Measures of Symmetry Symmetry Index Symmetry Ratio Statistical Methods James Richards modified by W Rose Symmetry Index SI when it 0 the gait is symmetrical Differences are reported against their average value If a large asymmetry is present the average value does not correctly reflec ID: 783350
Download The PPT/PDF document "Symmetry Definition: Both limbs are be..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Symmetry
Definition: Both limbs are behaving identicallyMeasures of Symmetry Symmetry IndexSymmetry RatioStatistical Methods
James Richards, modified by W. Rose
Slide2Symmetry Index
SI when it = 0, the gait is symmetricalDifferences are reported against their average value. If a large asymmetry is present, the average value does not correctly reflect the performance of either limbRobinson RO, Herzog W, Nigg BM. Use of force platform variables to quantify the effects of chiropractic manipulation on gait symmetry. J Manipulative Physiol Ther 1987;10(4):172–6.
Slide3Symmetry Ratio
Limitations: relatively small asymmetry and a failure to provide info regarding location of asymmetryLow sensitivitySeliktar R, Mizrahi J. Some gait characteristics of below-knee amputees and their reflection on the ground reaction forces. Eng Med 1986;15(1):27–34.
Slide4Statistical Measures of Symmetry
Correlation CoefficientsPrincipal Component AnalysisAnalysis of Variance
Use single points or limited set of points
Do not analyze the entire waveform
Sadeghi H, et al. Symmetry and limb dominance in able-bodied gait: a
review. Gait Posture 2000;12(1):34–45.
Sadeghi H, Allard P, Duhaime M. Functional gait asymmetry in ablebodied subjects. Hum Movement Sci 1997;16:243–58.
Slide5The measure of trend symmetry utilizes eigenvectors to compare time-normalized right leg and left leg gait cycles in the following manner. Each waveform is translated by subtracting its mean value from every value in the waveform.
for every ith pair of n rows of waveform data
Eigenvector Analysis
Slide6Eigenvector Analysis
Translated data points from the right and left waveforms are entered into a matrix (M), where each pair of points is a row. The rectangular matrix M is premultiplied by its transpose to form a 2x2 matrix
S:
S = M
T
M
The eigenvalues and eigenvectors of S are computed.
Slide7Eigenvector Analysis
To simplify the calculation process, we applied singular value decomposition (SVD) to the translated matrix M to determine the eigenvalues and eigenvectors of S=MTM,
since SVD performs the operations of multiplying M by its transpose and extracting the eigenvectors
. Note that the singular values of M are the non-negative square roots of the eigenvalues of S=M
T
M (as stated by
Labview
help).
Slide8Eigenvector Analysis
Each row of M is then rotated by (minus) the angle formed between the eigenvector and the X-axis, so that the points lie around the X-axis (Eq. (2)):
w
here
a
nd e
x
and
e
y
are the x and y components of the (largest) eigenvector of S, and a 4-quadrant inverse tangent function is used.
Slide9Eigenvector Analysis
The variability of the rotated points in the X and Y directions is then calculated. The Y-axis variability is the variability perpendicular to the eigenvector, and the X-axis variability is the variability along the eigenvector.
Compute the ratio
of the variability about the eigenvector
to the
variability along the
eigenvector. This number will always be between 0 and 1. The ratio is subtracted from 1, giving the Trend Symmetry, which will be between 1 and 0.
Slide10Eigenvector Analysis
Trend Symmetry = 1.0 indicates perfect symmetryTrend Symmetry = 0.0 indicates lack of symmetry.
The Trend Symmetry will be 1 if the ratio of
variabilities
is 0. This will occur if and only if the rotated points all lie on the X axis (which means the variability along Y is zero).
The Trend Symmetry will be 0 if the ratio of
variabilities
is 1. This will occur if the rotated points vary as much in the Y direction as they do in the X direction.
Slide11Additional
measures of symmetry: Range amplitude ratio quantifies the difference in range of motion of each limb, and is calculated by dividing the range of motion of the right limb from that of the left limb.
Range offset
, a measure of the differences in operating range of each limb, is calculated by subtracting the average of the right side waveform from the average of the left side waveform.
Slide12Eigenvector Analysis
Trend Symmetry: 0.948
Range Amplitude Ratio: 0.79, Range Offset:0
Slide13Eigenvector Analysis
Range Amplitude Ratio: 2.0
Trend Symmetry: 1.0, Range Offset: 19.45
Slide14Eigenvector Analysis
Range Offset: 10.0
Trend Symmetry: 1.0, Range Amplitude Ratio: 1.0
Slide15Eigenvector Analysis
Trend Symmetry: 0.979 Range Amplitude Ratio: 0.77 Range Offset: 2.9
°
Raw flexion/extension waveforms from an ankle
Slide16Eigenvector Analysis
Slide17Final Adjustment #1
Determining Phase Shift and the Maximum Trend Symmetry: Shift one waveform in 1-percent increments (e.g. sample 100 becomes sample 1, sample 1 becomes sample 2…) and recalculate the trend symmetry for each shift. The phase offset is the shift which produces the largest value for trend symmetry. The associated maximum trend symmetry value
is also noted.
Slide18Final Adjustment #2
Trend Symmetry (TS), as defined so far, is unaffected if one of the waves is multiplied by -1. Therefore Trend Symmetry, as computed, does not distinguish between symmetry and anti-symmetry. We can modify Trend Symmetry to distinguish between symmetric and anti-symmetric waveforms:
Slide19Final Adjustment #2
TS
mod
=
1.0 indicates perfect
symmetry.
TS
mod
= -1.0 indicates perfect
antisymmetry
.
TS
mod
= 0.0 indicates complete lack of symmetry.
Slide20Symmetry Example
Slide21Hip Joint
Trend Symmetry
Phase Shift (% Cycle
Max Trend Symmetry
Range Amplitude
Range Offset
95% CI
0.98 – 1.00
-3.1 – 2.9
0.99 – 1.00
0.88 - 1.16
-5.99 – 5.66
Unbraced
1.00
1
1.00
0.95
4.21
Braced
1.00
0
1.00
1.02
4.73
Amputee
1.00
-1
1.00
0.88
-0.72
Symmetry Example…Hip Joint
Braced
Amputee
Unbraced
Slide22Knee Joint
Trend Symmetry
Phase Shift (% Cycle
Max Trend Symmetry
Range Amplitude
Range Offset
95% CI
0.97 – 1.00
-2.6 – 2.5
0.99 – 1.00
0.87 - 1.16
-8.95 - 10.51
Unbraced
1.00
0
1.00
1.03
5.28
Braced
1.00
-1
1.00
0.99
6.40
Amputee
0.98
-1
0.99
0.91
4.15
Symmetry Example…Knee Joint
Braced
Amputee
Unbraced
Slide23Ankle Joint
Trend Symmetry
Phase Shift (% Cycle
Max Trend Symmetry
Range Amplitude
Range Offset
95% CI
0.94 – 1.00
-2.62 – 2.34
0.96 – 1.00
0.75 - 1.32
-6.4 – 7.0
Unbraced
0.98
-1
0.98
1.03
-2.96
Braced
0.73
-4
0.79
0.53
5.84
Amputee
0.58
4
0.61
1.27
0.48
Symmetry Example…Ankle Joint
Unbraced
Braced
Amputee
Slide24Normalcy Example
Slide25Hip Joint
Trend Normalcy
Phase Shift (% Cycle
Max Trend Normalcy
Range Amplitude
Range Offset
95% CI
0.98 – 1.00
-3.1 – 2.9
0.99 – 1.00
0.88 - 1.16
-5.99 – 5.66
Right hip
Unbraced
1.00
2
1.00
0.85
-14.91
Braced
0.99
3
1.00
0.90
-14.20
Amputee
0.97
-4
1.00
0.92
-8.08
Left hip
Unbraced
Braced
Amputee
Unbraced
Braced
Amputee
Slide26Hip Joint
Trend Normalcy
Phase Shift (% Cycle
Max Trend Normalcy
Range Amplitude
Range Offset
95% CI
0.98 – 1.00
-3.1 – 2.9
0.99 – 1.00
0.88 - 1.16
-5.99 – 5.66
Right hip
Unbraced
Braced
Amputee
Left hip
Unbraced
1.00
2
1.00
0.91
-19.28
Braced
0.99
4
1.00
0.91
-19.09
Amputee
0.99
-2
1.00
1.06
-7.52
Unbraced
Braced
Amputee
Slide27Knee Joint
Trend Normalcy
Phase Shift (% Cycle
Max Trend Normalcy
Range Amplitude
Range Offset
95% CI
0.97 – 1.00
-2.6 – 2.5
0.99 – 1.00
0.87 - 1.16
-8.95 – 10.51
Right knee
Unbraced
0.99
1
0.99
1.12
-11.89
Braced
0.98
3
0.99
1.07
-13.22
Amputee
0.96
-2
0.99
0.97
-7.45
Left knee
Unbraced
Braced
Amputee
Unbraced
Braced
Amputee
Slide28Knee Joint
Trend Normalcy
Phase Shift (% Cycle
Max Trend Normalcy
Range Amplitude
Range Offset
95% CI
0.97 – 1.00
-2.6 – 2.5
0.99 – 1.00
0.87 - 1.16
-8.95 – 10.51
Right knee
Unbraced
Braced
Amputee
Left knee
Unbraced
0.99
1
0.99
1.11
-16.35
Braced
0.97
4
0.99
1.10
-18.80
Amputee
0.98
-2
1.00
1.08
-10.78
Unbraced
Braced
Amputee
Slide29Ankle Joint
Trend Normalcy
Phase Shift (% Cycle
Max Trend Normalcy
Range Amplitude
Range Offset
95% CI
0.94 – 1.00
-2.62 – 2.34
0.96 – 1.00
0.75 - 1.32
-6.4 – 7.0
Right ankle
Unbraced
0.90
-2
0.94
1.48
1.33
Braced
0.65
-4
0.72
0.77
9.04
Amputee
0.80
-5
0.98
1.40
4.30
Left ankle
Unbraced
Braced
Amputee
Unbraced
Braced
Amputee
Slide30Ankle Joint
Trend Normalcy
Phase Shift (% Cycle
Max Trend Normalcy
Range Amplitude
Range Offset
95% CI
0.94 – 1.00
-2.62 – 2.34
0.96 – 1.00
0.75 - 1.32
-6.4 – 7.0
Right ankle
Unbraced
Braced
Amputee
Left ankle
Unbraced
0.93
-1
0.95
1.49
4.62
Braced
0.94
2
0.95
1.51
3.53
Amputee
0.11
-11
0.76
1.14
4.15
Unbraced
Braced
Amputee
Slide31Simpler way to compute Trend Symmetry
No need for SVD or Eigen-routines.Can be done in an Excel spreadsheet.Compute 2x2 covariance matrix:
Note s
12
=s
21
.
Eigenvalues of
S are the values of
which satisfy:
Slide32Simpler way to compute Trend Symmetry
Eigenvalues of S are the values of which satisfy :
Slide33Ratio of smaller to larger eigenvalue:
where
Then