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Different Methods of Calculating Body Sway Area

Different Methods of Calculating Body Sway Area. Thomas Wollseifen Brighton (UK), 9th-12th October 2011. Posturography. Postural instability Parkinson ‘ s disease Sleeping problems Elderly subjects. Centre of pressure. Body sway Length of path Velocity Distribution

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Different Methods of Calculating Body Sway Area

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  1. Different Methods of Calculating Body Sway Area Thomas Wollseifen Brighton (UK), 9th-12th October 2011

  2. Posturography • Postural instability • Parkinson‘s disease • Sleeping problems • Elderly subjects Centre of pressure • Body sway • Length of path • Velocity • Distribution • Body sway area Body sway path - trajectory

  3. How to calculate the body sway area? • Area of convex hull • 2) Area of ellipse – • Principal component analysis (PCA) • 3) Mean of circle areas

  4. 1) Convex hull convex not convex • Different algorithms: • Gift wrapping algorithm • Triangulation • etc.

  5. Convex hull - Algorithm • Triangulation of a point set (PROC G3GRID) • proc g3grid data=_ch00 • outtri=_ch01  triangulation of data set • out=_temp; • grid &y.*&x.=&z.; • run; • 2) Calculation of the determinant (of the convex hull = Area)

  6. Find the area of this (convex) polygon determinant (2,5) 7 6 5 4 (-4,3) 3 2 (5,1) 1 0 Y [cm] -1 -2 -3 -4 -5 -6 -7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 X [cm] data _det01; set &indata.; x2=lag(&x.); y2=lag(&y.); if not missing(x2) then d1=x2*&y.; else d1=0; if not missing(y2) then d2=y2*&x.; else d2=0; run; proc sql; create table _det02 as select * ,sum(d1) as sumd1 ,sum(d2) as sumd2 ,(calc. sumd2 – calc. sumd1)*0.5, as det from _det01 ; quit;

  7. 2) Principal component analysis (PCA) 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 Y [cm] Y [cm] 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 -7 -7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 X [cm] X [cm] e‘2 Find a „better“ bounding box! PCA e‘1 Idea: Reduce dimensionality of a point set by identifying the most significant directions (Principal components = Eigenvectors)

  8. Principal component analysis with PROC PRINCOMP proc princomp data=_pca01 COV OUT = _prins; var &y. &x.; ods output eigenvectors=_eigenvectors eigenvalues=_eigenvalues; run; Covariance Matrix y X Y 4.526000000 -2.408333333 X -2.408333333 5.195833333 Eigenvalues of the Covariance Matrix Eigenvalue Difference Proportion Cumulative 1 7.29242620 4.86301907 0.7501 0.7501 2 2.42940713 0.2499 1.0000

  9. From covariance matrix to Eigenvectors / Eigenvalues • Calculate 2x2 covariances cov(X,Y) of X,Y: • Prepare covariancematrix: Cov-Matrix • Calculate Eigenvalues and Eigenvectors of the covariance matrix:  x y Eigenvectore1(0.75, -0.66) Eigenvector e2(0.66, 0.75) Eigenvalue17.29 Eigenvalue22.42

  10. Find a bounding box - Algorithm 7 7 6 6 5 5 4 4 3 3 2 2 1 1 Y [cm] 0 0 Y [cm] -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 -7 -7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 X [cm] 2. – 6. 7. |e‘2| =2.78 Tranform & rotate Rotate box centroid |e‘1|=4.57 X [cm] • Calculate eigenvalues and eigenvectors e1 and e2 of covariance matrix (PROC PRINCOMP) • Calculate centroid of the point set • Translate point set such that the centroid is moved to the origin • Transform point set into polar coordinates (with angle  and radius r) • Rotate point set by angle of the 1st eigenvector • Calculate the bounding box and the length of the eigenvectors e’1 and e’2 • Re-rotate bounding box by ange of the1st eigenvector • 7. AreaPCA=

  11. 3) Mean circle area - Algorithm 1. Transformation into polar coordinates (r,) 2. Calculate the integer (INT) of each angle for each data point 3. Calculate max radius r for each angle (360 different angles possible) 4. For each remaining data point p‘ (r, ) calculate the circle area 5. Am=·r² 1.-4. A1=·r² ... ... A14=·r² 5. Am=·r²

  12. Mean circle area – Simple example 40.8 15.7 24.5 0 41.6 Polar coordinates Circle Area

  13. Comparison of methods Convex hull PCA method Circle method ACH=26.2 [cm²] ACM=37.1 [cm²] APCA=39.9 [cm²]

  14. Comparison of methods- different distributions Convex hull PCA method Circle method Normally distributed data ACH=8.8 ACM=3.6 APCA=11.7 „Flat data“ ACH=3.1 ACM=11.1 APCA=3.7

  15. Comparison of methods – different test data

  16. Summary    - -

  17. Questions &Answers Contact details PharmaNet / i3 Thomas.Wollseifen@i3global.com Taunusstrasse 9 65183 Wiesbaden Germany

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