Statistical Science

On the Relationship between the Theory of Cointegration and the Theory of Phase Synchronization

Rainer Dahlhaus, István Z. Kiss, and Jan C. Neddermeyer

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Abstract

The theory of cointegration has been a leading theory in econometrics with powerful applications to macroeconomics during the last decades. On the other hand, the theory of phase synchronization for weakly coupled complex oscillators has been one of the leading theories in physics for many years with many applications to different areas of science. For example, in neuroscience phase synchronization is regarded as essential for functional coupling of different brain regions. In an abstract sense, both theories describe the dynamic fluctuation around some equilibrium. In this paper, we point out that there exists a very close connection between both theories. Apart from phase jumps, a stochastic version of the Kuramoto equations can be approximated by a cointegrated system of difference equations. As one consequence, the rich theory on statistical inference for cointegrated systems can immediately be applied for statistical inference on phase synchronization based on empirical data. This includes tests for phase synchronization, tests for unidirectional coupling and the identification of the equilibrium from data including phase shifts. We study two examples on a unidirectionally coupled Rössler–Lorenz system and on electrochemical oscillators. The methods from cointegration may also be used to investigate phase synchronization in complex networks. Conversely, there are many interesting results on phase synchronization which may inspire new research on cointegration.

Article information

Source
Statist. Sci., Volume 33, Number 3 (2018), 334-357.

Dates
First available in Project Euclid: 13 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ss/1534147227

Digital Object Identifier
doi:10.1214/18-STS659

Mathematical Reviews number (MathSciNet)
MR3843380

Zentralblatt MATH identifier
06991124

Keywords
Cointegration phase synchronization weakly coupled oscillators driver response relationship Rössler–Lorenz system

Citation

Dahlhaus, Rainer; Kiss, István Z.; Neddermeyer, Jan C. On the Relationship between the Theory of Cointegration and the Theory of Phase Synchronization. Statist. Sci. 33 (2018), no. 3, 334--357. doi:10.1214/18-STS659. https://projecteuclid.org/euclid.ss/1534147227


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