Evaluation of Geometric Attractor Structure and Recurrence Analysis in Professional Dancers

BACKGROUND

Human motor systems contain nonlinear features. The purpose of this study was to evaluate the geometric structure of attractors and analyze recurrence in two different pirouettes (jazz and classic) performed by 15 professional dancers.

METHODS

The kinematics of the body's center of mass (CoM) and knee of the supporting leg (LKNE) during the pirouette were measured using the Vicon system. A time series of selected points were resampled, normalized, and randomly reordered. Then, every second time series was flipped to be combined with other time series and make a long time series out of the repetitions of a single task. The attractors were reconstructed, and the convex hull volumes (CHV) were counted for the CoM and LKNE for each pirouette in each direction. Recurrence quantification analysis (RQA) was used to extract additional information.

RESULTS

The CHVs calculated for the LKNE were significantly lower for the jazz pirouette. All RQA measures had the highest values for LKNE along the mediolateral axis for the jazz pirouette. This result underscores the high determinism, high motion recurrence, and complexity of this maneuver.

CONCLUSIONS

The findings offer new insight into the evaluation of the approximation of homogeneity in motion control. A high determinism indicates a highly stable and predictive motion trajectory.

Interpolating Strange Attractors via Fractional Brownian Bridges

We present a novel method for interpolating univariate time series data. The proposed method combines multi-point fractional Brownian bridges, a genetic algorithm, and Takens' theorem for reconstructing a phase space from univariate time series data. The basic idea is to first generate a population of different stochastically-interpolated time series data, and secondly, to use a genetic algorithm to find the pieces in the population which generate the smoothest reconstructed phase space trajectory. A smooth trajectory curve is hereby found to have a low variance of second derivatives along the curve. For simplicity, we refer to the developed method as PhaSpaSto-interpolation, which is an abbreviation for phase-space-trajectory-smoothing stochastic interpolation. The proposed approach is tested and validated with a univariate time series of the Lorenz system, five non-model data sets and compared to a cubic spline interpolation and a linear interpolation. We find that the criterion for smoothness guarantees low errors on known model and non-model data. Finally, we interpolate the discussed non-model data sets, and show the corresponding improved phase space portraits. The proposed method is useful for interpolating low-sampled time series data sets for, e.g., machine learning, regression analysis, or time series prediction approaches. Further, the results suggest that the variance of second derivatives along a given phase space trajectory is a valuable tool for phase space analysis of non-model time series data, and we expect it to be useful for future research.

High-Dimensional Phase Space Reconstruction with a Convolutional Neural Network for Structural Health Monitoring

In order to accurately diagnose the health of high-order statically indeterminate structures, most existing structural health monitoring (SHM) methods require multiple sensors to collect enough information. However, comprehensive data collection from multiple sensors for high degree-of-freedom structures is not typically available in practice. We propose a method that reconciles the two seemingly conflicting difficulties. Takens’ embedding theorem is used to augment the dimensions of data collected from a single sensor. Taking advantage of the success of machine learning in image classification, high-dimensional reconstructed attractors were converted into images and fed into a convolutional neural network (CNN). Attractor classification was performed for 10 damage cases of a 3-story shear frame structure. Numerical results show that the inherently high dimension of the CNN model allows the handling of higher dimensional data. Information on both the level and the location of damage was successfully embedded. The same methodology will allow the extraction of data with unsupervised CNN classification to be consistent with real use cases.

Extracting new information from old waveforms: Symmetric projection attractor reconstruction: Where maths meets medicine

NEW FINDINGS

What is the topic of this review? Symmetric Projection Attractor Reconstruction (SPAR) is a relatively new mathematical method that can extract additional information pertaining to the morphology and variability of physiological waveforms, such as arterial pulse pressure. Herein, we describe the potential utility of the method for more sensitive quantification of cardiovascular changes. What advances does it highlight? We use a simple example of a human tilt table to illustrate these concepts. SPAR can be used on any approximately periodic waveform and may add value to experimental and clinical settings, where such signals are collected routinely.

ABSTRACT

Periodic physiological waveform data, such as blood pressure, pulse oximetry and ECG, are routinely sampled between 100 and 1000 Hz in preclinical research and in the clinical setting from a wide variety of implantable, bedside and wearable monitoring devices. Despite the underlying numerical waveform data being captured at such high fidelity, conventional analysis tends to reside in reporting only averages of minimum, maximum, amplitude and rate, as single point averages. Although these averages are undoubtedly of value, simplification of the data in this way means that most of the available numerical data are discarded. In turn, this may lead to subtle physiological changes being missed when investigating the cardiovascular system over time. We have developed a mathematical method (symmetric projection attractor reconstruction) that uses all the numerical data, replotting and revisualizing them in a manner that allows unique quantification of multiple changes in waveform morphology and variability. We propose that the additional quantification of these features will allow the complex behaviour of the cardiovascular system to be mapped more sensitively in different physiological and pathophysiological settings.