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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J Biomech. Author manuscript; available in PMC 2010 June 19.
Published in final edited form as:
PMCID: PMC2718682
NIHMSID: NIHMS128473

Comparison of Different State Space Definitions for Local Dynamic Stability Analyses

Abstract

Measures of local dynamic stability, such as the local divergence exponent (λs*) quantify how quickly small perturbations deviate from an attractor that defines the motion. When the governing equations of motion are unknown, an attractor can be reconstructed by defining an appropriate state space. However, state space definitions are not unique and accepted methods for defining state spaces have not been established for biomechanical studies. This study first determined how different state space definitions affected λs* for the Lorenz attractor, since exact theoretical values were known a priori. Values of λs* exhibited errors < 10% for 7 of the 9 state spaces tested. State spaces containing redundant information performed the poorest. To examine these effects in a biomechanical context, 20 healthy subjects performed a repetitive sawing-like task for 5 minutes before and after fatigue. Local stability of pre- and post-fatigue shoulder movements was compared for 6 different state space definitions. Here, λs* decreased post-fatigue for all 6 state spaces. Differences were statistically significant for 3 of these state spaces. For state spaces defined using delay embedding, increasing the embedding dimension decreased λs* in both the Lorenz and experimental data. Overall, our findings suggest that direct numerical comparisons between studies that use different state space definitions should be made with caution. However, trends across experimental comparisons appear to persist. Biomechanical state spaces constructed using positions and velocities, or delay reconstruction of individual states, are likely to provide consistent results.

Introduction

Nonlinear dynamics methods for determining the local stability of kinematics have gained increased interest in recent literature. For experimental data, the governing equations are typically unknown. During most repetitive movements, the dynamics of such movements can be represented geometrically within an n-dimensional ‘state space’ where n is the number of state variables (Dingwell and Cusumano, 2000; Kantz and Schreiber, 2004). Typically, these experimental data exhibit the structure of an attractor, i.e., a sub-space of the n-dimensional state space to which neighboring trajectories converge (Strogatz, 1994; Dingwell and Kang, 2007).

Measurable biomechanical state variables typically include linear and/or angular displacements, velocities, and/or accelerations. State spaces may also be defined using delay embedding, which allows reconstruction from a single scalar recording, once an appropriate time lag and embedding dimension are determined (Kantz and Schreiber, 2004). This can greatly simplify data collection and may be advantageous for situations where data for some measurable dimensions are prone to error. However, since divergence in one dimension may be compensated by contraction in another (Granata and Gottipati, 2008), analyzing one single trajectory may not adequately represent the movement. One proposed way to compensate for this is to perform delay embedding on the Euclidean norm of the three Euler angles at a joint (Granata and Gottipati, 2008)1. Another option is to analyze a state space composed of all angles and angular velocities at that joint. However, this will typically include redundant information which could negatively impact the results. One way to define state variables with minimum redundancy is to perform a principal components analysis (PCA) on the data (Daffertshofer et al., 2004). Many state spaces have been applied in biomechanical stability analyses (Table 1). To date, however, researchers have not explored how different definitions of these state spaces affect local stability measures.

Table 1
Example State Space Definitions

This study determined how including different state variables and/or different numbers of variables in the state space altered short-term local divergence exponents, λs*, a measure of local dynamic instability (Dingwell and Cusumano, 2000). Data for which the true λs* was known a priori were analyzed to determine how different state space definitions affect the resulting values of λs*. Experimental data were then analyzed to determine how different state space definitions might affect the answers to research questions attempting to quantify differences in stability between conditions. The state space definitions examined here were chosen primarily from previously published studies (Table 1).

Methods

The Lorenz attractor (Strogatz, 1994) was used to test the effect of different state space definitions on λs* The Lorenz system is defined by three coupled nonlinear differential equations:

x˙=σ(yx)y˙=x(ρz)yz˙=xyβz
(1)

where σ, ρ, and β are fixed parameters that were set to σ = 16, ρ = 45.92 and β = 4 (Rosenstein et al., 1993). These equations were integrated in Matlab using a fourth order Runge-Kutta method (ode45) to generate 10 trials of 155 seconds at 100 Hz. The first 5 seconds were removed to eliminate transients. The first derivative of each trajectory was estimated using a three point difference formula. Uniformly distributed random noise was added to attain a minimum signal-to-noise ratio of 100:1, which is considered ‘moderate’ (Rosenstein et al., 1993). Because the Lorenz attractor for these parameter values is chaotic, λs* in this case defines the maximum finite-time Lyapunov exponent, which should be 1.50 (Rosenstein et al., 1993).

We defined 9 different state spaces for this Lorenz attractor. The “Full” state space was defined by the three x(t), y(t), and z(t) trajectories (Eq. 1). A “Redundant” state space was defined from these three trajectories and their derivatives:

RedundantSS(t)=[x(t),y(t),z(t),x˙(t),y˙(t),z˙(t)]
(2)

We performed principal components analysis (Daffertshofer et al., 2004) on both the Full and Redundant state spaces. In cases where state variables had different units, we normalized each to unit variance. To explain ≥ 95% of the variance, 2 principal components were needed for the Full state space and 4 were needed for the Redundant state space. However, since this system is 3-dimensional (Eq. 1), we tested state spaces composed of both 3 and 4 principal components (see Supplementary Material).

Four additional state spaces were defined using delay embedding (Kantz and Schreiber, 2004) of x(t), y(t), z(t) and r(t)=x2+y2+z2 (Granata and Gottipati, 2008). Multi-dimensional state spaces were reconstructed from each original time series and its time-delayed copies (Takens, 1981):

Q(t)=[q(t),q(t+T),q(t+2T),,q(t+(dE1)T)]
(3)

where Q(t) ε {X(t),Y(t),Z(t),R(t)} was the dE-dimensional state vector, q(t) ε {x(t), y(t), z(t),r(t)} was the original 1-dimensional data, T was the time delay, and dE was the embedding dimension (Fig. 1). Time delays were calculated as the first minimum of the average mutual information function of each signal (Fraser and Swinney, 1986).

Figure 1
A) Data for the Lorenz attractor is shown with σ= 16, ρ= 45.92, and β = 4. A 3-D was created from the x(t), y(t) and z(t) data. B) Using the method of delays, a state space can be created by reconstruction of a single trajectory, ...

Maximum local divergence exponents (λs*) were calculated from the exponential rates of divergence of small perturbations in state space (Rosenstein et al., 1993; Dingwell and Cusumano, 2000). Positive exponents indicate local instability. Larger exponents indicate greater sensitivity to local perturbations. Short-term λs* were estimated from:

λ*s=1Δtln[dj(t)]
(4)

where left angle bracketln[dj(t)]right angle bracket represents the mean log divergence for all pairs of nearest neighbors, j (Rosenstein et al., 1993), over 0 t 1 seconds. To test the effects of embedding dimension, we calculated λs* for X(t), Y(t), and Z(t) for dE ε {2, 3, 4,5, 6, 7,15, 20}. Kennel et al. (1992) recommend using a dE that is twice the true dimension (i.e., dE = 6). Rosenstein et al. (1993) suggest using a dE equal to the true dimension (i.e., dE = 3).

Experimental Data

Twenty healthy subjects (age 25 ± 2 years) participated after providing institutionally approved written informed consent. Subjects completed a repetitive sawing-like task for five minutes before and after performing a repetitive sagittal plane lifting task. This was designed to test the effect of local fatigue of the shoulder flexors on shoulder stability. The 3-D positions of 19 reflective markers placed on the right arm and trunk were recorded continuously during the sawing task at 120 Hz using an 8-camera Vicon system (Oxford Metrics, Oxford, UK). Marker data were filtered using a 5th order Butterworth filter with a cutoff frequency of 15 Hz. Rotational motions of the shoulder were defined using Euler angles (Wu et al., 2005) (see Supplementary Material).

We defined six state spaces for shoulder movement. The “Full” state space was composed of the three rotational angles and their angular velocities. Each state was then normalized to unit variance and PCA was performed. Four principal components explained > 95% of the data variance. The second state space consisted of these four principle components. Four state spaces were defined using delay embedding. Global false neighbors analyses (Kennel, 1992) suggested dE = 5 for all three rotational angles. Therefore, we tested dE ε {3, 4,5, 6, 7}. Short-term local divergence exponents (λs*, Eq. 4) were calculated over 0 t 1 cycle (Dingwell and Marin, 2006). Stability estimates are sensitive to time series length (Granata and England, 2006; Bruijn et al., In Press). Therefore, each time series was first normalized to exactly 36,000 points. This resulted in a < 1% change in the length of each time series. Pre- and post-fatigue λs* from each trial were compared using paired t-tests for each state space.

Results

For the Lorenz attractor, λs* differed depending on which state space was used (Fig. 2A; Table 2). Redundant state spaces containing derivatives performed the poorest (errors > 10%). Taking only three principal components of the redundant state space decreased this error from 20.4% to 7.53%. All methods correctly indicated that the Lorenz attractor is locally unstable (λs* > 0).

Figure 2
Modeling Results. A) The short-term local divergence exponent, λs*, was calculated for 10 simulations with nine different state space definitions. Error bars represent the mean ± 95% confidence interval for each method defined in Table ...
Table 2
Percent error between the known local divergence exponent for the Lorenz attractor, λs*=1.5, and that calculated from nine different state spaces. The trajectories x(t), y(t), and z(t) define the system. PC1 – PC4 are the first ...

As the embedding dimension was increased, λs* decreased (Fig. 2B). We expected that a minimum of 3 (Rosenstein et al., 1993) to 6 (Kennel et al., 1992) states would reconstruct the state space with minimal error. Large errors were obtained when either too many or too few states were used.

For the experimental data, λs* values also changed across different state spaces. Shoulder motion was always locally unstable, both pre- and post-fatigue. For all state spaces, λs* tended to decrease post-fatigue (Fig. 3A). This decrease was statistically significant for 3 of the 6 methods (Fig. 3A). As with the Lorenz attractor (Fig 2B), increasing the embedding dimension caused λs* to decrease (Fig. 3B). However, pre- versus post-fatigue differences remained.

Figure 3
Results Using Experimental Data. A) Short-term local divergence exponents, λs*, for pre- and post-fatigue conditions for six state space definitions: the full state space (‘Full SS’) defined by [θ1,θ2,θ ...

Discussion

The only formal requirement for a valid state space is that it uniquely defines the state of a system at all points in time (Kantz and Schreiber, 2004). Thus, any proposed state space is not unique. Here, we explored how several different state spaces affect local dynamic stability (λs*) calculations. The formulations presented here are by no means exhaustive. It is quite possible that other state space definitions might perform as well or better.

Reasonably accurate results were obtained when PCA was performed on the full Lorenz state space and when the correct number of principal components was chosen from the redundant state space. PCA may overestimate the true dimension of the system (Clewley et al., 2008) and poor results were in fact obtained when too many principal components were included (Table 2). PCA also requires prior normalization of states that have different units. These normalizations are not unique and can affect the value of λs*. We therefore do not recommend using PCA to define the state space.

While there were quantitative differences in λs* for the different experimental state spaces (Fig. 3), all methods demonstrated whether trajectories were locally more or less stable. Differences between experimental conditions persisted, although the statistical significance of these differences did vary. Similarly, previous work demonstrated qualitatively similar changes in walking stability with changes in speed (Dingwell and Marin, 2006; England and Granata, 2007; Kang and Dingwell, 2008), despite using different state space definitions. Thus, while it may be difficult to make direct numerical comparisons between studies that use different state space definitions, it seems that qualitative trends persist. Overall, our findings suggest that biomechanical state spaces constructed using positions and velocities, or delay reconstruction of individual states, are likely to provide more consistent results than those constructed using principal components. Efforts should also be made to avoid using redundant information wherever possible.

Supplementary Material

01

ACKNOWLEDGMENTS

Funding for this project was provided by grant #EB003425 from the National Institutes of Health.

Footnotes

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1The Euclidean norm of 3 Euler angles is not an appropriate measure of the “distance” between two angular orientations, as it would be for linear displacements. This measure should be interpreted here merely as a weighted average of the 3 joint angles.

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