Different Methods of Calculating Body Sway Area
Thomas Wollseifen
Brighton (UK), 9th-12th October 2011
1
Posturography
Postural instability
 Parkinson‘s disease
 Sleeping problems
 Elderly subjects
Centre of pressure
Subject=7
7
6
5
4
3
2
Y [cm]
1
0
-1
-2
-3
-4
-5
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Body sway
 Length of path
 Velocity
 Distribution
 Body sway area
X [cm]
Body sway path - trajectory
Copyright © 2011 i3 | CONFIDENTIAL
2
How to calculate the body sway area?
1) Area of convex hull
2) Area of ellipse –
Principal component analysis (PCA)
3) Mean of circle areas
Copyright © 2011 i3 | CONFIDENTIAL
3
1) Convex hull
Convex hull
Subject=7
Subject=7
7
7
6
6
5
5
4
4
3
3
2
Y [cm]
1
0
-1
convex
not convex
-2
-3
-3
-4
-4
-5
-5
-6
-6
Different algorithms:
-7
-7
-7
-7



Gift wrapping algorithm
Triangulation
etc.
Copyright © 2011 i3 | CONFIDENTIAL
-6
-6
-5
-5
-4
-4
-3
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
X [cm]
X [cm]
4
Convex hull - Algorithm
Convex hull
Subject=7
7
1) Triangulation of a point set (PROC G3GRID)
6
5
4
3
proc g3grid data=_ch00
outtri=_ch01
 triangulation of data set
out=_temp;
grid &y.*&x.=&z.;
run;
2
Y [cm]
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) Calculation of the determinant (of the convex hull = Area)
x y
1 1
x y
2 2
1  
1
Area

 x y  x y  x y    xn y  y x  y x  y x    yn x
1
1 2 2 3 3 4
1
Convex hull 2
2 1 2 2 3 3 4
xn yn


x y
1 1
Copyright © 2011 i3 | CONFIDENTIAL
5

Find the area of this (convex) polygon determinant
(2,5)
7
6
5
4
(-4,3)
3
2
(5,1)
Y [cm]
1
0
-1
-2
2 5
1 4 3 1
Area 
 6  (4)  25   20  15  2  15
2 5 1 2
2 5
-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;
Copyright © 2011 i3 | CONFIDENTIAL
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;
6
A PCA    | e1 |  | e2 |
7
7
6
6
5
5
4
4
3
3
2
2
Find a „better“
bounding box!
1
0
-1
-2
-3
e‘2
1
Y [cm]
Y [cm]
2) Principal component analysis (PCA)
0
e‘1
-1
-2
-3
-4
PCA
-5
-6
-4
-5
-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]
Idea:
Reduce dimensionality of a point set by identifying the most
significant directions (Principal components = Eigenvectors)
Copyright © 2011 i3 | CONFIDENTIAL
7
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
Cov(y,y)
Cov(y,x)
Cov(x,y) Cov(x,x)
Eigenvalues of the Covariance Matrix
Eigenvalue
1
7.29242620
2 Copyright ©2.42940713
2011 i3 | CONFIDENTIAL
Difference
4.86301907
Proportion Cumulative
0.7501
0.7501
0.2499
1.0000
8
From covariance matrix to Eigenvectors / Eigenvalues
 Calculate 2x2 covariances cov(X,Y) of X,Y:
1 n
cov( X , Y ) 
( xi  x )( yi  y )

n  1 i 1
 Prepare covariance matrix:
Cov-Matrix
Cov(y,y)
Y
X
Cov(y,x)
Y
4.53
-2.41
Cov(x,y) Cov(x,x)
x
-2.41
5.20
 Calculate Eigenvalues and Eigenvectors
of the covariance matrix:
4.53    2.41
det( A  E )  0   2.41 5.20    0
Eigenvalue1
Eigenvalue2
Copyright © 2011 i3 | CONFIDENTIAL
7.29
2.42
Eigenvector e1
Eigenvector e2
x
y
(0.75, -0.66)
(0.66, 0.75)
9
Find a bounding box - Algorithm
2. – 6.
Subject=7
7
7
7
6
Tranform & rotate
5
6
centroid
4
2
Y [cm]
1
0
-1
5
4
3
3
2
2
1
1
Y [cm]
3
0
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
-6
-6
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
X [cm]
|e‘1|=4.57
0
-1
-7
|e‘2| =2.78
6
Rotate box
5
4
Y [cm]
7.
-7
-7
-6
-5
-4
-3
-2
-1
0
X [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]
1. Calculate eigenvalues and eigenvectors e1 and e2 of covariance matrix
(PROC PRINCOMP)
2.
3.
4.
5.
6.
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 the 1st eigenvector
Copyright © 2011 i3 | CONFIDENTIAL
7. AreaPCA=
  | e1 |  | e2 |
10
3) Mean circle area - Algorithm
Subject=7
7
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
6
5
4
3
2
Y [cm]
1
0
-1
-2
-3
-4
-5
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
1.-4.
4
5
6
7
X [cm]
Mean circle area
Subject=7
7
A1= ·r²
...
...
A14= ·r²
6
5
4
3
2
Y [cm]
1
0
-1
-2
-3
-4
-5
area=37.10
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
5.
X [cm]
Mean circle
area
Subject=7
7
6
5
4
3
Am= ·r²
2
Y [cm]
1
0
-1
-2
-3
-4
-5
Copyright © 2011 i3 | CONFIDENTIAL
11
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
X [cm]
1
2
3
4
5
6
7
Mean circle area – Simple example
Mean circle sum
usubjid=1
Mean circle sum
usubjid=1
7
7
6
6
5
5
4
4
3
40.8
3
2
2
1
15.7
0
Y [cm]
Y [cm]
1
0
-1
0
24.5
-1
-2
-2
-3
-3
-4
-4
-5
-5
41.6
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
area=24.54
3
-6
-7
4
5
6
7
-7
-6
-5
-4
-3
-2
-1
X [cm]
Polar
coordinates
max_r
0
1
2
3
4
5
6
7
X [cm]
x
y
r
theta
Circle Area
pi
n
area_i
mean_area
0
0
0
0
0
3,141593
4
0
24,54
3,61
2
3
3,61
56
3,141593
4
40,84
24,54
2,24
-2
1
2,24
153
3,141593
4
15,71
24,54
3,64
1
-3,5
3,64
285
3,141593
4
41,63
24,54
Copyright © 2011 i3 | CONFIDENTIAL
12
Comparison of methods
Convex hull
PCA method
PCA ellipse
Subject=7
Convex hull
Subject=7
Mean circle area
Subject=7
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
Y [cm]
7
Y [cm]
Y [cm]
Circle method
0
0
-1
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
-6
-6
-6
-7
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
X [cm]
ACH=26.2 [cm²]
Copyright © 2011 i3 | CONFIDENTIAL
6
7
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
X [cm]
APCA=39.9 [cm²]
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
X [cm]
ACM=37.1 [cm²]
13
Comparison of methods - different distributions
Convex hull
PCA method
Subject=8
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
Y [cm]
7
0
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
-6
-6
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-7
-7
7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-7
ACH=8.8
5
4
4
4
3
3
3
2
2
2
1
1
1
0
Y [cm]
6
5
Y [cm]
6
5
0
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
-6
-6
-6
-7
-7
-1
0
1
2
X [cm]
ACH=3.1
Copyright © 2011 i3 | CONFIDENTIAL
3
4
5
6
7
0
1
2
3
4
5
6
7
5
6
7
0
-1
-2
-1
Subject=7
6
-3
-2
ACM=3.6
7
-4
-3
Subject=7
7
-5
-4
X [cm]
7
-6
-5
APCA=11.7
Subject=7
-7
-6
X [cm]
X [cm]
Y [cm]
0
-1
-7
„Flat data“
Subject=8
7
Y [cm]
Y [cm]
Subject=8
Normally
distributed
data
Circle method
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
X [cm]
APCA=3.7
6
7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
X [cm]
ACM=11.1
14
Comparison of methods – different test data
Test
data
Calculation
Method
N
Mean
Std
min
max
test01
1 Convex hull
10
1.78
1.22
0.1
3.3
0.1
3.9
test02
test03
test04
2 PCA ellipse
10
2.08
1.44
3 Mean circle
10
13.88
5.65
11.1
29.6
1 Convex hull
10
3.58
2.45
0.3
6.7
0.4
7.8
2 PCA ellipse
10
4.16
2.84
3 Mean circle
10
11.85
2.66
10.1
18.7
1 Convex hull
10
5.36
3.68
0.4
10.0
0.5
11.6
2 PCA ellipse
10
6.23
4.25
3 Mean circle
10
11.30
1.54
10.0
14.5
1 Convex hull
10
8.93
6.16
0.6
16.7
0.7
19.3
10.1
12.9
2 PCA ellipse
10
10.33
7.12
3 Mean circle
10
11.11
0.95
Copyright © 2011 i3 | CONFIDENTIAL
Pairwise
comparison
p-value
F-test
Normal
Distr.?
1-2: 0.9789
1-3:<0.0001
2-3:<0.0001
<0.0001
No
1-2: 0.8785
1-3:<0.0001
2-3:<0.0001
<0.0001
Yes
1-2: 0.8329
1-3: 0.0014
2-3: 0.0062
0.0010
No
1-2: 0.8367
1-3: 0.6492
2-3: 0.9444
0.6683
Yes
15
Summary
Convex hull
 Areaconvex hull < AreaPCA
Convex hull
Subject=7
7
6
5
4
3
1
Y [cm]

 Good approximation of the body sway area
 Data independent
 Encloses all data points
2
0
-1
-2
-3
-4
-5
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
1
2
3
4
5
6
7
X [cm]

 Magnitude and direction
 Data independent
 Comparable result as convex hull
 Not all data points area enclosed
6
5
4
3
2
1
Y [cm]
PCA ellipse
PCA ellipse
Subject=7
7
0
-1
-2
-3
-4
-5
-6
-7
-7
--
 Fast algorithm
 Crude approximation of the sway area
 Depends on data distribution
 Not all data points area enclosed
-5
-4
-3
-2
-1
0
4
5
6
7
X [cm]
Mean circle area
Subject=7
7
6
5
4
3
2
1
Y [cm]
Circle
method
-6
0
-1
-2
-3
-4
-5
-6
-7
-7
Copyright © 2011 i3 | CONFIDENTIAL
-6
-5
-4
-3
-2
-1
0
X [cm]
1
2
3
4
5
6
7
16
Questions &Answers
Contact details
PharmaNet / i3
Thomas.Wollseifen@i3global.com
Taunusstrasse 9
65183 Wiesbaden
Germany
Copyright © 2011 i3 | CONFIDENTIAL
17