2.1 Subjects and procedures

This study involved two groups of participants: 13 elite athletes who competed in the Beijing Winter Olympics or World Championships (age: 22.50 ± 4.5 years; height: 174.68 ± 2.2 cm; weight: 70.46 ± 4.2 kg), 12 expert athletes who participated in national competitions (age: 20.00 ± 2.8 years; height: 174.87 ± 4.4 cm; weight: 69.22 ± 6.7 kg). All participants were in good physical condition and free from sports injuries or other diseases in the six months prior to testing, ensuring stable performance during the tests. The athletes were not fatigued during the tests, which were scheduled on their rest days. The recruitment period for this study started on 15/03/2020 and ended on 30/05/2022. This study was approved by the ethics committee2018(09).

A portable balance device (Humac Balance, USA), measuring 65 cm × 40 cm with a sampling frequency of 100 Hz, was used as the stable support surface for participants. To introduce a destabilizing factor to the athletes’ proprioception, a foam pad measuring 65 cm × 40 cm and 50 mm in thickness was employed. The athletes’ Center of Pressure (COP) on both stable and unstable support surfaces was recorded using the balance device.

Participants were instructed to remove their shoes and socks before standing, aiming to "maintain body trunk stability and minimize movement as much as possible." A quiet environment was maintained throughout the test to ensure optimal testing conditions. Following a standardized protocol, participants were instructed to place their hands on their hips, aligning the bottom of their second metatarsal bone with the marker on the balance device.

Participants were instructed to focus their gaze on a white circle with a diameter of 5 cm, positioned at eye level on a wall 3 meters away, while avoiding any other movements. Throughout the experiment, participants maintained forward gaze, with a slight bend in the hip and knee joints. The static standing posture was tested under four different conditions: T1: Both legs standing on a firm surface; T2: Both legs standing on a foam surface; T3: Both legs standing on a firm surface with eyes closed; T4: Both legs standing on a foam surface with eyes closed. Each standing condition lasted for 30 seconds, with a 30-second seated rest period between conditions (Fig 1).

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Fig 1. Experimental set-up and data analysis.

(a) data acquisition process. In Step 1, We selected 13 elite freestyle skiing aerials athletes and 12 expert freestyle skiing aerials athletes. Step 2 involves conducting two experiments on all participants: First, 30-second balance recording test during various support leg conditions, and second, a 30-second test under varying surface conditions. Throughout this process, detailed Balance data for all participants are meticulously recorded. In Step 3, we primarily record the athletes’ displacement in the X and Y directions of cop the cop length, and the tilt angle throughout the entire testing process. (b) The data analysis process includes performing Signal Decomposition, Hilbert Transform, Sample Entropy, Total Complexity, Composite Multiscale Complexity Index, and machine learning on all collected balance data.

https://doi.org/10.1371/journal.pone.0315454.g001

2.2 Data description and processing

2.2.1 Balance data.

Center of pressure (COP) coordinates were analyzed in MATLAB 2022a (Mathworks, Natick, MA, USA) to calculate displacement, Length, related parameters in anterior-posterior, mediolateral, and Tilt angle (Table 1).

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Table 1. Balance parameters description.

https://doi.org/10.1371/journal.pone.0315454.t001

In traditional balance control assessments, static postural methods are typically employed to evaluate the displacement of the Center of Pressure (CoP). However, these methods often fail to account for the dynamic aspects of balance control during movement. To address this limitation, we introduced the tilt angle as a novel parameter, derived from the Humac balance system, to provide a more comprehensive evaluation of athletes’ balance abilities [19].

To evaluate postural stability, we calculate the Tilt Angle by analyzing the displacement of the COP from its equilibrium position. The COP is derived using the coordinates COP_x and COP_y, which represent the body’s pressure center on the horizontal plane. The resultant displacement, denoted as COP_Len. The Tilt Angle is then derived by using 55% of the patient’s height to approximate the location of the Center of Mass (COM), providing a quantitative measure of the body’s inclination from the vertical axis. This angle is a critical indicator of the patient’s balance and stability across various postural tasks.

2.2.2 Composite Multiscale Complexity Index (CMCI).

The input signal for the Complex Modulation and Coupling Index (CMCI) analysis consists of Center of Pressure (CoP) data recorded during static balance tests. Specifically, the CoP data was collected from participants standing on both stable and unstable surfaces, including a firm surface and foam, under various conditions (with eyes open and closed).

1. (1) Signal Decomposition:

The signal is decomposed into a finite number of Intrinsic Mode Functions (IMFs) using Empirical Mode Decomposition (EMD). This decomposition occurs through an iterative sifting process, where local maxima and minima are identified to generate upper and lower envelopes. The mean of these envelopes is then subtracted from the original signal, isolating each IMF. The sifting continues until a predefined stopping criterion is met, which is typically based on the standard deviation between successive siftings. To ensure that the IMFs reflect the most relevant oscillatory modes of the signal, mode truncation is applied by discarding IMFs that exhibit negligible energy or represent irrelevant frequency components. This truncation process helps eliminate noise and ensures that the retained IMFs capture meaningful signal characteristics [20].

The signal S(t) is decomposed into a finite number of Intrinsic Mode Functions (IMFs) using EMD. Each IMF represents a simple oscillatory mode embedded in the signal [21].

1. (2) Hilbert Transform:

The Hilbert transform is applied to each IMF IMF_i(t) to obtain its analytical signal. The analytical signal is represented as a complex function where a_i(t) is the instantaneous amplitude and φ_i(t) is the instantaneous phase of the i-th IMF.

1. (3) Sample Entropy:

Sample Entropy (SE) is a measure of signal complexity, reflecting the unpredictability or irregularity of time-series data. In this study, SE is calculated for both the instantaneous amplitude and phase of each Intrinsic Mode Function (IMF) obtained through the Empirical Mode Decomposition of the COP signal. The SE calculation involves defining two key parameters: the embedding dimension r. Where m = 2 represents the length of subsequences being compared, and r = 0.2 × std represents the similarity criterion, based on 20% of the standard deviation of the signal[22]. For each IMF, SE measures the likelihood that similar subsequences remain similar as the sequence extends by one additional point. It is computed using the following formula:

SE is calculated for both the instantaneous amplitude a_i(t) and the instantaneous phase φ_i(t) of each IMF. SE is a measure of uncertainty or complexity within a signal.

1. (4) Total Complexity:

The total complexity of each IMF is calculated by summing the weighted entropies of both the instantaneous amplitude and instantaneous phase. To ensure that both components contribute meaningfully to the overall complexity measure, appropriate weights must be assigned to the entropies of amplitude and phase. In this analysis, the weights for the amplitude and phase entropies are assumed to be equal as the goal is to assess the overall complexity of the signal without prioritizing one component over the other.

The total complexity for each IMF is computed by summing the weighted entropies of both the amplitude and phase. ω_i represents the weight associated with each entropy term.

1. (5) Composite Multiscale Complexity Index (CMCI):

The Composite Multiscale Complexity Index (CMCI) is determined by averaging the total complexities of all IMF components. This index provides a comprehensive measure of the signal’s overall complexity across multiple scales.

2.3 Classification techniques

We employed two distinct feature extraction techniques for classification: original time-domain features and those derived from the Composite Multiscale Complexity Index (CMCI). The original time-domain features included the mean, standard deviation, skewness, and kurtosis, which were extracted from balance signals across four tasks. CMCI features were derived by applying the CMCI process to individual variables, as described in section 2.2.2.

To analyze the performance in evaluating our predictive approach, we employed a standard train-test methodology. The dataset, comprising all collected balance parameters labeled across two groups—elite athletes and regular athletes—was randomly split into a training set (70% of the data) and a test set (30%). We ensured that the proportion of elite and regular athletes was balanced in both sets to maintain consistency.

Classification is utilized to ascertain the category (group or class) to which a new observation or instance belongs, based on a training dataset containing instances with known category memberships. Classification techniques typically encompass three approaches: statistical methods, machine learning techniques, and neural network methods [23–25]. Considering these approaches, we employed four common classifiers to predict and distinguish balance ability similarities among various athletes.

Ranking Forest: We utilized the Ranking Forest algorithm for its robustness in handling high-dimensional data and its capacity to capture complex relationships among features. This method utilizes bagging to construct and aggregate 50 decision trees, each built on random subsets of the training data and features. Ranking Forest is particularly effective in providing stable predictions and reducing the risk of overfitting, thus making it particularly suitable for our balance data, which can be highly variable [26].

Naive Bayes: Naive Bayes was chosen due to its simplicity and efficiency, particularly when dealing with high-dimensional feature spaces. Although it assumes that features are conditionally independent—an assumption that may not always hold true in balance data—its probabilistic nature establishes it as a strong baseline classifier. Naive Bayes is particularly useful for understanding the contribution of individual features to the classification task, offering insights into which balance parameters are most predictive [27].

Logistic Regression: Logistic Regression is a widely used algorithm for binary classification tasks, perfectly aligning with our need to classify athletes into two groups. It models the probability that a given input belongs to a particular class, making it valuable for interpreting the importance of different balance features. Logistic Regression is particularly useful for understanding linear relationships between the features and the outcome, rendering it a suitable choice for initial analysis and interpretation of the balance data [28].

Support Vector Machines (SVM): SVM is recognized for its effectiveness in high-dimensional spaces and its capacity to handle non-linear relationships through kernel functions. Given the complex and non-linear nature of balance data, this makes SVM well-suited to identify subtle distinctions between elite and regular athletes. By maximizing the margin between different classes, SVM establishes a robust classification framework, particularly in cases where class balance is critical [29].

3.1 Subject’s characteristics

Table 2 provides an overview of the balance performance characteristics of the participants in this study. The analysis revealed no statistically significant differences between elite and expert athletes in the balance parameters COPLen, COPX, COPY, and Tilt angle across tasks T1, T2, and T3. However, a significant difference was observed in the COPY parameter during task T4 (P<0.05). These findings suggest that traditional balance measurement parameters may not sufficiently differentiate between elite and expert athletes. Given the essential role of balance in athletic performance, distinguishing between these two groups is crucial. This raises the question of whether these parameters are adequate for accurately classifying and distinguishing the similar balance abilities of elite and expert athletes.

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Table 2. Balance performance of elite and expert athletes.

https://doi.org/10.1371/journal.pone.0315454.t002

3.2 Performance of computational models

The results of the models are presented in two distinct sections. First, we analyzed the classification performance of various classifiers on datasets processed in two ways, illustrating the true positive rate and false positive rate for each developed model within their receiver operating characteristic (ROC) space (Fig 2). Balancing the dataset using CMCI techniques typically results in improved classification performance. The cluster of CMCI-processed data resides in the upper left quadrant of the ROC space (above the reference line), indicating optimal classifier performance. In contrast, the performance of datasets utilizing only original data approaches or falls below the reference line.

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Fig 2. Model performance in ROC space across different algorithms and task conditions.

Performance of models in ROC space for datasets under all conditions (firm surface with eyes open and eyes closed, foam surface with eyes open and eyes closed), using all data processing methods (original time domain and CMCI method), and all classification methods (Bayesian, Ranking Tree, Logistic Regression, Support Vector Machine). (a, c, e, g) represent the performance of all classifiers using the original time domain processing method, while (b, d, f, h) represent the performance of all classifiers using the CMCI processing method. Each model’s name follows the following naming rules: The first uppercase letter represents the name of the classifier (e.g., SVM: S, Logistic Regression: L, Bayesian: B, Ranking Tree: R); the second word indicates the data processing method, where OR represents the original time domain processing method, and CMCI represents data processing using the CMCI method; the last part of the name indicates the variable names used.

https://doi.org/10.1371/journal.pone.0315454.g002

Subsequently, we conducted a comprehensive analysis of the classifiers, including their performance under various test conditions and data processing methods, demonstrating enhanced performance in ROC space (Fig 2). Specifically, models constructed using the Ranking Forest classifier in conjunction with CMCI parameters typically exhibited the highest performance, achieving a sensitivity of 0.95 and a specificity of 0.35. Subsequently, the Naive Bayes classifier demonstrated satisfactory performance under the eyes-closed condition, achieving a sensitivity of 0.85 and a specificity of 0.40. However, the impact of integrating CMCI with various classifiers is contingent upon the specific balance tasks and parameters, which will be elaborated upon in detail in Section 3.3.

3.3 Impact of task difficulty on classification accuracy

Based on the results presented in Section 3.2, we evaluated the classification performance of the CMCI in conjunction with the Ranking Forest method on data from four distinct balance tasks in ROC space (Fig 3), assessing how varying tasks influenced classification accuracy. Specifically, for Task T1, neither the time-domain feature extraction nor the CMCI-processed data yielded satisfactory classification results with the Ranking Forest method. Notably, the ROC analysis of time-domain features indicated performance akin to a random classifier, with AUC values fluctuating between 0.49 and 0.54. Nevertheless, as task difficulty increased, the classification performance of the CMCI-processed COPLen variable exhibited the highest accuracy when utilizing the Ranking Forest method (sensitivity = 0.95, specificity = 0.35). As task difficulty further escalated, even under the eyes-closed condition, CMCI-COPLen persisted in demonstrating robust classification performance (sensitivity = 0.75, specificity = 0.35). Ultimately, in Task 4, the most challenging condition, none of the raw data features yielded accurate classification, while only the CMCI-processed tilt angle distinguished itself (sensitivity = 0.9, specificity = 0.55).

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Fig 3. Ranking forest model performance in ROC space for all balance features.

By applying the Ranking Forest machine learning algorithm to the dataset under all conditions (eyes-open firm surface, eyes-closed firm surface, eyes-open foam surface, and eyes-closed foam surface), we evaluated the ROC space of the models for all balance features (both original parameters and those processed using the CMCI method). The models developed using the datasets processed with the CMCI technique exhibited superior performance. Each model’s name follows the naming convention described below: The first uppercase letters indicate the set of variables used, with OR = original features, CMCI = variables processed with the CMCI method, and (a, b, c, d) representing different balance tasks(T1-T4).

https://doi.org/10.1371/journal.pone.0315454.g003

3.4 Construct validity

The CMCI processing method selected features associated with COP displacement and tilt angle across four tasks (firm and foam surfaces with eyes open and closed). CMCI-processed features surpassed the original time-domain features (COPX, COPY, COPLen, and tilt angle) in classification accuracy, particularly in high-difficulty tasks, such as balancing on a foam surface, which disrupts proprioceptive feedback. This improvement aligns with theoretical expectations that more challenging balance tasks should yield more significant results in assessing postural stability. Conversely, the original features demonstrated poor performance under difficult conditions, indicating limitations in distinguishing between various balance states.

The superior performance of CMCI-processed features in high-difficulty tasks supports the construct validity of these features, whereas the inferior performance of the original features raises questions about their construct validity. This discrepancy highlights the critical role of effective feature selection and processing in accurately capturing the construct of postural balance.

4.1 Balance classification performance for specific tasks

In our study of elite and expert athletes, the differences in balance ability between the two groups are minimal. We found it nearly impossible to classify them accurately using any features extracted from the time domain. We detected only slight differences between the two groups in the anterior-posterior direction during Task T4, which increases the demands for classification methods and data processing. Extensive prior research on balance classification has demonstrated that traditional balance features are inadequate for achieving accurate classification results [33,34]. Furthermore, we observed that classifying balance in simple tasks presents significant challenges. This difficulty may stem from the lack of external disturbances during simple tasks, resulting in a more stable and regular balance, which makes it challenging to accurately distinguish between balance abilities.

However, as the difficulty of the balance tasks increased, we observed that various tasks and external factors had a significant impact on classification accuracy. For instance, the introduction of proprioceptive interference to the athletes enhanced the classification accuracy of combinations of time-domain features and tilt angle parameters. This finding suggests that proprioceptive interference is more effective than visual interference in detecting subtle balance differences among individuals. Moreover, the classifier’s performance was significantly improved when employing the innovative CMCI approach. This implies that traditional feature analysis alone is inadequate for distinguishing subtle differences in balance ability. The CMCI method appears more suitable for differentiating between populations with minimal balance differences, particularly among skill-based athletes, where such differences are minimal. Furthermore, the results of traditional statistical tests must be interpreted with caution. Tests such as rank-sum tests indicate significant differences between two distributions; however, statistical differences do not necessarily yield effective solutions for classification problems. Additional information is required to classify new participants as elite or expert athletes.

Finally, does increasing the difficulty of balance tasks always enhance classifier accuracy? The answer is no; increasing task difficulty does not always lead to improved classifier accuracy. Our balance tests encompass simple tasks (T1, T2) and more challenging tasks (T3, T4), yet we did not observe the highest classification accuracy in T3 and T4. Traditionally, it is believed that more challenging balance tasks improve classifier performance by revealing greater postural sway and irregular movements. However, our study revealed that when both visual and proprioceptive interference were present, the classification accuracy between elite and expert athletes was suboptimal. This may stem from the increased difficulty, which causes athletes to struggle to maintain stable balance, resulting in greater randomness and uncontrollability [35,36]. Consequently, the relationship between balance test difficulty and classifier accuracy is intricate, particularly in detecting subtle differences in balance ability.

4.2 Distinguishing balance ability through the tilt angle

Prior studies have indicated that combinations of multiple features can partially reflect postural instability [37,38]. Although individual features (such as medial-lateral sway) are useful, they often fail to accurately differentiate between various populations, consistent with our findings. We also examined the impact of single features on classifier performance. Common indicators traditionally employed to evaluate balance ability include anterior-posterior and medial-lateral center of pressure displacement, center of pressure movement path, and center of pressure velocity. In contrast, we propose the utilization of the body’s tilt angle as a novel parameter for assessing balance ability. Our findings indicate that the tilt angle parameter extracted from time-domain features consistently improved the performance of various classifiers across nearly all standing tasks. Consequently, future research aimed at differentiating and revealing balance differences among diverse populations should consider the tilt angle as a key metric, particularly across various standing tasks. The tilt angle may represent one of the more appropriate indicators for balance assessment.

Theoretically, this approach aligns with the notion that balance encompasses both static and dynamic aspects of postural control. Traditional features typically emphasize the displacement and movement of the center of pressure, primarily capturing dynamic changes while potentially overlooking static postural adjustments. Tilt angle, as a measure of the body’s alignment relative to the vertical, provides a direct assessment of static balance stability, complementing dynamic measures. This theoretical foundation supports the notion that incorporating tilt angle could yield a more comprehensive evaluation of balance, particularly in distinguishing populations with subtle differences in balance ability.

4.3 The power of non-linear models and CMCI in balance assessment

Previous research has also incorporated multiple indicators in their instability assessment models [39]. Reports indicate that employing machine learning methods to analyze multidimensional features can significantly enhance classification performance [40]. For example, earlier studies have employed ranking forests to differentiate between fallers and non-fallers based on static postural graphs, achieving an AUC of 0.75 [41]. These studies suggested that the algorithms relied on a straightforward linear combination of features. However, the nature of feature interrelationships can be highly complex, and linear models may not sufficiently capture these relationships. The inherently deep non-linear nature of ranking forest algorithms enables them to model more complex and generalized patterns [42]. In our study, we employed the CMCI method to extract features from the raw data and subsequently applied ranking forests for classification analysis. We discovered that the integration of CMCI with ranking forests significantly enhanced the classifier’s performance. Non-linear signal decomposition methods can extract features that reveal subtle variations in balance control, which are often challenging to detect using linear methods [43,44]. For instance, adaptive responses of the balance system to perturbations induce neuromuscular reactions across various frequency bands and time scales, which can be captured through non-linear feature extraction [45]. When these features are integrated into advanced classification models, they reveal deeper patterns related to balance control, thereby enhancing classification accuracy.

This enhancement can be attributed to the capacity of non-linear models to capture complex dependencies between features that linear models may overlook. CMCI, as a feature extraction method, identifies subtle variations in balance data that may not be apparent through traditional features. When combined with the non-linear capabilities of ranking forests, this approach enables a more nuanced understanding of balance instability, resulting in improved classification performance. These findings highlight the potential of integrating advanced feature extraction and non-linear modeling techniques to improve the accuracy and effectiveness of balance assessments.

4.4 Limitations

We assessed balance ability in elite and expert athletes across only four types of balance tasks. The study focused exclusively on two groups exhibiting minimal differences in balance ability. Future research should encompass a broader range of populations to enhance generalizability. Additionally, the selected indicators were based on commonly used metrics from prior literature and were not subjected to further screening. Future studies should incorporate a wider array of balance parameters to validate the classifier’s results and evaluate the feature recognition capabilities of CMCI for additional indicators.