Table 1.
Statistical analysis of elliptic drawings (1).
Figure 1.
An example of elliptic segmentation: Comparing the piecewise linear law (PLL) of log versus log
, versus the regular Power Law, where
is the velocity and
is the Euclidian curvature.
Empirical values (blue) of the pairs are compared to the piecewise regression lines of the PLL:
for
(red line), and
for
(green line), versus the regression line of the regular Power Law ( black). For all
. For this trajectory
, and
.
Table 2.
Statistical analysis of elliptic drawings (2): Log Velocity versus log Euclidian curvature, piecewise linear law (PLL) compared to the regular power law.
Figure 2.
Experimental data of elliptic drawings and regressions.
Log movement time (T) is plotted versus log perimeter (P). The regression lines between log T and Log P are shown for each subject ,
and
and each average speed condition (slow, natural and fast, red, green and black, respectively). Ellipses with different eccentricities are marked by different symbols (circle, cross and plus for narrow, medium eccentricity and circles, respectively). The regression parameters were calculated for all eccentricities together. The parameters of the regression lines are presented in Table 3.
Table 3.
Statistical analysis of elliptic drawings (3): Log time versus log perimeter.
Table 4.
Results of the tests for statistical significance of the presence of intervals.
Figure 3.
Drawing data: scores for the 4 models and for the different figural forms.
(A) Akaike's Information Criterion () scores averaged across subjects and repetitions for the 4 models for all drawn shapes. Also shown are the standard deviations (SDs)of the AIC scores. The lower the
score is, the better is the model. Red bars show the scores of the model of the combination of geometries (
); green bars, scores of the constrained minimum jerk model (
); yellow bars, the constant equi-affine speed model (
); cyan bars, scores of the constant affine velocity model (
). (B) Brown bars show the average probabilities (averaged across subjects and repetitions) that the combined model is better than the minimum-jerk model for the different shapes. Standard deviations are also shown. The probabilities were calculated according to the equation
, where
is the differences in scores between the two models. In both figure panels, the cloverleaves are marked by
in the order of ascending speed. The limaçon and the lemniscate are marked by
and by
, respectively, according to the ascending ratios of perimeters of the large to the small loops.
Figure 4.
Locomotion data: scores for the 4 models and for the different figural forms.
(A) Akaike's Information Criterion () scores, averaged across subjects and repetitions, for the 4 models for all the shapes of the locomotion data. SDs of the
scores are also shown. The figure in the upper row in panel (A) shows the
scores for the
(all data), while the figure in the lower row in panel (A) shows the
scores for the
data. The lower the
score is, the better is the model. Red bars show the scores of the combination of geometries model (
); green bars, scores of the constrained minimum-jerk model (
); yellow bars, the constant equi-affine speed model (
); cyan bars, the scores of the constant affine velocity model (
). (B) Brown bars show the average probabilities (averaged across subjects and repetitions) that the combined model is better than the minimum-jerk model for the different shapes. SDs are also shown. The probabilities were calculated according to the equation
, where
is the differences in scores between the two models. The figure on the left (panel (B)) shows the results for the
, while panel (B) on the right shows the results for the
. In both figure panels, the cloverleaves are marked by
. The limaçon and the lemniscate are marked by
and by
, respectively, according to the ascending ratios of perimeters of the large to the small loops.
Figure 5.
Examples from the drawing experiment.
Every row shows an example of the second repetition of a drawing trial. First row, drawing of a cloverleaf; second row, drawing of an oblate limaçon; third row, drawing of an asymmetric lemniscate. Panels (A), (D) and (G) show the paths drawn by the subject. The colors marked on the paths represent the Euclidian curvature. Blue segments have relatively low curvature (∼0), red segments have a higher curvature (∼0.75). Color scale is shown at the top of the panel. Panels (B), (E) and (H) show the velocity profiles of the drawing. Red, experimental velocity profile; blue, velocity profile predicted by the model of the combination of geometries. Panels (C), (F) and (I) show values of the functions. Red area, value of the
function; green area value of the
function; blue area, value of the
function. The values are aggregated one above the other such that their sum equals 1.
Figure 6.
Examples from the locomotion experiments.
Every row shows an example of the second repetition of a locomotion trial. First row, walking around a cloverleaf. Second row, walking along an oblate limaçon. Third row, walking around an asymmetric lemniscate. Panels (A), (D) and (G) show the paths drawn by the subject. The colors on the paths represent the Euclidian curvature; Blue, segments with a relatively low curvature (∼0); red, segments with a higher curvature (∼0.75). Color scale is shown in the panel. Panels (B), (E) and (H), the velocity profiles of the drawing. Red, experimental velocity profile; blue, the velocity profile predicted by the model of the combination of geometries. Panels (C), (F) and (I) show values of the functions. Red area, value of the
function; green area, value of the
function; blue area, value of the
function. The values are aggregated one above the other such that their sum equals 1.
Figure 7.
The scores of the 4 models for the different figural forms.
Summary of the scores for all 4 models for all the figural forms for both drawing and locomotion data. The bars represent mean scores ±SD averaged over all subjects and trials. Red, score obtained for the model of the combination of geometries (
); green, score of the constrained minimum-jerk model (
); yellow, score for the constant equi-affine velocity model (
); cyan, score for the constant affine velocity model (
). For the marking of the different forms see Figure 3.
Table 5.
The scores of the 4 models for the various figural forms.
Figure 8.
The mean values of the functions for the different figural forms.
The mean values of the and
functions averaged over trials and subjects, summarized over the templates of the different figural forms. In panel (A) the values of the
functions are aggregated and in panel (B) they are displayed separately with their corresponding SDs. For the marking of the different forms see Figure 3.
Table 6.
The mean values of the functions for the different figural forms.
Figure 9.
Representation of the values of the three functions during the different trials.
The distribution of the functions aggregated over all trials of the same figural form. A point within the triangle gives the values of the
,
and
functions where
. The values of
function for such a point are equal to the area delineated by the small triangle created by passing lines between this specific point and the two bottom vertices. The values of
are equal to the area delineated by the small triangle created by passing lines between this specific point and the left bottom and top vertices. The values of
function are equal to the area delineated by the small triangle created be passing lines between this point and the right bottom and top vertices. For example, a point on the triangle's edge marked by
is a point where
. For a point located at the top vertex
and
. In the center of the triangle
. The color of any point within the large triangle indicates the number of times that that specific combination of
function values was found. A white point shows a combination that did not appear in any of the trials. A dark blue point represents a combination occurring many times. Panel (A) contains all the trials of the drawing of cloverleaves. Panel (B) contains all the trials of the drawing of oblate limaçon. Panel (C) contains all the trials of the drawing of asymmetric lemniscate. Panel (D) contains all the trials of the locomotion of cloverleaves. Panel (E) contains all the trials of the locomotion of oblate limaçon. Panel (F) contains all the trials of the locomotion of asymmetric lemniscate.
Figure 10.
The score for all the coefficient combinations for equation 13.
The values of all the possible constant coefficients for the equation:
. A point within a triangle describes the values of
,
and
where
. The values of
equal the area delineated by the small triangle created by passing lines between this specific point and the two bottom vertices, where the area of the large triangle is equal to 1. The values of
are equal to the area delineated by the small triangle created by passing lines between this specific point to the left bottom and top vertices. The values of
are equal to the area delineated by the small triangle created by passing lines between the point to the right bottom and top vertices. For example, at a point on the edge of the triangle marked by
. For a point located on the top vertex
and
. In the center of the triangle
. The color of a point represents the value of the
score for the corresponding combination of the
values; the darker the color, the higher the value of the
scores. The red points are those with the highest
score. This value is given in red beside each triangle. Panel (A) contain the data of the locomotion of oblate limaçon. Panel (B) contain the data of the drawing of oblate limaçon. Panel (C) contain the data of the locomotion of asymmetric lemniscate. Panel (D) contain the data of the drawing of asymmetric lemniscate.
Figure 11.
Examples of the experimentally measured ratios of movement durations versus the experimentally measured ratios of Euclidian lengths.
The red dots represent the experimentally measured ratios of movement durations versus experimentally measured ratios of movement lengths
. The black line represents the function
for one of the set of constant B-s in the region of high
scores.