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When investigating the dynamics of three-dimensional multi-body biomechanical systems it is often difficult to derive spatiotemporally directed predictions regarding experimentally induced effects. A paradigm of ‘non-directed’ hypothesis testing has emerged in the literature as a result. Non-directed analyses typically consist of ad hoc scalar extraction, an approach which substantially simplifies the original, highly multivariate datasets (many time points, many vector components). This paper describes a commensurately multivariate method as an alternative to scalar extraction. The method, called ‘statistical parametric mapping’ (SPM), uses random field theory to objectively identify field regions which co-vary significantly with the experimental design. We compared SPM to scalar extraction by re-analyzing three publicly available datasets: 3D knee kinematics, a ten-muscle force system, and 3D ground reaction forces. Scalar extraction was found to bias the analyses of all three datasets by failing to consider sufficient portions of the dataset, and/or by failing to consider covariance amongst vector components. SPM overcame both problems by conducting hypothesis testing at the (massively multivariate) vector trajectory level, with random field corrections simultaneously accounting for temporal correlation and vector covariance. While SPM has been widely demonstrated to be effective for analyzing 3D scalar fields, the current results are the first to demonstrate its effectiveness for 1D vector field analysis. It was concluded that SPM offers a generalized, statistically comprehensive solution to scalar extraction’s over-simplification of vector trajectories, thereby making it useful for objectively guiding analyses of complex biomechanical systems.
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Vector field statistical analysis of kinematic and force trajectories
Todd C. Pataky1, Mark A. Robinson2, and Jos Vanrenterghem2
1Department of Bioengineering, Shinshu University, Japan
2Research Institute for Sport and Exercise Sciences, Liverpool John Moores University, UK
When investigating the dynamics of three-dimensional multi-body biomechanical systems it is often di-
cult to derive spatiotemporally directed predictions regarding experimentally induced eects. A paradigm
of ‘non-directed’ hypothesis testing has emerged in the literature as a result. Non-directed analyses typ-
ically consist of ad hoc scalar extraction, an approach which substantially simplifies the original, highly
multivariate datasets (many time points, many vector components). This paper describes a commen-
surately multivariate method as an alternative to scalar extraction. The method, called ‘statistical
parametric mapping’ (SPM), uses random field theory to objectively identify field regions which co-vary
significantly with the experimental design. We compared SPM to scalar extraction by re-analyzing three
publicly available datasets: 3D knee kinematics, a ten-muscle force system, and 3D ground reaction
forces. Scalar extraction was found to bias the analyses of all three datasets by failing to consider suf-
ficient portions of the dataset, and/or by failing to consider covariance amongst vector components.
SPM overcame both problems by conducting hypothesis testing at the (massively multivariate) vector
trajectory level, with random field corrections simultaneously accounting for temporal correlation and
vector covariance. While SPM has been widely demonstrated to be eective for analyzing 3D scalar
fields, the current results are the first to demonstrate its eectiveness for 1D vector field analysis. It was
concluded that SPM oers a generalized, statistically comprehensive solution to scalar extraction’s over-
simplification of vector trajectories, thereby making it useful for objectively guiding analyses of complex
biomechanical systems.
Keywords: biomechanics, random field theory, Statistical Parametric Mapping, multivariate statistics
Category Symbol Other Description
I i Vector components
J j Responses (i.e. experimental recordings)
K k Predictor variables
NExtracted scalars (e.g. maximum force)
Q q Field measurement nodes (e.g. 100 points in time)
Mean, variance:
yiy,s2Scalar response (with
yi(q)y(q), s2(q) Scalar field response (with field)
y(q)y(q), W(q) Vector field response (with covariance field)
Test statistics
tSPM{t}t(q)Student’s tstatistic
FSPM{F}F(q) Variance ratio (e.g. from ANOVA)
T2SPM{T2}T2(q)Hotelling’s T2statistic (vector equivalent of t)
RCanonical correlation coecient
Probability Type I error rate
pProbability value
CCA Canonical correlation analysis
EMG Electromyography
GRF Ground reaction force
PFP Patellofemoral pain
1 Introduction
Measurements of motion and the forces underlying that motion are fundamental to biomechanical ex-
perimentation. These measurements are often manifested as one-dimensional (1D) scalar trajectories yi(q),
where irepresents a particular physical body, joint, axis or direction, and where qrepresents 1D time or
space. Experiments typically involve repeated measurements of yi(q) followed by registration (i.e. homolo-
gously optimal temporal or spatial normalization) to a domain of 0–100% (Sadeghi et al., 2003). This paper
pertains to analysis of registered data yi(q).
Given that many potential sources of bias exist in yi(q) analysis (Rayner, 1985; James and Bates, 1997;
Mullineaux et al., 2001; Knudson, 2009), a non-trivial challenge is to employ statistical methods that are
consistent with one’s null hypothesis. Consider first ‘directed’ null hypotheses: those which claim response
equivalence in particular vector components i, and in particular points qor windows [q0,q1]:
Example ‘directed’ null hypothesis: Controls and Patients exhibit identical maximum knee flexion
during walking between 20% and 30% stance.
To test this hypothesis only maximum knee flexion should be assessed, and only in the specified time
window. Testing other time windows, joints, or joint axes in a post hoc sense would constitute bias. This is
because increasing the number of statistical tests increases our risk of incorrectly rejecting the null hypothesis
(see Supplementary Material – Appendix A). In other words, it is biased to expand the scope of one’s null
hypothesis after seeing the data. We refer to this type of bias as ‘post hoc regional focus bias’.
Next consider ‘non-directed’ null hypotheses: hypotheses which broadly claim kinematic or dynamic
response equivalence:
Example ‘non-directed’ null hypothesis: Controls and Patients exhibit identical hip and knee kine-
matics during stance phase.
To address this hypothesis both hip and knee joint rotations should be assessed, about all three orthogonal
spatial axes, and from 0% to 100% stance (i.e. the entire dataset yi(q)). It would be biased to assess only
maximum hip flexion, for example, in a post hoc sense but for the opposite reason: it is biased to reduce the
scope of one’s null hypothesis after seeing the data.
Non-directed hypotheses expose a second potential source of bias: covariance among the Ivector compo-
nents. Scalar analyses ignore covariance and are therefore coordinate-system dependent (see Supplementary
Material – Appendix B). This is important because a particular coordinate system — even one defined
anatomically and local to a moving segment — may not reflect underlying mechanical function (Kutch and
Valero-Cuevas, 2011). Joint rotations, for example, may not be independent because muscle lines of action
are generally not parallel to externally-defined axes (Jensen and Davy, 1975). Joint moments may also not
be independent because endpoint force control, for example, requires coordinated joint moment covariance
(Wang et al., 2000). Under a non-directed hypothesis this covariance must be analyzed because separate
analysis of the Icomponents is equivalent to an assumption of independence, an assumption which may not
be justified (see Supplementary Material – Appendix B). We refer to this source of bias as ‘inter-component
covariance bias’.
Both post hoc regional focus bias and inter-component covariance bias have been acknowledged previously
(Rayner, 1985; James and Bates, 1997; Mullineaux et al., 2001; Knudson, 2009). However, to our knowledge
no study has proposed a comprehensive solution.
The purpose of this paper is to show that a method called Statistical Parametric Mapping (SPM) (Friston
et al., 2007) greatly mitigates both bias sources. The method begins by regarding the data yi(q) as a vector
field y(q), a multi-component vector ywhose values change in time or space q(Fig.1). When regarding the
data in this manner, it is possible to use random field theory (RFT) (Adler and Taylor, 2007) to calculate
the probability that observed vector field changes resulted from chance vector field fluctuations.
We use SPM and RFT to conduct formalized hypothesis testing on three separate, publicly available
biomechanical vector field datasets. We then contrast these results with the traditional scalar extraction
approach. Based on statistical disagreement between the two methods we infer that, by definition, at least
one of the methods is biased. We finally use mathematical arguments (Supplementary Material) and logical
interpretations of the original data to conclude that scalar extraction constitutes a biased approach to non-
directed hypothesis testing, and that SPM overcomes these biases.
2 Methods
2.1 Datasets
We reanalyzed three publicly available datasets (Table 1):
Dataset A (Neptune et al., 1999) ( stance-phase lower extremity dy-
namics in ten subjects performing ballistic side-shue and v-cut tasks (Fig.2). Present focus was on
within-subject mean three dimensional knee rotations for the eight subjects whose data were labeled
unambiguously in the public dataset.
Dataset B (Besier et al., 2009) ( stance-phase knee-muscle
forces during walking and running in 16 Controls and 27 Patello-Femoral Pain (PFP) patients, as
estimated from EMG-driven forward-dynamics simulations. Present focus was on walking and abso-
lute forces (newtons) (Fig.3).
Dataset C (Dorn et al., 2012) ( one subject’s full-body kine-
matics and ground reaction forces (GRF) during running at four dierent speeds: 3.56, 5.20, 7.00, and
9.49 ms1. Present focus was on three-dimensional left-foot GRF (Fig.4), for which a total of eight
responses were available. We linearly interpolated the GRF data across stance phase to Q=100 time
These three datasets were chosen, first, to represent a range of biomechanical data modalities: kinematics,
modeled (internal) muscle forces, and external forces. Second, they were chosen to demonstrate how vector
field analysis applies to a range of statistical tests: (A) paired ttests, (B) two-sample ttests, and (C) linear
2.2 Traditional scalar extraction analysis
Two, ten, and three scalars were respectively extracted from the three datasets (Table 1). These particular
scalars were chosen either because they appeared to be most aected by the experiment (Datasets A and
C), or because they were physiologically relevant (Dataset B: maximum force). As indicated above, Dataset
A’s task eects were assessed using paired ttests, Dataset B’s group eects were assessed using two-sample
(independent) ttests, and Dataset C’s speed eects were assessed using linear regression.
Since we conducted one test for each scalar, we performed N=2, N=10 and N=3 tests on Datasets A, B
and C, respectively, where Nis the number of extracted scalars. To retain a family-wise Type I error rate
of =0.05 we adopted ˇ
Sid´ak thresholds of p=0.0253, p=0.0051, and p=0.0170 respectively, where the ˇ
threshold is:
pcritical =1(1 )1/N (1)
These scalar analyses superficially appear to be legitimate analysis options. However, through comparison
with the equivalent vector field analyses (§2.3), we will show how and why scalar extraction is biased for
non-directed null hypothesis testing.
2.3 Statistical Parametric Mapping (SPM)
SPM analyses (Friston et al., 2007) were conducted using vector field analogs to the aforementioned
univariate tests (§2.2). Before detailing SPM procedures, we note that they are conceptually identical to
univariate procedures: conducting a one-sample ttest on ten scalar values, for example, is nearly identical
to conducting a one-sample ttest on ten vector fields. The only dierences are that SPM: (i) considers
vector covariance when computing the test statistic, (ii) considers field smoothness and size when computing
the critical test statistic threshold, and (iii) considers random field behavior when computing pvalues (see
Appendix A and B – Supplementary Material).
Ultimately each SPM test results in a test statistic field (e.g. the tstatistic as a function of time), and
RFT is used to assess the significance of this statistical field. §2.3.1–§2.3.3 below detail test statistic field
computations for the current datasets, §2.3.4 describes RFT computations of critical test statistic values and
pvalues, and §2.3.5 suggests a post hoc procedure for scrutinizing vector field test results.
2.3.1 Paired Hotelling’s T2test (Dataset A)
SPM’s vector field analog to the paired ttest is the paired Hotelling’s T2test, which is given by the
one-sample T2statistic (Cao and Worsley, 1999):
SPM{T2}T2(q)=Jy(q)>W(q)1y(q) (2)
where Jis the number of vector fields (Table 1) and y(q) is the mean vector field or — in the case of a
paired test — the vector field dierence y(q) (see Supplementary Material – Appendix C). Wis the (II)
sample covariance matrix:
W(q)= 1
representing the variances-within and correlations-between vector components across the Jresponses (Sup-
plementary Material - Appendix D).
The notation “SPM{T2}” (Friston et al., 2007) indicates that the test statistic T2varies in continu-
ous time (or space), forming a temporal (or spatial) statistical ‘map’. To clarify: “SPM” refers to the
methodology, and “SPM{T2}” to a specific variable.
2.3.2 Two-sample Hotelling’s T2test (Dataset B)
SPM’s vector field analog to the two-sample ttest is the Hotelling’s T2test (Cao and Worsley, 1999):
SPM{T2}T2(q)= J1J2
where subscripts “1” and “2” index the two groups. Here Wis the pooled covariance matrix:
where the domain “(q)” is dropped for compactness.
2.3.3 Canonical correlation analysis (Dataset C)
SPM’s vector field analog to linear regression is canonical correlation analysis (CCA) (Hotelling, 1936;
Worsley et al., 2004). The goal of CCA is to determine the strength of linear correlation between a set of
predictor variables xj(K-component vectors) and a set of response variables yj(I-component vectors). We
provide a brief technical summary of CCA. An extended discussion is provided as Supplementary Material
(Appendix E).
Following Worsley et al. (2004), the test statistic of interest was the maximum canonical correlation (R),
a single correlation coecient which varies over q, and which transforms to the Fstatistic via the identity:
To compute R, one must first assemble three covariance matrices:
CXX —the(KK) predictors covariance matrix
CYY —the(II) responses covariance matrix
CXY —the(KI) predictor-response covariance matrix
The maximum canonical correlation (R) is the maximum eigenvalue of the (KK) canonical correlation
matrix (C) (Worsley et al., 2004):
XY (7)
An equivalent interpretation is that Ris the maximum correlation coecient obtainable when the pre-
dictor and response coordinate systems are permitted to mutually rotate. K=2 predictors (running speed
and an intercept) were employed to model the I=3 force vector components of Dataset C.
2.3.4 Statistical inference
To determine the significance of the aforementioned test statistic fields, first field smoothness was es-
timated from the temporal gradients of the residuals (Friston et al., 2007). Next, given this smoothness,
RFT (Adler and Taylor, 2007) was used to determine the critical test statistic threshold that retained a
family-wise error rate of =0.05 (Cao and Worsley, 1999; Worsley et al., 2004). Last, the probability with
which suprathreshold clusters could have been produced by chance (i.e. by random fields with the same
temporal smoothness) was calculated using analytic expectation (Cao and Worsley, 1999). In other words,
rather than controlling the false-positive rate at each point in time, we presently controlled the false-positive
rate of the test statistic field’s sample-rate invariant topological features (Friston et al., 2007). For additional
details refer to Appendix A (Supplementary Material).
2.3.5 Post hoc scalar field SPM
When testing non-directed hypotheses regarding biomechanical vector fields, we propose that SPM should
be implemented in a hierarchical manner, analogous to ANOVA with post hoc t testing. One should first
use SPM to analyze the entire vector field y(q), and particular vector components (scalar field yi(q)) should
only be tested, in a post hoc manner, if statistical significance is reached at the vector-field level.
Following vector field analyses, pos t hoc tests were conducted on each vector component separately (i.e.
on scalar fields). For scalar fields the aforementioned tests (§2.3.1–§2.3.3) reduce to: the paired tstatistic
(Dataset A), the two-sample tstatistic (Dataset B), and the linear regression tstatistic (Dataset C). Each
scalar field test produced one SPM{t}, whose significance was determined as described above §.2.3.1. To
maintain a family-wise error of =0.05, Sid´ak thresholds (Eqn.1) of p=0.0170, p=0.0051, and p=0.0170 were
used to correct for the I=3, I=10, and I=3 vector components of Datasets A, B, and C, respectively (Table
1). All aforementioned analyses were implemented in Python 2.7 using Enthought Canopy 1.0 (Enthought
Inc., Austin, USA).
3 Results
3.1 Dataset A: knee kinematics
The knee appeared to be comparatively more flexed (Fig.2a) and somewhat more externally rotated
(Fig.2c) in the side-shue vs. v-cut tasks, with slightly more abduction at 0% stance (Fig.2b). Statistical
tests on the extracted scalars found significant dierences between the two tasks for both maximal knee
flexion (t=3.093, p=0.018) and abduction at 0% stance (t=3.948, p=0.006).
SPM vector field analysis (Fig.5) found significant kinematic dierences between the two tasks at ap-
proximately 1%, 10%, 20%, 30-35% and 95-100% stance. Post hoc t tests revealed that the eects over
30-35% and 95-100% stance resulted primarily from increased flexion (p=0.015) and increased external ro-
tation (p=0.004), respectively, in the side-shue vs. v-cut tasks (Fig.6). Apparent discrepancies amongst
vector field SPM, scalar field SPM, and scalar extraction (both here and in the remainder of the Results),
are addressed in the Discussion.
3.2 Dataset B: muscle forces
Most muscles appeared to exhibit higher forces in PFP vs. Controls over most of stance (Fig.3). Nonethe-
less, none of the statistical tests on the extracted scalars reached significance; the medial gastrocnemius force
exhibited the strongest eect (t=2.617, p=0.013), but like the nine other muscles (t<1.91, p>0.063) this failed
to reach the ˇ
Sid´ak significance threshold of p=0.0051.
In contrast, SPM vector field analyses found significance for the entire stance phase (Fig.7; p<0.001).
Post hoc ttests on individual muscle trajectories found significantly greater forces in PFP only for the medial
gastrocnemius, and only over scattered time regions (maximum p=0.002) (Fig.8).
3.3 Dataset C: ground reaction forces
Forces appeared to increase systematically with running speed, particularly in the vicinity of 30% and
75% stance (Fig.4). Linear regression found that all three extracted scalars surpassed the ˇ
Sid´ak threshold
for significance (p=0.0170); analysis of maximum propulsion, vertical and lateral forces yielded r2=0.951,
0.691 and 0.737, and p=0.00004, 0.001, and 0.006, respectively.
SPM vector field analysis (Fig.9) found that GRF was significantly correlated with running speed in three
intervals with approximate windows of: 10–18%, 20–43% and 60–88% stance. Post hoc scalar field analysis
revealed that GRFxwas primarily responsible for the 10–18% and 60–88% eects (Fig.10a), and that GRFy
was primarily responsible for the 20–43% eect (Fig.10b).
4 Discussion
The current vector field SPM and scalar extraction results all agreed qualitatively with the data, yet the
two approaches yielded dierent results and even incompatible statistical conclusions. This, by definition,
indicates that at least one of the methods is biased. For non-directed hypotheses testing we contend that
scalar extraction is susceptible to two non-trivial bias sources:
1. Post hoc regional focus bias — Type I or Type II error (i.e. false positives or false negatives) resulting
from the failure to consider the entire measurement domain.
2. Inter-component covariation bias — Type I or Type II error resulting from the failure to consider the
covariance amongst vector components.
We further contend that vector field testing overcomes both bias sources because it uses the entire
measurement domain and all vector components to maintain a constant error rate of . The remainder of
the Discussion is devoted to justifying these claims.
4.1 Bias in scalar extraction analyses
Dataset A exhibited Type I error due to post hoc regional focus bias. Scalar extraction analysis of
maximum flexion (at 50% stance) reached significance (§3.1) but neither vector field analysis (Fig.5) nor
post hoc scalar field analysis (Fig.6a) reached significance in this field region. Similarly, scalar extraction
found a significant ab-/adduction eect at 0% stance, but SPM did not. These discrepancies are resolved
through multiple comparisons theory (Knudson, 2009); it is highly likely that at least one of Dataset A’s
303 vector field points will exceed an (uncorrected) threshold of p=0.05 simply by chance. By extracting
only scalars which appeared to exhibit maximum eects (Fig.2) we eectively conducted 303 tests and then
chose to report the results of only two.
The opposite eect (Type II error) was also present in Dataset A. Scalar extraction focussed on only two
scalars, and thus failed to identify the other eects present in the dataset (Fig.5), and in particular the large
late-stance internal/external rotation eect (Fig.6c). A simple example (Supplementary Material – Appendix
A) clarifies how it is possible for scalar extraction and SPM to yield opposite statistical conclusions, and
that the scalar extraction results can’t be trusted because they fail to honor the error rate.
Dataset B exhibited Type II error due to covariation bias: scalar extraction failed to reach significance
(§3.2) even though SPM found substantial evidence for muscular dierences between Controls and PFP
(Fig.7). This is resolved by correlation amongst muscles like the vasti (Fig.3). A simple example (Supple-
mentary Material – Appendix B) clarifies how it is possible for vector resultant changes to reach significance
when vector component changes do not. Scalar analysis of vector data cannot be trusted because it fails to
account for vector component covariance.
Scalar extraction analysis of Dataset C exhibited both Type I and Type II error due to regional focus
bias. Scalar extraction analyses of lateral forces exhibited Type I error because there is insucient field-wide
evidence to support its conclusion of significance (Fig.10c). Scalar extraction also exhibited Type II error by
failing to analyze braking forces and therefore failing to identify positive correlation between running speed
and braking forces at 15% stance (Fig.10a).
4.2 Bias in scalar field SPM analyses
Scalar field SPM solves regional focus bias (because it tests the entire domain q), but it remains susceptible
to covariance bias because it separately tests the Ivector components. Scalar field analysis of Dataset
A exhibited Type II error by failing to identify all field eects, and particularly the large early-stance
eect (Fig.5,6). Appendix B clarifies that this was caused by scalar field analysis’ failure to consider inter-
component covariance; it regards trajectory variance as a 1D time-varying ‘cloud’ (Fig.2) when in fact it is
an I-D time-varying hyper-ellipsoid (Fig.1) representing both within- and between-component (co)variance.
In Dataset B, vector field analysis reached significance (Fig.7), so scalar field analysis, had it not been
conducted in a post hoc manner, would have exhibited Type II error by underestimating the temporal scope
of eects (Fig.8). This is also explained by covariance (Appendix B); the eect was manifested more strongly
in the resultant 10-component muscle force vector than it was in each muscle independently.
In Dataset C vector field eect timing (Fig.9) agreed with scalar field eect timing (Fig.10), so the latter
would not have been biased had they not been conducted in a post hoc sense. Nevertheless, by failing to
consider covariance the scalar field results fail to capture the full temporal extent of the vector eects.
A separate but notable trend was that Dataset C’s covariance ellipses all tended to be narrow and to point
toward the origin (Fig.1). This suggests that vector magnitude was far more variable than vector direction.
A plausible mechanical explanation is friction: to avoid slipping normal forces must increase when tangential
forces increase. Regardless of the mechanism, this observation reinforces our contention that non-directed
hypotheses must consider vector changes.
4.3 SPM’s solution to regional focus and covariance bias
SPM solves both regional focus bias and covariance bias by considering the covariance of all vector com-
ponents (i) across the entire measurement domain (q), while simultaneously handling the inherent problem
of multiple comparisons (Knudson, 2009) in a theoretically robust manner. Specifically, SPM uses a RFT
correction (Adler and Taylor, 2007; Worsley et al., 2004) to ensure that no more than % of the points in the
(IQ) vector field reach significance simply by chance; this RFT correction is embodied in the thresholds
depicted in Figs.5-10.
Non-RFT corrections like the ˇ
Sid´ak correction (Eqn.1) can partially solve the problem of multiple compar-
isons, but only partially because they fail to consider the (spatiotemporal) smoothness of the measurement
domain q, and therefore overestimate the number of independent tests. This ultimately leads to an overly
conservative threshold (i.e. inflated Type II error rate) except for very rough fields (Friston et al., 2007).
Non-RFT corrections also fail to solve covariance bias because they assume that vector components vary
independently (Supplementary Material – Appendix B). While covariance bias could partially be solved with
a principal axis rotation prior to statistical testing (Cole et al., 1994; Knudson, 2009); we’d argue that: (i)
Hotelling’s T2and CCA are simpler solutions because their results are identical for all coordinate system
definitions, and (ii) principal axis rotations of only the response vectors do not necessarily maximize the
mutual correlation between predictors and responses (§2.3.3).
We acknowledge that many additional important sources of bias exist (James and Bates, 1997), (Mullineaux
et al., 2001), (Knudson, 2009). However, none of these is unique to SPM. Trajectory mis-registration (Sadeghi
et al., 2003) and unit normalization (e.g. absolute vs. relative muscle forces), for example, pose common
problems to scalar extraction and vector field analyses. We contend only that SPM addresses two bias
4.4 SPM generalizability
Although SPM was originally developed to analyze 3D brain function (Friston et al., 2007), it has been
shown that SPM is generalizable to a variety of biomechanical scalar datasets including 1D trajectories
(Pataky, 2012), 2D pressure fields and 3D strain fields (Pataky, 2010). The current study is the first, in
any scientific field, to have shown that SPM is also applicable to a large class of practical 1D vector field
problems. SPM theory suggests that generalizations to biomechanical vector/tensor fields in nD spaces are
also possible (Xie et al., 2010).
SPM encompasses the entire family of parametric hypothesis testing (Worsley et al., 2004; Friston et al.,
2007). It also accommodates all non-parametric variants (Nichols and Holmes, 2002; Lenhoet al., 1999),
which may be useful if one’s data do not adhere to the parametric assumption that the residuals are normally
distributed (Friston et al., 2007). This hypothesis testing generalization is apparent when one considers the
following hierarchy: vector fieldCCA simplifies to the vector field Hotelling’s T2test when the predictors are
binary (Worsley et al., 2004), which in turn simplifies to scalar field ttests when there is only one vector
component i, which in turn simplifies to the univariate Student’s ttest when the scalar field reduces to
a single point q. Thus SPM, through CCA, generalizes to all statistical tests of Idimensional vectors on
arbitrarily sized fields Qof arbitrary dimensionality (Worsley et al., 2004).
For readers interested in implementing SPM analyses, we note that constructing test statistic trajectories
is straightforward; it is trivial to combine mean and standard deviation trajectories to form an SPM{t}, for
example. The non-trivial step is statistical inference. As a first approximation it is easy to implement a
Sid´ak correction (Eqn.1), which will (very) conservatively reduce the Type I error rate, but which will also
unfortunately inflate the Type II error rate. For more precise control of both error rates (via RFT) the
reader is directed to the literature (Friston et al., 2007) and open source software packages (Pataky, 2012).
4.5 Conclusions
Ad hoc reduction of vector trajectories through scalar extraction can non-trivially bias non-directed
biomechanical hypothesis testing, most notably via regional focus and coordinate system bias sources. This
paper shows that SPM overcomes both sources of bias by treating the vector field as the fundamental, initially
indivisible unit of observation. Grounded in random field theory, SPM appears to be a useful, generalized
tool for the analysis of often-complex biomechanical datasets.
Financial support for this work was provided in part by JSPS Wakate B Grant#22700465.
Conflict of Interest
The authors report no conflict of interest, financial or otherwise.
Adler, R. J. and Taylor, J. E. 2007. Random Fields and Geometry, Springer-Verlag, New York.
Besier, T. F., Fredericson, M., Gold, G. E., Beaupre, G. S., and Delp, S. L. 2009. Knee muscle forces during walking and running
in patellofemoral pain patients and pain-free controls, Journal of Biomechanics 42(7), 898–905, data:
Cao, J. and Worsley, K. J. 1999. The detection of local shape changes via the geometry of Hotelling’s T2 fields, Annals of
Statistics 27(3), 925–942.
Cole, D. A., Maxwell, S. E., Arvey, R., and Salas, E. 1994. How the power of MANOVA can both increase and decrease as a
function of the intercorrelations among the dependent variables., Psychological Bulletin 115(3), 465–474.
Dorn, T. T., Schache, A. G., and Pandy, M. G. 2012. Muscular strategy shift in human running: dependence of running speed
on hip and ankle muscle performance., Journal of Experimental Biology 215, 1944–1956, data:
Friston, K. J., Ashburner, J. T., Kiebel, S. J., Nichols, T. E., and Penny, W. D. 2007. Statistical Parametric Mapping: The
Analysis of Functional Brain Images, Elsevier/Academic Press, Amsterdam.
Hotelling, H. 1936. Relations between two sets of variates, Biometrika 28(3), 321–377.
James, C. R. and Bates, B. T. 1997. Experimental and statistical design issues in human movement research, Measurement in
Physical Education and Exercise Science 1(1), 55–69.
Jensen, R. H. and Davy, D. T. 1975. An investigation of muscle lines of action about the hip: A centroid line approach vs the
straight line approach, Journal of Biomechanics 8(2), 103–110.
Knudson, D. 2009. Significant and meaningful eects in sports biomechanics research, Sports Biomechanics 8(1), 96–104.
Kutch, J. J. and Valero-Cuevas, F. J. 2011. Muscle redundancy does not imply robustness to muscle dysfunction, Journal of
Biomechanics 44(7), 1264–1270.
Lenho, M. W., Santer, T. J., Otis, J. C., Peterson, M. G., Williams, B. J., and Backus, S. I. 1999. Bootstrap prediction and
confidence bands: a superior statistical method for analysis of gait data, Gait and Posture 9, 10–17.
Mullineaux, D. R., Bartlett, R. M., and Bennett, S. 2001. Research design and statistics in biomechanics and motor control,
Journal of Sports Sciences 19(10), 739–760.
Neptune, R. R., Wright, I. C., and van den Bogert, A. J. 1999. Muscle coordination and function during cutting movements,
Medicine & Science in Sports & Exercise 31(2), 294–302, data:
Nichols, T. E. and Holmes, A. P. 2002. Nonparametric permutation tests for functional neuroimaging a primer with examples,
Human Brain Mapping 15(1), 1–25.
Pataky, T. C. 2010. Generalized n-dimensional biomechanical field analysis using statistical parametric mapping, Journal of
Biomechanics 43(10), 1976–1982.
Pataky, T. C . 2012. One-dimensional statistical parametric mapping in Python, Computer Methods in Biomechanics and
Biomedical Engineering 15(3), 295–301.
Rayner, J. M. 1985. Linear relations in biomechanics: the statistics of scaling functions, Journal of Zoology 206(3), 415–439.
Sadeghi, H., Mathieu, P. A., Sadeghi, S., and Labelle, H. 2003. Continuous curve registration as an intertrial gait variability
reduction technique, IEEE Transactions on Neural Systems and Rehabilitation Engineering 11(1), 24–30.
Wan g , X., Ve rri e st, J . P., Le b ret on- G ade gbek u , B., Te ssi e r, Y . , and Tr a sb o t , J. 20 00. E x p er i men tal i nv est iga t ion a nd bi o me-
chanical analysis of lower limb movements for clutch pedal operation, Ergonomics 43(9), 1405–1429.
Wor s ley, K . J ., Ta ylo r , J. E. , Toma i uol o, F. , a nd L e rch , J. 20 04. U n ifie d uni va r ia t e and m ult ivari ate r a ndo m fiel d t heo ry,
NeuroImage 23, S189–S195.
Xie, Y., Vemuri, B. C., and Ho, J. 2010. Statistical analysis of tensor fields, Medical Image Computing and Computer-Assisted
Intervention 13(1), 682–698.
Table 1: Dataset and scalar extraction overview. I,J,Qand Nare the numbers of vector com-
ponents, responses, time points, and extracted scalars, respectively. For vector field analyses,
post hoc scalar field analyses, and extracted scalar analyses we conducted one, Iand Ntests,
respectively. ˇ
Sid´ak thresholds of p=0.0253, p=0.0170 and p=0.0051 maintained a family-wise
error rate of =0.05 across 2, 3, and 10 tests, respectively (see Eqn.1).
I J Q N Extracted scalars
Dataset A 3 8 101 2 (1) Max. flexion (at 50% stance)
(2) Ad-abduction at 0% stance
Dataset B 10 43 100 10 Max. force for each muscle (J1=16, J2=27)
Dataset C 3 8 100 3
(1) Max. propulsion force (GRFx,75% stance)
(3) Max. vertical force (GRFy,30–50% stance)
(3) Max. lateral force (GRFz,15% stance)
Figure 1. Vector field schematic: a two-component vector varying in time. Depicted are mean
ground reaction force (GRF) vectors F = [Fx Fy]T from one subject during running (Dorn et al.,
2012), where +x and +y represent the anterior and vertical directions, respectively. These vectors,
when projected on the (Time, Fx) and (Time, Fy) planes, produce common GRF plots (see Fig.
4a,b); here vertical dotted lines depict standard deviation ‘clouds’. When F is projected on the
(Fx, Fy) plane these standard deviations are revealed to arise from covariance ellipses, where
ellipse orientation indicates the direction of maximum covariance between Fx and Fy (see
Supplementary Material - Appendix B).
Figure 2. Dataset A (Neptune et al., 1999) depicting knee kinematics in side-shuffle vs. v-cut
tasks. Cross-subject mean trajectories with standard deviation clouds (dark: side-shuffle, light: v-
cut) are depicted. Each of the eight subjects has three (scalar) trajectories yi(q) for each task, and
these were combined into a single (I=3, Q=101) vector field y(q) for each subject and each task.
Figure 3. Dataset B (Besier et al., 2009) depicting muscle forces during walking in Control vs.
Patello-Femoral Pain (PFP) subjects; 16 and 27 subjects, respectively. Cross-subject mean
trajectories with standard deviation clouds (dark: Control, light: PFP). These ten scalar
trajectories were combined into a single (I=10, Q=100) vector field y(q) for each subject.
Figure 4. Dataset C (Dorn et al., 2012) depicting ground reaction forces (GRF) during running/
sprinting at various speeds. Single-subject cross-trial means; standard deviation clouds are not
depicted in interest of visual clarity. These data form one (I=3, Q=100) vector field y(q) for each
Figure 5. Dataset A, Hotelling’s T2 trajectory (SPM{T2}). The horizontal dotted line indicates the
critical random field theory threshold of T2 = 29.39.
Figure 6. Dataset A, post hoc scalar field t tests (SPM{t}), depicting where side-shuffle angles
were greater (+) and less (-) than v-cut angles. At a Sidak threshold of p=0.017 (Eqn.1), the thin
dotted lines indicate the critical RFT thresholds for significance: |t| > 4.52, 5.24, 5.26 for (a), (b),
and (c) respectively. The thresholds are different because each vector component has different
temporal smoothness (Fig.2); less smooth trajectories have higher thresholds because there are
more ‘processes’ present between 0 and 100% time. Probability (p) values indicate the likelihood
with which each suprathreshold cluster is expected to have been produced by a random field
process with the same temporal smoothness.
Figure 7. Dataset B, Hotelling’s T2 trajectory (SPM{T2}), depicting where muscle forces differed
between Controls and PFP. The horizontal dotted line indicates the critical RFT threshold of T2 =
9.35. The entire trajectory has exceeded the threshold, so the single suprathreshold cluster has a
very low p value.
Figure 8. Dataset B, post hoc scalar trajectory t tests (SPM{t}), depicting where Control forces
were greater than (+) and less than (-) PFP forces. Thin dotted lines indicate the critical RFT
thresholds for significance.
Figure 9. Dataset C, canonical correlation analysis results, with SPM{F} depicting where ground
reaction forces were correlated with running speed. Critical RFT threshold: F = 38.1.
Figure 10. Dataset C, post hoc scalar trajectory linear regression tests (SPM{t}), depicting the
strength of positive (+) and negative (-) correlation between ground reaction forces (GRF) and
running speed.
Appendix A. Scalar extraction vs. scalar field statistics
The purpose of this Appendix is to demonstrate how scalar extraction can bias non-directed
hypothesis testing. To this end we developed and analyzed an arbitrary dataset (Fig.S1). We
caution readers that we have constructed these data specifically to demonstrate particular con-
cepts. The reader is therefore left to judge the relevance of this discussion to real (experimental)
The specific goal of this Appendix is to scrutinize the similarities and dierences between:
(a) a typical univariate two-sample ttest, and (b) a scalar field two-sample ttest.
Consider the simulated scalar field dataset in Fig.S1. In Fig.S1a, arbitrary true mean fields
are defined for two experimental conditions: “Cond A” and “Cond B”. The Cond B mean was
produced using a half sine cycle. The Cond A mean was produced by adding a small Gaussian
pulse (at time= 85%) to the Cond B mean. This Gaussian pulse is evident in the true mean
field dierence (Fig.S1b).
Figure S1: Simulated scalar field dataset depicting two experimental conditions: “Cond A”
and “Cond B” (arbitrary units).
We next simulate smooth random fields: five for each condition (Fig.S1c). These random
fields were constructed by generating ten fields, each containing 100 random, uncorrelated
and normally distributed numbers, then smoothing them using a Gaussian kernel. Adding
the random fields to the true field means (Fig.S1a) produced the final simulated responses
(Fig.S1d). For interpretive convenience, let us assume that these data represent joint flexion.
Imagine next that we wish to test the following (non-directed) null hypothesis: “Cond A
and Cond B yield identical kinematics”. Consider first scalar extraction: after observing the
data (Fig.S1d) one might decide to extract and analyze the maximum flexion, which occurs
near time = 50%:
yA=100.0 91.2 92.2 95.5 97.1
yB=97.2 101.9 104.8 106.3 111.7
A two-sample ttest on these data yields: t=3.16, p=0.013. We would reject the null
hypothesis at =0.05, and we would conclude that Cond B produces significantly greater
maximal flexion than Cond A.
An alternative is to use Statistical Parametric Mapping (SPM) (Fig.S2). The SPM pro-
cedures are conceptually identical to univariate procedures (Table S1). The only apparent
dierence is that SPM uses a dierent probability distribution (Steps 4 and 5). This probabil-
ity distribution is actually not dierent because it reduces to the univariate distribution when
Q=1 (i.e. if there is only one time point).
SPM results find significant dierences between the two conditions near time = 85% (Fig.S2d).
We would therefore reject our null hypothesis, with the caveat that significant dierences were
only found near time = 85%.
Although univariate ttesting and SPM ttesting are conceptually identical, they have yielded
(eectively) opposite results. The univariate test found significantly greater maximal flexion
in Cond B, but SPM found significantly greater flexion in Cond A (near time=85%).
Table S1: Comparison of computational steps for univariate and SPM two-sample ttests
(“” = standard deviation).
Step (a) Univariate two-sample ttest (b) SPM two-sample ttest Figure
1 Compute mean values yAand yB. Compute mean fields yA(q) and yB(q) S2(b)
2 Compute values sAand sB. Compute fields sA(q) and sB(q) S2(b)
3 Compute the ttest statistic:
Compute the ttest statistic field:
SPM{t}t(q)= yB(q)yA(q)
4 Conduct statistical inference. First use
and the univariate tdistribution to com-
pute tcritical.Ift>t
critical, then reject null
Conduct statistical inference. First use
and the random field theory tdistribution
to compute tcritical.IfSPM{t}exceeds
tcritical, then reject null hypothesis for the
suprathreshold region(s).
5 Compute exact pvalue using tand the uni-
variate tdistribution.
Compute exact pvalue(s) for each
suprathreshold cluster using cluster size
and random field theory distribution(s) for
Figure S2: Scalar field analysis using Statistical Parametric Mapping (SPM). In panel (d)
the thin dotted lines depict the critical random field theory threshold of |tcritical |=3.533. The
(incorrect) ˇ
Sid´ak threshold is |tcritical|=5.595.
This discrepancy can be resolved through standard probability theory regarding multiple
comparisons, through a consideration of ‘corrected’ and ‘uncorrected’ thresholds. First consider
conducting one statistical test at =0.05. The choice: =0.05” means that we are accepting
a 5% chance of incorrectly rejecting the null hypothesis, or, equivalently, a 5% chance of a
‘false positive’. If we conduct more than one test, there is a greater-than 5% chance of a false
positive. Specifically, if we conduct Nstatistical tests, the probability of at least one false
positive is given by the family-wise error rate :
=1(1 )N
For N=2 tests, there is a =9.75% chance that at least one test will produce a false positive.
For N=100 tests, =99.4%.
To protect against false positives, and to maintain a constant family-wise error rate of
=0.05, we must adopt a corrected threshold. One option is the ˇ
Sid´ak threshold:
pcritical =1(1 )1/N
For N=2 and N=100 tests, the ˇ
Sid´ak thresholds are pcritical=0.0253 and pcritical=0.000513,
Herein lies one problem: our scalar extraction analysis has used an uncorrected threshold
of pcritical=0.05. Even though we have formally conducted only one statistical test, the data
were extracted from a dataset that is 100 times as large. Since we observed the data before
choosing which scalar to extract, we eectively conducted N=100 tests, albeit visually, then
chose to focus on only one test. By failing to adopt a corrected threshold, we have biased our
Although the ˇ
Sid´ak correction helps to avoid false positives, it is not generally a good choice
because it assumes that there are 100 independent tests (i.e. one for each time point in our
dataset). The points in this dataset are clearly not independent because the curves are smooth,
changing only gradually over time. Thus the ˇ
Sid´ak correction is too severe, lowering well
below 0.05. An overly severe threshold produces the opposite bias: an increased chance of false
SPM employs a random field theory (RFT) correction to more accurately maintain =0.05.
The precise threshold is based not only on field size (Q=100), but also on field smoothness —
which is estimated from temporal derivatives. Computational details for RFT corrections are
provided in the SPM literature.
Unfortunately, even if our scalar analysis had employed a corrected threshold, it still would
have been biased, but for a separate reason. By focussing only on maximal flexion (which did
not appear in our null hypothesis), we have neglected to consider the signal at time = 85%,
and have therefore not detected the true field dierence (Fig.S1a). In contrast, SPM was able
to uncover the true signal because it both adopted a corrected threshold and considered the
entire field simultaneously (Fig.S1d).
The aforementioned sources of bias — (1) failing to adopt a corrected threshold, and (2)
failing to consider the entire field — are referred to collectively in the main manuscript as
‘regional focus bias’.
Last, we reiterate that this Appendix is relevant only to non-directed hypotheses. If we
had formulated a (directed) hypothesis regarding only maximal flexion — prior to observing
the data — then our scalar extraction analyses would not have been biased because our null
hypothesis would not have pertained to the entire time domain 0–100%.
In summary, regional focus bias can be avoided by:
1. Specifying a directed null hypothesis — before observing the data — and then extracting
only those scalars which are specified in the null hypothesis.
2. Analyzing the data using SPM or another field technique which both considers the entire
temporal domain and which adopts a corrected threshold.
Appendix B. Univariate vs. vector analysis
The purpose of this Appendix is to demonstrate how univariate testing of vector data can
bias non-directed hypothesis testing. To this end we developed and analyzed an arbitrary
dataset (Table S2). As in Appendix A, we caution readers that we have constructed these
data specifically to demonstrate particular concepts. The reader is therefore left to judge the
relevance of this discussion to real (experimental) datasets.
The specific goal of this Appendix is to compare and contrast the (univariate) ttest and
its (multivariate) vector equivalent: the Hotelling’s T2test.
Table S2: A simulated dataset exhibiting biased univariate testing. (a) Two-component force
vector responses F=[Fx,Fy]>. (b)-(d) Scalar (univariate) testing. (e)-(g) Vector (multivari-
ate) testing. Sources of bias and further details are discussed in the text. Technical overviews
of covariance matrices (W) and the Hotelling’s T2statistic are provided in Appendix D and
§2.3 (main manuscript), respectively.
Group A Group B Inter-Group
(a) Responses
FA1 = [159, 719]>FB1 = [143, 759]>
FA2 = [115, 762]>FB2 = [172, 734]>
FA3 = [177, 681]>FB3 = [161, 735]>
FA4 = [138, 694]>FB4 = [195, 733]>
FA5 = [98, 697]>FB5 = [168, 706]>
(b) Means (Fx)A= 137.4 (Fx)B= 167.8 Fx= 30.4
(Fy)A= 710.6 (Fy)B= 733.4 Fy= 22.8
(c) (sx)A= 28.6 (sx)B= 16.8 sx= 23.5
(sy)A= 28.5 (sy)B= 16.8 sy= 23.4
(d) ttests tx=1.832; px=0.104
ty=1.380; py=0.205
(e) Means FA= [137.4, 710.6]>FB= [167.8, 733.4]>F= [30.4, 22.8]>
(f) Covariance WA="817.8323.2
(g) T2test T2=7.113; p=0.028
In Table S2(a) above there are five force vector responses (F=[Fx,Fy]>) for each of two
groups: “A” and “B”. Their means and standard deviations are shown in Table S2(b)-(c). In
Table S2(d) we see that ttests pertaining to both Fxand Fyfail to reach significance; pvalues
are greater than (even an uncorrected) threshold of p=0.05. An adequate interpretation is
that the mean force component changes (Fxand Fy) are not unexpectedly large given their
respective variances (i.e. standard deviations: sxand sy).
We next jump ahead to the final results of the vector procedure in Table S2(g): here we
see that the Hotelling’s T2test reached significance (p=0.032). An adequate interpretation is
that the mean force vector change (F) was unexpectedly large given its (co)variance (W).
Let us now backtrack and consider why the univariate and vector procedures yield dierent
The first step of the vector procedure is to compute mean vectors; in Table S2(e) we can
see that the vector means have the same component values as the univariate means from Table
S2(b). However, there is already one critical discrepancy to note: the vector procedure assesses
F,whichistheresultant vector connecting the Group A and Group B means (Fig.S3).
From Pythagoras’ theorem:
it is clear that the magnitude of the resultant will always be greater than the magnitude of its
components — except in the experimentally unlikely cases of Fx=0 and/or Fy=0. This is
non-trivial for two reasons. First, since the vector procedure assesses the maximum dierence
between the two groups, it is more robust to Type II error than univariate procedures (note:
the univariate tests in Table S2 exhibit Type II error by failing to reach significance). Second,
the vector technique’s assessment of dierences is independent of the xy coordinate system
definition; whereas the component eects (Fxand Fy) can change when the xy coordinate
system definition changes, both the resultant and the variance along the resultant direction will
always have the same magnitude. This may have non-trivial implications for biomechanical
datasets that employ dicult-to-define coordinate systems (e.g. joint rotation axes).
Figure S3: Graphical depiction of the data from Table S2. Small circles depict individual
responses. Thick colored arrows depict the mean force vectors for the two groups. The thick
black arrow depicts the (vector) dierence between the two groups, and thin black lines indicate
its xand ycomponents. The ellipses depict within-group (co)variance; their principal axes (thin
dotted lines) are the eigenvectors of the covariance matrices in Table S2(f ). Here covariance
ellipse radii are scaled to two principal axis standard deviations (to encompass all responses).
The next step of the vector procedure is to compute covariance matrices W(Appendix D).
The diagonal elements of WAand WBin Table S2(f) are simply the variances (i.e. squared
standard deviations) s2
xand s2
yfrom Table S2(c). The o-diagonal terms are equal and represent
the covariance (i.e. correlation) between Fxand Fy.IfFxtends to increase when Fyincreases
then the o-diagonal terms would be positive, but in this case they are negative, indicating
that Fxtends to decrease when Fyincreases. This tendency can be seen in the raw data (small
circles) in Fig.S3.
The presence of non-zero o-diagonal terms thus has a critical implication: changes in Fx
and Fyare not independent. This is critical because univariate tests implicitly assume that Fx
and Fyare independent.
To appreciate this point it is useful to recognize that covariance matrices may be interpreted
geometrically as ellipses: the eigenvectors of Wrepresent the ellipse’s principal axes, and its
eigenvalues represent the variance along each principal direction. This is perfectly analogous to
inertia matrices: the eigenvectors of an inertia matrix define a body’s principal axes of inertia,
and eigenvalues specify the principal moments of inertia.
The importance of this geometric interpretation becomes clear when visualizing covariance
ellipses. In Fig.S3 we can see that the principal axes of the covariance matrices are not aligned
with the xy coordinate system, implying that changes in Fxand Fyare not independent.
Critically, we can also see that the direction of minimum variance is very similar to the direction
of F. Thus the standard deviations sxand sy(used in the univariate analyses) are larger
than the standard deviation in the direction of F.
In summary, vector statistical testing more objectively detects vector changes because : (a)
it is coordinate system-independent, (b) it considers both the maximum dierence between
groups (i.e. the resultant dierence) and the variation along this direction. This Appendix has
demonstrated how univariate testing of vector data can lead to Type II error. With a dierent
dataset it would also be possible to demonstrate Type I error, but in interest of space we end
here. The most important point, the main paper contends, is that non-directed hypothesis
testing mustn’t assume vector component independence.
Appendix C. Mean vector field calculation
An I-component vector ywhich varies over Qpoints in space or time may be regarded as
an (IQ) vector field response y(q). Given Jresponses, the mean vector field is:
y(q)= 1
yj(q) (C.1)
For the paired Hotelling’s T2test (Dataset A: §2.3.1, main manuscript), one must first
compute pairwise dierences:
yj(q)=yBj(q)yAj (q) (C.2)
where “A” and “B” represent the two tasks (v-cut and side-shue) and jindexes the subjects.
A paired Hotelling’s T2test is thus equivalent to a one-sample Hotelling’s T2test conducted
on the pairwise dierences y(q). The same is true in the univariate case: a paired ttest is
equivalent to a one-sample ttest on pairwise dierences.
Appendix D. Covariance matrices
Although the concepts presented below apply identically to vector fields, for brevity present
discussion is limited to simple vectors.
Consider a two-component force vector response F:
Fj=Fxj Fyj>(D.1)
where jindexes the responses, and there are a total of Jresponses. After computing the mean
force vector Fas:
the covariance matrix Wcan be assembled as follows:
Wxx Wxy
Wyx Wyy
where the elements of Ware:
Wxx =1
(Fxj Fx)2(D.4)
Wyy =1
(Fyj Fy)2(D.5)
Wxy =Wyx =1
(Fxj Fx)(Fyj Fy) (D.6)
Thus the diagonal elements Wxx and Wyy are the intra-component variances (i.e. squared
standard deviations), and the o-diagonal elements Wxy and Wyx are the inter-component
covariances between Fxand Fyover multiple responses. Importantly, changes in Fxand Fyare
completely uncorrelated if and only if Wxy=0.
One contention of this paper is that separate (univariate) analysis of Fxand Fyis biased
when testing non-directed hypotheses. The main reason is that Fxanalysis considers only Wxx
and Fyanalysis considers only Wyy. This is equivalent to assuming Wxy=0, an assumption
which may not be valid (Appendix B).
A geometric interpretation of Wis useful both for visualizing vector variance (Fig.S3) and
for appreciating canonical correlation analysis (Appendix E). Consider that Wrepresents an
ellipse whose geometry is defined by the solutions to the eigenvalue problem:
Wv =v(D.7)
Here vand are the eigenvectors and eigenvalues, respectively, and there are two unique
eigensolutions unless both (Wxx =Wyy) and (Wxy = 0), in which case there is only one
eigensolution and Wrepresents a circle. When there are two solutions the eigenvectors repre-
sent the ellipse axes (or equivalently: principal axes), and the eigenvalues represent the axes’
lengths (or variance in the direction of the principal axes). An equivalent interpretation is that
one eigenvector of Wrepresents the direction of maximum variance within the dataset. This
means that we can rotate our original coordinate system xy to a new coordinate system x0y0so
that variance along the new x0axis is the maximum possible variance obtainable for all possible
Appendix E. Canonical correlation analysis (CCA)
CCA aims to quantify the amount of variance that a multivariate predictor (i.e. vector)
Xcan explain in a multivariate response Y. One type of CCA useful for hypothesis testing is
to find the maximum possible correlation coecient that can be obtained when the coordinate
systems defining Xand Yare permitted to mutually rotate.
Consider a response variable Ythat describes three orthogonal force components F:
where “1”, “2” and “3” represent orthogonal axes and where jindexes a total of Jresponses.
Next consider a predictor variable Xthat describes the rotations about two orthogonal axes
at a given joint:
where “1” and “2” indicate the two joint axes. The relevant null hypothesis is: Xand Yare
not linearly related.
To test this hypothesis one needs to assemble three covariance matrices. The first is a (3
3) response covariance matrix WYY which describes variance within and the co-variation
between the three force components (see Appendix D). The second is a (2 2) predictor
covariance matrix WXX which describes the variance and covariance of the two joint angles.
The third is a (2 3) predictor-response covariance matrix WXY which describes how each of
the predictor variables co-varies with each of the response variables.
The predictor-response covariance matrix WXY is relevant to the null hypothesis because it
embodies the strength of linear correlation between Xand Y. For completion, in the example
above WXY has six elements, corresponding to:
1. The linear correlation between 1and F1
2. The linear correlation between 1and F2
3. The linear correlation between 1and F3
4. The linear correlation between 2and F1
5. The linear correlation between 2and F2
6. The linear correlation between 2and F3
Initially these correlations refer only to X’s and Y’s original coordinate systems. Since
arbitrary coordinate systems can bias non-directed hypothesis testing (Appendix B), we must
allow the coordinate systems to rotate in order to most objectively test our null hypothesis.
One CCA solution is to choose the Xand Ycoordinate systems that mutually maximize
a single correlation coecient. The logic is that all other coordinate systems underestimate
correlation strength. In other words, as the coordinate systems rotate the elements of WXY
change, and one (not necessarily unique) coordinate system combination maximizes an element
of WXY . CCA solves this problem eciently using the maximum eigenvalue of the canonical
correlation matrix (Eqn.7, main manuscript).
As an aside, we note that the K=2 model in the main manuscript is equivalent to a K=1
model (i.e. only a running speed regressor) because only one (diagonal) element of WXX is
non-zero. For generalizability the main manuscript treats CCA in its K>1 form.

Supplementary resource (1)

... For the analysis of normally distributed sEMG data, we utilized SPM analysis, and for non-normally distributed data, we used statistical non-parametric mapping (SnPM). In particular, the covariance between the EMG time series was elucidated using the SPM Hotelling's T 2 statistic, which is a vector-field equivalent to univariate tests such as the twosample t-test [32,33]. The sEMG data during the weight-acceptance phase were analyzed as a vector field I = 8, J = 13, Q = 101. ...
... Significant differences in sEMG amplitudes within the time-series data were defined when the time-varying SPM{t} or SnPM{t} values exceeded a critical threshold (t*). particular, the covariance between the EMG time series was elucidated using the SPM Hotelling's statistic, which is a vector-field equivalent to univariate tests such as the two-sample t-test [32,33]. The sEMG data during the weight-acceptance phase were analyzed as a vector field I = 8, J = 13, Q = 101. ...
Full-text available
Single-leg drop landing (SLDL) and jump landing (SLJL) are frequently used as assessment tools for identifying potential high-risk movement patterns; thus, understanding differences in neuromuscular responses between these types of landings is essential. This study aimed to compare lower extremity neuromuscular responses between the SLDL and SLJL. Thirteen female participants performed an SLDL and SLJL from a 30-cm box height. Vertical ground reaction force (vGRF), time to peak vGRF, and surface electromyography (sEMG) data were collected. Continuous neuromuscular responses, peak vGRF, and time to peak vGRF were compared between the tasks. Statistical parametric mapping (SPM) analysis demonstrated that the SLJL had a significantly higher sEMG activity in the rectus femoris (RF), vastus lateralis (VL) and vastus medialis (VM) within the first 10% of the landing phase compared with SLDL. At 20–30% of the landing phase, sEMGs in the RF and VL during the SLDL were significantly higher compared with SLJL (p < 0.05). A higher peak vGRF and shorter time to peak vGRF was observed during SLJL (p < 0.05). In conclusion, our findings highlight that SLJL exhibited greater RF, VL, and VM activities than SLDL at initial impact (10% landing), coinciding with a higher peak vGRF and shorter time to attain peak vGRF. Our findings support the role of the quadriceps as the primary energy dissipator during the SLJL.
... In this work, the evaluation of the different methods is done using the statistical parametric mapping (SPM) and the Bland-Altman (B-A) difference against the mean graph. Initially used for neuroimaging analyses (35) and later adapted to biomechanics (36), SPM considers covariance among vector components. Mean continua are extracted from the measurements of each method and the variance is studied. ...
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Introduction: Recent advances in Artificial Intelligence (AI) and Computer Vision (CV) have led to automated pose estimation algorithms using simple 2D videos. This has created the potential to perform kinematic measurements without the need for specialized, and often expensive, equipment. Even though there’s a growing body of literature on the development and validation of such algorithms for practical use, they haven’t been adopted by health professionals. As a result, manual video annotation tools remain pretty common. Part of the reason is that the pose estimation modules can be erratic, producing errors that are difficult to rectify. Because of that, health professionals prefer the use of tried and true methods despite the time and cost savings pose estimation can offer. Methods: In this work, the gait cycle of a sample of the elderly population on a split-belt treadmill is examined. The Openpose (OP) and Mediapipe (MP) AI pose estimation algorithms are compared to joint kinematics from a marker-based 3D motion capture system (Vicon), as well as from a video annotation tool designed for biomechanics (Kinovea). Bland-Altman (B-A) graphs and Statistical Parametric Mapping (SPM) are used to identify regions of statistically significant difference. Results: Results showed that pose estimation can achieve motion tracking comparable to marker-based systems but struggle to identify joints that exhibit small, but crucial motion. Discussion: Joints such as the ankle, can suffer from misidentification of their anatomical landmarks. Manual tools don’t have that problem, but the user will introduce a static offset across the measurements. It is proposed that an AI powered video annotation tool that allows the user to correct errors would bring the benefits of pose estimation to professionals at a low cost.
... The discrete value itself does not represent the complete description of a single variable over time nor taking spatiotemporal confounding effect into consideration [67]. In the last decade, Statistical Parametric Mapping (SPM) has been used within the biomechanics and human movement science community for uni/multivariate time series data (i.e., kinematics and kinetics) [68,69]. This method is advantageous for biomechanical analysis as it allows full-field non-direct hypotheses tests and visualization of the statistical results over time series [70]. ...
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The shoulder is a unique and complex joint in the human body with three bones and four joints, which makes it the most unstable joint in the body due to the amount of motion. To improve approaches toward understanding the performance of overhead throwing movements, this systematic review summarizes the type of analysis related to shoulder biomechanics involving overhead sporting motions. A search of seven databases identified 33 eligible studies, which were subsequently scored using the Modified Coleman Methodology Score scale. A total of nine articles from badminton, seven from baseball, five from volleyball, five from tennis, three from cricket, and one from softball were reviewed. All 33 studies evaluated shoulder kinematics and 12 of them also investigated the forces and torques (kinetics). The most common methods used were 3D motion analysis (76%), digital video cameras (15%), electromagnetic tracking system (6%), and finally 3% used IMU sensors. Overall, shoulder external rotation during the back swing, internal rotation, and elbow extension during the forward acceleration phase were the strongest predictors of high velocity overhead throwing movement. The findings provide some useful insights and guidance to researchers in their future contribution to the existing body of literature on shoulder overhead throwing movement biomechanics.
... SPM{t} was created by calculating the conventional univariate t-statistic at each muscle activation point [27]. Hotelling's T 2 test, the vector field equivalent to the twosample t-test [28,29] was used to check the magnitude and time period of the ipsilateral/bilateral EMG activation differences. The magnitude at each time point was calculated using the scalar output statistic, SPM{T2}. ...
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The step-aside movement, also known as the dodging step, is a common maneuver for avoiding obstacles while walking. However, differences in neural control mechanisms and ankle strategies compared to straight walking can pose a risk of falling. This study aimed to examine the differences in tibialis anterior (TA), peroneus longus (PL), and soleus (SOL) muscle contractions, foot center of pressure (CoP) displacement, and ground reaction force (GRF) generation between step-aside movement and straight walking to understand the mechanism behind step-aside movement during walking. Twenty healthy young male participants performed straight walking and step-aside movements at comfortable walking speeds. The participants’ muscle contractions, CoP displacement, and GRF were measured. The results show significant greater bilateral ankle muscle contractions during the push and loading phases of step-aside movement than during straight walking. Moreover, the CoP displacement, GRF generation mechanism, and timing differed from those observed during straight walking. These findings provide valuable insights for rehabilitation professionals in the development of clinical decisions for populations at a risk of falls and lacking gait stability.
... We investigated the hip, knee and ankle joint angles for the three anatomical planes and for the five loading conditions. Statistical parametric mapping was used to compare the normalized movement cycle from start to take-off rather than discrete values (Pataky et al., 2013). A two-way ANOVA (sex and load) with repeated measures on one parameter (load) was performed and pairwise paired t-tests were used as post hoc test with Bonferroni corrected p-values if the ANOVA detected significant effects (α = 0.05). ...
Conference Paper
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The aim of this study was to investigate the effect of sex and additional load during countermovement jumps on hip, knee, and ankle joint kinematics. For this purpose, a total of 25 female and male participants performed barbell-loaded countermovement jumps with an additional load of up to 80 % body mass. No significant main effect of sex on any of the kinematic parameters was found. The additional load resulted in a significant decrease in hip, knee, and ankle joint angles in the sagittal plane and an increase of the absolute performance time with increasing additional. The changes in kinematics of all three anatomical planes could increase the risk of injury and should therefore be closely monitored, especially in less experienced individuals performing countermovement jumps.
... Therefore, this study was motivated by the controversial information about whether children with DMD change their gait symmetry, and one aspect where this is lacking is previous symmetry analysis only concerning the univariate variables, which may cause a form of bias, i.e., the inter-component covariance bias [11]. For example, lateral excessive trunk motion and contralaterally pelvic obliquity have been reported as a compensatory result for hip abductor muscle weakness during walking in children with DMD [12]. ...
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Background: Gait is a complex, whole-body movement that requires the coordinated action of multiple joints and muscles of our musculoskeletal system. In the context of Duchenne muscular dystrophy (DMD), a disease characterized by progressive muscle weakness and joint contractures, previous studies have generally assumed symmetrical behavior of the lower limbs during gait. However, such a symmetric gait pattern of DMD was controversial. One aspect of this is criticized, because most of these studies have primarily focused on univariate variables, rather than on the coordination of multiple body segments and even less investigate gait symmetry under a motor synergy of view. Methods: We investigated the gait pattern of 20 patients with DMD, compared to 18 typical developing children (TD) through 3D Gait Analysis. Kinematic and muscle synergies were extracted with principal component analysis (PCA) and non-negative matrix factorization (NNMF), respectively. The synergies extracted from the left and right sides were compared with each other to obtain a symmetry value. In addition, bilateral spatiotemporal variables of gait, such as stride length, percentage of stance and swing phase, step length, and percentage of double support phase, were used for calculating the symmetry index (SI) to evaluate gait symmetry as well. Results: Compared with the TD group, the DMD group walked with decreased gait velocity (both p < 0.01), stride length (both p < 0.01), and step length (both p < 0.001). No significant difference was found between groups in SI of all spatiotemporal parameters extracted between the left and right lower limbs. In addition, the DMD group exhibited lower kinematic synergy symmetry values compared to the TD group (p < 0.001), while no such significant group difference was observed in symmetry values of muscle synergy. Conclusions: The findings of this study suggest that DMD influences, to some extent, the symmetry of synergistic movement of multiple segments of lower limbs, and thus kinematic synergy appears capable of discriminating gait asymmetry in children with DMD when conventional spatiotemporal parameters are unchanged.
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Introduction: Knee OA progression is related to medial knee contact forces, which can be altered by anatomical parameters of tibiofemoral alignment and contact point locations. There is limited and controversial literature on medial-lateral force distribution and the effect of anatomical parameters, especially in motor activities different from walking. We analyzed the effect of tibiofemoral alignment and contact point locations on knee contact forces, and the medial-lateral force distribution in knee OA subjects with varus malalignment during walking, stair ascending and stair descending. Methods: Fifty-one knee OA subjects with varus malalignment underwent weight-bearing radiographs and motion capture during walking, stair ascending and stair descending. We created a set of four musculoskeletal models per subject with increasing level of personalization, and calculated medial and lateral knee contact forces. To analyze the effect of the anatomical parameters, statistically-significant differences in knee contact forces among models were evaluated. Then, to analyze the force distribution, the medial-to-total contact force ratios were calculated from the fully-informed models. In addition, a multiple regression analysis was performed to evaluate correlations between forces and anatomical parameters. Results: The anatomical parameters significantly affected the knee contact forces. However, the contact points decreased medial forces and increased lateral forces and led to more marked variations compared to tibiofemoral alignment, which produced an opposite effect. The forces were less medially-distributed during stair negotiation, with medial-to-total ratios below 50% at force peaks. The anatomical parameters explained 30%–67% of the variability in the knee forces, where the medial contact points were the best predictors of medial contact forces. Discussion: Including personalized locations of contact points is crucial when analyzing knee contact forces in subjects with varus malalignment, and especially the medial contact points have a major effect on the forces rather than tibiofemoral alignment. Remarkably, the medial-lateral force distribution depends on the motor activity, where stair ascending and descending show increased lateral forces that lead to less medially-distributed loads compared to walking.
Background: The implementation of a strategy to control the hip angle during gait is important to avoid disease progression in patients with hip osteoarthritis (OA). Research question: Do patients with hip OA tend to stabilize their hip angles by a combination of whole-body movements during gait in terms of variability? Methods: A public gait dataset comprising 80 asymptomatic participants and 106 patients with hip OA was used. Uncontrolled manifold analysis was performed using the joint angles as elemental variables and the hip joint angles as performance variables. The synergy index ΔV, variances of elemental variables that did not affect the performance variable (VUCM) and of those that affected the performance variable (VORT), and index of covariation strategy (COV) were calculated in sagittal and frontal plane. A one-sample t-test for statistical parametric mapping (SPM) analysis was used for ΔV and COV. Two-sample t-tests of SPM analyses were used for ΔV, VUCM, and VORT to compare the two groups. Results: In both planes, the ΔV and COV were significantly larger than zero in both groups (p < 0.001). In the sagittal plane, the VORT was higher in the hip OA group than in the control group after 77 % of stance phase. In the frontal plane, the hip OA group had larger ΔV and VUCM after last half and last quartile of stance phase compared to the control group, respectively. The VORT was smaller in the hip OA group than in the control group. Significance: The hip angle was stabilized in the hip OA group in the frontal plane but insufficiently stabilized in the sagittal plane; however, the patients changed their hip angle during the early phase of stance. The combination of whole-body movements contributed to the stabilization of hip angle.
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Introduction Analysis of human locomotion is challenged by limitations in traditional numerical and statistical methods as applied to continuous time-series data. This challenge particularly affects understanding of how close limb prostheses are to mimicking anatomical motion. This study was the first to apply a technique called Dynamic Time Warping to measure the biomimesis of prosthetic knee motion in young children and addressed the following research questions: Is a combined dynamic time warping/root mean square analysis feasible for analyzing pediatric lower limb kinematics? When provided at an earlier age than traditional protocols dictate, can children with limb loss utilize an articulating prosthetic knee in a biomimetic manner? Methods Warp costs and amplitude differences were generated for knee flexion curves in a sample of ten children five years of age and younger: five with unilateral limb loss and five age-matched typically developing children. Separate comparisons were made for stance phase flexion and swing phase flexion via two-way ANOVAs between bilateral limbs in both groups, and between prosthetic knee vs. dominant anatomical knee in age-matched pairs between groups. Greater warp costs indicated greater temporal dissimilarities, and a follow-up root mean square assessed remaining amplitude dissimilarities. Bilateral results were assessed by age using linear regression. Results The technique was successfully applied in this population. Young children with limb loss used a prosthetic knee biomimetically in both stance and swing, with mean warp costs of 12.7 and 3.3, respectively. In the typically developing group, knee motion became more symmetrical with age, but there was no correlation in the limb loss group. In all comparisons, warp costs were significantly greater for stance phase than swing phase. Analyses were limited by the small sample size. Discussion This study has established that dynamic time warping with root mean square analysis can be used to compare the entirety of time-series curves generated in gait analysis. The study also provided clinically relevant insights on the development of mature knee flexion patterns during typical development, and the role of a pediatric prosthetic knee.
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Statistical parametric mapping (SPM) is a topological methodology for detecting field changes in smooth n-dimensional continua. Many classes of biomechanical data are smooth and contained within discrete bounds and as such are well suited to SPM analyses. The current paper accompanies release of 'SPM1D', a free and open-source Python package for conducting SPM analyses on a set of registered 1D curves. Three example applications are presented: (i) kinematics, (ii) ground reaction forces and (iii) contact pressure distribution in probabilistic finite element modelling. In addition to offering a high-level interface to a variety of common statistical tests like t tests, regression and ANOVA, SPM1D also emphasises fundamental concepts of SPM theory through stand-alone example scripts. Source code and documentation are available at:
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It is well-known that muscle redundancy grants the CNS numerous options to perform a task. Does muscle redundancy, however, allow sufficient robustness to compensate for loss or dysfunction of even a single muscle? Are all muscles equally redundant? We combined experimental and computational approaches to establish the limits of motor robustness for static force production. In computer-controlled cadaveric index fingers, we find that only a small subset (<5%) of feasible forces is robust to loss of any one muscle. Importantly, the loss of certain muscles compromises force production significantly more than others. Further computational modeling of a multi-joint, multi-muscle leg demonstrates that this severe lack of robustness generalizes to whole limbs. These results provide a biomechanical basis to begin to explain why redundant motor systems can be vulnerable to even mild neuromuscular pathology.
In an age where the amount of data collected from brain imaging is increasing constantly, it is of critical importance to analyse those data within an accepted framework to ensure proper integration and comparison of the information collected. This book describes the ideas and procedures that underlie the analysis of signals produced by the brain. The aim is to understand how the brain works, in terms of its functional architecture and dynamics. This book provides the background and methodology for the analysis of all types of brain imaging data, from functional magnetic resonance imaging to magnetoencephalography. Critically,Statistical Parametric Mappingprovides a widely accepted conceptual framework which allows treatment of all these different modalities. This rests on an understanding of the brain's functional anatomy and the way that measured signals are caused experimentally. The book takes the reader from the basic concepts underlying the analysis of neuroimaging data to cutting edge approaches that would be difficult to find in any other source. Critically, the material is presented in an incremental way so that the reader can understand the precedents for each new development. This book will be particularly useful to neuroscientists engaged in any form of brain mapping; who have to contend with the real-world problems of data analysis and understanding the techniques they are using. It is primarily a scientific treatment and a didactic introduction to the analysis of brain imaging data. It can be used as both a textbook for students and scientists starting to use the techniques, as well as a reference for practicing neuroscientists. The book also serves as a companion to the software packages that have been developed for brain imaging data analysis. * An essential reference and companion for users of the SPM software * Provides a complete description of the concepts and procedures entailed by the analysis of brain images * Offers full didactic treatment of the basic mathematics behind the analysis of brain imaging data * Stands as a compendium of all the advances in neuroimaging data analysis over the past decade * Adopts an easy to understand and incremental approach that takes the reader from basic statistics to state of the art approaches such as Variational Bayes * Structured treatment of data analysis issues that links different modalities and models * Includes a series of appendices and tutorial-style chapters that makes even the most sophisticated approaches accessible.
This article directly addresses explicit contradictions in the literature regarding the relation between the power of multivariate analysis of variance (MANOVA) and the intercorrelations among the dependent variables. Artificial data sets, as well as analytical methods, revealed that (1) power increases as correlations between dependent variables with large consistent effect sizes (that are in the same direction) move from near 1.0 toward –2.0, (2) power increases as correlations become more positive or more negative between dependent variables that have very different effect sizes (i.e., one large and one negligible), and (3) power increases as correlations between dependent variables with negligible effect sizes shift from positive to negative (assuming that there are dependent variables with large effect sizes still in the design). Implications for the reliability of dependent variables and strategies for selecting these variables in MANOVA designs are discussed. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The problem of fitting a linear relation to a bivariate data cluster obtained from morphometric measurement or from experiment is formulated rigorously, and a family of solutions (the general structural relation, g.s.r.) is derived. The regression, reduced major axis and major axis models are special cases of this model; it permits a more realistic treatment of the errors in the variates, in particular when the errors are correlated, which is particularly important in the many biological situations in which the variates contain uncontrolled real variation in addition to measurement errors. The analysis is particularly directed to the testing of hypotheses about scaling relations derived from biomechanical theory. By making different assumptions about the configuration of the errors, the g.s.r. can also be used to test for transposition allometry and for the significance of an estimated or hypothesized gradient. The model generalizes simply to multivariate problems. Application is demonstrated with examples drawn from the study of bird flight mechanics. Finally, it is demonstrated that since observed quantities correspond to peaks of adaptation or of selective fitness, scaling relations are determined primarily by scale variation of constraints on adaptation and behaviour, and are the result of a variety of interacting factors rather than a response to a single selective force described by one simple hypothesis.
Concepts of correlation and regression may be applied not only to ordinary one-dimensional variates but also to variates of two or more dimensions. Marksmen side by side firing simultaneous shots at targets, so that the deviations are in part due to independent individual errors and in part to common causes such as wind, provide a familiar introduction to the theory of correlation; but only the correlation of the horizontal components is ordinarily discussed, whereas the complex consisting of horizontal and vertical deviations may be even more interesting. The wind at two places may be compared, using both components of the velocity in each place. A fluctuating vector is thus matched at each moment with another fluctuating vector. The study of individual differences in mental and physical traits calls for a detailed study of the relations between sets of correlated variates. For example the scores on a number of mental tests may be compared with physical measurements on the same persons. The questions then arise of determining the number and nature of the independent relations of mind and body shown by these data to exist, and of extracting from the multiplicity of correlations in the system suitable characterizations of these independent relations. As another example, the inheritance of intelligence in rats might be studied by applying not one but s different mental tests to N mothers and to a daughter of each
Humans run faster by increasing a combination of stride length and stride frequency. In slow and medium-paced running, stride length is increased by exerting larger support forces during ground contact, whereas in fast running and sprinting, stride frequency is increased by swinging the legs more rapidly through the air. Many studies have investigated the mechanics of human running, yet little is known about how the individual leg muscles accelerate the joints and centre of mass during this task. The aim of this study was to describe and explain the synergistic actions of the individual leg muscles over a wide range of running speeds, from slow running to maximal sprinting. Experimental gait data from nine subjects were combined with a detailed computer model of the musculoskeletal system to determine the forces developed by the leg muscles at different running speeds. For speeds up to 7 m s(-1), the ankle plantarflexors, soleus and gastrocnemius, contributed most significantly to vertical support forces and hence increases in stride length. At speeds greater than 7 m s(-1), these muscles shortened at relatively high velocities and had less time to generate the forces needed for support. Thus, above 7 m s(-1), the strategy used to increase running speed shifted to the goal of increasing stride frequency. The hip muscles, primarily the iliopsoas, gluteus maximus and hamstrings, achieved this goal by accelerating the hip and knee joints more vigorously during swing. These findings provide insight into the strategies used by the leg muscles to maximise running performance and have implications for the design of athletic training programs.
Conference Paper
In this paper, we propose a Riemannian framework for statistical analysis of tensor fields. Existing approaches to this problem have been mainly voxel-based that overlook the correlation between tensors at different voxels. In our approach, the tensor fields are considered as points in a high-dimensional Riemannian product space and accordingly, we extend Principal Geodesic Analysis (PGA) to the product space. This provides us with a principled method for linearizing the problem, and coupled with the usual log-exp maps that relate points on manifold to tangent vectors, the global correlation of the tensor field can be captured using Principal Component Analysis in a tangent space. Using the proposed method, the modes of variation of tensor fields can be efficiently determined, and dimension reduction of the data is also easily implemented. Experimental results on characterizing the variation of a large set of tensor fields are presented in the paper, and results on classifying tensor fields using the proposed method are also reported. These preliminary experimental results demonstrate the advantages of our method over the voxel-based approach.