Statistical dependence: Beyond Pearson's $ρ$

Tjøstheim, Dag; Otneim, Håkon; Støve, Bård

doi:10.48550/arxiv.1809.10455

Cited by 5 publications

(5 citation statements)

References 110 publications

(180 reference statements)

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“…While independence between two features implies that the PCC is zero, the converse is generally false. The PCC, which is often used to analyze dependence, only tracks linear correlations and has other shortcomings such as sensitivity to outliers [113]. Any type of dependence between features can have a strong impact on the interpretation of the results of IML methods (see Sect.…”

Section: Confusing Linear Correlation With General Dependencementioning

confidence: 99%

“…Kernel-based measures, such as kernel canonical correlation analysis (KCCA) [6] or the Hilbert-Schmidt independence criterion (HSIC) [44], are commonly used. They have a solid theoretical foundation, are computationally feasible, and robust [113]. In addition, there are information-theoretical measures, such as (conditional) mutual information [24] or the maximal information coefficient (MIC) [93], that can however be difficult to estimate [9,116].…”

Section: Confusing Linear Correlation With General Dependencementioning

confidence: 99%

See 1 more Smart Citation

General Pitfalls of Model-Agnostic Interpretation Methods for Machine Learning Models

Molnar

König

Herbinger

et al. 2022

Lecture Notes in Computer Science

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An increasing number of model-agnostic interpretation techniques for machine learning (ML) models such as partial dependence plots (PDP), permutation feature importance (PFI) and Shapley values provide insightful model interpretations, but can lead to wrong conclusions if applied incorrectly. We highlight many general pitfalls of ML model interpretation, such as using interpretation techniques in the wrong context, interpreting models that do not generalize well, ignoring feature dependencies, interactions, uncertainty estimates and issues in high-dimensional settings, or making unjustified causal interpretations, and illustrate them with examples. We focus on pitfalls for global methods that describe the average model behavior, but many pitfalls also apply to local methods that explain individual predictions. Our paper addresses ML practitioners by raising awareness of pitfalls and identifying solutions for correct model interpretation, but also addresses ML researchers by discussing open issues for further research.

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Section: Confusing Linear Correlation With General Dependencementioning

confidence: 99%

Section: Confusing Linear Correlation With General Dependencementioning

confidence: 99%

General Pitfalls of Model-Agnostic Interpretation Methods for Machine Learning Models

Molnar

König

Herbinger

et al. 2022

Lecture Notes in Computer Science

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“…Examples for spaces of negative type are 𝐿 𝑝 spaces for 1 ≤ 𝑝 ≤ 2, ultrametric spaces, and weighted trees (Meckes 2013, eorem 3.6) Spearman's 𝜌, or Kendall's 𝜏), the statistical literature on how to best measure dependency in applications quickly becomes immensely broad and sca ered. Besides the approaches mentioned above, suggestions include maximal correlation coe cients (Gebelein 1941;Koyak 1987), rank or copula based methods (Schweizer and Wol 1981;De e et al 2013;Marti et al 2017), or various measures acting on the distribution of pairwise distances (Friedman and Rafsky 1983;Heller et al 2013), to only cite a few (for a more complete survey of established methods, see Tjøstheim et al 2018). Recently, optimal transport maps have been utilized to de ne multivariate rank statistics that allow for (asymptotically) distribution-free independence tests (Ghosal and Sen 2019;Shi et al 2020;Shi et al 2021).…”

Section: Mutual Informationmentioning

confidence: 99%

Transport Dependency: Optimal Transport Based Dependency Measures

Nies¹,

Staudt²,

Munk³

2021

Preprint

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Finding meaningful ways to determine the dependency between two random variables 𝜉 and 𝜁 is a timeless statistical endeavor with vast practical relevance. In recent years, several concepts that aim to extend classical means (such as the Pearson correlation or rank-based coe cients like Spearman's 𝜌) to more general spaces have been introduced and popularized, a well-known example being the distance correlation. In this article, we propose and study an alternative framework for measuring statistical dependency, the transport dependency 𝜏 ≥ 0, which relies on the notion of optimal transport and is applicable in general Polish spaces. It can be estimated consistently via the corresponding empirical measure, is versatile and adaptable to various scenarios by proper choices of the cost function, and intrinsically respects metric properties of the ground spaces. Notably, statistical independence is characterized by 𝜏 = 0, while large values of 𝜏 indicate highly regular relations between 𝜉 and 𝜁 . Indeed, for suitable base costs, 𝜏 is maximized if and only if 𝜁 can be expressed as 1-Lipschitz function of 𝜉 or vice versa. Based on sharp upper bounds, we exploit this characterization and de ne three distinct dependency coe cients (a-c) with values in [0, 1], each of which emphasizes di erent functional relations. ese transport correlations a ain the value 1 if and only if 𝜁 = 𝜑 (𝜉), where 𝜑 is a) a Lipschitz function, b) a measurable function, c) a multiple of an isometry. e properties of coe cient c) make it comparable to the distance correlation, while coe cient b) is a limit case of a) that was recently studied independently by Wiesel (2021). Numerical results suggest that the transport dependency is a robust quantity that e ciently discerns structure from noise in simple se ings, o en out-performing other commonly applied coe cients of dependency.

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“…There have been many methods employed and proposed, see e.g. [24], [43] and [29] for recent surveys. Usually these focus on the (functional) dependence of pairs of variables.…”

Section: Introductionmentioning

confidence: 99%

Dependence and dependence structures: estimation and visualization using the unifying concept of distance multivariance

Böttcher

2019

Open Statistics

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Distance multivariance is a multivariate dependence measure, which can detect dependencies between an arbitrary number of random vectors each of which can have a distinct dimension. Here we discuss several new aspects, present a concise overview and use it as the basis for several new results and concepts: In particular, we show that distance multivariance unifies (and extends) distance covariance and the Hilbert-Schmidt independence criterion HSIC, moreover also the classical linear dependence measures: covariance, Pearson's correlation and the RV coefficient appear as limiting cases. Based on distance multivariance several new measures are defined: a multicorrelation which satisfies a natural set of multivariate dependence measure axioms and m-multivariance which is a dependence measure yielding tests for pairwise independence and independence of higher order. These tests are computationally feasible and under very mild moment conditions they are consistent against all alternatives. Moreover, a general visualization scheme for higher order dependencies is proposed, including consistent estimators (based on distance multivariance) for the dependence structure.Many illustrative examples are provided. All functions for the use of distance multivariance in applications are published in the R-package multivariance.

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Statistical dependence: Beyond Pearson's $ρ$

Cited by 5 publications

References 110 publications

General Pitfalls of Model-Agnostic Interpretation Methods for Machine Learning Models

General Pitfalls of Model-Agnostic Interpretation Methods for Machine Learning Models

Transport Dependency: Optimal Transport Based Dependency Measures

Dependence and dependence structures: estimation and visualization using the unifying concept of distance multivariance

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