Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 6917658, 13 pages

https://doi.org/10.1155/2019/6917658

## Assessment of Local Dynamic Stability in Gait Based on Univariate and Multivariate Time Series

^{1}Institute of Informatics, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland^{2}Centre for Research and Development, Polish-Japanese Academy of Information Technology, Aleja Legionów 2, 41-902 Bytom, Poland

Correspondence should be addressed to Henryk Josiński; lp.lslop@iksnisoj.kyrneh

Received 22 March 2019; Revised 26 June 2019; Accepted 3 July 2019; Published 25 July 2019

Academic Editor: Marko Gosak

Copyright © 2019 Henryk Josiński et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The ability of the locomotor system to maintain continuous walking despite very small external or internal disturbances is called local dynamic stability (LDS). The importance of the LDS requires constantly working on different aspects of its assessment method which is based on the short-term largest Lyapunov exponent (LLE). A state space structure is a vital aspect of the LDS assessment because the algorithm of the LLE computation for experimental data requires a reconstruction of a state space trajectory. The gait kinematic data are usually one- or three-dimensional, which enables to construct a state space based on a uni- or multivariate time series. Furthermore, two variants of the short-term LLE are present in the literature which differ in length of a time span, over which the short-term LLE is computed. Both a state space structure and the consistency of the observations based on values of both short-term LLE variants were analyzed using time series representing the joint angles at ankle, knee, and hip joints. The short-term LLE was computed for individual joints in three state spaces constructed on the basis of either univariate or multivariate time series. Each state space revealed walkers’ locally unstable behavior as well as its attenuation in the current stride. The corresponding conclusions made on the basis of both short-term LLE variants were consistent in ca. 59% of cases determined by a joint and a state space. Moreover, the authors present an algorithm for estimation of the embedding dimension in the case of a multivariate gait time series.

#### 1. Introduction

Stability means the ability to return to a stable state after having been subjected to some form of perturbation. Focusing on gait, if infinitesimally small perturbations, naturally occurring tiny variations in the walking surface and/or natural noise in the neuromuscular system, are concerned, then the ability of the locomotor system to keep the gait smooth by attenuating them is called local dynamic stability (LDS) [1]. The aforementioned disturbances are the cause of slightly different conditions at the beginning of successive strides. As a consequence, the LDS can be assessed using a measure of the extreme sensitivity to initial conditions.

Gait stability is of great importance for older people who are considered prone to falls. It requires constantly working on different aspects of LDS assessment method, which is derived from the dynamical systems theory. The method is based on a trajectory in a state space which is reconstructed from time series generated by a dynamical system. The dynamical properties of a system in the true state space are preserved under the reconstruction process, which enables to analyze the system’s behavior using the reconstructed trajectory, with particular emphasis on system’s sensitivity to initial conditions.

The authors intended to investigate how a state space structure affects the LDS. The input data for constructing a state space were time series describing the movement at a single joint. The authors created state spaces on the basis of one- or three-dimensional time series for hip, knee, and ankle joints separately and used the reconstructed trajectory for the LDS assessment according to the approach briefly described in the next section.

#### 2. Materials and Methods

##### 2.1. Theoretical Background

A symptom of extreme sensitivity to initial conditions is the exponential rate of divergence of trajectories from their starting points which are located in a state space very close to each other. This rate, which is called the largest Lyapunov exponent (LLE), is defined as follows:where is the initial time instant and represents a distance between corresponding points on initially nearby trajectories at any time instant . From the perspective of gait analysis, a positive LLE value indicates locally unstable behavior (i.e., trajectories diverge; however, due to the presence of the attractor the distance between them cannot grow without limit). The higher the LLE, the greater the system’s sensitivity to extremely small perturbations during gait and thus the lower the LDS. The LLE, which estimates the local stability immediately after a potential perturbation, is called the short-term largest Lyapunov exponent. The short-term LLE is computed over a time span of a length either corresponding to one step [2–4] or one stride [1, 5, 6] using the Rosenstein algorithm [7]. The idea behind this method is that pairs of segments of the state space trajectory reconstructed on the basis of experimental data repeatedly imitate two initially neighboring trajectories, which makes it possible to trace the divergence of them. A comprehensive and precise description of the component methods leading finally to the determination of LLE was included in the work of Perc [8].

A state space structure is an important aspect of the LDS assessment. The reconstruction procedure is based on two parameters: *time delay * (*reconstruction delay*, *lag*) and *embedding dimension *. For a time series, which is composed of points , an -element vector of delay coordinates of the point on the reconstructed trajectory is given by , where (the reconstructed trajectory consists of points) [9].

For a multivariate time series, which is composed of univariate time series of equal length , the reconstruction parameters are defined by a time delay vector and an embedding dimension vector . Therefore, the state space dimensionality is a sum of all , , and the delay coordinates of the point form a vector where [10, 11].

##### 2.2. Review of the Previous Work

Concentrating on the importance of analyzing LDS and utilizing its results, some other pieces of research deserve a mention, e.g., Terrier et al. [12] investigated LDS in patients with chronic impairments after foot and ankle injuries, Bruijn et al. [13] discussed the relationship between gait stability and arm swing, and Dingwell and Marin [5], as well as England and Granata [6], analyzed the influence of gait speed on LDS. Several studies [14, 15] indicate that LDS is associated with the fall risk. Moreover, LDS may be used as a potential fall predictor to differentiate fall-prone adults [16]. LDS turned out to be sensitive to age-related degeneration [2]. The authors of [3] show that dance training might improve LDS of normal walking of the elderly. Another important aspect of LDS is its suitability for monitoring of geriatric or neurological pathologies in their early phases [2]. Comprehensive reviews of measures assessing the stability of human locomotion were prepared by Hamacher et al. [17], Bruijn et al. [18], and Van Emmerik et al. [19].

The discussion of state space structures in the context of motion data was initiated by Gates and Dingwell [20] who focused on univariate time series based on Euler angles describing rotational motion of a shoulder. The authors conclude that the comparison between outcomes for different state spaces should be made with caution; however, the trends identified in the analyzed data remain relevant. Besides, the authors do not recommend the tested PCA-based reduction of a state space dimensionality. In [21], a multivariate time series, which represented movement at hip, knee, and ankle joints in the sagittal plane, was applied to analyze quiet standing balance. As far as the total embedding dimension for a multivariate time series is concerned, Vlachos and Kugiumtzis [10] presented two modified variants (FNN1 and FNN2) of the false nearest neighbors method [22], which were adjusted to a multivariate time series. According to FNN1, the same embedding dimension is applied to all the component time series. The FNN2 method is an exhaustive algorithm. The third method proposed in [10] was based on the criterion of prediction error minimization (PEM), whereas Zhang et al. [23] suggested applying a maximal joint entropy criterion. The consequences of a fixed time delay and/or a fixed embedding dimension were investigated by van Schooten et al. [4]. Hamacher et al. [2] evaluated multiple state space definitions differing in signal type (linear acceleration and angular velocity), signal dimension (one-dimensional and three-dimensional), and location of an inertial sensor (trunk and forefoot). Piórek et al. [24] used a quaternion-based interpretation of body segments’ rotations and replaced a multivariate time series of Euler angles by a quaternion angle time series. Moreover, they showed a correlation between LLE values computed for time series consisting of (1) quaternion angles and (2) joint angles in a group of young individuals for hip, knee, and ankle joints in different variants of walking speed and ground inclination. The same set of experimental data was also analyzed using a new quaternion-based variant of the approximate entropy measure [25]. A systematic review of methodological approaches of the LLE quantification was prepared by Mehdizadeh [26].

It should also be pointed out that the range of applications of the LLE as a measure of sensitivity to infinitesimal changes in initial conditions goes beyond the gait analysis. For instance, Jagrič et al. [27] analyzed the irregularity in short electrocardiographic (ECG) recordings in a similar manner to predict successful defibrillation in patients with ventricular fibrillation. A higher level of irregularity was interpreted as an indicator of patients who may be subjected to effective defibrillation.

##### 2.3. The Goal of the Research

As mentioned above, the examined state space structures were constructed on the basis of time series built of joint angles at hip, knee, and ankle joints. Three time series, which are related to the given joint, represent specific types of movement in sagittal, frontal, and transverse planes. For instance, movements at hip joint are called flexion/extension, abduction/adduction, and internal/external rotation, respectively. However, analysis of human gait focuses often on the sagittal plane to which the vast majority of the work during gait is assigned (ca. 74%, 85%, and 93% in case of hip, knee, and ankle joints, respectively) [28]. All the planes can be included using a state space based on a multivariate times series. At the previous research stage, which was extensively described in [29], the authors only used one state space that was based on multivariate time series composed of experimental data recorded in the CAREN extended environment (http://www.motekforcelink.com/product/caren/). Various experiments’ scenarios (i.e., variants of gait) were proposed which differed from each other with respect to walking speed, platform slope, and optional external perturbation. The results presented here are based on the same set of experiments. However, this time both the comparison of the LDS in three pairs of the “opposed” scenarios (i.e., gait variants which differ in one of the aforementioned aspects) and the statistical analysis were made for ankle, knee, and hip joints separately. Moreover, two additional structures of a state space were taken into consideration. The state spaces based either on a multivariate times series or on a univariate time series, which represents joint angles in the sagittal plane, will be described in Section 2.4.

Finally, three state space structures, which were constructed for each joint separately, were used to verify if the differences in LLE values between the opposed scenarios are significant for individual joints.

The authors also present a modification of the LDS computation method, i.e., an algorithm for estimation of one of its crucial parameters, embedding dimension, for the case of a multivariate gait time series, in which the parameter is not estimated for each of the component time series separately, but holistically.

The following research questions are addressed in the paper:(i)Are there any significant differences in the local dynamic stability between compared gait variants, which can be revealed using the individual state spaces?(ii)Is the predominant role of sagittal plane preserved in a state space which is based on a multivariate time series?(iii)Does the length of the time span, over which the short-term LLE is computed, influence the difference in the local dynamic stability between compared gait variants?

##### 2.4. The Research Procedure

The research procedure was composed of the following steps:(1)Data acquisition and preprocessing.(2)Estimation of the reconstruction parameters.(3)Trajectory reconstruction.(4)Estimation of the short-term LLE, taking into consideration both the aforementioned time span variants.

Inspired by reports in the literature, the authors decided to incorporate three different state space structures into research. The *UniS* state space is reconstructed on the basis of a univariate time series describing a movement at a joint in the sagittal plane. The next two spaces—*MultiFull* and *MultiFNN*—are reconstructed on the basis of a multivariate time series which is composed of three univariate time series. Each of the series is related to movement at a given joint in one of the motion planes: sagittal, frontal, and transverse. The *MultiFull* space is built on the basis of three pairs of independently determined parameters , , resulting in space dimensionality . The average mutual information (AMI) method [30] was used in each case to determine time delays, whereas the false nearest neighbors (FNN) method was utilized to estimate embedding dimensions. The number of bins required by the AMI was determined according to the Sturges formula [31], and according to Kennel et al.’s example [22], the following values were assigned to the first () and the second () criterion of the FNN for designating a point as a “false” neighbor: , . However, in the case of *MultiFNN*, a variant of the FNN adjusted to multivariate time series was applied which takes into consideration the quota of work done during gait in individual motion planes. Based on this criterion, the planes are ordered descending as follows: sagittal, frontal, and transverse [28].

The dimensionality of *MultiFNN* space is determined holistically, i.e., on all three time series treated as a whole. Starting from 1, the embedding dimension is gradually increased by inserting successive elements to the vector of delay coordinates. The coordinates are taken from cyclically changed component time series , , and (the symbols “,” “,” and “” stand for sagittal, frontal, and transverse planes, respectively), with appropriate time delay for each time series. The time series describing a movement in the sagittal plane is the first one used in each cycle. The stop criterion is met when the percentage of the “false” nearest neighbors falls below a given threshold (e.g. 1%) (a neighbor of a given point in a space of dimensionality turns out to be “false,” when it is no longer a neighbor of in a space of dimensionality ). As a result, the number of coordinates taken from the time series cannot be lower than the number of coordinates from the series which, in turn, cannot be lower than the number of coordinates from the series : . By that means, the method to some extent takes into account the domination of an anterior-posterior movement in gait.

This research is a part of an extensive project carried out in cooperation with the University of the Third Age (U3A). The project focuses on elderly people that would like to remain active over the age of 65. Table 1 presents characteristics of 14 U3A students, who agreed to participate in the experiments (12 women, 2 men), including median, mean, and standard deviation (SD) values of age, height, weight, and the body mass index (BMI).