1. Symmetry Definition: Measures of Symmetry
James Richards, modified by W. Rose
Definition:
Both limbs are behaving identically
Measures of Symmetry
Symmetry Index
Symmetry Ratio
Statistical Methods
SI when it = 0, the gait is symmetrical
Using such an equation may have major limitations because differences are reported against their average value. For example, if a large asymmetry is present, the average value does not correctly reflect the performance of either limb
Symmetry Ratio
Limitations: relatively small asymmetry and a failure to provide info regarding location of asymmetry, low sensitivity

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 1987;10(4):172–6.

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 1986;15(1):27–34.

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 2000;12(1):34–45.
Sadeghi H, Allard P, Duhaime M. Functional gait asymmetry in ablebodied subjects. Hum Movement Sci 1997;16:243–58.

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 to form a 2x2 matrix S:
S = MTM
The eigenvalues and eigenvectors of S are computed.

7. Eigenvector 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=MTM (as stated by Labview help).

8. Eigenvector 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)):
where
𝜃= tan −1 𝑒 𝑥 , 𝑒 𝑦
and ex and ey are the x and y components of the (largest) eigenvector of S, and a 4-quadrant inverse tangent function is used.

9. Eigenvector 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.
𝑇𝑟𝑒𝑛𝑑 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦=1− 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑎𝑏𝑜𝑢𝑡 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑎𝑙𝑜𝑛𝑔 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟

10. Eigenvector Analysis
𝑇𝑟𝑒𝑛𝑑 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦=1− 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑎𝑏𝑜𝑢𝑡 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑎𝑙𝑜𝑛𝑔 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟
Trend Symmetry = 1.0 indicates perfect symmetry
Trend 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.

11. 𝑅.𝐴.𝑅.= 𝐿𝑒𝑓𝑡 𝑙𝑖𝑚𝑏 𝑀𝑎𝑥.−𝑀𝑖𝑛. 𝑅𝑖𝑔ℎ𝑡 𝑙𝑖𝑚𝑏 𝑀𝑎𝑥.−𝑀𝑖𝑛.
Additional 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.
𝑅.𝐴.𝑅.= 𝐿𝑒𝑓𝑡 𝑙𝑖𝑚𝑏 𝑀𝑎𝑥.−𝑀𝑖𝑛. 𝑅𝑖𝑔ℎ𝑡 𝑙𝑖𝑚𝑏 𝑀𝑎𝑥.−𝑀𝑖𝑛.
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.
𝑅𝑎𝑛𝑔𝑒 𝑂𝑓𝑓𝑠𝑒𝑡=𝑀𝑒𝑎𝑛 𝐿𝑒𝑓𝑡 −𝑀𝑒𝑎𝑛(𝑅𝑖𝑔ℎ𝑡)

12. Eigenvector Analysis
𝑇𝑟𝑒𝑛𝑑 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦=1− 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑎𝑏𝑜𝑢𝑡 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑎𝑙𝑜𝑛𝑔 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟
Trend Symmetry: 0.948
Range Amplitude Ratio: 0.79, Range Offset:0

13. Eigenvector Analysis 𝑅.𝐴.𝑅.= 𝐿𝑒𝑓𝑡 𝑙𝑖𝑚𝑏 𝑀𝑎𝑥.−𝑀𝑖𝑛. 𝑅𝑖𝑔ℎ𝑡 𝑙𝑖𝑚𝑏 𝑀𝑎𝑥.−𝑀𝑖𝑛.
Range Amplitude Ratio: 2.0
Trend Symmetry: 1.0, Range Offset: 19.45

14. Eigenvector Analysis 𝑅𝑎𝑛𝑔𝑒 𝑂𝑓𝑓𝑠𝑒𝑡=𝑀𝑒𝑎𝑛 𝐿𝑒𝑓𝑡 −𝑀𝑒𝑎𝑛(𝑅𝑖𝑔ℎ𝑡)
Range Offset: 10.0
Trend Symmetry: 1.0, Range Amplitude Ratio: 1.0

15. Eigenvector Analysis Raw flexion/extension waveforms from an ankle
Trend Symmetry: Range Amplitude Ratio: Range Offset: 2.9°

16. Eigenvector Analysis

17. Final 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.

18. Final 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:
𝑇𝑆 𝑚𝑜𝑑 = +𝑇𝑆 𝑖𝑓 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑙𝑜𝑝𝑒≥0 −𝑇𝑆 𝑖𝑓 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑙𝑜𝑝𝑒<0

19. Final Adjustment #2
𝑇𝑆 𝑚𝑜𝑑 = +𝑇𝑆 𝑖𝑓 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑙𝑜𝑝𝑒≥0 −𝑇𝑆 𝑖𝑓 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑙𝑜𝑝𝑒<0
TSmod = 1.0 indicates perfect symmetry.
TSmod = -1.0 indicates perfect antisymmetry.
TSmod = 0.0 indicates complete lack of symmetry.

20. Symmetry Example

21. Symmetry Example…Hip 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
-5.99 – 5.66
Unbraced
1.00 1
0.95
4.21
Braced
1.02
4.73
Amputee -1
0.88
-0.72

22. Symmetry Example…Knee 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
Unbraced
1.00
1.03
5.28
Braced -1
0.99
6.40
Amputee
0.98
0.91
4.15

23. Symmetry Example…Ankle Joint
Unbraced
Braced
Amputee
Ankle 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
-6.4 – 7.0
Unbraced
0.98 -1
1.03
-2.96
Braced
0.73 -4
0.79
0.53
5.84
Amputee
0.58 4
0.61
1.27
0.48

24. Normalcy Example

25. Hip Joint 95% CI Right hip Unbraced 1.00 2 0.85 -14.91 Braced 0.99 3
Amputee
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
-5.99 – 5.66
Right hip
Unbraced
1.00 2
0.85
-14.91
Braced
0.99 3
0.90
-14.20
Amputee
0.97 -4
0.92
-8.08
Left hip

26. Hip Joint 95% CI Right hip Unbraced Braced Amputee Left hip 1.00 2
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
-5.99 – 5.66
Right hip
Unbraced
Braced
Amputee
Left hip
1.00 2
0.91
-19.28
0.99 4
-19.09 -2
1.06
-7.52

27. Knee Joint 95% CI Right knee Unbraced 0.99 1 1.12 -11.89 Braced 0.98 3
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
-8.95 – 10.51
Right knee
Unbraced
0.99 1
1.12
-11.89
Braced
0.98 3
1.07
-13.22
Amputee
0.96 -2
0.97
-7.45
Left knee

28. Knee Joint 95% CI Right knee Unbraced Braced Amputee Left knee 0.99 1
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
-8.95 – 10.51
Right knee
Unbraced
Braced
Amputee
Left knee
0.99 1
1.11
-16.35
0.97 4
1.10
-18.80
0.98 -2
1.00
1.08
-10.78

29. Ankle Joint 95% CI Unbraced 0.90 -2 0.94 1.48 1.33 Braced 0.65 -4 0.72
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
-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

30. Ankle Joint 95% CI Unbraced Braced Amputee 0.93 -1 0.95 1.49 4.62 0.94
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
-6.4 – 7.0
Right ankle
Unbraced
Braced
Amputee
Left ankle
0.93 -1
0.95
1.49
4.62
0.94 2
1.51
3.53
0.11
-11
0.76
1.14
4.15

31. Simpler way to compute Trend Symmetry
No need for SVD or Eigen-routines.
Can be done in an Excel spreadsheet.
Compute 2x2 covariance matrix:
𝑆= 𝑥 𝑖 − 𝑥 𝑥 𝑖 − 𝑥 𝑦 𝑖 − 𝑦 𝑥 𝑖 − 𝑥 𝑦 𝑖 − 𝑦 𝑦 𝑖 − 𝑦 = 𝑠 11 𝑠 12 𝑠 21 𝑠 22
Note s12=s21.
Eigenvalues of S are the values of 𝜆 which satisfy:
𝑑𝑒𝑡 𝑆−𝜆𝐼 =𝑑𝑒𝑡 𝑠 11 −𝜆 𝑠 12 𝑠 12 𝑠 22 −𝜆 =0

32. Simpler way to compute Trend Symmetry
Eigenvalues of S are the values of 𝜆 which satisfy :
𝑑𝑒𝑡 𝑆−𝜆𝐼 =𝑑𝑒𝑡 𝑠 11 −𝜆 𝑠 12 𝑠 12 𝑠 22 −𝜆 =0
𝑠 11 −𝜆 𝑠 22 −𝜆 − 𝑠 12 2 =0
𝜆 2 − 𝑠 11 + 𝑠 22 𝜆+ 𝑠 11 𝑠 22 − 𝑠 =0
𝜆 1 , 𝜆 2 = 𝑠 11 + 𝑠 22 ± 𝑠 11 + 𝑠 −4 𝑠 11 𝑠 22 − 𝑠
𝜆 1 , 𝜆 2 = 𝑠 11 + 𝑠 22 ± 𝑠 𝑠 −2 𝑠 11 𝑠 𝑠

33. Ratio of smaller to larger eigenvalue:
𝑟𝑎𝑡𝑖𝑜= 𝜆 2 𝜆 1
where
𝜆 1 = 𝑠 11 + 𝑠 𝑠 𝑠 −2 𝑠 11 𝑠 𝑠
𝜆 2 = 𝑠 11 + 𝑠 22 − 𝑠 𝑠 −2 𝑠 11 𝑠 𝑠
Then
𝑇𝑟𝑒𝑛𝑑𝑆𝑦𝑚𝑚=1− 𝜆 2 𝜆 1
𝑇𝑟𝑒𝑛𝑑𝑆𝑦𝑚𝑚= 𝜆 1 − 𝜆 2 𝜆 1
𝑇𝑟𝑒𝑛𝑑𝑆𝑦𝑚𝑚= 𝑠 𝑠 −2 𝑠 11 𝑠 𝑠 𝜆 1