Trajectory classification through Freeman's curve encoding and entropic analysis

Abstract

The classification of trajectories in two dimensions was done through an entropic analysis of their coded representation. The steps include discretising the trajectory into an 8-symbol code using the Freeman procedure. The resulting sequence is amenable to entropic analysis. Kolmogorov-Sinai entropy, effective complexity measure and informational distance are used. Different classification schemes can be used based on the value of the entropy variables. Two examples are discussed to illustrate the approach: the Hénon-Heiles model, often used as a test bench for complexity analysis and a real experimental case of human posture analysis.

Fig 1. Trajectory encoding.

An (a) 8-character alphabet $\chi$ is defined for discrete direction, four edges, four diagonals, in the two-dimensional space. Freeman encoding starts with imposing a square grid of length l (b) over the trajectory. The intercept of the trajectory with the grid determines the closest corner of the grid (c) to be taken as reference points; (d) from the reference points, the segments are determined, and characters from the alphabet are assigned; (e) the character string of the whole trajectory follows.

Fig 2. Analysis procedure.

For a given trajectory, the Freeman coding is obtained after discretization and treated as a symbolic sequence over which the entropic analysis can be carried out.

Fig 3. Left: The energy landscape V(x,y) of the Hénon-Heiles potential.

Points (1), (2) and (3) correspond to energy isolines of e = 0.030, e = 0.111 and e = 0.167, respectively. Right: (a) the Poincaré section at the energy values given at the end of the row; (b) Two orbits, one in black and one in red, with slightly different initial points but the same energy. For increasing energy values, the orbits become more irregular; (c) The difference between the two orbits. As energy increases, the system is more sensible to the initial conditions. Notice that the spatial scales change from one row to the next.

Fig 4. Above: The mean informational distance between two trajectories with close initial conditions as a function of energy e (error bars too small to be represented).

Below: The ratio between the number of regular orbits ${\#}_r$ with respect to the total number of orbits ${\#}_T$ as a function of energy. The threshold used to distinguish between regular and chaotic trajectories was taken at 0.6, the value of $⟨d⟩$ where the slope changes. The graph shows the ability of the entropic analysis over the chain-coded trajectories to discriminate between regular and chaotic orbits.

Fig 5. Left: The effective measure complexity-entropy density map, the three colours represent different fixed energies 0.01 (blue), 0.111 (green) and 0.165 (red).

For each energy, 2100 orbits were calculated for initial random values. The spread of values resembles an arrowhead. The graph shows that a given entropy density can accommodate a range of structuring given by E. Right: The entropy density for different directions in space. The value of entropy density is proportional to the distance of the points to the centre. The angular direction in the plot is related to the direction of the perturbed initial condition from the well centre. The energy is fixed at 0.111. The maximum value of h is around 0.5. The middle graph corresponds to initial points starting at the well centre ($x=0,y=0$) but with nonzero initial momentum compatible with the energy choice and pointing in some angular direction. The right graph corresponds to initial points with zero momentum ($p_{x_0}=p_{y_0}=0$), but out of the well minimum. Again, the angular direction in the plot corresponds to the angular direction of the initial position out of the well centre. In both cases, directions of nearly zero and maximum entropy can be identified. The flower-shaped diagram has a six-fold symmetry, which is compatible with the HH-potential symmetry shown below the plots.

Fig 6. The centre of pressure is measured as a two dimensional trajectory (anterior-posterior and medial-lateral axes) while the subject is standing for 60 seconds over a measuring scale.

Four conditions were considered: with eyes closed and open, and standing over a firm surface or foam mat.

Fig 7. Pairplot visualization of entropy density h and effective measure complexity E for fallers and non-fallers under two representative conditions: above, eyes open on both types of surfaces; below, eyes closed on both types of surfaces.

Each off-diagonal subplot shows a scatter plot of one feature against the other (h vs. E), while the diagonal panels display the one-dimensional marginal distributions (histograms) of each feature. The scatter plots reveal heavy overlap between the two classes, with no clear linear separation, and the histograms confirm the similarity of the marginal distributions. These results support the hypothesis that class boundaries are non-linear, motivating the use of neural networks to capture more complex discriminative patterns.

Fig 8. The ROC-AUC curves for the two groups of subject: young and elderly.

The plots show the sensitivity (true positive rate) against the unspecificity (false positive rate) as the classification threshold changes. The area value of 0.88 for the young group shows a strong predictive capability, while the elderly group achieves a 0.80 value still indicates a good performance.