Evidence for power law localization in disordered systems

A scale-dependent measure of system dimensionality

A fundamental problem in science is uncovering the effective number of degrees of freedom in a complex system: its dimensionality. A system’s dimensionality depends on its spatiotemporal scale. Here, we introduce a scale-dependent generalization of a classic enumeration of latent variables, the participation ratio. We demonstrate how the scale-dependent participation ratio identifies the appropriate dimension at local, intermediate, and global scales in several systems such as the Lorenz attractor, hidden Markov models, and switching linear dynamical systems. We show analytically how, at different limiting scales, the scale-dependent participation ratio relates to well-established measures of dimensionality. This measure applied in neural population recordings across multiple brain areas and brain states shows fundamental trends in the dimensionality of neural activity—for example, in behaviorally engaged versus spontaneous states. Our novel method unifies widely used measures of dimensionality and applies broadly to multivariate data across several fields of science.

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The scale-dependent dimensionality unifies widely used measures of dimensionality

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Dynamical systems show distinct dimensionality properties at different scales

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The scale-dependent dimensionality allows us to identify critical scales of the system

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Fundamental trends in dimensionality of neural activity depend on the brain state

Data mining is based on the discovery of structure within data. However, such a structure is often complex. The fact that the properties of data distributions vary depending on the scale at which they are examined is a fundamental component of this complexity. For example, a manifold may appear smooth at small scales but jagged or even fractal at larger scales. This scale dependence is critical, yet it is commonly overlooked. We introduce a fundamental approach for analyzing the properties of data distributions at all scales. This single scale-dependent description enables simultaneous examination of how characteristics vary across all scales, offering insight into the structure of the data distribution. This will help us gain a better grasp of data structures and pave the way for future theoretical advances in data science.