Deep multi-modal learning for joint linear representation of nonlinear dynamical systems.

Dynamical systems pervasively seen in most real-life applications are complex and behave by following certain evolution rules or dynamical patterns, which are linear, non-linear, or stochastic. The underlying dynamics (or evolution rule) of such complex systems, if found, can be used for understanding the system behavior, and furthermore for system prediction and control. It is common to analyze the system's dynamics through observations in different modality approaches. For instance, to recognize patient deterioration in acute care, it usually relies on monitoring and analyzing vital signs and other observations, such as blood pressure, heart rate, respiration, and electroencephalography. These observations convey the information describing the same target system, but the dynamics is not able to be directly characterized due to high complexity of individual modality and maybe time-delay interactions among modalities. In this work, we suppose that the state behavior of a dynamical system follows an intrinsic dynamics shared among these modalities. We specifically propose a new deep auto-encoder framework using the Koopman operator theory to derive the joint linear dynamics for a target system in a space spanned by the intrinsic coordinates. The proposed method aims to reconstruct the original system states by learning the information provided among multiple modalities. Furthermore, with the derived intrinsic dynamics, our method is capable of restoring the missing observations within and across modalities, and used for predicting the future states of the system that follows the same evolution rule.

Recurrence Quantification Analysis of Ankle Kinematics During Gait in Individuals With Chronic Ankle Instability.

Purpose: An investigation of the ankle dynamics in a motor task may generate insights into the etiology of chronic ankle instability (CAI). This study presents a novel application of recurrence quantification analysis (RQA) to examine the ankle dynamics during walking. We hypothesized that CAI is associated with changes in the ankle dynamics as assessed by measures of determinism and laminarity using RQA. Methods: We recorded and analyzed the ankle position trajectories in the frontal and sagittal planes from 12 participants with CAI and 12 healthy controls during treadmill walking. We used time-delay embedding to reconstruct the position trajectories to a phase space that represents the states of the ankle dynamics. Based on the phase space trajectory, a recurrence plot was constructed and two RQA variables, the percent determinism (%DET) and the percent laminarity (%LAM), were derived from the recurrence plot to quantify the ankle dynamics. Results: In the frontal plane, the %LAM in the CAI group was significantly lower than that in the control group (p < 0.05. effect size = 0.86). This indicated that the ankle dynamics in individuals with CAI is less likely to remain in the same state. No significant results were found in the %DET or in the sagittal plane. Conclusion: A lower frontal-plane %LAM may reflect more frequent switching between different patterns of neuromuscular control states due to the instabilities associated with CAI. With further study and development, %LAM may have the potential to become a useful biomarker for CAI.