The PCMCI algorithm is proposed by Runge et al. [2019], aiming to detect time-lagged causal See Fig.1 for more detail. A simple proof is shown below through Markov assumption ( A2). 3 Figure 2: Partial causal graph for 3-variate time series Fig.2 shows a partial causal graph for a 3-variate time series with Semi-Stationary SCM. However, they may not share the same marginal distribution. Still in Fig.2, based on the definition of homogenous time partition, time partition subset Based on Eq.(12) and Eq.(17), we have: p(X Without loss of generality, we assume T is a multiple of δ all the time. A1-A7 and with an oracle (infinite sample size limit), we have that: null G = G (47) almost surely.

Discovering causal relations from observational time series without making the stationary assumption is a significant challenge.

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Crystal Structure Prediction (CSP) is crucial in various scientific disciplines.

However, existing ways of implementing the ME-HP for such data are either inflexible, as the exogenous (background) rate functions are typically constant and the endogenous (excitation) rate functions are specified parametrically, or inefficient, as inference usually relies on Markov chain Monte Carlo methods with high computational costs.

Periodic graphs are graphs consisting of repetitive local structures, such as crystal nets and polygon mesh. Their generative modeling has great potential in real-world applications such as material design and graphics synthesis. Classical models either rely on domain-specific predefined generation principles (e.g., in crystal net design), or follow geometry-based prescribed rules. Recently, deep generative models have shown great promise in automatically generating general graphs. However, their advancement into periodic graphs has not been well explored due to several key challenges in 1) maintaining graph periodicity; 2) disentangling local and global patterns; and 3) efficiency in learning repetitive patterns.

Rhythmic fluctuations in acoustic energy and accompanying neuronal excitations in cortical oscillations are characteristic of human speech, yet whether a corresponding rhythmicity inheres in the articulatory movements that generate speech remains unclear. The received understanding of speech movements as discrete, goal-oriented actions struggles to make contact with the rhythmicity findings. In this work, we demonstrate that an unintuitive -- but no less principled than the conventional -- representation for discrete movements reveals a pervasive limit cycle organization and unlocks the recovery of previously inaccessible rhythmic structure underlying the motor activity of speech. These results help resolve a time-honored tension between the ubiquity of biological rhythmicity and discreteness in speech, the quintessential human higher function, by revealing a rhythmic organization at the most fundamental level of individual articulatory actions.