Entropy and Fluctuation Analysis of Horse Riders During Extended and Collected Gait Motions

Abstract

This study investigates the relationship between phase characteristics of horse-riders and subjective harmony scores evaluated by dressage judges. We aim to enhance understanding of how motion patterns reflect harmony perception, hypothesizing that adaptive anti-persistent motion improves subjective evaluations. Nine elite dressage riders and their horses participated. Accelerometer data from the rider’s centre of gravity were collected during passage and extended trot. Signals were processed using detrended fluctuation analysis (H) and Shannon entropy (Ɛ). Subjective harmony scores were correlated with H and Ɛ using Spearman’s rank correlation. Significance was set at p < .05. In passage, H positively correlated with harmony scores (r = 0.77, p < .01, H = 1.07 ± 0.06), suggesting riders exhibiting greater persistence were evaluated as more harmonious. Conversely, in extended trot, H negatively correlated with scores (r = −0.71, p < .02, H = 1.04 ± 0.06), indicating anti-persistent motion correlated with higher ratings. Shannon entropy correlated negatively with scores in passage (r = −0.62, p < .03, Ɛ = 9.28 ± 0.25) and positively in extended trot (r = 0.61, p < .05, Ɛ = 9.46 ± 0.12). Harmony perception in dressage varies by gait. Collected movements benefit from stable and persistent motion, while extended gaits demand adaptive, anti-persistent patterns. Training should focus on stability for precision-oriented gaits and flexibility for dynamic tasks. Objective metrics like detrended fluctuation analysis and entropy may enhance harmony evaluation, though interpretation requires further refinement to address subjectivity in scoring practices.

Introduction

Equitation is a sport in which performance depends on the interaction between two complex systems (i.e. horse and rider). Their movements act as a feedback loop, each exerting an influence on the other. This is a so-called complex coupled system (Peham et al., 2001). It is a common experience among trainers and riders to find that a horse’s performance when tested at liberty often does not transfer to its performance under saddle. Sometimes experienced riders paired with capable horses may achieve inferior results than less proficient riders coupled with horses of more limited physical capabilities. This phenomenon, colloquially referred to as finding a ‘good/bad match’ can be explained by a fundamental characteristic of complex systems: emergence. Briefly, emergence is a property that implies new collective behaviours due to the interaction and self-organization among elements in the system, which cannot be produced by a single unit (Zheng, 2021). The uniqueness of equitation compared to other sports that are composed of two systems (i.e. cycling, skiing, and tennis) is that both parts of the coupled system are adaptive, decision-making systems. Consequently, the overall proficiency level of the dyad is not equal to the sum of the skill levels of its individual components and emergence (Haken, 1987; Kelso & Schöner, 1988) is more likely to interact with end-point performance, behaving in a non-linear manner.

Such non-linear behaviour is what explains why it has been repeatedly argued that the key performance feature of a horse-rider dyad is a subjective feature: harmony (Evans & Franklin, 2010; Hobbs et al., 2023; Peham et al., 2001). This non-linearity may also explain why static and dynamic posture of riders have been analysed at various gaits without consensus on what defines good motion patterns, sometimes leading to conflicting results (Baillet et al., 2017; Eckardt & Witte, 2016, 2017; Elmeua González & Šarabon, 2021; Hobbs et al., 2014, 2023; Kang et al., 2010; Münz et al., 2014; Terada, 2000; C. Wilkins, 2021; C. A. Wilkins et al., 2020; Witte et al., 2009). Moreover, the equipment that lies between the horse and the rider (i.e. saddle, bit, and bridle) adds an extra degree of complexity to the dyad.

To date the most used definition of harmony is that given by the Fédération Equestre Internationale (FEI) as stated in (Peham et al., 2001):

‘The object of dressage is the harmonious development of the physique and the ability of the horse. As a result, it makes the horse calm, supple, loose and flexible, but also confident, attentive and keen, thus achieving perfect understanding with his rider. The rider follows the movements of the horse smoothly and freely and the horse shows the harmony in its whole behaviour in the execution of different movements by its lightness and ease’.

It’s important to note that harmony is a largely subjective concept, leading to ongoing controversy in disciplines like dressage, where harmony is scored by a panel of judges and can influence the final competition outcome. Other disciplines, such as show jumping, are also indirectly affected by this subjective interpretation: while harmony does not directly impact show jumping results – since scoring in this discipline is objective – it can still influence how trainers coach their riders and how riders perceive and refine their riding technique.

Attempts to evaluate the characteristics of proficient riding technique have been presented in this article, but have repeatedly overlooked the non-linear characteristics of complex systems, potentially explaining the divergent and inconclusive results. Statistical methods like dimensionality reduction can post a useful approach when facing the motor redundancy paradox (Scholz & Schöner, 1999) and have been applied in equitation science (Elmeua González & Šarabon, 2020; Witte et al., 2009) though they are restricted to their inherently linear nature. Moreover, these methods are dependent on a reduction of the number of functional units identified to be less than or equal to the number of original units. Recently, application of time-variability analysis methods to sports technique evaluation has produced significant insights, showing how more simplistic approaches can unveil complex behaviours that emerge from multi-component systems (Bruno et al., 2011; Montull et al., 2020; Norris et al., 2016). In addition, entropy types of analysis – such as Shannon entropy – provide insights on the complexity of motion dynamics in the spatial domain – higher entropy values correspond to higher complexity – serving as a robust complement to DFA.

The aforementioned definition of harmony as proposed by the FEI acknowledges harmony as a multi-level feature that affects emotions, physiology, and biomechanics of the dyad, further supporting the notion that harmony’s structure can be invariant across different scales of interaction (i.e. harmony exhibits fractal characteristics).

Persistent oscillations in biological signals (i.e. when time-series points follow the trend of preceding points) have previously been associated to rigidity in adaptability, showing that biomechanical features may become more rigid with fatigue and lower skill level (Falaki et al., 2017; Garcia-Retortillo et al., 2023; Mateo-March et al., 2023; Montull et al., 2020). However, these studies have investigated phase characteristics of either non-coupled complex systems or coupled complex systems that involve a non-decisional counterpart, such as an automatic or reactive system that influences dynamics without active decision-making processes. Assessing harmony in equitation presents a methodological challenge, and even when results are obtained, their interpretation remains uncertain. Therefore, further research is needed to warrant accurate interpretation of the results.

The aim of this study is to investigate the relationship between phase characteristics of horse riders and subjective score evaluations provided by a panel of dressage judges, in order to enhance the understanding of how we subjectively perceive harmony in dressage rider’s motion. Our hypothesis was that horse riders with lower long-range correlation in their motion patterns will achieve better subjective evaluation scores, showing a better ability to adapt to the horse’s motion.

Methods

Subjects

Nine highly trained dressage riders (35 ± 5 y. o.) from the classical school of riding in Lipica (>20 h of training per week, competing at S level) volunteered to take part in this study, each one participating with one of their usual training horses. Nine Lipizzaner horses (12 ± 2 y. o.) participated as dyad counterparts. Each horse was trained 6 days per week, at S level, working on average 1 hour per day. All dyads had been training together for >4 years. Participants were asked to sign an informed consent after being verbally instructed on the study’s procedure. The National Medical Ethics Committee (Ministry of Health, Ljubljana) gave full approval to the project according to the Declaration of Helsinki (approval number: 0120-346/2018/3).

Design

A single-visit study design was implemented. All measures were taken on a 40 × 60 m dressage arena, with silica/geotextile footing. After obtaining the informed consent from the participants, and instrumentation of the horse and rider, riders were asked to perform a self-guided, on-horse warm-up. Then, riders were required to perform 20 strides in passage and 20 strides in extended trot down the centre line of the arena in both directions −10 strides coming from the right lead and 10 strides coming from the left lead. After eliminating the acceleration and deceleration phases by visual inspection, at least 1024 datapoints (∼7 sec) were left for analysis, which is the minimum required for a correct DFA calculation (Ihlen, 2012). Acceleration was continuously recorded at the centre of gravity (CoG) of the horse rider. Upon finalization, the dyads were required to cool down at walk and loose reins, and leave the arena.

Subjective Evaluation

All trials were recorded in the sagittal plane with a dedicated video camera (FDR-AX43 4K, Sony, Tokyo, Japan) at a sampling rate of 60 fps. Each video was independently provided to a panel of three judges for independent evaluation of horse-rider harmony, ensuring an unbiased assessment. The final score was the resulting mean of the three judges. All three judges had previously judged national events up to the Grand Prix level. Harmony was subjectively assessed and evaluated based on the harmony definition provided by the FEI (Internationale, 2022). The scoring system followed the official FEI scoring system ranging from 0 to 10, defined as follows: 0 = not performed, 1 = very bad, 2 = bad, 3 = fairly bad, 4 = insufficient, 5 = sufficient, 6 = satisfactory, 7 = fairly good, 8 = good, 9 = very good, 10 = excellent.

Acceleration Acquisition

A sweat-resistant, self-adhesive interface was used to place a wireless inertial measuring unit (IMU) (Delsys Trigno, Natick, MA, USA) as close as possible to the spinous process of the fifth lumbar vertebrae, but far enough from the rider’s breeches to avoid rubbing between the unit and the clothing. This placement ensured noise-free signal, while still providing a solid representation of the CoG and lumbo-pelvic hip complex’s motion. Acceleration was continuously recorded at a frequency of 148.15 Hz.

Signal Processing

Acceleration data was processed ex-situ using a custom-made MATLAB script (MATLAB R2021a, Natick, Massachusetts). To isolate the frequency components relevant to gait analysis, the data was filtered with a 0.05/15 Hz, second-order, bandpass Butterworth filter. This filtering procedure was conducted separately for the X, Y, and Z components of each accelerometer signal. To avoid phase distortions, the filtering employed the forward-backward filtering technique. After filtering, the resultant acceleration was calculated as the root-sum-of-squares of the filtered X, Y, and Z components. The resultant signal was normalized by dividing it by its standard deviation, ensuring consistent scaling across all data segments.

Data Analysis

Detrended Fluctuation Analysis

To determine if the time-series data was stationary, we employed the Augmented Dickey–Fuller test using MATLAB’s ‘adftest’ function. This test checks for the presence of a unit root, which indicates non-stationarity. A time series is considered stationary if the null hypothesis of the test is rejected, and the results of this test do not affect DFA protocol but can indeed affect interpretation (Chen et al., 2002).

Then, the filtered and normalized resultant acceleration of each rider and gait was processed individually with a custom-made MATLAB script adapted from an algorithm available elsewhere (Ihlen, 2012). Briefly, the algorithm follows a 5-step process: (i) the signal is detrended and its cumulative sum is computed, (ii) the cumulative data is segmented into windows of varying sizes, defined by logarithmically spaced scales ranging from 16 to 1024 samples, (iii) in each segment, a linear trend is fitted and the RMS from the fit is computed, (iv) the fluctuation function (F(s)) is calculated for each scale s as the average RMS across all segments, and (v) a log-log regression of F(s) against s yields the scaling exponent (H), representing the strength of temporal correlations in the signal.

Shannon Entropy

The filtered and normalized resultant acceleration of each rider and gait was used to compute Shannon entropy, a measure of signal complexity. Probabilities were derived by normalizing the signal values to sum to one. Shannon entropy (Ɛ) was calculated as described elsewhere (Ausloos, 2023; Feutrill & Roughan, 2021). Briefly, entropy is given by

$\mathrm{Ɛ}=\sum p_i·\hspace{0.17em}{\mathit{log}}_2\left(p_i\right)$

where p_i represents the probabilities of signal intensity values. When a system is completely deterministic (i.e. no entropy), Ɛ = 0 because

$\mathrm{Ɛ}={\displaystyle \sum }_{i=1}^n{p}_i·\hspace{0.17em}{\mathit{log}}_2\left(p_i\right)$

Meaning that all 1024 samples produce the same value and no uncertainty will be found in the system:

$\mathrm{Ɛ}=-\left(1·\hspace{0.17em}{\log }_2\left(1\right)\right)=0$

On the other hand, if all 1024 samples lead to different states, Ɛ = 10, meaning that 10 corresponds to the maximum complexity in the signal, because log₂(1024) = 10.

Statistical Analysis

Spearman’s rank correlation was used to assess the strength and significance of the monotonic relationship between the DFA scaling exponent values and the corresponding subjective evaluation scores. Significance threshold was set at p < .05. Only significant correlation coefficients were retained. Results are presented as mean ± SD.

To assess inter-rater reliability of the judges, the intraclass correlation coefficient (ICC) was calculated using a two-way random-effects model for single measures, denoted as ICC(2,1), which evaluates absolute agreement among raters. This was implemented using a custom MATLAB function. The ICC was computed for each movement, along with its 95% confidence interval (CI), F-statistic, degrees of freedom, and associated p-value.

Additionally, inter-rater correlation coefficients (Pearson’s r) were calculated for each pair of judges (Judge1–Judge2, Judge1–Judge3, Judge2–Judge3), and the mean of these correlations was reported. The coefficient of variation (CV) was computed for each performance as the ratio of the standard deviation to the mean score across judges, expressed as a percentage. Mean and standard deviation of CVs were reported for each movement.

Results

Figure 1 portrays the acceleration signal registered at the CoG of the lowest scored rider and the highest scored rider, both at extended trot and passage.

Figure 1.

Representative centre of gravity acceleration signal pattern of the highest and lowest scored riders in the extended trot and passage

Evaluation Scores

For the extended trot, the mean scores were 7.06 ± 1.30 for Judge 1, 7.06 ± 1.30 for Judge 2, and 6.89 ± 1.28 for Judge 3. For the passage, the corresponding mean scores were 6.33 ± 1.08 for Judge 1, 6.72 ± 1.45 for Judge 2, and 6.50 ± 1.34 for Judge 3.

Inter-rater agreement among the three judges was high for both gaits.

For the extended trot, ICC (2,1) = 0.91 (95 % CI [0.87–0.95], p < .001), with a mean coefficient of variation (CV) = 4.23 ± 5.30% and mean inter-rater correlation r = 0.91.

For the passage, ICC(2,1) = 0.86 (95 % CI [0.79–0.92], p < .001), mean CV = 6.68 ± 4.55%, and mean r = 0.90.

Detrended Fluctuation Analysis

The Augmented Dickey–Fuller revealed non-stationary characteristics of the signal, indicating that Hurst exponents (H) of 0.5 < H <1.5 are to be expected (Chen et al., 2002).

Spearman’s rank correlation revealed a strong, significant positive correlation between subjective evaluation scores and H exponents at the CoG level of the rider in passage (r = 0.77, p < .01; H = 1.07 ± 0.06). On the other hand, a strong, significant negative correlation was observed between subjective evaluation scores and H at the CoG level of the rider in the extended trot (r = 0.71, p < .02; H = 1.04 ± 0.06).

Shannon Entropy Analysis

Spearman’s rank correlation revealed a moderate, significant negative correlation between subjective evaluation scores and Shannon entropy (Ɛ) at the CoG level of the rider in passage (r = −0.62, p < .03; Ɛ = 9.28 ± 0.25). On the other hand, a moderate, significant positive correlation was observed between subjective evaluation scores and H at the CoG level of the rider in the extended trot (r = 0.61, p < .05; Ɛ = 9.46 ± 0.12).

Discussion

The present study aimed to explore the relationship between the phase characteristics of horse riders during different dressage movements and subjective evaluations of harmony, as judged by a panel of expert dressage judges. Our findings suggest that non-linear analyses, specifically DFA and Shannon entropy, reveal significant correlations with subjective harmony scores, thus providing new insights into the evaluation of rider performance.

A significantly strong positive correlation between subjective evaluation scores and H exponents at the CoG level of the rider in passage suggests that riders who exhibited greater long-range correlations (persistence) in their movement patterns during highly collected gaits were perceived as more harmonious by the judging panel. In contrast, during the extended trot, we found a significantly strong negative correlation between subjective evaluation scores and H exponents. This indicates that riders who received higher scores tended to show more anti-persistent CoG behaviour, reflecting a more dynamic and adaptive coordination pattern suited to the demands of the extended gait. Mean H and Ɛ values appear contradictory at first glance, showing a generalized persistent behaviour in all subjects (H >1), while showcasing near absolute randomness (10 > Ɛ >9). These results align with the rhythmic, yet complex nature of equitation, respectively. DFA and Shannon entropy represent different layers of analysis; while H captures the fractal nature of the motion, Ɛ reflects fine-scale variability.

Such high H exponents are not common but they are possible and typical of strong, persistent trends in non-stationary data (Chen et al., 2002). These results align well with those found in a study that investigated centre of pressure perturbations in various involuntary body sway situations on a force plate (Borg & Laxåback, 2010); however, recent findings on subjects finding balance over a slack-line partially differ from both our findings and the aforementioned study (Montull et al., 2020). Establishing a conceptual parallel, collected gaits may be analogous to the controlled conditions of involuntary body sway on a force plate, while extended gaits resemble the more dynamic, higher-amplitude balance task of slack-line walking. This framework allows our findings to be interpreted as a synthesis, aligning with the results of both prior studies under these analogous conditions.

It has previously been argued that flow in the context of sport and exercise – defined as a harmonious psychomotor state that enable exceptional performance (Csikszentmihalyi, 1990) – is based on adaptive motion patterns which exhibit anti-persistent behaviours (Montull et al., 2020). We therefore hypothesized that horse riders with lower long-range correlation in their motion patterns would achieve better subjective evaluation scores. However, our data suggest this hypothesis holds true only in the context of extended gaits, which may be more physically demanding but less precision-oriented than collected gaits. This observation challenges conventional definitions of flow and harmony, as the likely explanation for these discrepancies lies in the distinct demands of different movement tasks. Specifically, fine, precise, and low-amplitude tasks, such as collected gaits, may require lower entropy and greater persistence in motion patterns. Conversely, tasks with expansive dynamics, such as extended gaits, may benefit from increased entropy and anti-persistent behaviours, indicative of greater adaptability.

Synergistic motion depends on a well found balance between stability and flexibility (Latash et al., 2007). Stabilization of performance variables is seen as a goal that is achieved through flexibility. Co-variation patterns of the muscle synergies define how balanced flexibility and stability are. This definition aligns with the equilibrium-point hypothesis which states that the central nervous system is constantly parameterizing the neuromuscular system by setting and adjusting equilibrium points, setting thresholds for the motor-neuronal pools (Feldman & Levin, 2009).

This dichotomy underscores the complexity of achieving harmony in equestrian sports, highlighting the importance of context-specific strategies. It suggests that effective training and performance evaluation should consider the idiosyncrasy of each gait, promoting precision and stability in collected movements while encouraging adaptability and flexibility in extended movements. On the other hand, these results stress the context-dependent nature of harmony and flow. However, the reliance on subjective evaluations of harmony, as used in dressage scoring, raises important questions about bias and consistency. This is an intrinsic limitation of this study since we have assumed that better scores correspond to better performance, and while this holds true for a competition context it might not automatically translate into better scores corresponding to the horse moving optimally from a biomechanical point of view, as we have recently seen with rollkur or coercive hyperflexion at the poll level (von Borstel et al., 2009).

Conclusion

The application of simple, non-linear analyses has the potential to complement the subjective perception of harmony in equestrian sports, providing valuable insights to enhance both training methodologies and performance evaluation strategies.

More specifically, introducing a complex-systems perspective to training sessions can provide riders and coaches with objective feedback to assess adaptability and flow, identifying rigid or inconsistent movements to improve harmony between horse and rider.

In competition, such metrics could reduce bias in subjective scoring by supplementing the judges’ perspectives, enhancing consistency and fairness.

While the FEI provides a framework for assessing harmony, its interpretation remains subjective and may benefit from the integration and better understanding of objective metrics such as DFA and entropy analyses. Further research is needed to determine if such findings can be extrapolated to other precision-demanding sports and activities.