Determination of Body Sway Area by Fourier Analysis of its Contour

Transcription

1 PhUSE 213 Paper SP8 Deternaton of Body Sway Area by Fourer Analyss of ts Contour Abstract Thoas Wollsefen, InVentv Health Clncal, Eltvlle, Gerany Posturography s used to assess the steadness of the huan body by easurng the oveents of the centre of pressure (COP) of a standng subject on a force platfor (stabloetry). Ths paper presents a new ethod of calculatng the area of the centre of pressure (COP) trajectory (sway area). The outlne of COP area s deterned by detectng ponts that are furthest fro the centre n a gven angular nterval. A Fourer seres s ftted to ths outlne so that the ponts nsde and outsde are weghted dfferently. The SAS procedure PROC SPECTRA s used to calculate the Fourer coeffcents. Wth Fourer coeffcents a seres of sne and cosne waves that descrbe an approxaton of the body sway outlne s overlad. Fourer analyss of the sway area contour s hghly sutable for data nterpretaton. It gves not only the value of the sway area but also soe nforaton about ts shape. INTRODUCTION Posturography s a general ter that covers all technques used to quantfy postural control n uprght stance under n ether statc or dynac condtons. I wll only concentrate on statc posturography. Statc posturography s carred out by placng the patent n a standng posture on a fxed-nstruented platfor (force platfor) connected to senstve detectors (force and oveent transducers), that are able to detect tny oscllatons of the body. Posturography, also called test of balance, s a non-nvasve specalzed clncal assessent technque used to quantfy the central nervous systes (sensory/otor) nvolved n the control of posture and balance. The ablty to perfor routne actvtes of daly lfe requres stable control of posture and balance. A balance dsorder s a dsturbance that causes an ndvdual to feel unsteady, for exaple when standng or walkng. It ay be accopaned by feelngs of dzzness, or havng a sensaton of oveent, spnnng, or floatng. Balance s the result of several body systes workng together: the vsual syste (eyes), vestbular syste (ears) and proprocepton (the body's sense of where t s n space). Degeneraton or loss of functon n any of these systes can lead to balance defcts [1]. The assessent of posture has been studed by easurng center of pressure (COP). The COP s an ndrect easure of postural sway. The trajectory of the COP s ontored and can be dsplayed over a te nterval. The sway area that s traced by the trajectory s an effectve paraeter for easurng postural sway. In ths paper I have used an effcent ethod of coputaton of body sway contour (Fg. 1a-1c) of a gven sway path. Ths ethod was already ntroduced n [2] and copared wth the PCA ethod. The sway path s reduced to an outlne whch keeps only ponts wth a axu dstance to the orgn based on a decoposton of the area nto sectors (parttonng of a full revoluton nto a chosen nuber of ntervals. The outlne (Fg. 1b) s the bass of the resultng Fourer transforaton decoposton. I wll analyse the area of the center of pressure (COP) by depctng of the area contour and then ts descrpton n ters of Fourer coeffcents. The outlne wll be descrbed by a seres of Fourer coeffcents (Fg. 2). 1

2 PhUSE 213 PCA ellpse Subject=23 Subject=23 Subject= Fg. 1a: Posturography, sway path wth sulaton data - Fg. 1b: Contour reduced to concave polygon - Fg. 1c: Resultng contour calculated by FT At the end of the paper I wll copare the Fourer ethod wth two other effectve ethods - the Convex hull and PCA (prncpal coponent analyss) to calculate the area and contour of a gven set of ponts. The followng equaton descrbes the sway area outlne expressed n polar coordnates R ( ), where R s the radus t s the dstance fro the orgn of the coordnate syste to the outlne pont gven a polar angle. The equaton of the Fourer transfor (FT) decoposton of the seres gven an angle s. ax R R( ) A cos( ) B sn( ) 2 2 Where s the nuber of frequences n the Fourer decoposton. A and R( ) descrbes the functon of the radus R for a gven angle. coeffcents whch buld the weghts of the decoposton nto sne and cosne waves. (1) B are the Fourer SAS PROCEDURE PROC SPECTRA Proc SPECTRA perfors spectral and cross-spectral analyss of te seres. One can use the procedure to analyse the data for perodc or cyclc patterns Proc SPECTRA produces estates of the spectral and cross-spectral denstes of a ultvarate te seres. Estates of the spectral denstes are calculated usng a Fourer transforaton. It returns perodgras and cross-perodgras. I used SAS Proc SPECTRA to calculate the Fourer coeffcents [4]. proc spectra data=ft out=fourercoeff coef; var r ; by usubjd; run; Proc SPECTRA uses the fnte Fourer transfor to decopose data seres nto a su of sne and cosne waves of dfferent apltudes and wavelengths. In the VAR stateent one specfes the varables to be analysed. Fourer coeffcents are calculated and wrtten to the OUT=data set wth the COEF stateent. The Fourer coeffcents A and B are gven below n Fg. 2 as exaple as COS_1 and SIN_1 respectvely. 2

3 PhUSE 213 Fg. 2 Fourer coeffcents A (COS_1) and B (SIN_1) calculated wth Proc SPECTRA The area of the contour can be calculated fro the Fourer coeffcents by use of the followng equaton: A R R( ) drd 2 2 CACULATION OF THE SWAY AREA OUTLINE ax 2 2 A B Frstly we need to deterne a reduced pont set that can be used as an outlne of the sway path and whch wll be nserted n the Fourer transforaton. To calculate the sway area outlne, all data n ponts ( x, y ) are converted nto polar coordnates wth radus R and polar angle. The full revoluton (3 ) s then dvded nto chosen nuber of ntervals (3/ sectors). For good results at least 3 to ponts are usually sutable. The ore ponts used, the ore detals of the shape are shown. For each angular nterval the pont wth the largest dstance ' 2 (2) R s deeed the representatve pont on the outlne approxaton (Fg. 1b). These ponts buld up a concave polygon that wll be used as the startng pont of the Fourer transforaton. The next step s the deternaton of the sooth sway area wth Fourer transforaton. ALGORITHM FOR FOURIER TRANSFORM DECOMPOSITION I developed a acro to calculate the Fourer transfor decoposton of the contour. A Fourer transfor decoposton s calculated for the reduced pont set. Proc SPECTRA returns the (n/2)+1 (f n s even) or (n+1)/2 Fourer coeffcents f n s odd whch copose the Fourer seres. The seres s a su of sne and cosne ters. Each ter s ultpled by a specfc Fourer coeffcent. The ore ponts are ncluded, the closer the approxaton of the contour s to the prary pont set. The followng SAS code s the an part of the SAS acro %fourertransfor. It shows the calculaton of the radus R by Fourer transfor decoposton for each angle. The calculaton s nsde a loop fro to. The resultng R are suarzed for each. It also calculates the area A, of the contour gven by equaton (2): data ft; set ft; array r (&nax); array b (&nax); array a (&nax); array area (&nax); r_=; area_=; do ph=-&p to &p by &step_fourer ; r_=; area_=; do =2 to &nax by 1; r{}=a{}*cos(ph*)+b{}*sn(ph*); area{}=a{}**2 + b{}**2; area_=area_+area{}; r_=r_+r{}; 3

4 PhUSE 213 f =&nax then do; r_=r_+(a{1})/2; area_=&p*area_+&p*(a{1}/2)**2; output; end; end; end; run; The su of Fourer coeffcents s calculated by equaton (1). The acro %fourertransfor s adjustable for the nuber of angles that dvdes the full angle nto parttons and thus nto certan nuber of ponts whch consttute the frst approxaton of the outlne. Ths approxaton s then used as a startng set to calculate the Fourer transforaton and the resultng sooth sway outlne. The followng fgure (Fg. 3) shows dfferent approxatons of the contour of a gven pont set. For each Fourer transforaton a dfferent nuber of ponts was used to calculate the Fourer coeffcents. The ore ponts are used the better s the approxaton of the contour. A nu of 3 ponts are needed to calculate the Fourer coeffcents, otherwse Proc SPECTRA returns an error. Calculatng the coeffcents wth just 3 ponts generates a nearly perfect crcle. Introducng ore ponts n Proc SPECTRA results n ore Fourer coeffcents beng calculated and the resultng outlne shows ore detals of the gven sway outlne. In the exaple below I tred to nclude 9 to 9 ponts to be used to calculate the Fourer coeffcents. One can see the ore detals of the sway contour are calculated (ore ponts) the saller s the calculated area. Wth 3 ponts the area s axu d: FT wth 24 contour ponts Fg. 3) c: FT wth 18 contour ponts 3b: FT wth contour ponts - 3a: FT wth 9 contour ponts 3e: FT wth 3 contour ponts - 3f: FT wth 9 contour ponts Contour calculated wth Fourer coeffcents COMPARISON OF FT OUTLINE CALCULATION WITH CONVEX HULL AND PCA METHOD 4

5 PhUSE 213 In [3] I prevously dscussed the approxaton of the sway outlne va the convex hull and PCA ethods. Both ethods are sutable and relable to descrbe the sway outlne. They were used to calculate the area of the gven pont set and copared wth the ntroduced n ths paper. The convex hull s a fnte pont set. A pont q s vsble fro q when the connecton between p and q s copletely n the polygon (Fg. 4a-b). Fg. 4a) Pont q s not vsble fro p Fg. 4b) convex (a) and concave (b) pont set A polygon s convex f two ponts nsde the polygon also nclude the connecton between those two ponts. In Fg. 4b a convex (a) and a concave (b) polygon are shown. To calculate the convex hull I used trangulaton of the gven pont set. Proc G3GRID deternes a trangulaton of the pont set. The outer trangles descrbe the resultng convex hull (Fg. 5a). The convex hull of a pont set s always unque. In Fg. 5b we see a pcture of the pont set surrounded by a boundng box wth an ellpse nsde. Ths ellpse s calculated usng the PCA ethod. The PCA s a atheatcal procedure that uses an orthogonal transforaton to convert a set of observatons of possbly correlated varables nto a set of values of uncorrelated varables called prncpal coponents. In [3] I used Proc PRINCOMP to derve the prncpal coponents. The resultng egenvectors of the covarant atrx generate the vectors that for the ellpse wth a specfc an axs gven as the an angle of the ellpse. The PCA ethod also returns a unque ellpse. In Fg. 5a-c the results of the three ethods are shown wth the calculated sway outlne and the area that s surrounded by the polygon or ellpse. The area calculated by the depends on the nuber of ponts used for the calculaton. The ore ponts used, the saller s the area. The resultng sway outlne also shows ore detals f we use ore ponts. For the convex hull and PCA ethod the sway outlne s unque and only depends on the gven pont set. To copare the ethods I used sulaton data. Convex hull PCA ellpse Fg. 5a) Convex hull (area=21.3) Fg. 5b) PCA ethod (area=24.) Fg. 5c) FT (area=23.4) EFFECT OF OUTLIERS The followng fgures (Fg. a-c) show the nfluence of outlers on the resultng curve. The convex hull (Fg a) always encloses all data ponts. Outlers have a large nfluence on the area of the convex hull. The PCA ethod s also not nvarant aganst outlers (Fg. b) t also tres to enclose the data ponts and 5

6 PhUSE 213 the calculated ellpse becoes larger. In contrast the s ore robust aganst outlers. The nfluence of extree values on the FT area s saller than for the other two ethods. Convex hull Subject=14 PCA ellpse Subject=14 Subject= Fg. a) Convex hull (area=59.2) Fg. b) PCA ethod (area=7.5) - Fg. c) FT (area=51.8) CONCLUSION It was shown that the deternaton of the sway outlne wth Fourer analyss s sutable for data nterpretaton. The shape of the sway outlne depends on the nuber of ponts ncluded n the frst approxaton of the outlne polygon. The outlne reasonably descrbes the shape of the area. Durng coparson of the three ethods t was shown that the area calculated by Fourer transforaton s usually saller, because t descrbes ore detals wth concave ebayents, whereas the convex hull and PCA ethods always descrbe a convex course of the outlne. Convex hull always nclude all data ponts PCA and do not necessarly surround the pont set. One advantage of the PCA ethod s that t also returns an angle of the an axs of oveent. The thus deterned ellpse always les on ths an axs. The descrbes the actual outlne wth ore detals and can be adjusted based on the nuber of ponts used for calculaton. The could be refned by addng a step to nze the dstance between the calculated ponts of the contour and the experental ponts of the outlne. Ths wll result n a nzaton proble of the square su of the dfference between experental and calculated ponts. If the calculated pont s closer to the orgn than the experental pont, then the calculated pont s chosen as pont on the outlne or vce versa. All SAS progras entoned are avalable fro the author on request. REFERENCES [1] [2] Rugelj, D.; Sevsek, F.: Postural sway area of elderly subjects, WSEAS transactons on sgnal processng, 27, vol. 3, ss. 2, str [3] Wollsefen, T.: Dfferent Methods of Calculatng Body Sway Area, Pharaceutcal Prograng, vol. 5, nubers 1-2, /211, pp. 91-1(1). [4] sas.support.co, a_sect1.ht ACKNOWLEDGMENTS I would lke to thank Coln de Klerk (Prncpal Statstcal Prograer, nventv Health Clncal) for hs great support durng prograng and thoughtful revew of ths anuscrpt.

7 PhUSE 213 CONTACT INFORMATION Your coents and questons are valued and encouraged. Contact the author at: Thoas Wollsefen InVentv Health Clncal Grosse Hub 1d Eltvlle / 5344 Work Phone: +49 () Eal: thoas.wollsefen@nventvhealth.co Web: Brand and product naes are tradearks of ther respectve copanes. 7