The calculated dynamic biomechanical or muscle waveform data characterising lower limb function exists as temporal waveforms of normalised magnitude of the examined parameter against percentage of stance phase or gait cycle (from 0 – 100%). There are characteristics of this data that must considered for a meaningful analysis:
Firstly, there are a large amount of data owing to biomechanical interpretation. Each waveform is composed of 101 points, each of which describes kinematic movement and kinetic information, much of which may not be useful to group comparison. Due to the high dimensionality of the data, simple comparison of variance testing is not applicable. Secondly, there is considerable variability in the data which may be related to kinematic and kinetic differences that either occur between subjects, considered inherent inter- subject variability, or that occurs between groups which is of primary interest. Extracting salient information from that which is considered noise is challenging, and so methods to retain the most important discriminatory features are often explored (Chau, 2001). Typically, there are two approaches to evaluate temporal waveform data employed in biomechanics studies, both aimed at reducing the data into discrete summative measures for statistical testing. Parameterisation, which involves extracting discrete parameters such as waveform peaks (e.g. minimal and maximal values) and integrals (area under the curve), and multivariate methods that consider the entire waveform, such as principal component analysis (PCA) and factor analysis. Although discrete parameterisation generates an easily interpretable set of variables that describe key features of the waveform, much of the important and potentially discriminatory
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temporal information is discarded. Another limitation of parameterisation is that consistent quantification of comparable features is not always possible, due to the highly variable nature of human locomotion.
As discussed previously, the calculation of joint angles and moment waveforms can be prone to offsets due to errors in marker placement and force-plate calibration (2.4.2.7). Considering the last two points discussed above, it is only possible to detect meaningful differences when the inter-group differences are much more prominent than both inter- subject differences caused by offset error and inherent inter-subject variability, rendering it unreliable for detecting group differences.
Each point of a waveform is related to adjacent points in the parameter, as well as those of other waveforms within the same point of the gait cycle. Multivariate statistical techniques such as PCA take advantage of the collinear multidimensional nature of waveform data. Both supervised (compute using prior grouping information about the variables) or unsupervised methods exist, which both serve to reduce data and highlight patterns in potentially correlated multivariate data related to common modes, or ‘features’ of variance. PCA has advantages over other methods. Firstly, it transforms the data into a smaller set of linearly independent unique variables which can be interpreted, of which only a few are needed to adequately explain the original data. Secondly, it generates a set of ‘scores’ for each subject, based on their data relative to the model, allowing further exploratory analyses such as comparison of means (e.g. t-tests), regression, clustering methods as well as discriminatory analyses. Due to advantages and the large number of gait biomechanics studies employing PCA in current literature, it was used for the primary analysis within this thesis for analysis of biomechanics data. PCA has previously been utilized more than other multivariate methods, thus allowing for better comparability to previous studies.
2.4.1 Principal component analysis of biomechanical waveforms
Mathematically, PCA takes a number (p) of potentially correlated variables 𝑿 = 𝒙𝟏, 𝒙𝟐, … , 𝒙𝒑 from n observations and converts them into a reduced number ofindependent, uncorrelated variables 𝒁 = 𝒛𝟏, 𝒛𝟐, … , 𝒛𝒑 through orthogonal
transformations, which are arranged in a hierarchy of decreasing sample variances. The resulting variables, known as principal components (PCs), are summative measures related to the original shape of the examined waveform that are representative of common variances of the waveform. Within this study, PCA was applied to biomechanical
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and muscle activation waveforms within Inspect 3D, software developed by C-motion, who also developed the biomechanical modelling software (Visual 3D) discussed previously. Their applied method was developed with Dr. Kevin Deluzio, who first introduced the application within the musculoskeletal biomechanics field as a data reduction tool to investigate differences between OA and control subject gait (Deluzio et al., 1997). It has since been used extensively in the field for analysis of biomechanical and muscle activation waveforms (Hubley-Kozey et al., 2008, Landry et al., 2007, Chau, 2001). A method for PCA was also adopted by Jones (2004) at Cardiff University alongside a classification method based on Dempster-Shafer Theory of Evidence, which has successfully been used to detect improvement of OA function following total knee replacement (TKR) surgery (Jones et al., 2006).
Since the method used presented within this thesis was that integrated within Inspect 3D, the methods described below are representative, and further extended and adapted from those outlined in (Robertson et al., 2013) of which Deluzio’s methods are based:
Calculating principal components
Since within this study the original data exists as time-series data, it can be represented as a matrix
𝐗 = [
𝒙
𝟏𝟏⋯
𝒙
𝟏𝒑⋮
⋱
⋮
𝒙
𝒏𝟏⋯ 𝒙
𝒏𝒑]
(1.0)
whereby n is the number of subjects to be included in the model, and p is the number of time-points (samples), which in this study are 101 samples normalised to percentage of the gait cycle/stance phase (0 – 100%). PCA is applied to the columns of X so that the correlated variables are the p normalised samples observed on n subjects.The next step is to calculate the covariance matrix S, in order to express the variance structure contained within the original data matrix. This is necessary to understand the variance in the waveform over time and how subject waveforms vary from each other.
S = [
𝑠
11⋯
𝑠
1𝑝⋮
⋱
⋮
𝑠
𝑝1⋯ 𝑠
𝑝𝑝]
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By calculating the mean of the i th column of X, followed by the average squared distance between the mean and all n waveform values at that instantaneous point in time, the diagonal factors