How to estimate the correlation dimension of high-dimensional signals?

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.

Using curvature to select the time lag for delay reconstruction

We propose a curvature-based approach for choosing a good value for the time-delay parameter τ in delay reconstructions. The idea is based on the effects of the delay on the geometry of the reconstructions. If the delay is too small, the reconstructed dynamics are flattened along the main diagonal of the embedding space; too-large delays, on the other hand, can overfold the dynamics. Calculating the curvature of a two-dimensional delay reconstruction is an effective way to identify these extremes and to find a middle ground between them: both the sharp reversals at the extremes of an insufficiently unfolded reconstruction and the bends in an overfolded one create spikes in the curvature. We operationalize this observation by computing the mean Menger curvature of a trajectory segment on 2D reconstructions as a function of time delay. We show that the minimum of these values gives an effective heuristic for choosing the time delay. In addition, we show that this curvature-based heuristic is useful even in cases where the customary approach, which uses average mutual information, fails-e.g., noisy or filtered data.

High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpiński resonators: Effect of fractal depth and irregularity

Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic dynamics even in the context of an elementary, single-transistor oscillator circuit. Sierpiński gaskets of variable depth are constructed using discrete capacitors and inductors, whose values are scaled according to a simple sequence. It is found that in regular fractals of this kind, each iteration effectively adds a conjugate pole/zero pair, yielding gradually more complex and broader frequency responses, which can also be implemented as much smaller Foster equivalent networks. The resonators are instanced in the circuit as one-port devices, replacing the inductors found in the initial version of the oscillator. By means of a highly simplified numerical model, it is shown that increasing the fractal depth elevates the dimension of the chaotic dynamics, leading to high-order hyperchaos. This result is overall confirmed by SPICE simulations and experiments, which however also reveal that the non-ideal behavior of physical components hinders obtaining high-dimensional dynamics. The issue could be practically mitigated by building the Foster equivalent networks rather than the verbatim fractals. Furthermore, it is shown that considerably more complex resonances, and consequently richer dynamics, can be obtained by rendering the fractal resonators irregular through reshuffling the inductors, or even by inserting a limited number of focal imperfections. The present results draw attention to the potential usefulness of fractal resonators for generating high-dimensional chaotic dynamics, and underline the importance of irregularities and component non-idealities.