The Arrow of Time in Multivariate Time Series

By Stefan Bauer, Bernhard Schölkopf, Jan Peters. 2016

33 citations recorded by the source corpus.

We prove that a time series satisfying a (linear) multivariate autoregressive moving average (VARMA) model satisfies the same model assumption in the reversed time direction, too, if all innovations are normally distributed. This reversibility breaks down if the innovations are non-Gaussian. This means that under the assumption of a VARMA process with non-Gaussian noise, the arrow of time becomes detectable. Our work thereby provides a theoretic justification of an algorithm that has been used for inferring the direction of video snippets. We present a slightly modified practical algorithm that estimates the time direction for a given sample and prove its consistency. We further investigate how the performance of the algorithm depends on sample size, number of dimensions of the time series and the order of the process. An application to real world data from economics shows that considering multivariate processes instead of univariate processes can be beneficial for estimating the time direction. Our result extends earlier work on univariate time series. It relates to the concept of causal inference, where recent methods exploit non-Gaussianity of the error terms for causal structure learning.

Subfields: Statistics and Probability, Signal Processing, Time-Series & Forecasting

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Citations in the corpus

Toward Causal Representation Learning (2021, background): “On the other hand, although methods for learning causal structure from observations exist [18], [37], [83], [113], [123], [139], [161], [174]–[176], [188]–[190], [229], [237], [246], [279], learning causal relations frequently requires collecting data from multiple environments or the ability to per”
Towards Causal Representation Learning (2021, background): “[237, 188, 229, 113, 174, 187, 139, 17, 246, 277, 175, 123, 186, 176, 36, 82, 161], learning causal relations frequently requires collecting data from multiple environments, or the ability to perform interventions [251].”
Causal Models for Dynamical Systems (2020, background): “Peters et al. [2009], Bauer et al. [2016] discuss the relation between causality and the arrow of time.”
Causality for Machine Learning (2019, background): “…X,Y generated by an additive noise model, one cannot fit an additive noise model in the opposite direction (i.e., with the roles of X and Y interchanged) (Hoyer et al., 2009; Mooij et al., 2009; Peters et al., 2014; Kpotufe et al., 2014; Bauer et al., 2016), cf. also the work of Sun et al. (2006).”
SAITS: Self-Attention-based Imputation for Time Series (2022, background): “Multivariate time-series data is ubiquitous in many application domains, for instance, transportation [1, 2], economics [3, 4], healthcare [5, 6, 7], and meteorology [8, 9, 10].”
BRITS: Bidirectional Recurrent Imputation for Time Series (2018, background): “Multivariate time series data are abundant in many application areas, such as financial marketing [5, 4], health-care [10, 22], meteorology [31, 26], and traffic engineering [29, 35].”
Causal Inference via Kernel Deviance Measures (2018, methodology): “On the other hand, the method developed in [Bauer et al., 2016] that models the data as an autoregressive moving average model with non-Gaussian noise correctly identiﬁes the causal direction.”