Basic linear models

Basic linear models with no latent confounders
(acyclic models, time series, cyclic models)

Acyclic models

Model

  • G. Park. Identifiability of Additive Noise Models Using Conditional Variances. Journal of Machine Learning Research, 21: 1--34, 2020.
    [pdf] [Google scholar]

  • G. Park and Y. Kim. Identifiability of Gaussian Structural Equation Models with Homogeneous and Heterogeneous Error Variances. Journal of the Korean Statistical Society, xx(xx): xx--xx, 2020.
    [pdf] [Google scholar]

  • S. Yu, M. Drton, A. Shojaie. Directed Graphical Models and Causal Discovery for Zero-Inflated Data. Arxiv preprint arXiv:2004.04150, 2020.
    [pdf] [Google schlar]

  • N. Gnecco, N. Meinshausen, J. Peters, S. Engelke. Causal discovery in heavy-tailed models. Arxiv preprint arXiv:1908.05097, 2019.
    [pdf] [Google scholar]

  • W. Wiedermann and A. von Eye. Testing the causal direction of mediation effects in randomized intervention studies. Prevention Science, xx(xx): xx--xx, 2018.
    [pdf] [Google scholar]

  • A Ghoshal and J Honorio. Learning identifiable Gaussian Bayesian networks in polynomial time and sample complexity. Arxiv preprint arXiv:1703.01196, 2017.
    [pdf] [Google scholar]

  • A. Ghoshal and J. Honorio. Learning linear structural equation models in polynomial time and sample complexity. Arxiv preprint arXiv:1707.04673, 2017.
    [pdf] [Google scholar]

  • W. Wiedermann and A. von Eye. Direction of effects in mediation analysis. Psychological Methods, 20(2): 221--244, 2015.
    [pdf] [Google scholar]

  • W. Wiedermann and A. von Eye. Direction of effects in multiple linear regression models. Multivariate Behavioral Research, 50(1): 23--40, 2015.
    [pdf] [Google scholar]

  • A. Sokol, M. H. Maathuis and B. Falkeborg. Quantifying identifiability in independent component analysis. Electronic Journal of Statistics, 8: 1438--1459, 2014.
    [pdf] [Google scholar]

  • J. Peters and P. Bühlmann. Identifiability of Gaussian structural equation models with equal error variances. Biometrika, 101(1): 219--228, 2014.
    [pdf] [Google scholar]

  • D. Entner and P. O. Hoyer. Estimating a causal order among groups of variables in linear models. In Proc. 22nd International Conference on Artificial Neural Networks (ICANN2012), pp. 83--90, Lausanne, Switzerland, 2012.
    [pdf] [code] [Google scholar]

  • P. O. Hoyer, A. Hyvärinen, R. Scheines, P. Spirtes, J. Ramsey, G. Lacerda, and S. Shimizu. Causal discovery of linear acyclic models with arbitrary distributions. In Proc. 24th Conf. on Uncertainty in Artificial Intelligence (UAI2008), pp. 282-289, Helsinki, Finland, 2008.
    [pdf] [TETRAD IV] [Google scholar]

  • S. Shimizu, P. O. Hoyer, A. Hyvärinen and A. Kerminen. A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7(Oct): 2003--2030, 2006.
    [pdf] [erratum] [Matlab/Octave code] [Google scholar]
    R code by Patrik O. Hoyer and Antti Hyttinen
    R code by Doris Entner
    R package: pcalg by Kalisch et al.
    Python code
    Also implemented in TETRAD IV.

  • S. Shimizu, A. Hyvärinen, Y. Kano and P. O. Hoyer. Discovery of non-gaussian linear causal models using ICA. In Proc. 21st Conf. on Uncertainty in Artificial Intelligence (UAI2005), pp. 525-533, Edinburgh, Scotland, 2005.
    [pdf] [Google scholar]

  • Y. Dodge and V. Rousson. On asymmetric properties of the correlation coefficient in the regression setting.
    The American Statistician, 55(1): 51--54, 2001.
    [pdf] [Google scholar]

  • Y. Dodge and V. Rousson. Direction dependence in a regression line. Communications in Statistics - Theory and Methods, 29(9-10): 1957--1972, 2000.
    [pdf] [Google scholar]

Estimation

  • NEW H. Kawaguchi. Application of quantum computing to a linear non-Gaussian acyclic model for novel medical knowledge discovery. arXiv preprint arXiv:2110.04485, 2021.
    [pdf] [Google schlar]

  • NEW A. Shahbazinia, S. Salehkaleybar, M. Hashemi. ParaLiNGAM: Parallel Causal Structure Learning for Linear non-Gaussian Acyclic Models. Arxiv preprint arXiv:2109.13993, 2021.
    [pdf] [Google schlar]

  • K. Harada, H. Fujisawa. Sparse estimation of Linear Non-Gaussian Acyclic Model for Causal Discovery. Neurocomputing, xx(x): xx-xx, 2021.
    [pdf] [Google scholar]

  • G. Park, S. Moon, J.-J. Jeon. Learning a High-dimensional Linear Structural Equation Model via l1-Regularized Regression. Journal of Machine Learning Research, xx(x): xx-xx, 2021.
    [pdf] [Google scholar]

  • M. Kaiser, M. Sipos. Unsuitability of NOTEARS for Causal Graph Discovery. Arxiv preprint arXiv:2104.05441, 2021.
    [pdf] [Google schlar]

  • Z. Fang, S. Zhu, J. Zhang, Y. Liu, Z. Chen, Y. He. Low Rank Directed Acyclic Graphs and Causal Structure Learning. Arxiv preprint arXiv:2006.05691, 2020.
    [pdf] [Google schlar]

  • K. Harada, and H. Fujiasawa. Estimation of Structural Causal Model via Sparsely Mixing Independent Component Analysis. arXiv:2009.03077, 2020.
    [pdf] [Google schlar]

  • I. Ng, A. E. Ghassami, K. Zhang. On the Role of Sparsity and DAG Constraints for Learning Linear DAGs. Arxiv preprint arXiv:2006.10201, 2020.
    [pdf] [Google schlar]

  • G. Park, and Y. Kim. Learning high-dimensional Gaussian linear structural equation models with heterogeneous error variances. Computational Statistics & Data Analysis, xx(x): xx-xx, 2020.
    [pdf] [Google scholar]

  • H. Zhang, S. Zhou, C. Yan, J. Guan, X. Wang, J. Zhang, and J. Huan. Learning Causal Structures Based on Divide and Conquer. IEEE Transactions on Cybernetics, xx(x): xx-xx, 2020.
    [pdf] [Google scholar]

  • Y. S. Wang and M. Drton. High-dimensional causal discovery under non-Gaussianity. Biometrika, xx(x): xx-xx, 2020.
    [pdf] [code] [Google scholar]

  • G Mai, Y Hong, P Chen, K Chen, H Huang, G Zheng. Distinguish Markov Equivalence Classes from Large-Scale Linear Non-Gaussian Data. IEEE Access, xx(x): xx-xx, 2020.
    [pdf] [Google scholar]

  • C. Yan and S. Zhou. Effective and Scalable Causal Partitioning Based on Low-order Conditional Independent Tests. Neurocomputing, xx(x): xx-xx, 2020.
    [pdf] [Google scholar]

  • Y. Zeng, Z. Hao, R. Cai, F. Xie, L. Ou, R. Huang. A causal discovery algorithm based on the prior selection of leaf nodes. Neural Networks, xx(x): xx-xx, 2020.
    [pdf] [Google scholar]

  • F. Xie, R. Cai, Y. Zeng, Z. Hao. Causal Discovery of Linear Non-Gaussian Acyclic Model with Small Samples. In Proc. 9th Intelligence Science and Big Data Engineering. Big Data and Machine Learning (IScIDE 2019), pp. 381-393, 2019.
    [pdf] [Google scholar]

  • H. Zhang, S. Zhou, J. Guan, J. L. Huan. Measuring Conditional Independence by Independent Residuals for Causal Discovery. ACM Transactions on Intelligent Systems and Technology (TIST) , 10(5): xx-xx, 2019.
    [pdf] [Google scholar]

  • H. Zhang, S. Zhou, C. Yan, J. Guan, X. Wang. Recursively Learning Causal Structures Using Regression-Based Conditional Independence Test. In Proc. 33nd AAAI Conference on Artificial Intelligence (AAAI2019), pp. xx-xx, 2019.
    [pdf] [Google scholar]

  • X. Zheng, B. Aragam, P. K. Ravikumar, and E. P. Xing. DAGs with NO TEARS: Continuous Optimization for Structure Learning. In Advances in Neural Information Processing Systems 32 (NIPS2018), pp. xx-xx, 2018.
    [pdf] [Google scholar]

  • J. Yang, N. Li, N. An, Y. Chen, and G. Alterovitz. An efficient causal structure learning algorithm for linear arbitrarily distributed continuous data. The Journal of Supercomputing, pp. xx-xx, 2018.
    [pdf] [Google scholar]

  • W. Wiedermann and X Li. Direction dependence analysis: A framework to test the direction of effects in linear models with an implementation in SPSS. Behavior Research Methods, pp. xx-xx, 2018.
    [pdf] [Google scholar]

  • R. Cai, J. Qiao, Z. Zhang, and Z. Hao. SELF: Structural equational likelihood framework for causal discovery. In Proc. 32nd AAAI Conference on Artificial Intelligence (AAAI2018), pp. xx-xx, 2018.
    [pdf] [Google scholar]

  • R. Cai, F. Xie, W. Chen, and Z. Hao. An efficient kurtosis-based causal discovery method for linear non-Gaussian acyclic data. In Proc. 2017 IEEE/ACM 25th International Symposium on Quality of Service: 208-216, 2017.
    [pdf] [Google scholar]

  • W. Wiedermann. Decisions concerning the direction of effects in linear regression models using fourth central moments. In Dependent Data in Social Sciences Research, pp. 149-169, 2015.
    [pdf] [Google scholar]

  • W. Wiedermann and M. Hagmann. Asymmetric properties of the Pearson correlation coefficient: Correlation as the negative association between linear regression residuals. Communications in Statistics - Theory and Methods, xx(xx): xx--xx, 2015. In press.
    [pdf] [Google scholar]

  • W. Wiedermann and A. von Eye. Direction-dependence analysis: A confirmatory approach for testing directional theories. International Journal of Behavioral Development, xx(xx): xx--xx, 2015. In press.
    [pdf] [Google scholar]

  • F. Thoemmes. Empirical evaluation of directional-dependence tests. International Journal of Behavioral Development, xx(xx): xx--xx, 2015. In press.
    [pdf] [Google scholar]

  • P-L. Loh and P. Bühlmann. High-dimensional learning of linear causal networks via inverse covariance estimation. Journal of Machine Learning Research, 15(Oct):3065−3105, 2014.
    [pdf] [Google scholar]

  • W. Wiedermann, M. Hagmann and A. von Eye. Significance tests to determine the direction of effects in linear regression models. British Journal of Mathematical and Statistical Psychology, 68(1): 116--141, 2015.
    [pdf] [Google scholar]

  • D. Feng, F. Chen and W. Xu. Learning linear non-Gaussian networks: A new view from matrix identification. Journal of Experimental & Theoretical Artificial Intelligence, 25(2): 251--271, 2013.
    [pdf] [Google scholar]

  • A. Hyvärinen and S. M. Smith. Pairwise likelihood ratios for estimation of non-Gaussian structural equation models. Journal of Machine Learning Research, 14(Jan): 111--152, 2013.
    [pdf] [Matlab code] [Google scholar]

  • A. Hyvärinen. Pairwise measures of causal direction in linear non-Gaussian acyclic models. In JMLR Workshop and Conference Proceedings (Proc. 2nd Asian Conference on Machine Learning, ACML2010), 13: 1-16, 2010.
    [pdf] [Google scholar]

  • Y. Dodge and I. Yadegari. On direction of dependence. Metrika, 72: 139--150, 2010.
    [pdf] [Google scholar]

  • R. Henao and O. Winther. Sparse linear identifiable multivariate modeling. Journal of Machine Learning Research, 12(Mar): 863--905, 2011.
    [pdf] [code] [Google scholar]

  • R. Henao and O. Winther. Bayesian sparse factor models and DAGs inference and comparison. In Advances in Neural Information Processing Systems 22 (NIPS2009), pp. 736-744, 2010.
    [pdf] [Google scholar]

  • P. O. Hoyer and A. Hyttinen. Bayesian discovery of linear acyclic causal models. In Proc. 25th Conf. on Uncertainty in Artificial Intelligence (UAI2009), pp. 240-248, Montreal, Canada, 2009.
    [pdf] [code] [Google scholar]

  • S. Shimizu, T. Inazumi, Y. Sogawa, A. Hyvärinen, Y. Kawahara, T. Washio, P. O. Hoyer and K. Bollen. DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model. Journal of Machine Learning Research, 12(Apr): 1225--1248, 2011.
    [pdf] [Matlab/Python code] [R code by Genta Kikuchi] [Double-pendulum data] [General social survey] [Google scholar]

  • T. Inazumi, S. Shimizu and T. Washio. Use of prior knowledge in a non-Gaussian method for learning linear structural equation models. In Proc. 9th International Conference on Latent Variable Analysis and Signal Separation (LVA/ICA2010), Saint-Malo, France, pp.221--228, 2010.
    [pdf] [Matlab code] [Google scholar]

  • Y. Sogawa, S. Shimizu, Y. Kawahara and T. Washio. An experimental comparison of linear non-Gaussian causal discovery methods and their variants. In Proc. Int. Joint Conference on Neural Networks (IJCNN2010), pp. 768--775, Barcelona, Spain, 2010.
    [pdf] [Matlab code] [Google scholar]

  • S. Shimizu, A. Hyvärinen, Y. Kawahara and T. Washio. A direct method for estimating a causal ordering in a linear non-Gaussian acyclic model. In Proc. 25th Conf. on Uncertainty in Artificial Intelligence (UAI2009), pp. 506-513, Montreal, Canada, 2009.
    [pdf] [notes] [erratum] [Matlab code] [Google scholar]

  • Y. Sogawa, S. Shimizu, T. Shimamura, A. Hyvärinen, T. Washio and S. Imoto. Estimating exogenous variables in data with more variables than observations. Neural Networks, 24(8): 875-880, 2011 (Selected papers from ICANN2010).
    [pdf] [Matlab code] [Google scholar]

  • Y. Sogawa, S. Shimizu, A. Hyvärinen, T. Washio, T. Shimamura and S. Imoto. Discovery of exogenous variables in data with more variables than observations. In Proc. International Conference on Artificial Neural Networks (ICANN2010), pp.67-76, Thessaloniki, Greece, 2010.
    [pdf] [Matlab code] [Google scholar]

  • K. Ozaki, K. Nakamura and H. Murohashi. A multilevel model using 2nd and 3rd order moments. Proceedings of the Institute of Statistical Mathematics, 58(2): 207--221, 2010. (In Japanese)
    [pdf] [Google scholar]

  • K. Zhang, H. Peng, L. Chan and A. Hyvärinen. ICA with sparse connections: Revisited. In Proc. 8th Int. Conf. on Independent Component Analysis and Signal Separation (ICA2009), pp. 195-202, Paraty, Brazil, 2009.
    [pdf] [Google scholar]

  • A. B. Nielsen and L. K. Hansen. Structure learning by pruning in independent component analysis. Neurocomputing, 71: 2281--2290, 2008.
    [pdf] [Google scholar]

  • K. Zhang and L. Chan. ICA with sparse connections. In Proc. 7th Conf. on Intelligent Data Engineering and Automated Learning (IDEAL2006), pp. 530-537, Burgos, Spain, 2006.
    [pdf] [Google scholar]

  • S. Shimizu and Y. Kano. Use of non-normality in structural equation modeling: Application to direction of causation. Journal of Statistical Planning and Inference, 138: 3483--3491, 2008.
    [pdf] [Google scholar]

Time series

Structural vector autoregressive models

  • G. Bormetti, F. Corsi. A Lucas Critique Compliant SVAR model with Observation-driven Time-varying Parameters. Arxiv preprint arXiv:2107.05263, 2021.
    [pdf] [Google scholar]

  • M. Lanne, J. Luoto. GMM Estimation of Non-Gaussian Structural Vector Autoregression. Journal of Business & Economic Statistics, 39(1): 69-81, 2021.
    [pdf] [Google scholar]

  • S. M. Zema. Non-Normal Identification for Price Discovery in High-Frequency Financial Markets. LEM Working Paper Series, ISSN(ONLINE) 2284-0400, 2020.
    [pdf] [Google scholar]

  • C. Velasco. Identification and estimation of Structural VARMA models using higher order dynamics. Arxiv preprint arXiv:2009.04428, 2020.
    [pdf] [Google scholar]

  • P. Xie, J. Ye, J. Wang. Volatility Estimation of Multivariate ARMA-GARCH Model. Journal of Harbin Institute of Technology (New Series), 27(1): 36-43, 2020.
    [pdf] [Google scholar]

  • R. Pamfil, N. Sriwattanaworachai, S. Desai, P. Pilgerstorfer, P. Beaumont, K. Georgatzis, B. Aragam. DYNOTEARS: Structure Learning from Time-Series Data. JMLR Workshop and Conference Proceedings, AISTATS 2020 (Proc. 23th International Conference on Artificial Intelligence and Statistics), x: xx-xx, 2020.
    [pdf] [Google scholar]

  • B. Huang, K. Zhang, J. Zhang, J. Ramsey, B. Schölkopf. Causal Discovery and Hidden Driving Force Estimation from Nonstationary/Heterogeneous Data. Arxiv preprint arXiv:1903.01672, 2019.
    [pdf] [Google scholar]

  • M. Lanne, J. Luoto. Identification of Economic Shocks by Inequality Constraints in Bayesian Structural Vector Autoregression. Oxford Bulletin of Economics and Statistics, xx(xx): xx-xx, 2019.
    [pdf] [Google scholar]

  • B. Huang, K. Zhang, M. Gong, C. Glymour. Causal Discovery and Forecasting in Nonstationary Environments with State-Space Models. In Proc. 36rd International Conference on Machine Learning (ICML2019), pp. xx-xx, Long Beach, California, 2019.
    [pdf] [Google scholar]

  • A. Tank, E. Fox, and A. Shojaie. Identifiability and estimation of structural vector autoregressive models for subsampled and mixed-frequency time series. Biometrika, 106(2), 433-452, 2019.
    [pdf] [Google scholar]

  • C. Gouriéroux, A. Monfort, and J.-P. Rennec. Identification and estimation in non-fundamental structural VARMA models. xx, 2017.
    [pdf] [Google scholar]

  • M. Lanne, M. Meitz and P. Saikkonen. Identification and estimation of non-Gaussian structural vector autoregressions. Journal of Econometrics, 196: 288-304, 2017.
    [pdf] [Google scholar]

  • M. Gong, K. Zhang, B. Schölkopf, C. Glymour, and D. Tao. Causal discovery from temporally aggregated time series. In Proc. 33rd Conference on Uncertainty in Artificial Intelligence (UAI2017), pp. xx-xx, Sydney, Australia, 2017.
    [pdf] [Google scholar]

  • T. H. Hai. Estimation of volatility causality in structural autoregressions with heteroskedasticity using independent component analysis. Statistical Papers, xx(xx): xx-xx, 2017.
    [pdf] [Google scholar]

  • S. Bauer, B. Schölkopf, and J. Peters. The arrow of time in multivariate time series. Arxiv preprint arXiv:1603.00784, 2016.
    [pdf] [Google scholar]

  • C. Gouriéroux, A. Monforta, and J.-P. Renne. Statistical inference for independent component analysis: Application to structural VAR models. Journal of Econometrics, xx(xx): xx-xx, 201x. Accepted.
    [pdf] [Google scholar]

  • W. Wiedermann and A. von Eye. Directional dependence in the analysis of single subjects. Journal of Person-Oriented Research, 2(1-2): 20-33, 2016.
    [pdf] [Google scholar]

  • M. Kalli and J. E. Griffin. Bayesian nonparametric vector autoregressive models. xx, 2016.
    [pdf] [Google scholar]

  • M. Lanne and J. Luot. Data-driven inference on sign restrictions in Bayesian structural vector autoregression. CREATES Research Paper 2016-4, 2016.
    [pdf] [Google scholar]

  • M. Lanne, M. Meitz, and P. Saikkon. Identification and estimation of non-Gaussian structural vector autoregressions. CREATES Research Paper 2015-1, 2015.
    [pdf] [Google scholar]

  • B. Huang, K. Zhang, and B. Schölkopf. Identification of time-dependent causal model: a Gaussian process treatment. In Proc. 24th International Joint Conference on Artificial Intelligence (IJCAI2015), pp. xx-xx, Buenos Aires, Argentina, 2015.
    [pdf] [Google scholar]

  • M. Gong, K. Zhang, B. Schölkopf, D. Tao, and P. Geiger. Discovering temporal causal relations from subsampled data. In Proc. 32nd International Conference on Machine Learning (ICML2015), pp. xx-xx, Lille, France, 2015.
    [pdf] [Google scholar]

  • L. Schiatti, G. Nollo, G. Rossato, and L. Faes. Extended Granger causality: A new tool to identify the structure of physiological networks. Physiological Measurement, 36: 827-843, 2015.
    [pdf] [Google scholar]

  • M. Lanne and P. Saikkonen. Noncausal vector autoregression. Econometric Theory, 29(3): 447-481, 2013.
    [pdf] [Google scholar]

  • Z. Chen, K. Zhang and L. Chan. Causal discovery with scale-mixture model for spatiotemporal variance dependencies. In Advances in Neural Information Processing Systems 25 (NIPS2012), pp. xx-xx, 2012.
    [pdf] [Google scholar]

  • W. Gao and H. Yang. Identifying structural VAR model with latent variables using overcomplete ICA. Far East Journal of Theoretical Statistics, 40(1): 31-44, 2012.
    [pdf] [Google scholar]

  • Y. Kawahara, S. Shimizu and T. Washio. Analyzing relationships among ARMA processes based on non-Gaussianity of external influences. Neurocomputing, 74(12-13): 2212-2221, 2011.
    [pdf] [Related code] [Python code by T. Ikeuchi and G. Haraoka] [real data] [Google scholar]

  • M. Lanne and H. Lütkepohl. Structural vector autoregressions with nonnormal residuals. Journal of Business & Economic Statistics, 28(1): 159-168, 2010.
    [pdf] [Google scholar]

  • L. Faes, S. Erla, A. Porta and G. Nollo. A framework for assessing frequency domain causality in physiological time series with instantaneous effects. Philosophical Transactions of the Royal Society A, 371: 20110618, 2013.
    [pdf] [Matlab code] [Google scholar]

  • L. Faes, S. Erla, E. Tranquillini, D. Orrico and G. Nollo. An identifiable model to assess frequency-domain Granger causality in the presence of significant instantaneous interactions. In Proc. 32nd Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS2010), pp.1699-1702, Buenos Aires, Argentina, 2010.
    [pdf] [Google scholar]

  • A. Hyvärinen, K. Zhang, S. Shimizu, P. O. Hoyer. Estimation of a structural vector autoregressive model using non-Gaussianity. Journal of Machine Learning Research, 11(May): 1709−1731, 2010.
    [pdf] [Google scholar]
    R code by Doris Entner
    Matlab code by Luca Faes
    Python code by T. Ikeuchi and G. Haraoka

  • A. Hyvärinen, S. Shimizu and P. O. Hoyer. Causal modelling combining instantaneous and lagged effects: an identifiable model based on non-Gaussianity. In Proc. Int. Conf. on Machine Learning (ICML2008), pp. 424-431, Helsinki, Finland, 2008.
    [pdf] [videolecture] [Google scholar]

Others

  • K. Du, Y. Xiang . Causal Inference Using Linear Time-Varying Filters with Additive Noise. Arxiv preprint arXiv:2012.13025, 2020.
    [pdf] [Google scholar]

  • D. Malinsky, and P. Spirtes. Causal Structure Learning from Multivariate Time Series in Settings with Unmeasured Confounding. In Proc. 2018 ACM SIGKDD Workshop on causal discovery (CD2018), pp. xx-xx, London, UK, 2015.
    [pdf] [Google scholar]

  • P. Geiger, K. Zhang, M. Gong, D. Janzing, and B. Schölkopf. Causal inference by identification of vector autoregressive processes with hidden components. In Proc. 32nd International Conference on Machine Learning (ICML2015), pp. xx-xx, Lille, France, 2015.
    [pdf] [Google scholar]

  • P. Morales-Mombiela, D. Hernández-Lobato and A. Suárez. Statistical tests for the detection of the arrow of time in vector autoregressive models. In Proc. 23rd International Joint Conference on Artificial Intelligence (IJCAI2013), pp. 1544-1550, Beijing, China, 2013.
    [pdf] [Google scholar]

  • J. M. Hernández-Lobato, P. Morales-Mombiela and A. Suárez. Gaussianity measures for detecting the direction of causal time series. In Proc. 22nd International Joint Conference on Artificial Intelligence (IJCAI2011), pp. 1318-1323, Barcelona, Catalonia (Spain), 2011.
    [pdf] [Google scholar]

  • R. Henao and O. Winther. Sparse linear identifiable multivariate modeling. Journal of Machine Learning Research, 12(Mar): 863--905, 2011.
    [pdf] [code] [Google scholar]

  • D. Janzing. On the entropy production of time series with unidirectional linearity. Journal of Statistical Physics, 138(4-5): 767-779, 2010.
    [pdf] [Google scholar]

  • J. Peters, D. Janzing, A. Gretton and B. Schölkopf. Detecting the direction of causal time series. In Proc. 26th Int. Conf. on Machine Learning (ICML2009), pp. 801-808, Montreal, Canada, 2009.
    [pdf] [code] [Google scholar]

Cyclic models

  • A. Hyvärinen and S. M. Smith. Pairwise likelihood ratios for estimation of non-Gaussian structural equation models. Journal of Machine Learning Research, 14(Jan): 111--152, 2013.
    [pdf] [Matlab code] [Google scholar]

  • J. Hirayama and A. Hyvärinen. Structural equations and divisive normalization for energy-dependent component analysis. In Advances in Neural Information Processing Systems 24 (NIPS2011), pp. xx-xx, 2011.
    [pdf] [Google scholar]

  • G. Lacerda, P. Spirtes, J. Ramsey and P. O. Hoyer. Discovering cyclic causal models by independent components analysis. In Proc. 24th Conf. on Uncertainty in Artificial Intelligence (UAI2008), pp. 366-374, Helsinki, Finland, 2008.
    [pdf] [videolecture] [TETRAD IV] [Google scholar]