Show Summary Details
More options …

# Journal of Causal Inference

Ed. by Imai, Kosuke / Pearl, Judea / Petersen, Maya Liv / Sekhon, Jasjeet / van der Laan, Mark J.

2 Issues per year

Online
ISSN
2193-3685
See all formats and pricing
More options …
Volume 6, Issue 1

# Detecting Confounding in Multivariate Linear Models via Spectral Analysis

Dominik Janzing
• Corresponding author
• Deaprtment ‘Empirical Inference’,Max Planck Institute for Intelligent Systems,Spemannstr. 36, 70569 Tübingen,Germany
• Email
• Other articles by this author:
/ Bernhard Schölkopf
• Deaprtment ‘Empirical Inference’,Max Planck Institute for Intelligent Systems,Tübingen,Germany
• Other articles by this author:
Published Online: 2017-10-28 | DOI: https://doi.org/10.1515/jci-2017-0013

## Abstract

We study a model where one target variable $Y$ is correlated with a vector $\mathbf{\text{X}}:=\left({X}_{1},\dots ,{X}_{d}\right)$ of predictor variables being potential causes of $Y$. We describe a method that infers to what extent the statistical dependences between $\mathbf{\text{X}}$ and $Y$ are due to the influence of $\mathbf{\text{X}}$ on $Y$ and to what extent due to a hidden common cause (confounder) of $\mathbf{\text{X}}$ and $Y$. The method relies on concentration of measure results for large dimensions $d$ and an independence assumption stating that, in the absence of confounding, the vector of regression coefficients describing the influence of each $\mathbf{\text{X}}$ on $Y$ typically has ‘generic orientation’ relative to the eigenspaces of the covariance matrix of $\mathbf{\text{X}}$. For the special case of a scalar confounder we show that confounding typically spoils this generic orientation in a characteristic way that can be used to quantitatively estimate the amount of confounding (subject to our idealized model assumptions).

## References

• [1]

Reichenbach H. The direction of time. Berkeley: University of California Press, 1956.Google Scholar

• [2]

Pearl J. Causality: Models, reasoning, and inference. Cambridge University Press, 2000.Google Scholar

• [3]

Spirtes P, Glymour C, Scheines R. Causation, Prediction, and Search (Lecture notes in statistics). New York, NY: Springer-Verlag, 1993.Google Scholar

• [4]

Bowden R, Turkington D. Instrumental variables. Cambridge: Cambridge University Press, 1984.Google Scholar

• [5]

Hoyer P, Shimizu S, Kerminen A, Palviainen M. Estimation of causal effects using linear non-gaussian causal models with hidden variables. Int J Approx Reason. 2008;49:362–378.

• [6]

Janzing D, Peters J, Mooij J, Schölkopf B. Identifying latent confounders using additive noise models. In: Ng A, Bilmes J, editor. Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence (UAI 2009). Corvallis, OR, USA: AUAI Press, 2009:249–257.Google Scholar

• [7]

Janzing D, Sgouritsa E, Stegle O, Peters P, Schölkopf B. Detecting low-complexity unobserved causes. In: Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence (UAI 2011). Available at: http://uai.sis.pitt.edu/papers/11/p383-janzing.pdf.Google Scholar

• [8]

Janzing D, Balduzzi D, Grosse-Wentrup M, Schölkopf B. Quantifying causal influences. Ann Stat. 2013;41:2324–2358.

• [9]

Janzing D, Schölkopf B. Causal inference using the algorithmic Markov condition. IEEE Trans Inf Theo. 2010;56:5168–5194.

• [10]

Lemeire J, Janzing D. Replacing causal faithfulness with algorithmic independence of conditionals. Minds Mach. 2012;23:227–249.

• [11]

Li M, Vitányi P. An Introduction to Kolmogorov Complexity and its Applications. New York: Springer, 1997 (3rd edition: 2008).Google Scholar

• [12]

Janzing D, Steudel B. Justifying additive-noise-based causal discovery via algorithmic information theory. Open Syst Inf Dynam. 2010;17:189–212.

• [13]

Meek C. Strong completeness and faithfulness in Bayesian networks. In: Proceedings of 11th Uncertainty in Artificial Intelligence (UAI). Montreal, Canada: Morgan Kaufmann, 1995:411–418.

• [14]

Uhler C, Raskutti G, Bühlmann P, Yu B. Geometry of the faithfulness assumption in causal inference. Ann Stat. 2013;41:436–463.

• [15]

Kato T. Perturbation theory for linear operators. Berlin: Springer, 1996.Google Scholar

• [16]

Murphy G. ${C}^{\ast }$-algebras and operator theory. Boston: Academic Press, 1990.Google Scholar

• [17]

Reed M, Simon B. Functional Analysis. San Diego, California: Academic Press, 1980.Google Scholar

• [18]

Janzing D, Hoyer P, Schölkopf B. Telling cause from effect based on high-dimensional observations. In: Proceedings of the 27th International Conference on Machine Learning (ICML 2010), Haifa, Israel, 06, 2010:479–486.Google Scholar

• [19]

Zscheischler J, Janzing D, Zhang K. Testing whether linear equations are causal: A free probability theory approach. In: Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence (UAI 2011), 2011. Available at: http://uai.sis.pitt.edu/papers/11/p839-zscheischler.pdf.Google Scholar

• [20]

Voiculescu D, editor. Free probability theory, volume 12 of Fields Institute Communications. American Mathematical Society, 1997.Google Scholar

• [21]

Chandrasekaran V, Parrilo P, Willsky A. Latent variable graphical model selection via convex optimization. Ann Stat. 2012;40:1935–1967.

• [22]

Datta BN. Numerical Linear Algebra and Applications. Philadelphia, USA: Society for Industrial and Applied Mathematics, 2010.Google Scholar

• [23]

Cima J, Matheson A, Ross W. The Cauchy Transform. Mathematical Surveys and Monographs 125. American Mathematical Society, 2006.Google Scholar

• [24]

Simon B. Spectral analysis of rank one perturbations and applications. Lectur given at the Vancouver Summer School in Mathematical Physics (1993). Available at: 1994.

• [25]

Simon B. Trace ideals and their applications. Providence, RI: American Mathematical Society, 2005.Google Scholar

• [26]

Kiselev A, Simon B. Rank one perturbations with infinitesimal coupling. J Funct Anal. 1995;130:345–356.

• [27]

Albeverio S, Konstantinov A, Koshmanenko V. The Aronszajn-Donoghue theory for rank one perturbations of the ${H}_{-2}$-class. Integral Equ Operat Theo. 2004;50:1–8.

• [28]

Albeverio S, Kurasov P. Rank one perturbations, approximations, and selfadjoint extensions. J Func Anal. 1997;148:152–169.

• [29]

Bartlett MS. An inverse matrix adjustment arising in discriminant analysis. Ann. Math. Statist. 1951;22:107–111.

• [30]

Mingo J, Speicher R. Free probability and random matrices. New York: Springer, 2017.Google Scholar

• [31]

Bercovici H, Voiculescu D. Free convolution of measures with unbounded supports. Ind Univ Math J. 1993;42:733–773.

• [32]

Rudelson M. Random vectors in the isotropic position. J Func Anal. 1999;164:60–72.

• [33]

Vershynin R.. How close is the sample covariance matrix to the actual covariance matrix? J Theo Probab. 2012;25:655–686.

• [34]

Karlin S, Rinott Y. Classes of orderings of measures and related correlation inequalities. I. multivariate totally positive distributions. J Multiv Anal. 1980;10:467–498.Google Scholar

• [35]

Fallat S, Lauritzen S, Sadeghi K, Uhler C, Wermuth N, Zwiernik P. Total positivity in markov structures. To appear in Annals of Statistics, 2016.Google Scholar

• [36]

Lichman M. UCI machine learning repository. Available at: http://archive.ics.uci.edu/ml, 2013.Google Scholar

• [37]

City of Chicago. Data portal: Chicago poverty and crime. Available at: https://data.cityofchicago.org/Health-Human-Services/Chicago-poverty-and-crime/fwns-pcmk.

• [38]

Yeh C. Concrete compressive strength data set. https://archive.ics.uci.edu/ml/datasets/Concrete+Compressive+ Strength.

• [39]

Yeh I-C. Modeling of strength of high performance concrete using artificial neural networks. Cement Concrete Res. 1998.Google Scholar

• [40]

Schölkopf B, Smola A. Learning with kernels. Cambridge, MA: MIT Press, 2002.Google Scholar

• [41]

Gretton A, Herbrich R, Smola A, Bousquet O, Schölkopf B. Kernel methods for measuring independence. J Mach Learn Res. 2005;6:2075–2129.Google Scholar

• [42]

Speicher R. Free probability theory and non-crossing partitions. LOTHAR. COMB. 1997;39.Google Scholar

Revised: 2017-06-14

Accepted: 2017-09-21

Published Online: 2017-10-28

Published in Print: 2018-03-26

Citation Information: Journal of Causal Inference, Volume 6, Issue 1, 20170013, ISSN (Online) 2193-3685,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.