Estimating exogenous variables in data with more variables than observations

Sogawa, Yasuhiro; Shimizu, Shohei; Shimamura, Teppei; Hyvärinen, Aapo; Washio, Takashi; Imoto, Seiya

doi:10.1016/j.neunet.2011.05.017

Cited by 12 publications

(4 citation statements)

References 17 publications

(20 reference statements)

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“…Though the Gaussian approximation has been a common approach [19], real-world data could be considered more or less non-Gaussian. In fact, non-Gaussian data appear in many applications including bioinformatics [20,21].…”

Section: Modelmentioning

confidence: 99%

Learning LiNGAM based on data with more variables than observations

Shimizu¹

2012

Preprint

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A very important topic in systems biology is developing statistical methods that automatically find causal relations in gene regulatory networks with no prior knowledge of causal connectivity. Many methods have been developed for time series data. However, discovery methods based on steady-state data are often necessary and preferable since obtaining time series data can be more expensive and/or infeasible for many biological systems. A conventional approach is causal Bayesian networks. However, estimation of Bayesian networks is ill-posed. In many cases it cannot uniquely identify the underlying causal network and only gives a large class of equivalent causal networks that cannot be distinguished between based on the data distribution. We propose a new discovery algorithm for uniquely identifying the underlying causal network of genes. To the best of our knowledge, the proposed method is the first algorithm for learning gene networks based on a fully identifiable causal model called LiNGAM. We here compare our algorithm with competing algorithms using artificially-generated data, although it is definitely better to test it based on real microarray gene expression data.

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Section: Modelmentioning

confidence: 99%

Learning LiNGAM based on data with more variables than observations

Shimizu¹

2012

Preprint

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“…The singular Wishart distribution has been studied for instance by Uhlig (1994), Janik & Nowak (2003), Srivastava (2003), Yu & Zhang (2002) and Yu, Ryu & Park (2014). As pointed out by Dudoit, Fridlyand & Speed (2002) and Sogawa, Shimizu & Shimamura (2011), cases where the number of variables,

p

, exceeds the number of observations frequently occur in gene expression datasets, for instance. Ratnarajah (2005) made use of the complex singular Wishart distribution to address a problem involving multiple‐antenna systems.…”

Section: Introductionmentioning

confidence: 99%

On the singular gamma, Wishart, and beta matrix‐variate density functions

Mathai

Provost

2022

Can J Statistics

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When a p × p real positive definite matrix 𝑆 follows a Wishart or, more generally, a matrix-variate gamma distribution with shape parameter 𝛼 and positive definite scale parameter matrix B, one can represent 𝑆 as XX ′ for some matrix X of dimension p × q. When p > q, 𝑆 has a singular distribution whose properties can be studied via the density function of X. It will be shown that when X follows a matrix-variate extended Gaussian distribution, the density function of the resulting singular gamma distribution can be obtained by making use of successive transformations and their associated Jacobians. The singular Wishart distribution will then be obtained as a particular case. The marginal and conditional density functions resulting from an arbitrary partitioning of X will be considered as well. The same technique will also be applied to the derivation of the density functions of real and complex singular type-1 and type-2 beta-distributed matrices. It so happens that the proposed approach, which is based on manipulations involving the wedge products of certain differential elements, generally proves more efficient than the intricate procedures that have hitherto been employed in the literature.Résumé: Lorsqu'une matrice réelle définie positive 𝑆 (p × p) suit une loi de Wishart ou, plus généralement, une distribution gamma de paramètre de forme 𝛼 et de paramètre d'échelle une matrice définie positive B, il est possible de représenter 𝑆 sous forme XX ′ où X est une matrice de dimension p × q. Lorsque p > q, 𝑆 a une distribution singulière dont les propriétés peuvent être étudiées via la fonction de densité de X. Les auteurs de cet article montrent que lorsque X suit une distribution gaussienne étendue à variable matricielle, la fonction de densité de la distribution gamma singulière résultante peut être obtenue en utilisant des transformations successives et leurs Jacobiens associés. La distribution de Wishart singulière en découle comme un cas particulier. Le cas de fonctions de densité marginale et conditionnelle résultant d'un partitionnement arbitraire de X a également été traité. La même technique a été utilisée pour déterminer les fonctions de densité de matrices singulières réelles et complexes de lois beta de type 1 et de type 2. Ainsi l'approche proposée, basée sur des manipulations mettant en jeu les produits extérieurs de certains éléments différentiels, s'avère généralement plus efficace que les procédures complexes utilisées jusqu'à présent dans la littérature.

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“…A major reason for this disadvantage is that this method explicitly or implicitly assumes the Gaussianity of data, and typically utilizes only the covariance structures of data for estimating causal relations. However, in many applications, it is common for non-Gaussian data to be obtained (Micceri, 1989;Hyvärinen et al, 2001;Smith et al, 2011;Sogawa et al, 2011;Moneta et al, 2013), which means that more information can be contained in the data distribution than in the covariance matrix. Bentler (1983) proposed making use of non-Gaussianity of data for estimating structural equation models, although this had not been extensively studied until recently.…”

Section: Introductionmentioning

confidence: 99%

Lingam: Non-Gaussian Methods for Estimating Causal Structures

Shimizu

2014

Behaviormetrika

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In many empirical sciences, the causal mechanisms underlying various phenomena need to be studied. Structural equation modeling is a general framework used for multivariate analysis, and provides a powerful method for studying causal mechanisms. However, in many cases, classical structural equation modeling is not capable of estimating the causal directions of variables. This is because it explicitly or implicitly assumes Gaussianity of data and typically utilizes only the covariance structure of data. In many applications, however, non-Gaussian data are often obtained, which means that more information may be contained in the data distribution than the covariance matrix is capable of containing. Thus, many new methods have recently been proposed for utilizing the non-Gaussian structure of data and estimating the causal directions of variables. In this paper, we provide an overview of such recent developments in causal inference, and focus in particular on the non-Gaussian methods known as LiNGAM.

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Estimating exogenous variables in data with more variables than observations

Cited by 12 publications

References 17 publications

Learning LiNGAM based on data with more variables than observations

Learning LiNGAM based on data with more variables than observations

On the singular gamma, Wishart, and beta matrix‐variate density functions

Lingam: Non-Gaussian Methods for Estimating Causal Structures

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