Criticality in a multisignal system using principal component analysis

In systems with dynamical transitions, criticality is usually defined by the behavior of suitable individual variables of the system. In the case of time series, the usual procedure involves the analysis of the statistical properties of the selected variable as a function of a control parameter in both the time and frequency domains. An interesting question, however, is how to identify criticality when multiple simultaneous signals are required to provide a reliable representation of the system, especially when the signals exhibit different dynamics and do not individually display the characteristic signs of criticality. In that situation, a technique that analyzes the collective behavior of the signals is necessary. In this work we show that the eigenvalues and eigenvectors obtained from principal components analysis (PCA) can be used as a way to identify collective criticality. To do this, we construct a multilayer Ising model comprised of coupled two-dimensional Ising lattices that have distinct critical temperatures when isolated. We apply PCA to the collection of magnetization signals for a range of global temperatures and study the resulting eigenvalues. We find that there exists a single global temperature at which the eigenvalue spectrum follows a power law, and identify this as an indicator of "multicriticality" for the system. We then apply the technique to electroencephalographic recordings of brain activity, as this is a prime example of multiple signals with distinct individual dynamics. The analysis reveals a power-law eigenspectrum, adding further evidence to the brain criticality hypothesis. We also show that the eigenvectors can be used to distinguish the recordings in the resting state from those during a cognitive task, and that there is important information contained in all eigenvectors, not just the first few dominant ones, establishing that PCA has great utility beyond dimensionality reduction.