Calculation of the statokinesigram area (SKG)

Association Française de Posturologie, Eight Lessons of Posturology

AnnexeD of the SECOND LESSON, Third Edition PARIS 1993

by Pierre-Marie Gagey

Guy Bizzo, Luc Bonnier, René Gentaz, Patrick Guillaume, Claudie Marucchi

& Philippe Villeneuve

Introduction

A statokinesigram is the layout of a line connecting the successive positions of the center of pressure during the recording, it is thus not a geometrical figure, a statokinesigram therefore does not have area with the meaning of the mathematical term. However authors, for a long time, measured by various processes the "area" of this "scrawl". As, obviously, they thus sought to quantify the dispersion of the successive positions of the center of pressure, Takagi proposed to measure this dispersion by a statistical parameter: the area of the ellipse of confidence containing 90% of the sampled positions of the center of pressure (Tagaki A., 1985). Takagi showed that the value of this new parameter was close to the value of the old "area of the SKG" when the duration of the recording exceeded 30 seconds.

The calculation of this area of the 90% confidence ellipse presented here is faster than the calculation of the old "area", and especially this parameter is more coherent with what it wishes to quantify, it avoids in particular the insoluble problem posed in the old method by the "loops": when the subject, during a short time, had deviated considerably from his mean position, the layout of the statokinesigram recorded this variation as a loop of which nobody never knew if they had or not to be entered in the measurement of "area".

The principal stages of the calculation of the area of the confidence ellipse are as follows:

a) Study of the slope of the main axis of the ellipse, knowing that its point of intersection with the small axis has as co-ordinates:

b) Determining the length of main axis (GG') and of the small axis (PP'), knowing that its with 2 degrees of freedom for a risk alpha=0,1 equals 4,6:

wehre V1 : variance of GG'

where V2 : variance of PP'

The whole of these calculations makes it possible to publish:

- the slope of the main axis,

- the length of the axes,

- the area of the ellipse, which advantageously replaces the old "area of the SKG".

Usually we will speak about "Area" to indicate the area of the 90% confidence ellipse.

Calculation of the area of the 90% confidence ellipse

In a cloud of N points the Euclidean distance from any point P(x_{i}, y_{i}) to a line Y=AX+B is:

PH=PQcos p PH=(y _{i}-y_{q})cos p |
||||||

If the line goes through the point M, mean of the set of N points | ||||||

The sum of them is :

Noticinig that the variance of the set of points in relation to x et y and their covariance are, respectively::

V = [A^{2}Vx - 2ACxy + vy] / (A^{2} +1 )

Therefore this variance V is a function of the slope of the curve:

V=f(A)

This kind of function presents a maximum and a minimum when its first derivative is null:

dV/dA = V' = [2CxyA^{2} + 2(vx - vy)A - 2Cxy] / (A^{2} + 1)^{2}

As (A^{2}+1) is never nul, V' is null if:

A^{2}+[(Vx - Vy)/Cxy]A - 1 = 0

As

[(Vx - Vy)^{2} / (Cxy)^{2}] + 4

is always positive this quadratic equation has always two roots:

The roots of this equation, A1 et A2, give the maximum and the minimum of the function: V = f(A)

- If Cxy < 0 the function increases from | to the inferior root, therefore the function V = f(A) goes through a minimum for the value of the superior root, | |||||||

- If Cxy > 0 The function decreases from | to the inferior root, therefore the function V = f(A) goes through a minimum for the value of the inferior root, | |||||||

- If Cxy = 0 the function becomes:: | ||||||||

dV/dA = 2 (Vx - Vy)A / (A^{2}+1)^{2}

As it is very unlikely that vx = vy we may admit that the only root of the equation is A=0.

Therefore it is possible to calculate directly the slope of the main axis of the ellipse.

The of the distribution of N points can be expressed::

*1) As a function of *

*2) As a function of *

The _{} with two degrees of freedom at the risk alpha = 0,1 (90% of the points inside the ellipse) equals 4,6, the length of the half-axes of the confidenceellipse is given by the formula::

And the area of the 90% confidence ellipse is :

REFERENCES

Takagi A., Fujimura E., Suehiro S. (1985) A new method of statokinesigram area measurement. Application of a statistically calculated ellipse. In Igarashi M., Black O. (Eds) Vestibular and visual control on posture and locomotor equilibrium. Karger (Bâle): 74-79.

Acknowledgments to Professor Jean-Pierre NAKACHE from the Département de Biophysique et de Biomécanique de l'Université Pierre et Marie CURIE, Paris, for his help.