Fig. 7.

HAVOK analysis applied to a number examples, including analytical systems (Lorenz and Rössler), stochastic magnetic field reversal, and systems characterized from real-world data (electrocardiogram, electroencephalogram, and measles outbreaks). The model is extremely accurate for the first four analytical cases, providing faithful attractor reconstruction and predicting dominant transient and intermittent events. Similarly, in the case of measles outbreaks, the forcing signal is potentially predictive of large transients corresponding to outbreaks. The examples are characterized by nearly symmetric forcing distributions with fat tails (Gaussian forcing is shown in black dashed line), corresponding to rare forcing events. Nonlinear measurements y(t) = g(x(t)) may be used in (4) to enhance features of the embedded attractor. This HAVOK analysis builds on the decades of existing time-delay embedding literature by providing accurate intermittently forced linear regression models for chaotic dynamics. Credit for images in the left column: (Earth’s magnetic field) Zureks on Wikimedia Commons; (human heart) Public domain; (human brain) Sanger Brown M.D. on Wikimedia Commons; (measles) CDC/Cynthia S. Goldsmith, William Bellini, Ph.D