0.1, 0.2 for Setups 1-4; outlier detection: oracle-selected from {0.1,0.3,0.5,1}
    B1 controls the range of β over which local depth is integrated; it is a user-chosen tuning parameter, selected by cross-validation in classification and by maximizing test precision in outlier detection.
  • Minimum neighborhood size n0 = 3 (for spatial depth)
    Sets the lower endpoint of the integration grid via β1=n0/n; a technical convention stated in Remark 2.1 and used in the SILD summation.
  • Weighting measure W over (0,1] = Uniform density on (B0,B1] by default
    W is a modeling choice that decides how much each locality level matters; the theory allows arbitrary probability measures, and the applications use uniform weights.
assumptions (7)
  • domain assumption Assumption 1: D satisfies Definition 1.1 with T the similarity transformations and is weakly continuous
    Stated in Section 3; used to transfer consistency and continuity results from β-LD to β-ILD.
  • domain assumption P is absolutely continuous
    Assumed for consistency, continuity, and centrality results; needed for conditional distributions P^β_x to be well-defined.
  • domain assumption W is absolutely continuous with bounded measurable density on (0,1]
    Assumed at the start of Section 3 to justify DCT-based proofs of Properties 1 and 2.
  • domain assumption Quasi-concavity of D in Property 2
    Stated in Property 2; gives convex central regions, needed for the continuity proof.
  • ad hoc to paper Strict monotonicity and convex central regions for Hausdorff convergence (Dyckerhoff [38])
    Invoked in Proof of Property 2, Step 2, but not listed among the paper's assumptions; stronger than the quasi-concavity assumption stated in Property 2.
  • domain assumption Uniform weak continuity and c_D independent of P for Property 4
    Stated in Property 4; used to extend the extreme-locality centrality of β-LD from Paindaveine and Van Bever to β-ILD.
  • standard math Consistency and β-continuity of β-LD from Paindaveine and Van Bever [1] (Theorem 3.1, Lemma A.1)
    Foundational results used to prove Properties 1 and 4 via DCT; accepted from the literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of $\beta$-integrated local depth and corresponding partitioned local depth representation." pith.science (2026). https://pith.science/paper/XUSZYBCE

@misc{pith2026250614108,
  author       = {Pith},
  title        = {Pith review of: $\beta$-integrated local depth and corresponding partitioned local depth representation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XUSZYBCE}},
  note         = {Machine review of arXiv:2506.14108}
}
abstract

A novel local depth definition, $\beta$-integrated local depth ($\beta$-ILD), is proposed as a generalization of the local depth introduced by Paindaveine and Van Bever \cite{paindaveine2013depth}, designed to quantify the local centrality of data points. $\beta$-ILD inherits desirable properties from global data depth and remains robust across varying locality levels. A partitioning approach for $\beta$-ILD is introduced, leading to the construction of a matrix that quantifies the contribution of one point to another's local depth, providing a new interpretable measure of local centrality. These concepts are applied to classification and outlier detection tasks, demonstrating significant improvements in the performance of depth-based algorithms.

Figures

Figures reproduced from arXiv: 2506.14108 by the authors.

Figure 1
Figure 1. Examples of LCD and β-ILD values in the simulated smiley dataset: (a) LCD (b) β-ILD. The development of local depth motivates us to propose an extension of previous local depth definitions called β-integrated local depth (β-ILD). This new contribution builds on β-LD and is designed to be more robust to changes in the locality parameter. Moreover, β-ILD can also resemble the performance of LCD by considering all poin… view at source ↗
Figure 2
Figure 2. Local projection depth for a selected point in a toy example at three different [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Comparison between β-ILD and β-LD across different β values (or B in β-ILD). The depth values are computed for the target point with respect to the data points shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Graphs of examples for four setups. (a): Setup 1; (b): Setup 2; (c): Setup 3. (d): Setup 4. For methods requiring parameter selection, 5-fold cross-validation (CV) is used to determine the optimal values for the locality parameters (β for β-LD and B for B-ILD and B-PIL…
Figure 5
Figure 5. Figure 5: Boxplots of classifier accuracies for 4 setups based on 100 simulations. Classifiers include: max depth (MD), max LD with β set to 0.1, 0.3, 0.5 and cross-validated (LD(0.1), LD(0.3), LD(0.5), LD(cv)), max Full-ILD (ILD), max ILD with B set to 0.1, 0.3, 0.5 and cross￾v…
Figure 6
Figure 6. Figure 6: Graphs of examples for two toy examples with red points representing outliers. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Boxplots of precision in anomaly detections for two toy examples based on [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 34 canonical work pages

  1. [38]

    Dyckerhoff, Convergence of depths and depth-trimmed regions, arXiv preprint arXiv:1611.08721 (2016)

    R. Dyckerhoff, Convergence of depths and depth-trimmed regions, arXiv preprint arXiv:1611.08721 (2016)

    arXiv 2016

  2. [1]

    Paindaveine, G

    D. Paindaveine, G. Van Bever, From depth to local depth: a focus on centrality, Journal of the American Statis- tical Association 108 (503) (2013) 1105–1119

    work page 2013

  3. [2]

    T. J. W., Mathematics and the picturing of data, Proceedings of the International Congress of Mathematicians, Vancouver, 1975 2 (1975) 523–531. URLhttps://cir.nii.ac.jp/crid/1573950399770196096

    arXiv 1975

  4. [3]

    R. Y . Liu, On a notion of data depth based on random simplices, The Annals of Statistics (1990) 405–414

    work page 1990

  5. [4]

    Serfling, A depth function and a scale curve based on spatial quantiles, in: Statistical Data Analysis Based on the L 1-Norm and Related Methods, Springer, 2002, pp

    R. Serfling, A depth function and a scale curve based on spatial quantiles, in: Statistical Data Analysis Based on the L 1-Norm and Related Methods, Springer, 2002, pp. 25–38

    work page 2002

  6. [5]

    Y . Zuo, R. Serfling, General notions of statistical depth function, Annals of statistics (2000) 461–482

    work page 2000

  7. [6]

    Cuevas, R

    A. Cuevas, R. Fraiman, On depth measures and dual statistics. a methodology for dealing with general data, Journal of Multivariate Analysis 100 (4) (2009) 753–766

    work page 2009

  8. [7]

    Ramsay, S

    K. Ramsay, S. Durocher, A. Leblanc, Integrated rank-weighted depth, Journal of Multivariate Analysis 173 (2019) 51–69

    work page 2019

Show all 39 references
  1. [8]

    Mosler, P

    K. Mosler, P. Mozharovskyi, Choosing among notions of multivariate depth statistics, Statistical Science 37 (3) (2022) 348–368

    2022

  2. [9]

    Y . Chen, X. Dang, H. Peng, H. L. Bart, Outlier detection with the kernelized spatial depth function, IEEE Transactions on Pattern Analysis and Machine Intelligence 31 (2) (2008) 288–305

    2008

  3. [10]

    Dutta, S

    S. Dutta, S. Sarkar, A. K. Ghosh, Multi-scale classification using localized spatial depth, Journal of Machine Learning Research 17 (217) (2016) 1–30. 22

    2016

  4. [11]

    Agostinelli, M

    C. Agostinelli, M. Romanazzi, Local depth, Journal of Statistical Planning and Inference 141 (2) (2011) 817– 830

    2011

  5. [12]

    Fernandez-Piana, M

    L. Fernandez-Piana, M. Svarc, An integrated local depth measure, AStA Advances in Statistical Analysis 106 (2022) 175–197. doi:10.1007/s10182-021-00424-6

    2022 doi

  6. [13]

    K. S. Berenhaut, K. E. Moore, R. L. Melvin, A social perspective on perceived distances reveals deep community structure, Proceedings of the National Academy of Sciences 119 (4) (2022) e2003634119

    2022

  7. [14]

    Z. Liu, R. Modarres, Lens data depth and median, Journal of Nonparametric Statistics 23 (4) (2011) 1063–1074

    2011

  8. [15]

    Kleindessner, U

    M. Kleindessner, U. V on Luxburg, Lens depth function and k-relative neighborhood graph: versatile tools for ordinal data analysis, Journal of Machine Learning Research 18 (58) (2017) 1–52

    2017

  9. [16]

    M. Yang, R. Modarres,β-skeleton depth functions and medians, Communications in Statistics - Theory and Methods 47 (20) (2018) 5127–5143. doi:10.1080/03610926.2017.1386320. URLhttps://doi.org/10.1080/03610926.2017.1386320

    2018

  10. [17]

    X. Liu, Y . Zuo, Z. Wang, Exactly computing bivariate projection depth contours and median, Computational Statistics & Data Analysis 60 (2013) 1–11

    2013

  11. [18]

    X. Liu, Y . Zuo, Computing projection depth and its associated estimators, Statistics and Computing 24 (1) (2014) 51–63

    2014

  12. [19]

    S. Nagy, I. Gijbels, M. Omelka, D. Hlubinka, Integrated depth for functional data: statistical properties and consistency, ESAIM: Probability and Statistics 20 (2016) 95–130

    2016

  13. [20]

    Paindaveine, G

    D. Paindaveine, G. Van Bever, Nonparametrically consistent depth-based classifiers, Bernoulli (2015) 62–82

    2015

  14. [21]

    S. Wang, A. Leblanc, P. D. McNicholas, Depth-based local center clustering: A framework for handling different clustering scenarios, arXiv preprint arXiv:2505.09516 (2025)

    2025

  15. [22]

    A. K. Ghosh, P. Chaudhuri, On maximum depth and related classifiers, Scandinavian Journal of Statistics 32 (2) (2005) 327–350

    2005

  16. [23]

    J. Li, J. A. Cuesta-Albertos, R. Y . Liu, Dd-classifier: Nonparametric classification procedure based on dd-plot, Journal of the American statistical association 107 (498) (2012) 737–753

    2012

  17. [24]

    Lange, K

    T. Lange, K. Mosler, P. Mozharovskyi, Fast nonparametric classification based on data depth, Statistical Papers 55 (2014) 49–69

    2014

  18. [25]

    Jeong, J

    M.-H. Jeong, J. Yin, S. Wang, Outlier detection and comparison of origin-destination flows using data depth, in: 10th International Conference on Geographic Information Science (GIScience 2018), Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2018

    2018

  19. [26]

    Mozharovskyi, R

    P. Mozharovskyi, R. Valla, Anomaly detection using data depth: multivariate case, arXiv preprint arXiv:2210.02851 (2022)

    2022 arXiv

  20. [27]

    R. Y . Liu, J. M. Parelius, K. Singh, Multivariate analysis by data depth: descriptive statistics, graphics and inference,(with discussion and a rejoinder by liu and singh), The annals of statistics 27 (3) (1999) 783–858. 23

    1999

  21. [28]

    Pokotylo, P

    O. Pokotylo, P. Mozharovskyi, R. Dyckerhoff, Depth and depth-based classification with r-package ddalpha, arXiv preprint arXiv:1608.04109 (2016)

    2016 arXiv

  22. [29]

    G. O. Campos, A. Zimek, J. Sander, R. J. Campello, B. Micenková, E. Schubert, I. Assent, M. E. Houle, On the evaluation of unsupervised outlier detection: measures, datasets, and an empirical study, Data mining and knowledge discovery 30 (2016) 891–927

    2016

  23. [30]

    M. M. Breunig, H.-P. Kriegel, R. T. Ng, J. Sander, Lof: identifying density-based local outliers, in: Proceedings of the 2000 ACM SIGMOD international conference on Management of data, 2000, pp. 93–104

    2000

  24. [31]

    F. T. Liu, K. M. Ting, Z.-H. Zhou, Isolation forest, in: 2008 eighth ieee international conference on data mining, IEEE, 2008, pp. 413–422

    2008

  25. [32]

    Huang, Q

    J. Huang, Q. Zhu, L. Yang, J. Feng, A non-parameter outlier detection algorithm based on natural neighbor, Knowledge-Based Systems 92 (2016) 71–77

    2016

  26. [33]

    B. Tang, H. He, A local density-based approach for outlier detection, Neurocomputing 241 (2017) 171–180

    2017

  27. [34]

    Hahsler, M

    M. Hahsler, M. Piekenbrock, dbscan: Density-Based Spatial Clustering of Applications with Noise (DBSCAN) and Related Algorithms, r package version 1.2-0 (2024). URLhttps://CRAN.R-project.org/package=dbscan

    2024

  28. [35]

    Cortes, isotree: Isolation-Based Outlier Detection, r package version 0.6.1-1 (2024)

    D. Cortes, isotree: Isolation-Based Outlier Detection, r package version 0.6.1-1 (2024). URLhttps://CRAN.R-project.org/package=isotree

    2024

  29. [36]

    J. H. Madsen, DDoutlier: Distance and Density-Based Outlier Detection, r package version 0.1.0 (2018). URLhttps://CRAN.R-project.org/package=DDoutlier

    2018

  30. [37]

    Hofert, I

    M. Hofert, I. Kojadinovic, M. Maechler, J. Yan, copula: Multivariate Dependence with Copulas, r package ver- sion 1.1-4 (2024). URLhttps://CRAN.R-project.org/package=copula

    2024

  31. [39]

    Y . Zuo, R. Serfling, Structural properties and convergence results for contours of sample statistical depth func- tions, Annals of Statistics (2000) 483–499. 24

    2000

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.