68,176 research outputs found
Multivariate Granger Causality and Generalized Variance
Granger causality analysis is a popular method for inference on directed
interactions in complex systems of many variables. A shortcoming of the
standard framework for Granger causality is that it only allows for examination
of interactions between single (univariate) variables within a system, perhaps
conditioned on other variables. However, interactions do not necessarily take
place between single variables, but may occur among groups, or "ensembles", of
variables. In this study we establish a principled framework for Granger
causality in the context of causal interactions among two or more multivariate
sets of variables. Building on Geweke's seminal 1982 work, we offer new
justifications for one particular form of multivariate Granger causality based
on the generalized variances of residual errors. Taken together, our results
support a comprehensive and theoretically consistent extension of Granger
causality to the multivariate case. Treated individually, they highlight
several specific advantages of the generalized variance measure, which we
illustrate using applications in neuroscience as an example. We further show
how the measure can be used to define "partial" Granger causality in the
multivariate context and we also motivate reformulations of "causal density"
and "Granger autonomy". Our results are directly applicable to experimental
data and promise to reveal new types of functional relations in complex
systems, neural and otherwise.Comment: added 1 reference, minor change to discussion, typos corrected; 28
pages, 3 figures, 1 table, LaTe
Challenges of Big Data Analysis
Big Data bring new opportunities to modern society and challenges to data
scientists. On one hand, Big Data hold great promises for discovering subtle
population patterns and heterogeneities that are not possible with small-scale
data. On the other hand, the massive sample size and high dimensionality of Big
Data introduce unique computational and statistical challenges, including
scalability and storage bottleneck, noise accumulation, spurious correlation,
incidental endogeneity, and measurement errors. These challenges are
distinguished and require new computational and statistical paradigm. This
article give overviews on the salient features of Big Data and how these
features impact on paradigm change on statistical and computational methods as
well as computing architectures. We also provide various new perspectives on
the Big Data analysis and computation. In particular, we emphasis on the
viability of the sparsest solution in high-confidence set and point out that
exogeneous assumptions in most statistical methods for Big Data can not be
validated due to incidental endogeneity. They can lead to wrong statistical
inferences and consequently wrong scientific conclusions
Parameter estimation of ODE's via nonparametric estimators
Ordinary differential equations (ODE's) are widespread models in physics,
chemistry and biology. In particular, this mathematical formalism is used for
describing the evolution of complex systems and it might consist of
high-dimensional sets of coupled nonlinear differential equations. In this
setting, we propose a general method for estimating the parameters indexing
ODE's from times series. Our method is able to alleviate the computational
difficulties encountered by the classical parametric methods. These
difficulties are due to the implicit definition of the model. We propose the
use of a nonparametric estimator of regression functions as a first-step in the
construction of an M-estimator, and we show the consistency of the derived
estimator under general conditions. In the case of spline estimators, we prove
asymptotic normality, and that the rate of convergence is the usual
-rate for parametric estimators. Some perspectives of refinements of
this new family of parametric estimators are given.Comment: Published in at http://dx.doi.org/10.1214/07-EJS132 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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