Using Relative Motion Plots to Measure Changes in Intra-Limb and Inter-Limb Coordination

This article was downloaded by: [University of Minnesota Libraries, Twin Cities] On: 02 September 2013, At: 23:03 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Motor Behavior Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/vjmb20 Using Relative Motion Plots to Measure Changes in Intra-Limb and Inter-Limb Coordination W. A. Sparrowa, E. Donovana, R. van Emmerikb & E. B. Barryc a Faculty of Special Education and Paramedical Studies Victoria College, Burwood Campus b Physical Education Department University of Illinois at Urbana-Champaien c Physical Education Department Phillip Institute of Technology, Bundoora Campus Published online: 13 Aug 2013. To cite this article: W. A. Sparrow, E. Donovan, R. van Emmerik & E. B. Barry (1987) Using Relative Motion Plots to Measure Changes in Intra-Limb and Inter-Limb Coordination, Journal of Motor Behavior, 19:1, 115-129, DOI: 10.1080/00222895.1987.10735403 To link to this article: http://dx.doi.org/10.1080/00222895.1987.10735403 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions ' •· journal of Motor Behavior 1987, Vol. 19, No. 1, 115-129 Using Relative Motion Plots to Measure Changes in Intra-Limb and I Limb Coordination W. A. Sparrow and E. Donovan R. van Emmerik Faculty of Special Education Physical Education Department and Paramedical Studies University of Illinois Victoria College, Burwood Campus at Urbana-Champaign E. B. Barry Physical Education Department Phillip Institute of Technology, Bundoora Campus ABSTRACT. Methods for determining the degree of similarity between relative motion plots are examined and computational methods outlined. Hypothetical examples are provided to simply illustrate the function of selected indices of pat- tern shape, size, and orientation. Methods of using a composite of these meas- ures to assess asymmetry, abnormality, or refinements in motor function·are dis- cussed. Statistical procedures for determining the reliability of assessments of change in relative motions are presented. A modification to Freeman's (1961) pattern-recognition method is suggested as a more parsimonious application to angle-angle data derived in human movement research. RESEARCHERS IN MANY human movement domains share a com- mon need; that is, to describe accurately changes in the movement pat- tern of one limb segment in relation to another or to compare the mo- tion of different limbs. Inter-limb and intra-limb coordination can be op- erationalized using the relative pattern of angular displacement over time of limb segments. The method of angle-angle graphical representa- tion was devised by Grieve (1968) for use as a method to analyze walk- ing patterns. Movement patterns described in this way are called rela- tive motion plots or angle-angle plots because the diagrams obtained show change in segment angles over the course of one movement cycle. Our aim is to suggest some improvements over previous attempts made to quantify the difference in coordination patterns as reflected in rela- We would like to thank two anonymous reviewers for valuable advice on improvements to the draft manuscript. Requests for reprints sho~ld ~e addressed to W. A. Sparrow, Victoria College, Burwood Campus, V1ctorra 3125, Australia. 115 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 W. A Sparrow, E. Donovan, R. van Emmerick, & E. B. Barry tive motion plots. We describe computational procedures clearly and in sufficient detail to allow a straightforward transformation to computa- tional algorithms. Illustrations of changes in limb topology will be made using simple geometric shapes. This is done to simplify description of the concepts involved and to emphasize the generality of the numerical methods. The concepts relate to ways of evaluating the difference be- tween relative motion plots, and the measurement techniques are appli- cable to any relative motion plot comparisons. Relative motion plots have been mainly used to describe the charac- teristics of both normal and pathological gait (e.g., Barry, 1982; Char- teris, 1982; Miller, 1981; Shapiro, Zernicke, Gregor, & Diestel, 1981). In gait studies, angles subtended at the hip, knee, and ankle are computed from smoothed x, y coordinate data obtained using goniometers, high speed cameras, or other motion analysis techniques. The relative mo- tion plot is therefore constructed from known coordinates representing the relative positions of the limb segments at each sampled interval of the stride cycle. Given the diagram of angle-angle relationships it is then necessary to compare the patterns to assess changes in movement pro- duction due to such factors as learning, growth, recovery from injury, or change in speed of movement execution. Major contributions to solving the problem of quantifying the degree of similarity between relative motion plots have been made by Whiting and Zernicke (1982) and Hershler and Milner (1980). Whiting and Zer- nicke (1982) illustrated the use of a chain-encoding method described by Freeman (1961 ). This procedure involves superimposing a grid on the relative-motion plot and transforming the line into digital elements approximating the change in direction of the curve (Figure 1 ). Using the set of chain elements obtained, Whiting and Zernicke (1982) followed Freeman's (1961) procedure and cross-correlated pairs of integer chains ... to obtain a recognition coefficient (R), which is the peak value of the cross-correlation function (Rxy). The value of R gives one measure of the degree to which the shape of one relative motion plot differs from another. If the integer chains repre- senting the angle between adjacent data points for any two relative mo- tion plots are identical then R = 1.0, indicating perfect similarity be- tween the two diagrams. Conversely, as the angle-angle patterns of the limb segments become more dissimilar R approaches -1.0, showing that the angles between adjacent data points are increasingly out of phase. Freeman's (1961) method of calculating R has limitations with re- spect to its suitability for data obtained in human movement research. Before describing another way of calculating R, we will discuss its limita- tions in relation to additional measures of the similarity between relative motion plots. Although Whiting and Zernicke (1982) refer to the possible use of pat- tern centroids, areas, heights, widths, and lengths, as measures of simi- larity, they were not investigated. Hershler and Milner (1980) do give a detailed account of the use of some of these variables for interpreting the degree of similarity between angle-angle diagrams. One omission 116 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 5 • • 6 • • • • • • • • • 6 • • 7 • • • • • • • • • •• • ••• 4 6 • • • • • • • • • • • 7 • • • • o I • 0 o o 4 2 7 6 • 0 ••••••• Relative Motion Plotting Techniques A • • • • : 6 .• · 5 •• 0 2 2 1 • • 2 2 : 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • :a • • • • Figure 1. Example of a chain encoded curve. The actual curve (broken line) is repre- sented by a 17-element chain beginning at the initium (point A) and ending at the termi- nus (point 8). The encoded curve, expressed in terms of digits (0-7), is C(AB) = 654356- 67670122122. Thus, a unique set of 17 chain elements approximates the shape of the original analog curve. (From "Correlation of Movement Patterns via Pattern Recogni- tion" by W. C. Whiting and R. F. Zernicke, Journal of Motor Behavior, 1982, 14, (2), 135-142. Reprinted with permission of the Helen Dwight Reid Educational Foundation. Published by Heldref Publications, 4000 Albemarle St., N.W., Washington, D.C. 20016. Copyright ©1987.) from their paper, however, was a detailed description of the use of pat- tern centroid. Barry (1982) discusses pattern centroid, gives a method for its calculation, and provides normative data for its location in normal walking. We have chosen to describe the more complex of these meas- ures, specifically the pattern centroid (C), area (A), perimeter (P), and the dimensionless ratio PA = PI-/A. The latter, Hershler and Milner (1980) suggest, is a useful quantifier of the shape of the angle-angle dia- gram. The other measures of pattern congruence have properties that are illustrated ahd discussed later. Pattern orientation is also introduced as an additional measure of angle-angle diagram similarity. Given that the coordinates defining the angle-angle diagram are known, it is unnecessary to use the chain encoding procedure described by Freeman (1961) (Figure 1). Figure 2 (a) shows a section of line draw- ing encoded using Freeman's (1961) grid-intersect quantization tech- nique. In Figure 2 (b), a representation of the type of data constituting an angle-angle diagram is shown. The data points in Figure 2 (b) are une- qually spaced, making it extremely difficult to computationally superim- pose a grid on the data set of coordinate pairs. In addition, Freeman's (1961) technique requires that the figures to be cross-correlated are in 117 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 W. A Sparrow, E. Donovan, R. van Em me rick, & E. B. Barry the form of line drawings. To do this it is necessary to produce a figure by joining up the data points from which it is to be constructed; for ex- ample, in the same way that a contour line on a map would be drawn through points of equal height. As illustrated in Figure 2 (c), this proce- dure is unnecessary because it is easy enough to calculate the angle be- tween each segment and the horizontal axis so that no continuous line need be fitted to the data. Freeman's (1961) motivation for the sug- gested encoding technique was one of computing efficiency. We have treated large data sets, using the method described above, on a modern mainframe computer (VAX 11/750) and performed the lengthy cross- correlation procedure, without imposing any appreciable burden on the system. Our method of using the angle between each segment and the horizontal axis in conjunction with the segment lengths also lends it- self to straightforward computation of centroid, area, perimeter, and orientation. When using Freeman's (1961) chain encoding method, the valueR gives an accurate representation of the congruency between two fig- ures. The cross-correlation function described by Freeman (1961) is • g1ven: . 1 n Rxy(J) = .E(x; • Yi+J) n •=1 (1 .0) j = 0, ± 1, ± 2, ... , ± n where the product x1•Yi+J is equal to the cosine of the angle between the ith segment of shape x and the i + jth segment of the shape y. Formula 1.0 only gives a necessary and sufficient condition for identifying identi- cal figures when the chain encoding technique has been used. This means that a maximum value of Rxy of unity would show that the two figures were of equal size, shape, and orientation. This is true using Freeman's (1961) chain encoding method because his procedure deter- mines a specific length for each chain encoded segment. The angle be- tween any pair of segments determines the length of these segments. The segments shown in Figure 2 (c) are, however, of unspecified length. This is due to the irregular spacing of the data points comprising the angle-angle diagram. This being the case, in order to provide a nec- essary and sufficient condition for specifying identically shaped figures, a modification must be made to Formula 1.0. The revised formula for the cross-correlation function is given below (Formula 2.0). Having made this revision to take into account the segment length and angle it is still the case that if the shape of the figures is identical (i.e., the angles between the segments; the segment lengths and the orientations are equal) then R = 1.0. When R = -1.0 the patterns would be 180° out of phase. This would have the effect of rotating a pattern through 180° about the initium. R would therefore change systematically according to the degree of rotation about the initium. It is true that the maximum value of Rxy can be unity only when the orientation, length of sides, and angle between segments are equal. It is 118 • .. ' ' ' D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 • • • • • • • • • 0 • • y • • • y • • • • • • • t • • • • • • • • • • • • • {.x" 91 ' ' lt/{92 I I I I I~ I I I I I 1~94 k, ' '',,f'95 Relative Motion Plotting Techniques • • 0 • • • • • • • • X X Figure 2. (a) Chain encoded of line drawing as in figure 11 specifications are un- known. Dot matrix represents the superimposed grid. (b) Section of line drawing similar to ansiNnsle diasram. Constructed from known x1 y coordinates representing a fixed proportion of cycle time. (c) Section of line drawing encoded by calculating the absolute value of the angle subtended by each segment. 119 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 W. A Sparrow, E. Donovan, R. van Emmerick, & E. B. Barry interesting to consider, however, the effect of manipulating these pa- rameters independently. Figure 3 illustrates various hypothetical combi- nations of pattern similarity where some parameters are manipulated with the rest held constant. Figure 3 (a) shows that two identically shaped angle-angle diagrams (i.e., R and PA are equal) have the same coordination function even when the centroids are not coincident. Similarly, the range of limb segment excursion represented by changes in angles X and Y would be equal in both cases, as would the relative position of the limb segments at any point in the movement cycle. It is clear, however, that the initial and final limb positions in diagrams a and b reflect a different kind of movement. In Figure 3 (b), the diagrams would be described as having the same "shape" but the value of R would not equal unity because of the differ- ence in side length. A re-scaling of the relationship between X andY has taken place. A change in movement pattern analogous to physical growth or an increase in flexibility has occurred, such that the movement pat- tern is qualitatively the same in (a) and (b) but the range of motion has been affected. Figure 3 (c) represents movements qualitatively identical with respect to the pattern of coordination, while varying with respect to absolute movement extent and initial and final conditions of limb position. Three different shapes having the same perimeter but varying in area are presented in Figure 3 (d). This implies a change in PA or the "fuzzi- ness" of the shape. If the shapes were cross-correlated using the for- mula below, R would be less than unity. Interpreting the change in PA and R is more problematic than in the case of the other variables. The question is whether the change in "shape" represented by Rand PA, is sufficient to indicate a fundamental change in the coordinate function relating the relative motion of the limb segments. Alternatively, varia- tion in Rand PA may suggest a less well-controlled relationship between the limb segments, with the basic coordination function unchanged. Finally, Figure 3 (e) shows a change in the orientation of the angle- angle diagram, as represented by the angle subtended by the least squares regression line. In this example, area, perimeter, PA, and the centroid are identical for the two diagrams. The limb segments repre- sented here would be moving out of phase, describing a qualitatively different coordination pattern undetected by most of the other measures of pattern similarity. The R value would also indicate a change in orienta- tion of an identically shaped plot but the regression line gives a more con- venient and easily interpretable measure of change in orientation. Computational Methods Cross-correlation. Table 1 shows hypothetical data for the angle between each segment and the horizontal (8x,i and 8y) for the two oc- tagons shown in Figure 4. The octagons X andY have segments that sub- tend the same angle with the horizontal (8x, and 8y) but have sides of 120 j ·- •• ,. D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 10 20 30 40 50 60 60 10 20 y30 40 50 60 X 10 20 30 40 50 60 I I ~-I I I I I I I ---- ____ j C I ------- 10 20 I I I I r-_1_, I I _____ IC X 30 40 50 60 I I I I I I I - ____ JC Ia I I I I I I I ---------- __ J c x100 90 60 a b 70 b --· --· ~a 70 60 50 c 40 30 20 10 60 - 10 20 30 40 50 Relative Motion Plotting Techniques X w 20 30 40 50 60 70 60 60 ~ I I I I I I I I I I I I 60 ------- ___ .JCa/Cb V70 60 90 100 X 10 20 30 40 50 60 70 80 90 10 20 y 30 40 • 50 60 70 X - - c - - W2030~5060708090m W2030405060708060100 y y figure 3. Hypothetical angle-angle diagrams showing the relationship between various pattern parameters. (a) Centroids changed. (b) Perimeter and area changed. (c) Area, perimeter and centroid changed. (d) Area, perimeter/.tarea, and R changed. (e) Orien- tation changed. 121 • D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 W. A Sparrow, E. Donovan, R. van Emmerick, & E. B. Barry unequal length. Applying the revised version of Freeman's (1961) cross- correlation formula the cross-correlation function can be written: . _ 1 " _ • I xi I (2.0) where I xi I s 1 Yi+j I Table 1 Hypothetical Data For Two Identical Relative Motion Plots Showing the Relationship Between Rxy and Angle Between Adjacent Data Points (9x . and 9y ) and the Segmentlengths (I id and I y,.1 I ) " ',.1 Plot (x) Plot (y) Y.·I+J 45 45 0 0 315 315 270 270 225 225 180 180 135 135 90 90 Rxy j = 1 1/.J'}. 45 1/.J'}. -315 1/.J'}. 45 1/.J'}. 45 1/.J'}. 45 1/.J/. 45 1/.J'}. 45 1/.J'}. 45 Rxy j = 3 lx, I y,.j I .::l9 1/2 -225 1/2 -225 1/2 135 1/2 135 1/2 135 1/2 135 1/2 135 1/2 -225 Rxy 122 xd .j8 ~ 3 2 .j8 ~ 3 2 .j8 ~ 3 2 .j8 ~ 3 2 -x, Cos.::le • --,':_:-'-:-- y,.j 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 .so - Cos.::le • x, - Y1•1 -1/2 -1/2 -1/2 -1/2 -1/2 -1/2 -1/2 -1/2 -.50 j = 0 x,l - .::l9 Cos.::le • x, - • j = 5 1 I ,fl. -135 1 /,fl. -135 1 I ,fl. 225 1 I ,fl. 225 1 I ,fl. 225 1 I ,fl. -135 1 I ,fl. -135 1 I ,fl. -135 Rxy Relative Motion Plotting Techniques Table 1 (continued) lxd Cos.:le • 1 _ I Yt+j -1/2 -1/2 -1/2 1 I ,fl. 1 /,fl. 1 I ,fl. 1 /,fl. 1 I ,fl. 1 /,fl. 1 I ,fl. 1 /,fl. Rxy -1/2 -1/2 -1/2 -1/2 -1/2 -.50 j = 7 -45 -45 315 -45 -45 -45 -45 -45 2/3 2/3 2/3 2/3 2/3 213 2/3 2/3 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 .50 j = 6 -90 0 -90 0 270 0 270 0 -90 0 -90 0 -90 0 -90 0 0 and Rxy has n- 1 values for j = 0, 1, 2 ... n -1. 8,,; and eY.i+j are the an- gles from the horizontal axis of the ith and i + jth segments from the X and Y figures respectively. Further, I X; I and I Yi+j I are the lengths of the ith and the i + jth segments from the figures X and Y being compared. Note that I X;! must be less than or equal to I Yi+j I for all segments if a recognition coefficient of unity is to be a sufficient and necessary condi- tion for identical figures. It is necessary for the computer algorithm to test for the condition that the numerator of the quotient of the segment lengths be less than or equal to the denominator. It makes no difference to the interpretation of Rxy(j) whether I X; I is the numerator or is the de- nominator of the quotient, provided that the greater length is always the denominator. Table 1 shows that even though the angles between each segment and the horizontal are equal, by virtue of the discrepancy in segment lengths the maximum value of Rxy is less than unity. In this case, the maximum value of Rxy is .67, that is, R = .67. Note that if two figures had the same angle between segments, equal segment lengths but different orientations, the cross-correlation coefficient would not give a value of Rxy equal to unity. Centroid. The rectangular coordinates for the pattern centroid (C) are obtained according to the formula: 123 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 ~ N ~ 82(0) -- 03 (315) 82 83 3 \.'\.. 2 -.I8"'- - / "'- 91 (45) ~_:).- ('.,._ ~- 94(270) /-JUi -J18 91 ...:.L _L 94 3 31 2 21 98 (90) ~--\_ {)._~ 95(225) 981;-l {J_~_ 85 -JB/ "'--J1s -J18/ 3 fA 96(180) ""' / 2 -96 97 -- Figure 4. Two octagons with sides of unequal length, showing hypothetical data for the application of the cross-correlation function (Formula 2.0 and Table 1) to determine figure similarity from the recognition coefficient R (the peak value of Rxy). The angles, 8 1 to 8 8 , are the same for both figures whereas the length of sides, shown inside the figure, are different. • ' l. ' . :::: • )> If) " "' ~ 0 ~ - m • 0 0 :::> 0 < "' :::> - Al • < "' :::> m 3 3 ttl ~ -· l"'l ,.. - "' m • C:l • C:l "' ~ -< D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 ' Relative Motion Plotting Techniques C(x,y) = n n . (3.0) • Area and perimeter. Using a linear approximation we can use adja- cent data points to calculate areas and lengths (Figure 5). Using the co- sine formula: b2 2 2 Case, = ' + c, - at 2b,c, Where: a, = d[(x,.t, y,.t), (x,, y,)] b, = d[(x,, y,), (x, y)] (4.0) (4.1) (4.2) c, = d[(xi+l' y,.l) (x, y)] (4.3) and d(w,z) denotes the Euclidian distance on the plane between the pa- renthesized coordinates, the angles subtended from the centroid by ad- jacent data points can be calculated from Formula 4.0. The area approx- imation of the sector is: A . = 1 b.c.Sine. I I I I 0 ' (5.0) 2 Therefore the total area of the figure for n data points is: n Area = EA... (5.1) i=l ' Formula 5.1 holds only if the centroid lies within the figure, and with highly irregular angle-angle patterns it may be necessary to test whether this is the case. It is also likely that the motion plots will not be com- pletely closed, that is, the first data point and the last data point will not be coincident. We suggest simply joining the first and last data points with a straight line, accepting the fact that a small error (not calculated) will result in the calculation of area and perimeter. The perimeter of the figure is: n Perimeter = Ea,. (6.0) i=l Orientation. The orientation of the angle-angle diagram can be ob- tained using a linear regression of y on x. The angle of orientation in re- lation to the horizontal through the pattern centroid is therefore the arc- tangent of the slope of the regression line. Figure 3 (e) illustrates a change in orientation of an angle-angle pattern in which pattern a is turned 150° clockwise. Statistical Procedures We have focused on various measures designed to assess similarity in pattern shape (R and PA); two parameters that indicate quantitative 125 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 W. A Sparrow, E. Donovan, R. van Emmerick, & E. B. Barry changes in pattern dimensions (A and P) have been considered and ori- entation and centroid have been used to describe how change in pat- tern "location" implies a change in the pattern of coordination. Once these indices have been calculated, there remains the problem of com- paring them to assess the degree of similarity between movement pat- terns. The value of R, for example, provides a precise measure of simi- larity directly. It gives a real number between -1.0 and + 1.0 represent- ing the divergence in shape of the compared diagrams. Measures of area, perimeter, centroid, perimeter/..jarea, and orientation, do not pro- vide such a measure, therefore some way of comparing these indices from two or more relative motion plots is needed. The purpose of this section is to describe methods for obtaining and evaluating statistically a composite measure of these latter pattern indices. Additionally, we ex- tend our earlier discussion of correlation measures to include autocorre- lation, and consider some statistical issues associated with application of correlation measures to movement data. Asymmetry and normality indices. A method of comparing these pat- tern recognition indices has been devised by Barry (1982). His interest was in assessing abnormality and asymmetry in human gait. For the pres- ent discussion, the nature of the application is secondary to the princi- ple that Barry's (1982) "functional asymmetry index" (f.a.i.) and "func- tional normality index" (f.n.i.) provide a single quantitative measure of pattern similarity. Barry's (1982) functional indices derive from the arithmetic mean of four of the measures of pattern similarity. They are the angle-angle plot perimeter (P), area (A), centroid radius (Cr), and centroid angle (C9). Rather than using the mean of the raw scores, the ratio of the individual's score to that of a mean derived from normative data was used. Similarly, asymmetry was denoted by the ratio of values for the left and right limb X 0 10 20 y 30 (x,y} 60 - 70 figure 5. Schematic representation of the procedure for calculating area and perimeter of the angle-angle diagram. 126 0 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 Relative Motion Plotting Techniques Table 2 Normality (Subject to Normal Values Ratio) • ' Perimeter ratio Area ratio Centroid radius ratio and Asymmetry and Left Limb Ratio) For Relative Motion Plot Comparisons p NRp = p A NRA = A -r NR- = r -r - Normality measures norm norm norm • ARp = ARA = AR, = Asymmetry measures PR PL AR AL -fR -rL - Centroid angle ratio e ARe= eR NR 8 = -e norm eL Note. NR = Normality ratio, AR = Asymmetry ratio. segment plots (Table 2). The normality and asymmetry measures shown in Table 2 were then combined to form the f.a.i. and f.n.i., where: and f.n.i. = -1 (NRP + NRA + NR, + NR£) 4 f.a.i. = 1 (ARp + ARA + AR-, + AR£). · 4 The maximum value of the indices is 1 or 100% and the minimum value is 0 or 0%. Accordingly individuals were assessed based on their devia- tion from either the contralateral limb (asymmetry) or their deviation from the normative data (normality). This procedure provides a conven- ient method for angle-angle pattern comparisons, but there are two ma- jor limitations to this approach. The probability of misclassification is un- known and it is unclear whether the four measures used are equally in- fluential in differentiating between the angle-angle patterns. Discriminant analysis. To overcome the limitations described above, discriminant analysis can be used to derive an index that distinguishes to the greatest possible extent the difference between the two populations of angle-angle diagrams. This technique also provides a means of deter- mining the probability with which the difference between the relative notions is statistically reliable. If, for example, we wish to classify an indi- vidual as having normal or abnormal gait or, alternatively, monitor changes in angle-angle relationships as a function of practice, it is possible to construct a single index to distinguish between the two populations. 127 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 W. A Sparrow, E. Donovan, R. van Emmerick, & E. B. Barry The linear discriminant function comprises a linear combination of several measures (such as the measures of pattern similarity), providing an index representing the difference between the populations under consideration. Estimates of misclassification probabilities can be deter- mined given the statistical characteristics of the samples. For a detailed treatment of discriminant analysis the reader should consult Morrison (1976) or Tatsuoka (1971). Correlational measures. The section on computational methods above described the use of cross-correlation to determine the degree of corre- spondence between angle-angle diagrams. In this case, the angles be- tween successive data points on two angle-angle diagrams were cross- correlated. It is also possible to describe the relationship between suc- cessive terms in a single time-varying series. The procedure to do this is called "auto-correlation." With motor-skills researchers increasingly fo- cusing attention on comparison of movement patterns, it is timely to pass comment on the use of auto-correlation and cross-correlation as methods of evaluating time-varying data other than limb segment angles. In this section, we consider the general problem of obtaining reliable measures of the degree of similarity between sets of time-varying data. Correlational measures have been used to describe the relationship between successive terms in a time-varying series (autocorrelation) and for distinguishing between two time-varying series of kinematic varia- bles (cross-correlation). Den Brinker and Van Hekken (1982), for exam- ple, describe modifications to slalom-ski type movements as a function of practice, using both these measures to evaluate changes in velocity and displacement data. Equation 1.0 above gives the cross-correlation function for the angle-angle plot data. Similarly, it is possible to derive the autocorrelation function according to the formula: n i=t '' '''1 X i+j (7.0) where I X; I s I X;+j I and j = 0, 1, 2 ... n - 1. A question remaining, however, is whether the correlation measures are significantly different from zero. This would give a statistically relia- ble test of the change in the function over time or evaluate the differ- ence between two different samples of angle-angle data. To obtain a sta- tistical test Den Brinker and Van Hekker (1982) used Fisher's Z-transfor- mation of the sample cross-correlation coefficients. This procedure, however, violates the assumption appropriate to the use of Fisher's Z-transformation. The sample correlation coefficient (r), which is trans- formed using Fisher's procedure, is calculated on the assumption that the sample data are independent and are drawn from populations of normally distributed variables (Glass & Stanley, 1970). Data sampled from a time-dependent series are dependent unless the series is white noise or a similarly random variable, and it is not safe to assume that they are normally distributed. The problem remains, therefore, in devis- 128 r D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 Relative Motion Plotting Techniques ing valid sample statistics for testing the statistical significance of the auto-correlation and cross-correlation functions. Discussion The relative motion plot technique is a valuable tool in motor perform- ance research. Whiting and Zernicke (1982) brought this to the atten- tion of motor-skills researchers but there is little evidence that relative motion plots have been fully exploited in the study of motor skill learn- ing and development. The contemporary process orientation to motor skill performance demands powerful quantitative measures of change in coordination and control variables as a function of practice, physical growth, or change in task conditions. The present paper provides a methodological overview for applying relative motion plots to a variety of problems in human movement research, and highlights several asso- ciated problems in deriving statistically reliable measures of change in movement kinematics. REFERENCES Barry, E. B. (1982). Characterisations of gait. Unpublished master's thesis, University of New South Wales. Beauchamp, K., & Yuen, C. (1979). Digital methods for signal analysis. London: George Allen & Unwin. Charteris, j. (1982). Human gait cyclograms: Conventions, speed relationships and clinical applications. journal of Rehabilitation Research, 5, 507-518. Den Brinker, B. P. L. M., & Van Hekken, M. F. (1982). The analysis of slalom-ski type move- ments using a ski-simulator apparatus. Human Movement Science, 1, 91-108. freeman, H. (1961 ). A technique for the classification and recognition of geometric patterns. Proceedings of the 3rd International Congress on Cybernetics. Namur, Belgium. Glass, G. V., & Stanley, j. C. (1970). Statistical methods in education and psychology {pp. I 55- 176). New jersey, Englewood Cliffs: Prentice Hall. Grieve, D. W. (1968). Gait patterns and the speed of walking. Biomedical Engineering, 3, 119-122. Hershler, C., & Milner, M. (1980). Angle-angle diagrams in the assessment of locomotion. American journal of Physical Medicine, 59, 109-125. Miller, D. I. (1981 ). Biomechanical considerations in lower extremity amputee running and sports performance. Australian journal of Sports Medicine, 13, 55-67. Morrison, D. F. (1976). Multivariate statistical methods (2nd Edn). Sydney: McGraw-Hill. Shapiro, D. C., Zernicke, R. F., Gregor, R. j., & Diestel, j. D. (1981). Evidence for general- ized motor programs using gait pattern analysis. journal of Motor Behavior, 13, 33-47. Tatsuoka, M. M. (1971). Multivariate analysis; Techniques for educational and psychologi- cal research. New York: john Wiley & Sons. Whiting, W. C., & Zernicke, R. F. (1982). Correlation of movement patterns via pattern rec- ognition. journal of Motor Behavior, 14, 135-142. Submitted September 12, 1985 Revision submitted August 7, 1986 129 D ow nl oa de d by [U niv ers ity of M inn es ota L ibr ari es , T wi n C iti es ] a t 2 3:0 3 0 2 S ep tem be r 2 01 3 00000115 00000116 00000117 00000118 00000119 00000120 00000121 00000122 00000123 00000124 00000125 00000126 00000127 00000128 00000129