In quantitative gait analysis, variability is commonly understood to be the fluctuation in the value of a kinematic (e.g. joint angle), kinetic (e.g. ground reaction force), spatio-temporal (e.g. stride interval) or electromyographic measurement. This fluctuation may be observed in repeated measurements over time, across or within individuals or raters, or between different measurement, intervention or health conditions. In this paper, we will focus on the variability in two types of data: unidimensional gait variables and single-cycle, prototypical gait curves, as these are the most common abstractions of spatio-temporal, kinematic and kinetic data, typically collected within a gait laboratory.

Many different analytical methods have been proposed for estimating the variability in gait variables. The most widely used measures are those relating to the second moment of the underlying probability distribution of the gait variable of interest. Examples include, standard deviation (e.g., 1234), coefficient of variation (e.g., 5678) and coefficient of multiple correlation (e.g., 910). Other less conventional variability measures have also been suggested. For example, Kurz et al. demonstrated an information-theoretic measure of variability, where increased uncertainty in joint range-of-motion (ROM), and hence entropy, reflected augmented variability in joint ROM 11.

For gauging variability among gait curves, some distance-based measures have been put forth, including the mean distance from all curves to the mean curve in raw 3-dimensional spatial data 12, the point-by-point intercurve ranges averaged across the gait cycle 13 and the norm of the difference between coordinate vectors representing upper and lower standard deviation curves in a vector space spanned by a polynomial basis 14. Instead of reporting a single number, an alternative and popular approach to ascertain curve variability has been to peg prediction bands around a group of curves. Recent research on this topic has demonstrated that bootstrap-derived prediction bands provide higher coverage than conventional standard deviation bands 151617.

Additionally, various summary statistics, such as the intra-class correlation coefficient 8 and Pearson correlation coefficient 18, for estimating gait measurement reliability, repeatability or reproducibility have been deployed in the assessment of methodological, environmental and instrumentation or device-induced variability. Principal components and multiple correspondence analyses have also been applied in the quantification of variability in both gait variables and curves, as retained variance and inertia, respectively, in low dimensional projections of the original data 19.

As depicted in Figure 1, the numerous sources of variability in gait measurements can be loosely categorized as either internal or external to the individual being observed 20.

The magnitude of variability and its alteration bears significant clinical value, having been linked to the health of many biological systems. Particularly in human locomotion, the loss of natural fractal variability in stride dynamics has been demonstrated in advanced aging 32 and in the presence of neurological pathologies such as Parkinson's disease 42, and amyotrophic lateral sclerosis 42. In some cases, this fractal variability is correlated to disease severity 32. Variability may also serve as a useful indicator of the risk of falls 43 and the ability to adapt to changing conditions while walking 44. Stride-to-stride temporal variability may be useful in studying the developmental stride dynamics in children 45. Natural variability has been implicated as a protective mechanism against repetitive impact forces during running 14 and possibly a key ingredient for energy efficient and stable gait 46. Variability is not always informative and useful and in fact may lead to discrepancies in treatment recommendations. For example, due to variability in static range-of-motion and kinematic measurements, Noonan et al. found that different treatments were recommended for 9 out of 11 patients with cerebral palsy, examined at four different medical centres 13.

Given the ubiquity and health relevance of variability in gait measurements, it is critical that we summarize and compare gait data in a way that reflects the true nature of their variability. Despite the apparent simplicity of these tasks, if not conducted prudently, the derived results may be misleading, as we will exemplify. In fact, there are to date many open questions relating to the analysis of quantitative gait data, such as the elusive problem of systematically comparing two families of curves.

The objectives of this paper are twofold. First, we aim to review some of the analytical issues commonly encountered in the summary and comparison of gait data variables and curves, as a result of variability. Our second goal is to demonstrate some practical solutions to the selected challenges, using real empirical data. These solutions largely draw upon successful methods reported in the statistics literature. The remainder of the paper addresses these objectives under two major headings, one on gait variables and the other on gait curves. The paper closes with some suggestions for the summary and comparison of gait data and directions for future research on this topic.

It has been recently demonstrated that typical location and spread estimators used in quantitative gait data analysis, i.e. mean and variance, are highly susceptible to small quantities of contaminant data 48. Indeed, a few spurious or atypical measurements can unduly inflate non-robust estimates of gait variability. The challenge in the summary of highly variable univariate gait data lies in reporting location and spread, faithful to the underlying data distribution and minimally influenced by extraordinary observations.

Here, we focus on the issue of inflated variability and non-robust estimation by examining four different spread estimators, applied to stride period data from a child with spastic diplegic cerebral palsy. As stated above, the coefficient of variation and standard deviation are routinely employed in the summary of gait variables. Given a sample of N observations of a gait variable X, i.e., {x₁,..., x_N}, the coefficient of variation is defined as,

where the numerator is simply the sample standard deviation and the denominator, , is the sample mean. We also include two other estimators, although seldom used in gait analysis, to illustrate the qualitative differences in estimator robustness. The interquartile range of the sample is defined as

IQR(X) = x_0.75 - x_0.25     (2)

where x_0.75 and x_0.25 are the 75% and 25% quantiles. The q-quantile is defined as where as usual, F_X is the probability distribution of X. Equivalently, the q-quantile is the value, x_q, of the random variable where . That is, q × 100 percent of the random variable values lie below x_q. We also introduce the median absolute deviation 49,

MAD(X) = med (|X - med(X)|)     (3)

where med(X) is the median of the sample, or the 50% quantile as defined above. This last estimator is, as the name implies, the median of the absolute difference between the sample values and their median value. We are interested in studying how these different estimators perform when estimating the spread in a gait variable, the observations of which may contain outlying values or contaminants. In the left pane of Figure 2, we show a set of stride period data recorded from a child with spastic diplegia. The top graph shows the raw data with a number of obvious outliers with atypically long stride times. We adopted a common outlier definition, labeling points more than 1.5 interquartile ranges away from the sample median as extreme values. According to this definition there were 21 outlying observations. In the bottom graph, the outliers have been removed. The bar graph on the right-hand side of Figure 2 portrays the spread estimates of the stride period data, computed with each estimator introduced above, with and without the outliers.

We note immediately that the spread estimates in the presence of outliers are higher. The standard deviation and coefficient of variation change the most, dropping 42 and 36 percent in value, respectively, upon outlier removal. This observation is particularly important in the comparison of gait variables, as inflated variability estimates will diminish the probability of detecting significant differences when they do in fact exist. In contrast, the interquartile range and median absolute deviation, only change by 21 and 11%, respectively. We see that these latter estimates are more statistically stable, in that they are not as greatly influenced by the presence of extreme observations.

To more fully comprehend estimator robustness or lack thereof, the field of robust statistics offers a valuable tool called influence functions, which as the name implies, summarizes the influence of local contaminations on estimated values. Their use in gait analysis was first introduced in the context of stride frequency estimation 48.

We first introduce the concept of a functional, which can be understood as a real-valued function on a vector space of probability distributions 50. In the present context, functionals allow us to think of an estimator as a function of a probability distribution. For example, for the interquartile range, the functional is simply, .

Let the mixture distribution F_{z, ε} describe data governed by distribution F but contaminated by a sample z, with probability ε. The influence function at the contamination z is defined as

where T(·) is the functional for the estimator of interest. The influence function for a particular estimator measures the incremental change in the estimator, in the presence of large samples, due to a contamination at z. Clearly, if the impact of this contaminant on the estimated value is minimal, then the estimator is locally robust at z. Influence functions can be analytically derived for a variety of common gait estimators (see for example, 48), including those mentioned above. For the sake of analytical simplicity and practical convenience, we will instead use finite sample sensitivity curves, SC(z), which can be defined as,

SC(z) = (N + 1){T(x₁,..., x_N, z) - T(x₁,..., x_N)}     (5)

where as above, T(·) is the functional for the estimator in question, and z is the contaminant observation. When N → ∞ the sensitivity curve converges to the influence function for many estimators. Like the asymptotic influence functions, sensitivity curves describe the local impact of a contamination z on the estimator value. For the purposes of computer simulation, the functional T(x₁,..., x_N, z) and T(x₁,..., x_N) are simply the evaluations of the estimator of interest at the augmented and original samples, respectively. Figure 3 depicts the sensitivity curves for the estimators introduced in the stride period example. To generate these curves, we used the cleansed stride period data (without outliers) and incrementally added a deviant stride period from 0.5 below the lowest sample value to 0.5 above the highest sample value. The sample mean for this data was 1.41 seconds.

We observe that both standard deviation and coefficient of variation have quadratic sensitivity curves with vertices close to the sample mean. In other words, as contaminants take on extreme low or high values, the estimated values are unbounded. Clearly, these two estimators are not robust, explaining their high sensitivity to the outliers in the stride period data. In contrast, both the interquartile range and median absolute deviation have bounded sensitivity curves, in the form of step functions. The median absolute deviation is actually not sensitive to contaminant values above 1.1 seconds whereas the interquartile range has a constant sensitivity to contaminant values over 1.6. Since most of the outliers in the stride period data were well above the mean, this difference explains the considerably lower sensitivity of the median absolute deviation to outlier influence.

From this example, we appreciate that estimators of gait variable spread (i.e. variability) should be selected with prudence. The popular but non-robust variability measures of standard deviation and coefficient of variation both have 0 breakdown points 51, meaning that only a single extreme value is required to drive the estimators to infinity. Indeed, as seen in Figure 2, the presence of a small fraction of outliers can unduly inflate our estimates of gait variability. Outlier management 52, with methods such as outlier factors 53 or frequent itemsets 54, represents one possible strategy to reduce unwanted variability when using these non-robust estimators. Apart from the addition of a computational step, this strategy introduces the undesirable effects of outlier smearing and masking 55, which need to be carefully addressed.

In contrast, outliers need not be explicitly identified with robust estimation, hence circumventing the above complications and abbreviating computation. The interquartile range and median absolute deviation, have breakdown points of 0.25 and 0.5, respectively 51. Practically, this means that these estimators will remain stable (bounded) until the proportion of outliers reaches 25% and 50% of the sample size, respectively. To circumvent explicit outlier detection and its associated issues altogether, and in the presence of noisy data, which often result from spatio-temporal recordings and parameterizations of kinematic and kinetic curves, robust estimators may thus be preferable in the summary of gait variables.

Even in the absence of outliers, univariate gait data may not adhere to a simple, unimodal gaussian distribution. In fact, distributions of gait measurements and derived parameters may be naturally skewed, leptokurtic or multimodal 56. Neglecting these possibilities, we may summarize gait data with location and spread values which do not reflect the underlying data distribution.

It has been recognized that within a sample of single-cycle gait curves, there is both amplitude and phase variation 717273. Typically, when we describe variability in gait curves, we refer to amplitude variability. However, unchecked phase variation, that is the temporal misalignment of curves, can often lead to inflated amplitude variability estimates 7273. Computing cross-sectional averages over a family of malaligned gait curves can lead to the cancellation of critical shape characteristics and landmarks 74. This issue presents a significant challenge when summarizing a series of curves for clinical interpretation and treatment planning. On the one hand, the presentation of a large number of different curves can be overwhelmingly difficult to assimilate. On the other hand, a prototypical average curve which does not reflect the features of the individual curves is equally uninformative.

Curve registration 71 is loosely the process of temporally aligning a set of curves. More precisely, it is the alignment of curves by minimizing discrepancies from an iteratively estimated sample mean or by allineating specific curve landmarks. Sadeghi et al. demonstrated the use of curve registration, particularly to reduce intersubject variability in angular displacement, moment and power curves 7273. Additionally, they reported that curve characteristics, namely, first and second derivatives and harmonic content were preserved while peak hip angular displacement and power increased upon registration 72. This latter finding confirms that averaging unregistered curves may eliminate useful information.

Judging by the few gait papers employing curve registration, the method appears largely unknown among the quantitative gait analysis community. Here, we briefly outline the the global registration criterion method 7175.

Since each gait curve is a discrete set of points, it is useful to estimate a smooth sample function for each observed sample curve. Given the periodic nature of gait curves, the Fourier transform provides an adequate functional representation of each curve. The basic principle is then to repeatedly align a set of sample functions to an iteratively estimated mean function. The agreement between a sample function and the mean function can be measured by a sum-of-squared error criterion. The goal of registration is to find a set of temporal shift functions such that the evaluation of each sample function at the transformed temporal values minimizes the sum-of-squared error criterion. The sample mean is re-estimated at each iteration with the current set of time-warped curves. As an optimization problem, the curve registration procedure is the iterative minimization of the sum-of-squared criterion J,

where N is the number of sample curves, T is the time interval of relevance, w_i(·) is the time-warping function and is the iteratively estimated mean based on the current time-warped curves X_i (w_i (s)). For greater methodological details, the reader is referred to 717275. This global registration criterion method is only one of several possibilities for curve alignment. Related methods which are applicable to gait data include dynamic time warping based on identified curve landmarks 41 and latency corrected ensemble averaging 28.

We exemplify the impact of accounting for undesirable phase variation using ankle angular displacement data from a child with spastic diplegla. The top left graph of Figure 6 depicts the unregistered curves, exhibiting excessive dorsiflexion throughout the gait cycle and the absence of the initial valley during loading response. Below this graph are the aligned curves. Note particularly the alignment of the large valley at pre-swing and the peak in swing phase.

The right column of Figure 6 indicates that the differences in the mean and standard deviation curves before and after registration are non-trivial, with maximum changes of +15% and -51%, respectively. The post-registration mean curve not only exhibits heightened but shifted peaks (3 – 5% of the gait cycle). This observation suggests that simple cross-sectional averaging without alignment may not only diminish useful curve features but can also inadvertently misrepresent the temporal position of key landmarks. Inaccurate identification of these landmarks, such as the minimum dorsiflexion at the onset of swing phase in this example, could be problematic when attempting to coordinate spatio-temporal and EMG recordings with kinematic curves. The bottom right graph shows a dramatic decrease in variability after registration, particularly in terminal stance. This finding is in line with the tendency towards variability reduction reported by Sadeghi et al. 72.

While curve registration is useful for mitigating unwanted phase variation in gait curves, there may be instances where phase variability is itself of interest 3. In such instances, curve registration can still be useful in providing information about the relative temporal phase shifts among curves. Because curve registration actually changes the temporal location of data, it should not be applied in studies concerned with temporal stride dynamic characterizations, such as scaling exponents 21 or Lyapunov exponents 44. At present, only a few gait studies have applied curve registration to manage undesired phase variability. However, the evidence in those studies, along with the example above, supports further research and exploratory application of curve registration to fully grasp its merits and limitations in quantitative gait data analyses. For now, curve registration appears to be the most viable solution to the challenge of summarizing a family of temporally misaligned gait curves. In the ensuing sections, we will demonstrate how curve registration can be used advantageously, in conjunction with other methods to address other curve summary and comparison challenges.

We have already seen that curve registration can mitigate amplitude variability in a family of gait curves. The robust measurement of variability in gait curves is itself a non-trivial challenge. One may need to estimate the variability in a group of curves for the purposes of classifying a new observation as belonging to the same population, or not 15. Alternatively, knowledge of the variability among curves can help in the statistical comparison of two populations of curves 16, say arising from two different subject groups or pre- and post-intervention.

As in gait variables, the challenge lies in robustly estimating the spread of a sample of gait curves and to avoid fallacious under or overestimation. The intuitive and perhaps most popular way of estimating curve variability is the calculation of the standard deviation across the sample of curves, for each point in the gait cycle. This yields upper, U_X, and lower bands, L_X, around the sample of curves, i.e.

U_X (t) = μ_X (t) + σ_X (t) t = 1,..., 100

L_X (t) = μ_X (t) - σ_X (t)     (10)

where , for t = 1,..., 100, is the sample mean curve. Lenhoff et al. argued, by way of empirical examples and systematic cross-validation, that standard deviation bands provide inadequate coverage of the sample curves 15. They instead supported the use of bootstrap prediction bands 76 which, in their study, provided close to the targeted 90% coverage of the sample curves. Two subsequent studies 1617 have adopted the 90% bootstrap bands for the classification of new curves and for the comparison between groups of curves. The usefulness of bootstrap prediction bands for clinical identification of pathological deviations in kinematic curves has also been demonstrated 77. In this section, we provide further evidence to support the use of bootstrap prediction bands and argue that they are more stable than standard deviation bands.

The basic idea of the bootstrap method is to create a large number of bootstrap subsets by resampling the curves X_j, j = 1,..., N with replacement. For each subset, the bootstrap mean and standard deviation are calculated. One then checks how many of the sample curves are "covered" by the bootstrap standard deviation bands. A curve is considered covered, if its maximum absolute standardized difference from the bootstrap mean is less than the bootstrap constant C. The number of covered curves averaged over all the bootstrap subsets then yields the coverage probability for the given bootstrap constant, C. The upper and lower bootstrap prediction bands can then be written as,

The reader is referred to 15 for details for practical computer implementation of the above procedure.

To exemplify issues of robust spread estimation, we consider knee angle curves from a child with spastic diplegia. Initially standard deviation and bootstrap bands are computed for the data prior to curve registration. The maximum absolute deviation from the sample mean curve is reported in Table 3. For both methods, the maximum spread decreases significantly upon registration, suggesting that there is significant inflated variability in the unaligned curve sample. Once the curves are aligned, one suspicious curve, plotted as a thin dashed line in Figure 7, becomes evident. The standard deviation bands around the sample with and without this outlying curve are shown on the left side of Figure 7. The maximum spread, that is max_t and C, for standard deviation and bootstrap bands, respectively, are labeled on each graph. We see that by removing the outlying curve, both the standard deviation and bootstrap bands become narrower. In fact, as seen in Table 3, the maximum standard deviation decreases by a dramatic 27%. Thus it appears that the variability among a group of curves, as estimated by both standard deviation and bootstrapping, can be minimized by curve registration and further reduced by the subsequent removal of outlying curves.

To further understand the robustness properties of the two spread estimators, we generate sensitivity curves using the 45 knee angle curves introduced in Figure 4. These curves are first registered to minimize unwanted phase variability. In the case of gait curves, the contaminant is not a single point, but an entire curve. For convenience, we choose the following contaminant,

where δ ∈ ℝ and δ_min ≤ δ ≤ δ_max. In other words, the contaminant is just a shifted version of the sample mean curve, . For simulating the sensitivity curve, we choose δ_min = -50 and δ_max = 50, recognizing that in practice, we would never observe deviations of this magnitude. This large range does however, gives us a more complete picture of the sensitivity curves. We proceed to define the sensitivity curves for the standard deviation and bootstrap estimates as follows,

where is the variance of the uncontaminated sample and

is the variance of the contaminated sample. In the above, is the mean curve of the contaminated sample. The notations C_X and C_{X, z} represent the bootstrap constants determined using the original and contaminated data, respectively. In other words, these sensitivity curves will reflect the influence of a contaminant curve, z(t), on the maximum estimated spread across a group of curves, over the gait cycle. Figure 8 summarizes the results of evaluating (15) over the simulated contaminants defined in (13).

We note that, as in the univariate case, the standard deviation exhibits quadratic sensitivity with vertex at the zero deviation curve. This parabolic sensitivity curve indicates that the standard deviation bands are not locally robust to contaminant curves. In contrast, the sensitivity curve for the bootstrap bands is not smooth and quartic in nature. The lack of smoothness is due to the random resampling inherent in the bootstrap method, such that with each contaminant curve, slightly different bootstrap samples are used in estimating the 90% prediction bands. Initially, as the contaminant curve deviates from the mean curve, the sensitivity is negative, meaning that the width of the estimated bands are smaller than those for the uncontaminated data. Indeed, the actual value of the bootstrap constant initially decreases, likely to counter the accompanying sharp increase in the standard deviation bands. In other words, as the standard deviation bands widen, a smaller bootstrap constant is required to cover 90% of the sample curves. However, as the contaminant curve deviates farther from the mean, the slope of standard deviation sensitivity increases in magnitude more slowly. With a smaller change in standard deviation band per unit of deviation of the contaminant curve, the bootstrap constant necessarily increases to maintain 90% coverage. This reasoning accounts for the subsequent increase in the tails of the bootstrap sensitivity curve. Finally, we note that overall, the bootstrap sensitivity curve, although apparently unbounded, traverses a much smaller range than the standard deviation curve. This would suggest that with the kinematic data employed in this example, the bootstrap coverage bands enjoy greater stability than their highly sensitive standard deviation cousins.

In brief, the foregoing discussion further supports the use of bootstrap coverage bands in robustly summarizing the variability within a family of gait curves. Moreover, curve registration and outlier removal can further tighten the location of the prediction bands.

It is common to compare specific landmarks or features of gait curves to gauge the impact of an intervention or to determine differences among different subject populations. However, the identification of curve features is inherently problematic. Indeed, the multiplicity of peaks and valleys across two different groups of curves may be inconsistent. As an example, Figure 9 portrays the vertical ground reaction force of an able-bodied child on the left with the typical loading response peak, mid-stance valley and terminal stance peak 78. On the right is the vertical ground reaction force from the intact side of a child with an above-knee amputation, walking with a prosthetic lower limb. Note that there are at least three distinct peaks in the mean curve (thin dotted line). Attempts to compare the able-bodied and amputee force profiles are encumbered by the unclear choice of corresponding peaks. This is a common pitfall of comparing gait curves on the basis of specific landmarks.

The wavelet transform has been touted as a useful method for uncovering intrinsic trends in data 7980. Hence, it may be possible to extract an underlying low frequency trend from the amputee force curve for the sake of striking a comparison with the able-bodied curve. To this end, we decomposed the mean force curve for the child with amputation using a 4 – level coiflet wavelet transform 81. We reconstructed the force curve using only the approximation coefficients. The resulting trend line is plotted on the right graph of Figure 9 as the thick dashed line and more closely resembles the expected force profile. Extraction of the extrema yields plausible peak locations at 17% and 44% of the gait cycle and a valley at 30%. These locations are comparable to those for the able-bodied child (peaks at 12% and 44% and valley at 26% of the gait cycle), but suggest a slightly extended loading response phase.

The extraction of the trend line in this example illustrates that in some curves, the desired landmarks may be concealed by the fluctuations of higher frequency signal components and hence may be salvageable. However, even when landmarks are clearly identifiable among curves, they reflect only a very microscopic view of the entire curve. For example, two curves could have identical landmarks, but pronounced differences in shape characteristics. We therefore do not advocate the isolated use of simple parameterizations or landmarks for routine comparison of curves. Rather, the comparison of two sets of curves should be based on the entire curve and not isolated parameterizations. We suggest however, that landmark analysis and simple parameterizations can be meaningful as a post-hoc procedure, i.e. investigating how curves are similar or different, only after statistically significant differences among curves or lack thereof have been established. We therefore suggest to first statistically compare entire gait curves as unified objects, and reserve landmarks for post-hoc analysis. In the following section, we describe how such a statistical test may be carried out.

If gait curves were strictly deterministic, one could simply define a distance measure between two curves and be done. However, due to stride-to-stride variability, an extension of the univariate statistical test is needed, to determine if one set of curves could have arisen from the same statistical distribution as another. Alternatively, one could test whether the average difference between two sets of curves is approximately zero, within the critical values of an expected distribution of differences. The fundamental challenge is to compare families of gait curves as coherent entities rather than as unconnected, independent points. One way to consider curves as a whole rather than as disjoint points is to give them an appropriate functional representation. One can then compare the functional representations of the curves. Exploiting this principle, Fan and Lin 70 proposed a general method for comparing two sets of discrete time-sampled curves. In their method, the discrete Fourier transform of the standardized difference between the mean curves of two families of curves is computed. Only selected low frequency components of the transform, which encompass the majority of signal energy, are retained. These coefficients are then subjected to the adaptive Neyman test which yields the probability that the two families of curves have similar means. To the best of our knowledge, the adaptive Neyman statistic 69 has not yet been applied in the gait literature for the comparison of empirical gait curves. We therefore outline below, in some detail, the proposed procedure that we have adapted from Fan and Lin 70. Suppose that we would like to compare two families of gait curves, {X_j (t), j = 1,..., N_X} and {Y_j (t), j = 1,..., N_Y}, with t = 1,..., 100. The null hypothesis is that the difference between the means of the two families of curves is zero. In the random function formulation given by Equation (8), we can write, H₀ : f_X (t) - f_Y (t) = 0, where f_X (t) and f_Y (t) are the true underlying mean curves. For notational convenience, we will let t = 0,..., T - 1, where we have chosen T = 100 in the previous examples. The main steps of the test are as follows.

1. Compute the sample mean curves, μ_X (t) and μ_Y (t), where and likewise for μ_Y (t)

2. Compute the sample variance curves, and , where and likewise for .

3. Align the two mean curves using the global registration criterion method. Denote the registered curves as and . This step does not appear in the original formulation of Fan and Lin 70.

4. Compute the standardized difference Z(t) between the registered means,

5. Compute the discrete Fourier decomposition, , of the standardized difference,

where k = 0,..., T/2, Real(·) and Imag(·) denote the real and imaginary components of the complex Fourier coefficient , respectively, and k denotes the Fourier frequency.

6. Form a new vector of coefficients E, of length T + 1, by pairing real and imaginary coefficients of the complex Fourier coefficients, , as follows,

7. Estimate the adaptive Neyman statistic, T_AN(E) for the vector defined above. This proceeds in two steps.

(a) Determine the optimal the number of coefficients to retain to maximize , where E_i are the elements of the vector defined above and 1 <m <T + 1. This optimal value of m, denoted , maximizes the power of the adaptive Neyman statistic 70. The maximum statistic value is written as,

where Var(E²), is the variance of the square of the elements of E obtained in step 6.

(b) Let K = ln(T ln T). Compute the following final transformed test statistic value 70,

Here, we have explicited indicated that the statistic has been computed for the vector E of Fourier coefficients. Asymptotically, this statistic has an exponential of an exponential distribution 69, that is, P(T_AN ≤ x) → exp(-exp(-x)), as T becomes arbitrarily large.

8. Estimate the p-value of the computed test statistic value, T_AN(E), by Monte Carlo simulation of a large number, say 10⁶, of vectors, Y_i, i = 1,..., 10⁶, each of length T and whose elements are drawn from a standard normal distribution, i.e. Y_i ~ (0, 1), ∀i. The rationale is that when two sets of curves arise from the same random function, the standardized differences of their Fourier coefficients are normally distributed around 0. For each normal vector, Y_i, evaluate T_AN(Y_i) as in step 7 above. When the null hypothesis of no differences is true, the probability of observing an adpative neyman statistic as extreme as T_AN(E) is estimated as,

where H(·) is the heaviside function, where H(x) = 1 only if x > 0 and is 0 otherwise. In the examples below, we simulated 10⁶ such vectors to estimate the probability of observing T_AN.

We exemplify the above procedure with two kinematic data sets taken from a child with an above-knee amputation, wearing two different types of prosthetic devices. Of interest is whether the use of a swing phase control mechanism within the prosthetic knee affects gait. Figure 10 depicts the ankle angular displacements with (dashed line) and without (solid line) the swing phase control mechanisms. By visual inspection, the curves look similar in magnitude and exhibit a slight phase difference. We would expect statistical testing to conclude that the curves are not different. The top right graph is the standardized difference between the registered mean curves. Note that most values fluctuate around 0. The bottom right graph is a stem plot of selected Fourier coefficients of the Fourier transform of the standardized difference. Note the concentration of energy in the low frequencies and the distribution of coefficients above and below 0. It is thus not surprising that the result of the adaptive Neyman test statistic, yielded T_AN = -9.01 which corresponds to p = 1 for = 2. In other words, it is very likely that the two groups of curves came from the same distribution of random functions. In the present context, the swing phase mechanism had no effect on the ankle angular displacement.

To illustrate the statistical detection of differences, we draw upon a second example involving kinematic curves from an adult subject pre- and post-surgical replacement of the ankle. In Figure 11, the pre-surgery curves do not have well resolved peaks or valleys whereas post-surgically, distinct peaks and valleys emerge with substantial magnitude. On the basis of this visual inspection, one would anticipate that statistical testing should indicate that the pre- and post-surgery curves are indeed different. The standardized difference between the registered mean curves exhibits relatively large fluctuations around 0 and the retained Fourier coefficients are nearly all positive, resulting in a positively skewed coefficient distribution. The adaptive Neyman statistic value for these coefficients is T_AN = 5.99 corresponding to p = 2.6 × 10^-5 with = 6. Hence, the statistical test indicates that there is strong evidence for rejecting the null hypothesis. It appears that surgery has significantly altered the gait curves. Once significant statistical difference has been established, one can then seek to identify specific characteristics which differentiate the two sets of curves. For example, the post-surgical curves exhibit a well-defined valley, towards plantar flexion at toe-off and a strong first dorsiflexion peak in terminal stance. Both of these extrema are absent in the presurgery curves.

Note that we have not said anything about the requisite sample sizes for the statistical comparison of gait curves. Clearly, as in unidimensional power analysis 65, the required sample size depends on the effect size, significance level and specified power. To the best of our knowledge, no power-sample size tables have been derived for the adaptive Neyman statistic at the time of writing. For insights on the topic, the interested reader can refer to authoritative works 6582 on power analysis in the univariate case. The statistical testing demonstrated here can be extended to compare more than two groups of curves, using high-dimensional analysis of variance 70. Further, when the standardized difference curve is not smooth, wavelet denoising can be used to identify the frequency bands where the majority of signal energy is concentrated 69. The adaptive Neyman statistic introduced here is only one of several possibilities for objectively and rigorously testing differences among curves. Other alternatives include an ANOVA test for functional data 83 and functional canonical correlation analysis 84. The procedure outlined in this section formalizes the comparison of gait curves as coherent entities. The method provides a means of statistically confirming overall similarities and differences that we may detect by visual inspection, but may have difficulty quantifying with conventional time and frequency domain parameterizations.