A comparative study on approximate entropy measure and poincaré plot indexes of minimum foot clearance variability in the elderly during walking
© Khandoker et al; licensee BioMed Central Ltd. 2008
Received: 29 January 2007
Accepted: 02 February 2008
Published: 02 February 2008
Trip-related falls which is a major problem in the elderly population, might be linked to declines in the balance control function due to ageing. Minimum foot clearance (MFC) which provides a more sensitive measure of the motor function of the locomotor system, has been identified as a potential gait parameter associated with trip-related falls in older population. This paper proposes nonlinear indexes (approximate entropy (ApEn) and Poincaré plot indexes) of MFC variability and investigates the relationship of MFC with derived indexes of elderly gait patterns. The main aim is to find MFC variability indexes that well correlate with balance impairments.
MFC data during treadmill walking for 14 healthy elderly and 10 elderly participants with balance problems and a history of falls (falls risk) were analysed using a PEAK-2D motion analysis system. ApEn and Poincaré plot indexes of all MFC data sets were calculated and compared.
Significant relationships of mean MFC with Poincaré plot indexes (SD1, SD2) and ApEn (r = 0.70, p < 0.05; r = 0.86, p < 0.01; r = 0.74, p < 0.05) were found in the falls-risk elderly group. On the other hand, such relationships were absent in the healthy elderly group. In contrast, the ApEn values of MFC data series were significantly (p < 0.05) correlated with Poincaré plot indexes of MFC in the healthy elderly group, whereas correlations were absent in the falls-risk group. The ApEn values in the falls-risk group (mean ApEn = 0.18 ± 0.03) was significantly (p < 0.05) higher than that in the healthy group (mean ApEn = 0.13 ± 0.13). The higher ApEn values in the falls-risk group might indicate increased irregularities and randomness in their gait patterns and an indication of loss of gait control mechanism. ApEn values of randomly shuffled MFC data of falls risk subjects did not show any significant relationship with mean MFC.
Results have implication for quantifying gait dynamics in normal and pathological conditions, thus could be useful for the early diagnosis of at-risk gait. Further research should provide important information on whether falls prevention intervention can improve the gait performance of falls risk elderly by monitoring the change in MFC variability indexes.
Older population make up a large and increasing percentage of the population. As people grow older they are increasingly at risk of falling and consequent injuries. Approximately 30% of people over 65 fall each year, and for those over 75 the rates are higher. Between 20% and 30% of those who fall suffer injuries that reduce mobility and independence and increase the risk of premature death .
Human walking is a highly automated, rhythmic motor behaviour that is mostly controlled by subcortical locomotor brain regions. Gait analysis refers to the measurement and analysis of human walking patterns. One major aim of studying gait characteristics is to identify gait variables that reflect gait degeneration due to ageing with linkages to the causes of falls. This would help to undertake appropriate measures to prevent falls.
In our previous study , we studied the MFC variability and statistics for young and elderly females and described the changes of MFC central tendency and variability as one of the possible strategies by elderly individuals to minimize tripping. Analysis of linear statistics does not directly address their complexity and thus may potentially miss useful inherent information. Since the underlying mechanism involved in the human locomotor control has been reported to be mainly complex and nonlinear [5–7], the application of nonlinear technique seems appropriate. In this study, we, therefore, investigate the two types of nonlinear variability indexes (Approximate entropy and Poincaré plot indexes) of MFC to be able to perform a diagnostic function to distinguish walking patterns of elderly subjects with a history of balance impairments and falls from that of healthy peers.
Approximate entropy (ApEn), a mathematical approach to quantify the complexity and regularity of a system, has been introduced by Pincus , based on a novel systematic biological theory [8, 9]. Such theory has suggested that healthy dynamic stability arises from the combination of specific feedback mechanisms and spontaneous properties of interconnected networks, and the weak connection between systems or within system is the mechanism of disease, which is characterized by an increased irregularity of the time series [9, 10]. Therefore, ApEn was considered to provide a direct measurement of feedback and connection, and a low ApEn value often indicates predictability and high regularity of time series data, whereas a high ApEn value indicates unpredictability and random variation . Previous studies  on the entropy of human gait in multiple scales discussed the scaling effect of entropy on various walking patterns, indicating the changes of multiscale entropy values with slow, normal and fast walking.
Poincaré plot is a geometrical representation of a time series into a Cartesian plane, where the values of each pair of successive elements of the time series define a point in the plot. Indexes derived from Poincaré plot of minimum foot clearance (MFC) were used to classify young-old gait types in our previous study .
With an aim to find a better marker of gait dynamics due to balance impairments, we apply ApEn analysis method to the MFC gait data obtained from elderly subjects with and without balance problem, and compare the results with those obtained using Poincaré plot indexes analysis.
MFC Gait Data
Subject Characteristics, mean (± SD)
Healthy(n = 14)
Falls risk(n = 10)
Age ( years )
71.0 (± 2.1)
72.2 (± 3.1)
Height ( cm )
170 (± 11)
166 (± 12)
Weight ( kg )
63.2 (± 14.3)
66.9 (± 8.6)
Estimation of ApEn of MFC
In our study, we use data set of 400 adjacent MFC data points. We divide the data set into smaller sets of length, i.e., m = 2. This amounts to 200 smaller sub sets. The next step is to determine the number of subsets that are within the criterion of similarity d = 15% of the standard deviation of 400 MFC points. Then we repeat the same process for the second subset till each subset is compared with the rest of the data set. This process computes part of equation (1) and N-m+1 = 400-2+1 = 399. We repeat the same process for m = 3. Approximate entropy is then calculated using equation (1).
MFC Poincaré plots
All data were presented as mean ± SD. Associations between parameters and indexes were determined using Pearson's r. Student's (independent samples) t-test was used in order to compare the differences between the groups. In order to provide the relative importance of single index in discriminating two types of gait patterns, receiver-operating characteristics (ROC) curve analysis was used [13, 14], with the areas under the curves for each measure represented by ROCarea. An ROCarea value of 0.5 means that the distributions of the variables are similar in both populations. Conversely, an ROCarea value of 1.0 means that the distributions of the variables of two populations do not overlap at all. A threshold value was applied such that any value below the threshold was assigned into a healthy category whereas a value equal to or above the threshold was assigned into falls risk category. True positive or sensitivity is defined as a measure of the ability of a single parameter to identify a falls risk gait, whereas false positive or specificity is a measure to detect healthy gait characteristics. ROC curve plots true positive against false positive as the threshold decision level is varied. The area under ROC curve was approximated numerically using the trapezoidal rules as described in [13, 14]. The best accuracy, sensitivity and specificity obtained at a particular threshold for all features were also calculated with ROC areas. All data analyses were performed off-line, using custom software programs written for MATLAB (The Mathworks, Natick, MA).
Surrogate data analysis
To prove any intrinsic relationship of locomotor control system with ApEn, we followed a method of surrogate data analysis introduced by Theiler et al. . For each MFC series of falls risk subjects, 10 surrogate MFC series was obtained by randomly shuffling the original series. Each surrogate data sets had the identical MFC distribution (i.e., same mean, SD, and higher moments) as the original data sets and differed only in the sequential ordering of MFC series. Then the mean of the surrogate ApEn values were then calculated for the 10 surrogate data sets and compared to the ApEn of the original data set.
Mean ± standard deviation of parameters for healthy and falls-risk elderly subjects.
Heatlhy (n = 14)
Falls-risk (n = 10)
1.65 ± 0.75
2.01 ± 0.51
0.35 ± 0.13
0.48 ± 0.16
0.51 ± 0.19
0.72 ± 0.25
0.89 ± 0.32
1.15 ± 0.40
0.64 ± 0.13
0.64 ± 0.12
0.13 ± 0.13
0.18 ± 0.03
Correlation coefficients among measures of MFC in healthy elderly subjects
Correlation coefficients among measures of MFC in falls risk elderly subjects
Relationship between Poincaré plot indexes and mean MFC
Relationship between ApEn and mean MFC
The correlation coefficient of mean MFC with ApEn in the falls-risk group (r = 0.74) was significantly (p < 0.05) higher than that in the healthy group (r = 0.14). Panel F in Figure 3 illustrates significantly positive correlation (r = 0.74, p < 0.05) between ApEn and mean MFC measures in the falls risk group, however, such correlation was absent in the healthy elderly group (panel C in Figure 3).
Relationship between ApEn and Poincaré plot indexes
Correlation analysis also showed that ApEn was significantly inversely correlated with SD1 and SD2 (r = -0.68, P < 0.05; r = -0.74, p < 0.05) except SD1/SD2 (r = 0.38, p > 0.05) in the healthy elderly group. On the other hand, no significant (p > 0.05) but positive correlations were found between ApEn and SD1 & SD2 (r = 0.49, r = 0.59) in the falls-risk group. The relationship of ApEn with SD1/SD2 in falls-risk group was also insignificant but inverse (r = -0.28, p > 0.05).
ApEn of surrogate MFC data
ROC curve analysis
The results of this study highlight the implications of nonlinear variability indexes that have been utilized to characterize MFC signals of the elderly subjects during walking. Poincaré plot geometry and ApEn analysis of MFC gait data of elderly subjects provide useful information regarding identification of gait characteristics due to balance impairments in the elderly.
MFC data and statistics
In this study, MFC data from steady-state gait have been used to characterize gait patterns. There are two major reasons for this. Firstly, MFC provides a more sensitive measure of motor function of the locomotor system compared to some gross overall kinematic descriptions of gait such as joint angular changes or stride phase times, secondly its close linkage with tripping falls [2, 16]. Furthermore, long-term MFC data, as used in this study, are required so that variability indexes of MFC having long range correlation could be captured representative of the real gait performance. In our previous study  on MFC variability statistics for young/old gait patterns, we showed that MFC variability in the elderly is higher than that in the young subjects. Results from this study suggest that MFC variability in the healthy elderly is lower than that in the falls risk elderly. Higher mean MFC in the falls risk elderly group supports our previous findings  which showed that increasing the MFC height is one of the possible strategies used by elderly individuals to minimize tripping.
MFC Poincaré plot indexes
Our results demonstrated that gait pathology due to balance impairments was reflected in altered MFC Poincaré plots (Figure 2D) and indexes extracted from these plots are effective in differentiating healthy and falls-prone gaits. Poincaré plot geometry was used in our earlier study for young-old gait pattern classification . In this study, it has been extended to identifying elderly with a history of falls and balance problems. The pattern of MFC Poincaré plots and the increased range of SD1 and SD2 values are unique for particular type of gait abnormality like balance impairments. As both SD1 and SD2 are increased due to balance impairments (Table 3 &4) SD1/SD2 are not different between the two groups. Thus the indexes derived from this geometry may be considered as a characteristic parameter of diagnostic importance in clinical gait analysis. Nonlinear dynamics  considers the Poincaré plot as the two-dimensional (2-D) reconstructed MFC phase-space, which is a projection of the reconstructed attractor describing the dynamics of the locomotor system.
ApEn analysis for MFC data
The importance of ApEn lies in the fact that it is a measure of disorder or randomness in the MFC signals. Higher ApEn values displayed in the falls-risk group might be an indication of randomness in the walking pattern of falls-risk elderly. On the other hand for healthy elderly subjects where MFC signals are more regular, ApEn has lower values. The value of ApEn reflecting the degree of irregularity, randomness and complexity of the MFC time series data, could therefore, indicate the degree of stability in the control of foot motion over the ground. In contrast, however, Goldberger  proposed that increased regularity of signals represents a 'decomplexification' of illness, citing numerous examples of illness states with increased regularity of rhythms. For example, Cheyne-Stokes respiration, Parkinsonian gait, loss of EEG variability, preterminal cardiac oscillations, neutrophil count in chronic myelogenous leukaemia and fever in Hodgkin's disease all exhibit periodic, more regular variation in the dynamics of disease states. In contrast to the 'decomplexification' hypothesis, Vaillancourt and Newell [19, 20] noted increased complexity and increased approximate entropy in several disease states, including acromegaly and Cushing's disease, and hypothesized that disease may manifest with increased or decreased complexity, depending on the underlying dimension of the intrinsic dynamic (e.g. oscillating versus fixed point).
It is the first time that ApEn analysis has been used to characterize MFC signals. Therefore, values obtained in this study cannot be compared with other studies. However, a previous study involving stride interval gait time series, Costa et al  applied multi-scale entropy (MSE) for analysing gait with different speeds and studied the scaling effect on sample entropy for different walking rates. In that study, sample entropy (SampEn) in which self matches are excluded in the analysis, on multiple scales in normal spontaneous walking time series was found to be the highest value (i.e., highest complexity)when compared to slow and fast walking and also to walking paced by a metronome . Although both SampEn and ApEn quantify the regularity of a time series, methods of calculation are different . In our study, ApEn values of MFC in normal walking have been found to be higher in falls risk subjects than in healthy subjects. A principal advantage in the application of ApEn to biological signals is that ApEn statistics may be calculated for relatively short series of data which makes it a desirable application for routine diagnosis of possible gait impairment.
Correlation analysis was designed to quantify the relationship of mean MFC with Poincaré plot indexes and ApEn values, and the relationships among these measures. Significantly positive correlations of mean MFC with SD1, SD2 and ApEn values in the falls risk subjects might indicate that MFC variability and its randomness significantly increase with an increase of mean MFC in falls risk gait. On the other hand, insignificant correlations (Table 3) in the healthy subjects indicate that MFC variability and its randomness insignificantly increase with an increase of mean MFC. Besides, it is also interesting to note that inverse correlations between SD1, SD2 and ApEn values were present in healthy subjects indicating that the more the variability the less the randomness (i.e. lower ApEn) in their gait (Table 3 &4). In contrast, an insignificant but positive correlations were found in falls risk subjects. One possible interpretation may be that higher SD1 and SD2 values, which correspond to higher short term and long term variability respectively, of falls risk subjects imply more random gait (i.e. higher ApEn) due to impaired balance control system. On the other hand, the increase of SD1 and SD2 values render more regular gait (i.e. lower ApEn) in the gait pattern of healthy elderly subjects. These results are interesting but it needs to be further investigated in a larger and more diverse sample of healthy and falls risk elderly adults.
Surrogate data analysis
The use of surrogate data was aimed at destroying the underlying control mechanism and to increase the degree of randomness. Absence of correlation of mean MFC with ApEn and increased values of ApEn in the surrogate MFC data (shown in Figure 4) proved the presence of a particular locomotor control mechanism in the falls-risk elderly. Therefore, it could be inferred that MFC in the elderly walkers is not randomly executed from stride-to-stride rather it follows the fact that MTC output in such ageing gait is modulated by some other unknown mechanism which remains to be explored. These findings seem to support previous studies that have investigated complexity break down within both temporal and spatial  time series data amongst older adults and pathological groups.
ROC curves and decision
Although both Poincaré plot indexes and ApEn were effective in discriminating the gait characteristics patterns, larger area under ROC curves for ApEn (Figure 5) suggested that ApEn could perform better than Poincaré plot indexes in classifying gait pattern. One possible reason why a nonlinear index like ApEn could be a more effective gait identifier might be that neural control mechanism of healthy human gait is nonlinear and hence, correlated with indexes derived from nonlinear analysis. This result could be useful in designing an automated gait pattern recognition model using nonlinear MFC variability indexes as input features.
More research is needed to compare the prognostic value and clinical utility of the various statistical and new MFC variability measures before an ideal index can be introduced for clinical intervention purposes. Before the measurement of MFC variability can be considered to be of any clinical value, however, therapeutic interventions (e.g., exercise program to improve balance) are needed in the subjects who present with abnormal values (e.g., high ApEn values, higher MFC variability). Further validation should provide important information on whether falls prevention intervention can improve the gait performance of falls risk elderly by monitoring the change in linear and nonlinear MFC variability indexes. Different walking speeds may alter the MFC fluctuation magnitude which provides an alternative approach for future investigation of the relationship between ApEn and mean of MFC time series data.
Early detection of gait pattern changes due to ageing and balance impairments using indexes derived from Poincaré plot geometry and ApEn analysis of MFC might provide the opportunity to initiate pre-emptive measures to be undertaken to avoid injurious falls. Also, such nonlinear index could potentially be used as gait diagnostic marker in clinical situation. Further investigation should be carried out to validate the associations of derived nonlinear MFC variability indexes with balance impairments in the falls risk subjects undergoing falls prevention intervention.
MFC gait data for this study were taken from Victoria University (VU) Biomechanics database. Several people have contributed to the creation of the gait database. The authors wish to acknowledge contributions of various people to build this database, especially Simon Taylor of the VU Biomechanics Unit. This work was partially supported by an Australian Research Council (ARC) Linkage grant (LP0454378).
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