Graham Leverick Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada
Tony Szturm School of Medical Rehabilitation, University of Manitoba, Winnipeg, MB R3E 0T6, Canada e-mail: tony,[email protected]
Christine Q. Wu Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada e-mai I: christine.wu@uman itoba.ca
Using Entropy M easu res to C haracterize H um an Locom otion Entropy measures have been widely used to quantify the complexity of theoretical and experimental dynamical systems. In this paper, the value of using entropy measures to characterize human locomotion is demonstrated based on their construct validity, predic tive validity in a simple model of human walking and convergent validity in an experi mental study. Results show that four of the five considered entropy measures increase meaningfully with the increased probability of falling in a simple passive bipedal walker model. The same four entropy measures also experienced statistically significant increases in response to increasing age and gait impairment caused by cognitive interfer ence in an experimental study. Of the considered entropy measures, the proposed quan tized dynamical entropy (QDE) and quantization-based approximation o f sample entropy (QASE) offered the best combination of sensitivity to changes in gait dynamics and com putational efficiency. Based on these results, entropy appears to be a viable candidate for assessing the stability of human locomotion. [DOI: 10.1115/1.4028410] Keywords: entropy, gait analysis, stability, age effect, dual-task effects
1 Introduction Developing measures for assessing the quality and stability of human walking has been a subject of particular research interest [1-8]. There are a wide range of applications for measures of the stability of human locomotion including identifying aging individ uals who may be prone to falls and assessing rehabilitation patients’ progress. Pragmatically, stable human walking can be defined as walking that “does not lead to falls in spite of perturbations” [3]. In a sense, this definition of stability is consist ent with Lyapunov’s stability theory, wherein a system’s response is considered stable if it remains bounded to the desired motion (in the case of walking, a limit cycle). Two distinct categories of gait perturbations have been identified, which require different control strategies [3]. Small perturbations arise from small incon sistencies in the walking surface and noise. While a single small perturbation does not immediately destabilize gait, if let uncor rected, it can lead to an increasing deviation from the desired gait pattern, which can cause a fall. Large perturbations can immedi ately destabilize gait and require a change of behavior to correct. While robustness to both types of perturbations is important for healthy walking, the majority of measures of gait stability have been developed based on gait patterns affected by only small per turbations [3]. This is partially due to the logistical challenge of consistently introducing large perturbations to gait patterns. There is an inherent challenge in attempting to characterize gait stability based on measurements taken in the presence of only small perturbations. The probability of falling caused by small perturbations is very low for most individuals. Therefore, meas urements are typically taken during successful walking without falls causing the long term system stability to not be particularly indicative of the probability of falls outside of the trial (as sug gested by the limited success of zL [3]). While the short term in stantaneous Lyapunov exponents suggested by Dingwell et al. [5] have demonstrated themselves as a better indicator of gait stability [3], by only considering short trajectories, they seem to reflect a concept, which is independent of classical asymptotic stability theory. In Ref. [3], Bruijn provides a very thorough review of existing measures for characterizing the stability of human locomotion. Manuscript received January 27, 2014; final manuscript received August 7, 2014; accepted manuscript posted August 27, 2014; published online October 17, 2014. Assoc. Editor: Paul Rullkoetter.
Journal of Biomechanical Engineering
While Bruijn’s coverage is very inclusive, he deliberately excludes entropy measures owning to a lack of studies that adequately address their validity as a measure of the stability of human walking [3], There have been two prominent methodolo gies used in previous studies, which have applied entropy meas ures to human walking. One methodology is to calculate the entropy of linear accelerations measured approximately at the center of mass [9-11]. A number of other studies use entropy measures like a statistical tool to assess variations in spatial-temporal variables like stride interval [12-14] and mini mum toe clearance [15,16], Although these studies seem to sug gest that lower complexity may correlate with healthier walking, the methodologies used are not well established, and the strength of the findings are not highly convincing. While entropy measures have not appeared convincingly in studies of human walking, entropy measures are well developed as a measure of both theoretical and experimental dynamical systems. They have been demonstrated to be successful for char acterizing a wide range of different systems including speech sig nals and measurements of the brain and heart [17]. There are a number of different entropy measures that have been developed to characterize the complexity of dynamical systems. These were mostly developed to improve calculations of short noisy time series derived from experimentation [18]. Of the developed en tropy measures, sample entropy (SampEn) is arguably the most prominently used in literature. SampEn is a modified version of Pincus’s ApEn, which was designed to reduce statistical bias [19], Its calculation relies on comparing the self-similarity of the time series for two different embedding dimensions; m and m + 1. The time series’ self-similarity is assessed by counting the number of times a vector of m (or m + 1) consecutive points match other vectors of consecutive points in the time series. The result is an entropy measure that converges with relatively few data points [19] and exhibits high robustness to observational noise [18], Fuzzy entropy (FuzzyEn) is a further modification of SampEn, which adds a fuzzy function to the comparison of vectors of con secutive points and generalizes those vectors to reduce the impact of measurement drift [20]. Both SampEn and FuzzyEn are quite sensitive to changes in dynamics and robust to observational noise, but have rather low computational efficiency due to the large number of comparisons required in their calculation [18]. Permutation entropy (PE) is another prominently used entropy measure, which is completely distinct from SampEn and FuzzyEn.
Copyright © 2014 by ASME
DECEMBER 2014 , Vol. 136 / 121002-1
PE was developed by Brandt as a novel approach for characteriz ing the complexity of experimental time series [17]. PE uses the shape of neighboring points to assess complexity. If there are only a few different shapes in the time series and they appear regularly (like a sine wave), the time series has low PE. Conversely, a time series which contains many different shapes, which appear without regularity (white noise, for example) has high PE [4]. PE features good computational properties, but can react poorly to observational noise in some circumstances [18]. In response to the need for a computationally efficient entropy measure, which is also robust to observational noise, two new en tropy algorithms were proposed recently by the authors; QDE and a QASE [18]. Both of these entropy measures rely on a coarse quantization of the time series to simplify the classification and comparison of dynamical features. This procedure improves com putational efficiency by eliminating the need for computationally demanding intra time series comparisons. QDE is a novel entropy measure, which uses the quantization bin numbers of consecutive points in the time series to assign dynamical features a unique identifier. It then uses the relative frequency of these vector identi fiers to assess the complexity of the time series. QASE uses the developed method of identifying dynamical features by the quan tization bin numbers of their points to approximate SampEn in a computationally efficient manner. Both methods were thoroughly tested in Refs. [18] and [21] and are found to offer considerable computational savings with good robustness to noise and fast convergence. This paper has two primary objectives: (1) to establish the va lidity of using entropy measures to characterize the health and sta bility of human walking and (2) to demonstrate that the two new entropy measures proposed by the authors offer performance advantages over existing entropy measures. The validity of using entropy measures to characterize human walking will be based on the four levels of validity of a measure of human walking, which are well established in literature [3]: (1) Construct validity. Does the derivation of the measure plau sibly suggest a relationship with the probability of falling? (2) Predictive validity in a simple model: Can the measure pre dict the probability of falling for a simple model of human walking? (3) Convergent validity in experimental studies: Can the measure reflect an experimentally induced changed in the stability of human walking? (4) Predictive validity in observational studies: Can the mea sure predict the probability of falling based on prospective or retrospective observational studies? In this paper, the value of using entropy measures to assess the stability of human walking is demonstrated based on three of these four levels of validity; construct validity, predictive validity in a simple model and convergent validity in an experimental study. The forth level of validity (predictive validity in observatio nal studies) is beyond the scope of this paper, but is planned in future work. The first measure of validity cannot be tested experi mentally. In this paper, the construct validity of entropy measures is addressed through discussion of their derivation and features. The second level of validity is established by studying the model of a passive dynamic walker walking down an incline. The proba bility of the passive dynamic walker falling was varied by adding different sized bumps and dips in the surface of the incline, and the relationship between the probability of the walker falling and the measured entropy was investigated. The third level of validity is investigated based on an experimental study featuring two groups of 30 participants; active healthy young adults (mean age 25 yr, SD 3.2) and healthy, community-dwelling older adults (mean age 63 yr, SD 4.4). The gait pattern of each participant was measured in both a baseline walking (BL) scenario and dualtasking (DT) case where participants’ gait patterns will be inter rupted through the addition of a secondary task. The effect of the two different age groups and two different walking conditions on
121002-2 /
Vol. 136, DECEMBER 2014
the measured entropy was assessed. In order to demonstrate the performance advantages of the two new entropy measures pro posed by the authors (QDE and QASE), their performance is com pared to three well established entropy measures: SampEn, FuzzyEn, and PE. This comparison is based on the performance of the entropy measures in the passive dynamic walker model as well as in the experimental study of human walking.
2
Methodology
This section begins with the mathematical preliminaries of the two new entropy measures proposed by the authors: QDE and QASE. Following this, the passive dynamic walker model is intro duced. This is followed by a discussion of experimental methodol ogy used in the study of human walking. 2.1 Development of Entropy Measures. QDE and QASE were developed to improve computational efficiency of entropy calculations while retaining good robustness to observational noise and fast convergence. Sections 2.1.1 and 2.1.2 summarize the calculation of these two new entropy measures. For further details on their calculation, performance, and features, refer Refs. [18] and [21], 2.1.1 QDE. The calculation of QDE is based on performing a relatively coarse quantization of a scalar time series, X (whose data points are defined by x(i), i = l ,2 , ...,N). The quantized time series Xq is determined using the strictly positive parameter r, which defines the size of the quantization bins X - inf (X)
r
(i)
where inf(A) is the infimum (minimum) of the time series X and [•J is the floor function. The floor function of a variable y, [yj, is defined as the largest integer not greater than y. Next, consider a subset of neighboring points from the quantized time series, whose length is given by the embedding dimension m: xq(i), xq(i + 1),... ,xq(i + m —1). Each subset of m points can be assigned a vector identifier, q>t, whose elements are exclusively whole numbers (a feature of the quantized time series) Vi = [Xq(/),*q(i + 1), . . . , X q ( i + m - 1)]
(2)
By considering all sublets of m neighboring points in the time series, xq(i), xq(i+ 1),..., xq(i + m - 1), where 1 < i < N —m + 1, one can define the number of occurrences of each identifier as Q((Pi). The number of occurrences of each identifier in the time series is used to define the probability of encountering a given identifier, p(i)/(N - m + 1). QDE, H(m,r), is then defined as H(m, r) = - ^2p{Vi^og2P{Vi)
(3)
v
QDE can also be presented in its “entropy per symbol” form as given by h(m,r) = H(m,r)/m. The resulting entropy measure features high computational efficiency and good robustness to observational noise [18]. A time series where the same dynamical features (as defined by their vector identifiers) appear repeatedly has low QDE, whereas a time series containing a wide range of different dynamical features has high QDE. 2.1.2 QASE. QASE [18] relies on the same method of time series quantization used by QDE, in order to generate a computa tionally efficient approximation of SampEn. To calculate QASE, one uses the number of occurrences of each vector identifier, Q{cpm), to define terms Am(r) and Bm(r) B™{r) = { N - r n ) - 'Y J Q i v j f (P _m lZ \
(4)
Tony Szturm School of Medical Rehabilitation, University of Manitoba, Winnipeg, MB R3E 0T6, Canada e-mail: tony,[email protected]
Christine Q. Wu Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada e-mai I: christine.wu@uman itoba.ca
Using Entropy M easu res to C haracterize H um an Locom otion Entropy measures have been widely used to quantify the complexity of theoretical and experimental dynamical systems. In this paper, the value of using entropy measures to characterize human locomotion is demonstrated based on their construct validity, predic tive validity in a simple model of human walking and convergent validity in an experi mental study. Results show that four of the five considered entropy measures increase meaningfully with the increased probability of falling in a simple passive bipedal walker model. The same four entropy measures also experienced statistically significant increases in response to increasing age and gait impairment caused by cognitive interfer ence in an experimental study. Of the considered entropy measures, the proposed quan tized dynamical entropy (QDE) and quantization-based approximation o f sample entropy (QASE) offered the best combination of sensitivity to changes in gait dynamics and com putational efficiency. Based on these results, entropy appears to be a viable candidate for assessing the stability of human locomotion. [DOI: 10.1115/1.4028410] Keywords: entropy, gait analysis, stability, age effect, dual-task effects
1 Introduction Developing measures for assessing the quality and stability of human walking has been a subject of particular research interest [1-8]. There are a wide range of applications for measures of the stability of human locomotion including identifying aging individ uals who may be prone to falls and assessing rehabilitation patients’ progress. Pragmatically, stable human walking can be defined as walking that “does not lead to falls in spite of perturbations” [3]. In a sense, this definition of stability is consist ent with Lyapunov’s stability theory, wherein a system’s response is considered stable if it remains bounded to the desired motion (in the case of walking, a limit cycle). Two distinct categories of gait perturbations have been identified, which require different control strategies [3]. Small perturbations arise from small incon sistencies in the walking surface and noise. While a single small perturbation does not immediately destabilize gait, if let uncor rected, it can lead to an increasing deviation from the desired gait pattern, which can cause a fall. Large perturbations can immedi ately destabilize gait and require a change of behavior to correct. While robustness to both types of perturbations is important for healthy walking, the majority of measures of gait stability have been developed based on gait patterns affected by only small per turbations [3]. This is partially due to the logistical challenge of consistently introducing large perturbations to gait patterns. There is an inherent challenge in attempting to characterize gait stability based on measurements taken in the presence of only small perturbations. The probability of falling caused by small perturbations is very low for most individuals. Therefore, meas urements are typically taken during successful walking without falls causing the long term system stability to not be particularly indicative of the probability of falls outside of the trial (as sug gested by the limited success of zL [3]). While the short term in stantaneous Lyapunov exponents suggested by Dingwell et al. [5] have demonstrated themselves as a better indicator of gait stability [3], by only considering short trajectories, they seem to reflect a concept, which is independent of classical asymptotic stability theory. In Ref. [3], Bruijn provides a very thorough review of existing measures for characterizing the stability of human locomotion. Manuscript received January 27, 2014; final manuscript received August 7, 2014; accepted manuscript posted August 27, 2014; published online October 17, 2014. Assoc. Editor: Paul Rullkoetter.
Journal of Biomechanical Engineering
While Bruijn’s coverage is very inclusive, he deliberately excludes entropy measures owning to a lack of studies that adequately address their validity as a measure of the stability of human walking [3], There have been two prominent methodolo gies used in previous studies, which have applied entropy meas ures to human walking. One methodology is to calculate the entropy of linear accelerations measured approximately at the center of mass [9-11]. A number of other studies use entropy measures like a statistical tool to assess variations in spatial-temporal variables like stride interval [12-14] and mini mum toe clearance [15,16], Although these studies seem to sug gest that lower complexity may correlate with healthier walking, the methodologies used are not well established, and the strength of the findings are not highly convincing. While entropy measures have not appeared convincingly in studies of human walking, entropy measures are well developed as a measure of both theoretical and experimental dynamical systems. They have been demonstrated to be successful for char acterizing a wide range of different systems including speech sig nals and measurements of the brain and heart [17]. There are a number of different entropy measures that have been developed to characterize the complexity of dynamical systems. These were mostly developed to improve calculations of short noisy time series derived from experimentation [18]. Of the developed en tropy measures, sample entropy (SampEn) is arguably the most prominently used in literature. SampEn is a modified version of Pincus’s ApEn, which was designed to reduce statistical bias [19], Its calculation relies on comparing the self-similarity of the time series for two different embedding dimensions; m and m + 1. The time series’ self-similarity is assessed by counting the number of times a vector of m (or m + 1) consecutive points match other vectors of consecutive points in the time series. The result is an entropy measure that converges with relatively few data points [19] and exhibits high robustness to observational noise [18], Fuzzy entropy (FuzzyEn) is a further modification of SampEn, which adds a fuzzy function to the comparison of vectors of con secutive points and generalizes those vectors to reduce the impact of measurement drift [20]. Both SampEn and FuzzyEn are quite sensitive to changes in dynamics and robust to observational noise, but have rather low computational efficiency due to the large number of comparisons required in their calculation [18]. Permutation entropy (PE) is another prominently used entropy measure, which is completely distinct from SampEn and FuzzyEn.
Copyright © 2014 by ASME
DECEMBER 2014 , Vol. 136 / 121002-1
PE was developed by Brandt as a novel approach for characteriz ing the complexity of experimental time series [17]. PE uses the shape of neighboring points to assess complexity. If there are only a few different shapes in the time series and they appear regularly (like a sine wave), the time series has low PE. Conversely, a time series which contains many different shapes, which appear without regularity (white noise, for example) has high PE [4]. PE features good computational properties, but can react poorly to observational noise in some circumstances [18]. In response to the need for a computationally efficient entropy measure, which is also robust to observational noise, two new en tropy algorithms were proposed recently by the authors; QDE and a QASE [18]. Both of these entropy measures rely on a coarse quantization of the time series to simplify the classification and comparison of dynamical features. This procedure improves com putational efficiency by eliminating the need for computationally demanding intra time series comparisons. QDE is a novel entropy measure, which uses the quantization bin numbers of consecutive points in the time series to assign dynamical features a unique identifier. It then uses the relative frequency of these vector identi fiers to assess the complexity of the time series. QASE uses the developed method of identifying dynamical features by the quan tization bin numbers of their points to approximate SampEn in a computationally efficient manner. Both methods were thoroughly tested in Refs. [18] and [21] and are found to offer considerable computational savings with good robustness to noise and fast convergence. This paper has two primary objectives: (1) to establish the va lidity of using entropy measures to characterize the health and sta bility of human walking and (2) to demonstrate that the two new entropy measures proposed by the authors offer performance advantages over existing entropy measures. The validity of using entropy measures to characterize human walking will be based on the four levels of validity of a measure of human walking, which are well established in literature [3]: (1) Construct validity. Does the derivation of the measure plau sibly suggest a relationship with the probability of falling? (2) Predictive validity in a simple model: Can the measure pre dict the probability of falling for a simple model of human walking? (3) Convergent validity in experimental studies: Can the measure reflect an experimentally induced changed in the stability of human walking? (4) Predictive validity in observational studies: Can the mea sure predict the probability of falling based on prospective or retrospective observational studies? In this paper, the value of using entropy measures to assess the stability of human walking is demonstrated based on three of these four levels of validity; construct validity, predictive validity in a simple model and convergent validity in an experimental study. The forth level of validity (predictive validity in observatio nal studies) is beyond the scope of this paper, but is planned in future work. The first measure of validity cannot be tested experi mentally. In this paper, the construct validity of entropy measures is addressed through discussion of their derivation and features. The second level of validity is established by studying the model of a passive dynamic walker walking down an incline. The proba bility of the passive dynamic walker falling was varied by adding different sized bumps and dips in the surface of the incline, and the relationship between the probability of the walker falling and the measured entropy was investigated. The third level of validity is investigated based on an experimental study featuring two groups of 30 participants; active healthy young adults (mean age 25 yr, SD 3.2) and healthy, community-dwelling older adults (mean age 63 yr, SD 4.4). The gait pattern of each participant was measured in both a baseline walking (BL) scenario and dualtasking (DT) case where participants’ gait patterns will be inter rupted through the addition of a secondary task. The effect of the two different age groups and two different walking conditions on
121002-2 /
Vol. 136, DECEMBER 2014
the measured entropy was assessed. In order to demonstrate the performance advantages of the two new entropy measures pro posed by the authors (QDE and QASE), their performance is com pared to three well established entropy measures: SampEn, FuzzyEn, and PE. This comparison is based on the performance of the entropy measures in the passive dynamic walker model as well as in the experimental study of human walking.
2
Methodology
This section begins with the mathematical preliminaries of the two new entropy measures proposed by the authors: QDE and QASE. Following this, the passive dynamic walker model is intro duced. This is followed by a discussion of experimental methodol ogy used in the study of human walking. 2.1 Development of Entropy Measures. QDE and QASE were developed to improve computational efficiency of entropy calculations while retaining good robustness to observational noise and fast convergence. Sections 2.1.1 and 2.1.2 summarize the calculation of these two new entropy measures. For further details on their calculation, performance, and features, refer Refs. [18] and [21], 2.1.1 QDE. The calculation of QDE is based on performing a relatively coarse quantization of a scalar time series, X (whose data points are defined by x(i), i = l ,2 , ...,N). The quantized time series Xq is determined using the strictly positive parameter r, which defines the size of the quantization bins X - inf (X)
r
(i)
where inf(A) is the infimum (minimum) of the time series X and [•J is the floor function. The floor function of a variable y, [yj, is defined as the largest integer not greater than y. Next, consider a subset of neighboring points from the quantized time series, whose length is given by the embedding dimension m: xq(i), xq(i + 1),... ,xq(i + m —1). Each subset of m points can be assigned a vector identifier, q>t, whose elements are exclusively whole numbers (a feature of the quantized time series) Vi = [Xq(/),*q(i + 1), . . . , X q ( i + m - 1)]
(2)
By considering all sublets of m neighboring points in the time series, xq(i), xq(i+ 1),..., xq(i + m - 1), where 1 < i < N —m + 1, one can define the number of occurrences of each identifier as Q((Pi). The number of occurrences of each identifier in the time series is used to define the probability of encountering a given identifier, p(i)/(N - m + 1). QDE, H(m,r), is then defined as H(m, r) = - ^2p{Vi^og2P{Vi)
(3)
v
QDE can also be presented in its “entropy per symbol” form as given by h(m,r) = H(m,r)/m. The resulting entropy measure features high computational efficiency and good robustness to observational noise [18]. A time series where the same dynamical features (as defined by their vector identifiers) appear repeatedly has low QDE, whereas a time series containing a wide range of different dynamical features has high QDE. 2.1.2 QASE. QASE [18] relies on the same method of time series quantization used by QDE, in order to generate a computa tionally efficient approximation of SampEn. To calculate QASE, one uses the number of occurrences of each vector identifier, Q{cpm), to define terms Am(r) and Bm(r) B™{r) = { N - r n ) - 'Y J Q i v j f (P _m lZ \
(4)
