Symmetry

1
Symmetry

• Definition
• Both limbs are behaving identically
• Measures of Symmetry
• Symmetry Index
• Symmetry Ratio
• Statistical Methods

2
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 reflect the
  performance of either limb
• Robinson RO, Herzog W, Nigg BM. Use of force
  platform variables to quantify the effects of
  chiropractic manipulation on gait symmetry. J
  Manipulative Physiol Ther 198710(4)1726.

3
Symmetry Ratio

• Limitations relatively small asymmetry and a
  failure to provide info regarding location of
  asymmetry
• Low sensitivity
• Seliktar R, Mizrahi J. Some gait characteristics
  of below-knee amputees and their reflection on
  the ground reaction forces. Eng Med
  198615(1)2734.

4
Statistical Measures of Symmetry

• Correlation Coefficients
• Principal Component Analysis
• Analysis 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
200012(1)3445.
Sadeghi H, Allard P, Duhaime M. Functional gait
asymmetry in ablebodied subjects. Hum Movement
Sci 19971624358.
5
Eigenvector Analysis
The 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
6
Eigenvector 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 (MTM)
to form a square matrix S, and the eigenvectors
are derived from the square matrix S. To simplify
the calculation process, we applied a singular
value decomposition (SVD) to the translated
matrix M to determine the eigenvectors, since SVD
performs the operations of multiplying M by its
transpose and extracting the eigenvectors.
7
Eigenvector Analysis
Each row of M is then rotated by the angle formed
between the eigenvector and the X-axis (u) so
that the points lie around the X-axis (Eq. (2))
8
Eigenvector Analysis
The variability of the points is then calculated
along the X and Y-axes, where the Y-axis
variability is the variability about the
eigenvector, and the X-axis variability is the
variability along the eigenvector. The trend
symmetry value is calculated by taking the ratio
of the variability about the eigenvector the
variability along the eigenvector, and
subtracting it from 1.0. A value of 1.0
indicates perfect symmetry, and a value of 0.0
indicates asymmetry.
9
Eigenvector Analysis

• We also calculated two additional measures of
  symmetry between waveforms.
• 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.

10
Eigenvector Analysis
Expressed as ratio of the variance about
eigenvector to the variance along the eigenvector
Trend Symmetry 0.948
Range Amplitude Ratio 0.79, Range Offset0
11
Eigenvector Analysis
Expressed as a ratio of the range of motion of
the left limb to that of the right limb
Range Amplitude Ratio 2.0
Trend Symmetry 1.0, Range Offset 19.45
12
Eigenvector Analysis
Calculated by subtracting the average of the
right side waveform from the average of the left
side waveform
Range Offset 10.0
Trend Symmetry 1.0, Range Amplitude Ratio 1.0
13
Eigenvector Analysis
Raw flexion/extension waveforms from an ankle
Trend Symmetry 0.979 Range Amplitude
Ratio 0.77 Range Offset 2.9
14
Eigenvector Analysis
15
Final Adjustment 1

• Determining Phase Shift and the Maximum Trend
  Symmetry Index
• Shift one waveform in 1-percent increments (e.g.
  sample 100 becomes sample 1, sample 1 becomes
  sample 2) and recalculate the trend similarity
  for each shift. The phase shift is determined by
  identifying the index at which the smallest value
  for trend similarity occurs. The maximum trend
  similarity value independent of original phase
  position is also identified in this process.

16
Final Adjustment 2

• Modifications to Trend Symmetry Index to
  accommodate mirrored waveforms
• Assign the sign of the eigenvector slope to the
  TSI value. A modified TSI value of 1.0 indicates
  perfect symmetry in like oriented waveforms,
  while a TSI value of -1 indicates perfect
  symmetry in reflected waveforms. A TSI value of
  0.0 still indicates asymmetry.

17
Symmetry Example
18
Symmetry ExampleHip Joint
Unbraced
Braced
Amputee
Hip 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
19
Symmetry ExampleKnee Joint
Unbraced
Braced
Amputee
Knee 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
20
Symmetry ExampleAnkle Joint
Unbraced
Braced
Amputee
Ankle Joint Trend Symmetry Phase Shift ( Cycle Min 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
21
Normalcy Example
22
Braced
Amputee
Unbraced
Hip 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
23
Braced
Amputee
Unbraced
Hip 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
24
Unbraced
Braced
Amputee
Knee 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
25
Unbraced
Braced
Amputee
Knee 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
26
Unbraced
Braced
Amputee
Ankle 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
27
Unbraced
Braced
Amputee
Ankle 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