Trajectory analysis through entropy characterization over coded representation

TL;DR

This work tackles entropic characterization of continuous trajectories by discretizing them with Freeman coding to obtain a finite-symbol representation. It defines KS-entropy density $h$, effective measure complexity $E$, and fractal dimension $D$ as descriptors computed from the coded strings via Lempel-Ziv factorization, enabling a unified, information-theoretic view of trajectory structure. The authors validate the approach across three domains—Parkinsonian gait, fall risk from posture data, and regular/irregular motion in the Hénon-Heiles model—demonstrating robust discrimination and insight into pattern formation. With 2D/3D applications using alphabets of size 8 and 26, the approach provides a compact set of interpretable features that can support classification and analysis across domains.

Abstract

Any continuous curve in a higher dimensional space can be considered a trajectory that can be parameterized by a single variable, usually taken as time. It is well known that a continuous curve can have a fractional dimensionality, which can be estimated using already standard algorithms. However, characterizing a trajectory from an entropic perspective is far less developed. The search for such characterization leads us to use chain coding to discretize the description of a curve. Calculating the entropy density and entropy-related magnitudes from the resulting finite alphabet code becomes straightforward. In such a way, the entropy of a trajectory can be defined and used as an effective tool to assert creativity and pattern formation from a Shannon perspective. Applying the procedure to actual experimental physiological data and modelled trajectories of astronomical dynamics proved the robustness of the entropic characterization in a wealth of trajectories of different origins and the insight that can be gained from its use.

Figures (5)

• Figure 1: Trajectory encoding. A trajectory can be encoded as a string using a finite alphabet $\chi$ such as the (a) Freeman alphabet in two dimensions. Freeman encoding (b) starts with imposing a square grid of length $l$ (1) over the trajectory. The intercept of the trajectory with the grid (2) determines the closest corner of the grid (3) to be taken as reference points; (4) from the reference points, the segments are determined, and characters from the alphabet area are assigned (See Supporting Information for code). The character string (5) follows. The length of the coded string for different grid sizes can be used to determine the Hasudoff-Bersicovitch (c) fractal dimension $D$. Following equation (\ref{['eq:fractal']}), the fractal dimension can be estimated from the slope of the log-log plot of the code length $N(l)$ vs $1/l$. The procedure was applied to a quadratic Koch curve where the fractal dimension is $\log 5/\log 3$. The estimated fractal dimension has an error of $0.8\%$ with respect to the true value. The fractal dimension of other fractal curves was also estimated with similar results.
• Figure 2: Classification scheme. The trajectory data is discretized using Freeman coding. From the character stream, the entropy density ($h$), the effective measure complexity ($E$), and the fractal dimension ($D$) are estimated. (a) $(h, E, D)$ for each trajectory is fed into a machine learning (ML) procedure and the classification assignment in the training stage. (b) The trained ML is used for classifying other trajectories. Again, the trajectory is coded, and from the character stream, $(h,E,D)$ is estimated and fed into the classifier, which assigns a label to the data.
• Figure 3: Parkinson's disease analysis through gait measurements. Pressure in both feet was recorded as a function of time and taken as a trajectory. (a) Shows the plot of two trajectories, one for a control healthy subject (CO) and one for a subject with Parkinson's disease (PD). $RF$ and $LF$ refers to right and left foot, respectively. Subjects are walking at a normal pace for two minutes. Subject trajectories were discretized using Freeman coding and entropic analysis, and estimating the entropy density ($h$), effective measure complexity ($E$), and fractal dimension ($D$) was performed. From the entropic data, the classification of subjects into young and old was attempted (b). In order to perform the classification, the entropic magnitude was used on a neural network. The subjects were divided into two subsets: one for training, with the same number of young subjects and old patients, and a second subset for the classification trial. This was repeated $100$ times, choosing the training and trial subset randomly. The training set was never larger than $1/3$ of the total number of subjects. Four experiments were carried out with the neural network. In three experiments, two of the three magnitudes were used for training and classification; in a fourth experiment, all three magnitudes were used for training. For each trained neural network, the trial subjects' entropic measures were fed into them for classification. The mean (upper) and best-case (lower) results are presented. The same procedure was followed for classifying the set in CO and PD subjects (c). For the training, the subset was chosen with the same number of healthy and Parkinson's subjects. Subjects classified as PD by two or more trained neural networks were labelled as such. The mean (upper) and best-case (lower) results are shown. In both (b) and (c), the dashed lines are the fraction of the given subjects for each type.
• Figure 4: Fall occurrence through human posture measurements The centre of pressure of standing subjects with different health backgrounds is recorded for 60 seconds. Four different conditions are set for the experiments. The subject is with its eye open and on a firm surface or a foam soft surface. The same is done with the eyes closed. The measurements result in trajectories, which are coded into a Freeman string. From the character string, the entropy density ($h$), effective measure complexity ($E$) and fractal dimension ($D$) are estimated. The record of falls in the last 12 months is known for each subject. The ability to assert if a subject has fallen from the $(h, E, D)$ measurement is tested. We proceed as in the Parkinson's case. The subject set is split into two non-overlapping sets, one for training and one for trial. As described before, the training set is used to train a neural network. The trial set is then used to test the classification ability. The middle bar plot is the mean success rate for each condition over the trial set. The dashed line is the fraction of subjects with at least one falling episode. The success rate is measured for the combined fall and non-fall subjects, for the correct determination of the subjects with at least one falling event, and for subjects with no falling event. In all conditions, the success rate is far above the dashed line. The same result is shown for the best case in the classification procedure in the lower bar plot.
• Figure 5: Regular and irregular orbits in the Hénon-Heiles potential.Left: Effective measure complexity analysis of orbits in the Hénon-Heiles potential. $\langle E \rangle$ is the mean value of the effective measure complexity. At low energies, regular orbits are almost the total of all orbits; after a transient, at roughly $0.030$ the effective measure complexity settles in a plateau in the interval $[0.030, 0.111]$ at a mean value of $0.726\pm 0.002$ where almost all orbits are still regular. For energy values larger than $0.111$, the mean entropy density increases monotonically until an energy of $1/6$, which corresponds to the escape energy. This the region where irregular orbits starts to dominate. Right: In the inner energy well, below $1/6$ there is a wealth of different orbits depending on the initial conditions. Taking initial starting from the rest condition and different starting position, the entropy density was calculated from the chain coded trajectories and shown as a color label at each point in the well. The inner contour (in white) corresponds to an energy value of $0.01$. For energies below this value, all trajectories show trivial behavior from the predictability point of view. In the intermediate region between both contour lines, corresponding to the plateau, unpredictability increase but still with values of $h$ below $0.35$, the trajectories are more creative while still being regular. For larger energy values, irregular orbits start to dominate shown as the red regions in the plot. In certain symmetric directions, orbits are periodic and the corresponding entropy density is near zero almost in the whole energy range, this can be seen as straight black lines.