1 Introductıon

Aging is associated with various physical and physiological changes, including a decline in balance, gait, and neuromuscular function, which can lead to the risk of falls and other mobility-related problems. Age-related changes may also impact the sensory inputs of older people, such as the somatosensory, visual, vestibular, and proprioceptive systems. These changes could have a significant impact on balance and postural control [1]. The examination of equilibrium characteristics and comparative analysis of underlying balance attribute patterns can yield considerable advantages in terms of fall prediction and risk reduction. Postural sway dynamics can fluctuate over multiple timescales. The degree of multiscale complexity of postural control and stability can change with aging or disease. Various methods, including clinical tests, self-report measures, and objective assessments, can be used to measure age-related changes in balance in older adults. To overcome the lack of reliable clinical tests for the evaluation of balance and posture disorders, quantifiable analyses of postural control tend to be used through analysis of the trajectory of the center of pressure (COP). CoP provides information about the distribution and magnitude of ground reaction forces during various tasks and movements [2]. The force plate is perhaps the most common equipment used to measure standing balance and gait. In clinical practice, kinematics and kinetics metrics derived from CoP time series can provide valuable information and indicators for physicians and other healthcare professionals to track changes in balance and postural stability over time and to evaluate the effectiveness of interventions such as exercise programs or assistive devices. However, assessment of body balance control can be beneficial for evaluating the tendency to fall. Quantitatively determining age-related changes in balance control can be facilitated primarily by fundamental measurements of CoP, which are directly related to the ability to maintain a stable body posture. Previous studies have demonstrated a relationship between static body balance measurements and the risk of falls [3].

To support clinicians in their practice, machine learning (ML) can offer valuable clinical decision-support tools to improve efficiency, enhance diagnostic accuracy, and provide more personalized and effective care to patients. ML can also provide objective and standardized assessments of balance and performance, removing potential biases associated with subjective human assessments, by classifying CoP signals corresponding to different types of gait patterns or balance impairments [4, 5]. Several studies have used ML classification based on CoP for older adults [4,5,6]. In these studies, various ML models were used for CoP classification including decision trees, the k-nearest neighbor (k-NN) algorithm, support vector machines (SVMs), the random forest (RF), naive Bayes (NB), linear regression, and multilayer perceptron (MLP) models. SVMs and linear regression models are the most commonly used to classify age-related postural sways. In the field of postural control analysis, recent advances in ML have also demonstrated remarkable potential for detecting and characterizing age-related changes. While traditional methods form the basis for understanding equilibrium dynamics, new approaches have delved deeper into nonlinear complexity, emphasizing the importance of considering nonequilibrium sample conditions and multiple aspects of the data [7, 8]. It is important to note that the choice of the most suitable algorithm depends on the specific characteristics of the dataset, research question, and goals of the analysis. While these ML models offer advantages, other models may be more appropriate for classifying age-related changes in postural sway. To understand which ML model produces more accurate predictions and achieves better results, comparing the models for classification problems enables us to evaluate their performance, select the most appropriate one, assess their generalization ability, and drive optimization and improvement efforts. Partial least squares-discriminant analysis (PLS-DA) is a statistical method used for classification and discrimination tasks. It is widely used in various fields, including chemometrics, bioinformatics, and social sciences, for tasks such as disease classification, quality control, and pattern recognition [9,10,11,12]. It is especially useful when handling multicollinearity, limited sample sizes, high-dimensional data, model interpretability, and robustness to outliers when compared to other classification methods. Therefore, this study aims to develop a PLS-DA model. To the best of our knowledge, the application of PLS-DA methods to classify age-related CoP features has not been reported in the literature.

It is also important to first preprocess the CoP data to extract relevant features or character conditional analyses of CoPs that have generally been based on linear mathematics. The time- and frequency-domain features of the CoP data typically capture statistical characteristics, such as means or root mean square (RMS) values, Fourier analysis, standard deviations, CoP sway ranges, or percentiles of the CoP displacement or velocity over a specific time interval. These features are useful for describing the overall behavior of the system during static and dynamic postural tasks. However, they do not capture the complex nonlinear dynamics of postural control because the dynamics of the CoP in the context of postural control are a complex collection of nonlinear systems in both space and time. Traditionally, postural oscillations were thought to be random and unstructured. However, recent studies have shown that these fluctuations may have a time-dependent structure [12, 13]. This is supported by evidence of nonlinear properties in postural stability. Under specific conditions, these properties result from the intricate interactions of multiple factors (e.g., sensory input from visual, vestibular, proprioceptive, and somatosensory systems), feedback loops, adaptability, complex muscular control as nonlinear actuators, environmental interaction (e.g., rigid and compliant surface), fractal properties, nonlinear control mechanisms, and sensorimotor integration [13, 14]. All of these factors do not act linearly but interact in a nonlinear manner to maintain balance. Complex systems have the potential to generate output signals containing temporal structures embedded within what appear to be random oscillations. Understanding and modeling these complexities is essential for accurately assessing and characterizing postural control, especially in the context of age-related changes. The traditional linear methods mainly rely on simplifying assumptions and linear relationships, and may not fully address the complexity of postural sway patterns. However, entropy measurements represent a class of nonlinear measures that characterize orderliness in the temporal output of a complex system. When applied to COP data, entropy measurements can quantify irregularity, complexity, and unpredictability in time series data, making it particularly well suited to capture nonlinear and complex behaviors of postural sway patterns. For example, traditional methods often rely on summary statistics like means, standard deviations, or spectral power, which provide a single number to describe the entire COP time series. These summary statistics are linear and do not capture the intricate patterns and variations within the data. Consequently, recent developments have described the application of the dynamic approach to the evaluation of CoP signals. Nonlinear analysis can provide valuable features because it captures complex nonlinear relationships between the variables in the CoP time series, such as the complexity, regularity, or fractal properties of the CoP signals [15, 16]. Furthermore, nonlinear features are sensitive to changes in underlying control mechanisms and can capture subtle changes in postural control that may not be evident from time-domain features alone. Entropy is a nonlinear method used in CoP analysis to quantify the complexity or irregularity of the postural sway pattern by considering patterns in data points [15]. Entropy measurements calculate the likelihood of similar sequences of data points occurring in the time series, accounting for both local and global patterns. In traditional analysis of two different CoP time series, the mean, RMS, or SD value may be almost similar or there may be no statistically significant differences. However, the entropy values of the same two-time series may be different or their differences may be statistically significant [16]. For example, higher entropy values can indicate greater irregularity and complexity in the COP time series. PLS-DA and hybrid feature sets come into play to capture these intricate patterns and relationships within the CoP data. In the literature, several studies have reported ML classification using the entropy of CoP signals to assess balance control in older people [18, 19]. These studies showed that entropy measures of CoP signals were also useful for detecting changes in postural control associated with aging and disease. However, it should be noted that each study used different methodologies, sample sizes, and numbers of feature sets. Therefore, considering the hybrid approach (combined time-domain and nonlinear features) is important for obtaining a more comprehensive understanding of the underlying control mechanisms and developing interventions to detect changes in postural control associated with aging. Furthermore, hybrid enrichment could provide several advantages for classification tasks, including improved accuracy, robustness, interpretability, and flexibility [20, 21]. In particular, for hybrid features, the number of features in an ML algorithm can affect the classification performance. Too few features may result in information loss, while a larger feature can lead to: (i) complex models that are harder to interpret, making it challenging to understand the underlying decision-making process; (ii) increased risk of over-fitting, especially if the dataset is small relative to the number of features; (iii) potentially redundant or irrelevant features, which may require additional preprocessing or feature selection techniques to reduce noise and improve performance; and (iv) increased dimensionality of the problem, making it more challenging to process and analyze the data. Determining an optimal balance between model simplicity and performance is crucial for achieving accurate and robust predictions of age-related changes. To address these limitations, careful feature sets with a characterized complex balance and postural control system should be employed to ensure that the reduced feature set captures the most relevant and discriminative information. However, the methodologies focusing on balance and falls are deemed inadequate, underscoring the inherent challenges associated with the characterization of human balance [22]. For the classification of CoP-based hybrid features in older people, further research is required to establish the effectiveness of these approaches.

Therefore, to uncover age-related changes in postural sway patterns, this study aims to use a smaller (reduced) hybrid feature set by combining traditional statistical methods with the nonlinear dynamics of a CoP time series for the PLS-DA classifier. A small hybrid feature set including only sway area, sway velocity, sample entropy (SampEn), and fuzzy entropy (FuzzyEn) has not been tested, for the conditions "eyes open" and "eyes closed" on both “rigid” and “compliant surfaces,” specifically comparing results for old and young individuals. This study hypothesized that a PLS-DA algorithm trained on a reduced small hybrid feature set would achieve improved classification performance in discriminating age-related changes in eyes open and closed on rigid and compliant surfaces. From a clinical perspective, the proposed ML algorithm using a small hybrid feature set can provide insight into capturing the complex dynamics of postural control, which can be affected by changes in neuromuscular function, sensorimotor integration, and environmental demands, and can be effective for analyzing data collected over extended periods (e.g., long-term monitoring) and guiding the development of more effective interventions for older adults.

2 Methods

2.1 Data

A public COP dataset was used [23]. In total, 163 participants (young = 87 and old = 76) was enrolled in this dataset. Table 1 shows the demographic caharecteristics of participants. The data were recorded at quiet standing throughout four condition combinations: eyes open, eyes closed on a firm surface, and a compliant surface over 60 s at a sampling frequency of 100 Hz and filtered at 10 Hz (dual-pass fourth-order low-pass Butterworth). All details and more information on the data acquisition protocol can be found in [23].

Table 1 Demographic characteristics of subject
Full size table

2.2 Feature extraction

For the reduced hybrid feature set, the most sensitive measures were selected to discriminate age-related changes in balance [24]. Therefore, the two traditional linear mathematical measures in the time domain (sway area and sway velocity) and two entropy parameters (SampEn and FuzzyEn) as nonlinear parameters were extracted directly from postural algorithms. Sway velocity and entropy parameters were calculated in both coordinates of the anterior–posterior (A-P) and medial–lateral (M-L) directions, and seven features were used for the proposed PLS-DA model.

2.2.1 Time-domain CoP parameters

CoP measures provide estimations of features derived from time-domain analysis of COP data as a function of time, considering their projection onto the coordinates of A-P (Y) and M-L (X) directions. These features directly consider the temporal evolution of the COP coordinates (X and Y) as the variable "n" iterates through time steps. Each term in the summation represents the absolute difference in COP position at one time step compared to the previous time step. This analysis is inherently tied to the sequence of data points collected over time and focuses on how the COP coordinates change over time. The sway length of the sway path is estimated by the sum of the distances between consecutive points on the CoP path in the A-P and M-L directions in Eqs. 1 and 2.

$${\mathrm{Sway Length}}_{\_Y}=\sum_{n}\left|{X}_{n+1}-{X}_{n}\right|$$
(1)
$${\mathrm{Sway Length}}_{\_X}=\sum_{n}\left|{Y}_{n+1}-{Y}_{n}\right|$$
(2)

where “n” represents the time index or discrete time step in the sequence of COP data points in the X (horizontal) and Y (vertical) directions. \({X}_{n}\) and \({Y}_{n}\) centered on the M-L and A-P coordinates, respectively.

Sway velocity represents the rate of change of COP position. It quantifies how much distance (in this case, COP displacement) changes per unit of time (T). The sway velocities in both the A-P and M-L directions were obtained by dividing the total length by the duration of quiet standing using Eqs. 3 and 4.

$${\mathrm{Sway Velocity}}_{\_Y}= {\mathrm{Sway Length}}_{\_Y}/T$$
(3)
$${\mathrm{Sway Velocity}}_{\_X}= {\mathrm{Sway Length}}_{\_X}/T$$
(4)

where “X" represents the horizontal component of COP displacement, "Y" represents the vertical component of COP displacement, and T is the total duration of the signal (s). The sway area per second was computed by adding the area of the triangles whose vertices were two consecutive points of the COP trajectory and the mean position of COP [18] using Eq. 5.

$${\text{Sway}} {\text{Area}}=\frac{1}{2T}\sum_{n}^{N-1}\left|{X}_{n+1}{Y}_{n}-{X}_{n}{Y}_{n+1}\right|$$
(5)

2.2.2 Sample entropy (sampEn)

SampEn is a measure derived from approximate entropy to quantify the complexity or irregularity of a time series dataset [25]. It is particularly useful in analyzing biomedical signals and other types of time-dependent data. The main idea behind SampEn is to divide the data series into blocks (template vectors) and compare their similarities (possible vectors) to measure correlation, persistence, or regularity [19]. Mathematically, SampEn is the negative value of the logarithm of the conditional probability that two similar sequences of “m” points remain similar at the next point “m + 1” by counting each template vector over all the other possible vectors without allowing it to self-count. To calculate SampEn, CoP data series were called “x” time series. To create the sequence of all possible vectors of length m, a template of length “m” was selected and compared with all possible vectors of the same length within the tolerance level “r” in each CoP time series under analysis. Typically, m is set between one and two times the expected dimensionality of the system, whereas r is often a fraction (usually 0.1–0.25) of the standard deviation of the data.

In this study, a CoP dataset of length N (6000 samples at 100 Hz = 60 s) is obtained as x(1), x(2), …, x(N). Two parameters: embedding dimensions (m) and (r). The main input parameters were set as m = 2 with m ≤ N and r = 0.2*SD, namely the embedding dimension “m” and the tolerance or similarity criterion “r.” A subsequence of length “m” in the time series data was defined as x(i), x(i + 1), …, x(i + m − 1). The distance between two subsequences (x(i), …, x(i + m − 1)) and (x(j), …, x(j + m − 1)) is defined as the maximum absolute difference between corresponding elements: d([x(i), …, x(i + m − 1)], [x(j), …, x(j + m − 1)]) = max{|x(i + k)—x(j + k)|, k = 0, 1, …, m − 1} to determine whether the difference between all positions of the vectors is less than or equal to r (d[x(i), x(j)] = max|x(i) − x(j)|≤ r. A match occurs when two subsequences (x(i), …, x(i + m − 1)) and (x(j), …, x(j + m − 1)) have a distance of less than or equal to r. T e Chebyshev distance was calculated as d[x(i), x(j)] = maxk = 1,2,…,m(|x(i + k − 1) − x(j + k − 1)|). In the CoP time series, x(1) was first compared with x(2), then x(1) with x(3), and continued in the same manner without allowing self-counting (j ≠ i). Once all possible vectors were compared, x(2) was set as the template, and the same process was performed. SampEn is then given by the following formula using Eq. 6.

$${\text{SampEn}}\left(m,r,N\right)=-{\text{log}}\left[\frac{{\sum }_{i=1}^{N-m}\sum_{j=1,j\ne i}^{N-m}[\mathrm{count that} d[\left|{x}_{m+1}\left(j\right)-{x}_{m+1}\left(i\right)\right|<r]}{{\sum }_{i=1}^{N-m}\sum_{j=1,j\ne i}^{N-m}[\mathrm{count that} d[\left|{x}_{m}\left(j\right)-{x}_{m}\left(i\right)\right|<r]}\right]$$
(6)

As shown in Eq. 6, SampEn does not allow self-counting (j ≠ i). The sum of all the template vectors is within the logarithm of SampEn. This implies that SampEn considers the complete series, and if a template finds a match, SampEn is already defined. Once all the vectors were checked, then \({A}_{i}^{m}\left(r\right)\) and \({B}_{i}^{m}\left(r\right)\) \({\text{counted}}\) the probabilities, which were determined by summation over the whole series. We determined the conditional probability functions as the ratio of “matches[i]/possibles[i]” between 0 and 1 because the number of matches was always less than or equal to the number of possible vectors. SampEn was calculated simply as the negative logarithm of these probabilities, making direct use of the correlation integral, as shown in Eq. 7.

$${\text{SampEn}}\left(m,r,N\right)=-log\left[\frac{{A}^{m}\left(r\right)}{{B}^{m}\left(r\right)}\right]$$
(7)

where m is the length of the template (length of the block of different vector comparisons), N is the length of the CoP data, and r represents the threshold parameter (noise filter). \({A}_{i}^{m}\left(r\right)\) is the probability that two sequences are similar for “m + 1” points (matches), while \({B}_{i}^{m}\)(r) is the probability that two sequences are similar for “m” points (possible), within the resolution “r.” The Am(r)/Bm(r) ratio represents the conditional probability.

Regarding the algorithmic complexity of Eq. 7, the mathematical function involves a summation of data points, where each data point is used to compute a cross product and an absolute value. Therefore, the primary operation within the loop is the calculation of the cross product and absolute value. This loop runs from n = 0 to n = N − 1, meaning it iterates through all data points once. Inside the loop, f the cross product (Xn + 1⋅Yn − Xn⋅Yn + 1) is calculated, and then the absolute value of the cross product (|…|) is calculated. The dominant operation inside the loop is the calculation of the cross product and absolute value, which can be considered as constant-time operations (O(1)). Since these operations are performed for each of the N data points, the overall complexity is O(N). As a result, the algorithmic complexity of calculating the sway area using the provided formula is linear with respect to the number of data points, resulting in an O(N) complexity.

2.2.3 Fuzzy entropy (FuzzyEn)

FuzzyEn was methodologically based on SampEn [26]. Given a sequence of numbers x = {x(1), x(2),..., x(N)} of length N (60 s in this study), and form vectors using Eqs. 8 and 9.

$$ X_{i}^{m} = \left\{ {x\left( i \right), x\left( {i + 1} \right), . . . , x\left( {i + m - 1} \right)} \right\} {-} x0\left( i \right), i = 1, \ldots .,N - m + 1 $$
(8)
$$ D_{ij}^{m} = \mu \left( {d_{ij}^{m} ,r} \right) $$
(9)

where \({d}_{ij}^{m}\) is the maximum absolute difference of the corresponding scalar components of \({X}_{i}^{m}\) and \({X}_{j}^{m}.\) The percentage of vectors \({X}_{j}^{m}\) that were within “r” of \({X}_{i}^{m}\) with the average degree of membership that offers reliability, especially for short-length data using Eqs. 10 and 11.

$${B}^{m}\left(r\right)={(N-m)}^{-1}\sum_{{\text{j}}=1}^{{\text{N}}-{\text{m}}}\sum_{{\text{j}}=1,{\text{j}}\ne {\text{i}}}^{{\text{N}}-{\text{m}}}{D}_{ij}^{m}$$
(10)
$${A}^{m}\left(r\right)={(N-m)}^{-1}\sum_{{\text{j}}=1}^{{\text{N}}-{\text{m}}}\sum_{{\text{j}}=1,{\text{j}}\ne {\text{i}}}^{{\text{N}}-{\text{m}}+1}{D}_{ij}^{m+1}$$
(11)

FuzzyEn can be estimated using Eq. 12.

$${\text{FuzzyEn}}\left(m,r,N\right)=ln\left[\frac{{A}^{m}\left(r\right)}{{B}^{m}\left(r\right)}\right]$$
(12)

In this study,. FuzzyEn was calculated at m = 2, r = 0.2*SD based on prior studies. Gaussian-like membership function was selected because it aligns with the type of patterns we want to capture. The fuzzy membership function was calculated by Eq. 13.

$${e}^{-{\text{ln}}(2)({x/y)}^{2}}$$
(13)

In Eq. 13, the exponential function e−ln(2)(x/y) [2] results in a rapid decay of membership values as ∣x/y∣ increases. This means that it assigns higher membership values to data points that are closer to the center (x = 0) and lower membership values to data points that are farther away. The term ln(2) in the exponent introduces a rapid decay of membership values as ∣x/y∣ increases. The quadratic term in the exponent (x/y) [2] amplifies differences between x and y, which penalizes values of x/y that deviate from zero. This effectively emphasizes points closer to the center (x = 0). In the term (x/y)2, in practice, we typically chose a nonzero value for “y” to ensure that the function remains well defined. The choice of “y” was based on the desired shape of the membership function's curve.

2.3 Partial least squares-discriminant analysis (PLS-DA)

PLS-DA is a supervised learning algorithm used for classification, but it is specifically designed for high-dimensional datasets with many features [27]. PLS-DA has the advantage of being able to handle multicollinearity among the features and can extract the most relevant features for classification. It combines the principles of principal component analysis and linear discriminant analysis to identify the underlying latent variables that explain the maximum variation in the data and maximize the separation between the classes. PLS-DA may assume multivariate normality within each class and is not as sensitive to the assumptions of normality or homoscedasticity as univariate parametric tests. That is, we can say that it is a dimensionality reduction method that focuses on maximizing covariance between the predictors (features) and the response variable (class labels) while minimizing the effects of multicollinearity. It can be relatively robust to moderate departures from normality like the LDA algorithm. PLS-DA works by projecting the original data into a lower-dimensional space using a set of latent variables and then applying LDA to the projected data to find the linear discriminant function that best separates the classes. For the proposed PLS-DA model, the dataset of 152 subjects was split into training and testing subsets, which were implemented to assess the generalization performance. Separate datasets were created for old and young participants, considering eye conditions (eyes open and closed) on both rigid and compliant surfaces. Each dataset was split into training and cross-validation datasets.

The PLS-DA model mathematically finds a projection that maximizes between-class separation on the global Euclidean scale. X is an n × p matrix representing the predictor variables (independent variables), with n observations and p variables. Y is an n × q matrix that represents the response variables (dependent variables) with q classes. Both X and Y are mean-centered. PLS-DA transforms the predictor variables X and response variables Y into a set of latent variables (LVs) using the following form by Eqs. 14 and 15.

$$X={{\text{TP}}}^{T}+E$$
(14)
$$Y={{\text{UQ}}}^{T}+F$$
(15)

where T and U represent the score matrices that are calculated for each LV and represent the transformed data in the LV space. The goal was to find a set of LVs that maximize the covariance between X and Y to achieve effective class separation. T was obtained by multiplying predictor variable X by weight matrix W (T = XW). P and Q are loading matrices of X and Y. While P loading matrice was obtained by performing weighted linear regressions between X and T, Q loading matrice was obtained by performing weighted linear regressions between Y and U. The loading matrix P was calculated for each LV. The loading matrices represent the importance or contribution of each variable in X to the corresponding latent variables. E and F are the matrices corresponding to the residual matrices of X and Y, respectively. A weight matrix W with “w” and “c” vectors was computed for each LV that was obtained by performing a weighted linear regression between X and Y. The weight matrix represents the importance or contribution of each variable in X to LV.

The number of LVs was determined using two classes (old vs. young) in the response variable. The LVs were constructed by maximizing the covariance between X and Y using the following form by Eq. 16.

$${\left[{\text{Cov}} (t,u)\right]}^{2 }= {{\text{max}}}_{\Vert w\Vert =\Vert c\Vert }{\left[{\text{cov}} ({X}_{w},{Y}_{c})\right]}^{2}$$
(16)

where Cov (T and U) is the sample covariance between the t- and u-score vectors extracted from matrices T and U. The first LV1 was obtained by finding the linear combination of X that maximized the covariance with the linear combination of Y. The second latent variable LV2 was obtained by finding the linear combination of X that was orthogonal to LV1 and maximizing the covariance with the linear combination of Y, and so on. This process was continued until a predefined number of LVs was reached. For discriminant analysis, once the LVs were obtained, the traditional LDA method was applied to classify and discriminate data. To objectively test the generalization performance of the proposed models in predicting balance performance for older people, the hyperparameters of the PLS-DA were performed by selecting two LVs and tenfold stratified cross-validation for internal validation. Samples were usually divided into cross-validation groups based on the Venetian blinds approach. For each sample in the test dataset, the posterior probability of belonging to each class is calculated using Bayes' theorem.

Regarding the PLS-DA algorithm, to calculate the global algorithmic complexity, it was considered several factors including number of latent variables (LVs = 2), number of features (p = 7), size of the dataset (n = the dataset contains 152 subjects, and each subject corresponds to 60 s of data, resulting in a total of 6000 sample points), and cross-validation (k-fold = tenfold stratified cross-validation). During training, PLS-DA computes the LVs, and the number of LVs (2 in this case) affects the computational complexity. Each LV requires iterative optimization, which involves matrix operations. With tenfold cross-validation, the training process is repeated 10 times, each time with a different subset of the data. This introduces an additional factor of 10 in the computational complexity. For each case (for example eyes open on firm surface), there is 643 sample points. We have total four cases (eyes open and close on rigid and compliant surface) and there is totally 2572 sample points, and PLS-DA processes these data points for each fold in cross-validation. Therefore, the size of the dataset contributes to the overall complexity. There are seven features (p) that PLS-DA must operate on these features for each LV and during each fold of cross-validation. Considering these factors, the global algorithmic complexity of PLS-DA can be approximated as general complexity ≈ number of LVs (2) × cross-validation folds (10) × size of the data (n, 2572) × number of features (p, 7). A general estimation of the global algorithmic complexity is calculated as complexity ≈ (2 × 10 × 2572 × 7) computational operations.

2.4 Classification performance of PLS-DA algorithm

The classification results obtained for old and young individuals across different eye conditions (eyes open and closed) on rigid and compliant surfaces were compared. To evaluate the classification performance, a confusion matrix and receiver operating characteristic (ROC) curve were used. Specificity, sensitivity, precision, and accuracy were derived from the confusion matrix, as shown in Table 2.

Table 2 Confusion matrix with two classes
Full size table

where TP (true positive) is the number of old subjects correctly classified as old, TN (true negative) is the number of young subjects correctly classified as young, FN (false negative) is the number of old subjects incorrectly classified as young, and FP (false positive) is the number of young subjects incorrectly classified as old. Sensitivity, specificity, precision, and accuracy were calculated from the confusion matrix as sensitivity = TP/(TP + FN), specificity = TN/(FP + TN), precision = TP/(TP + FN), and accuracy = (TP + TN)/(TP + TN + FP + FN), respectively. The nonerror rate (NER) was calculated as the arithmetic mean of all class nonerror rates, which was calculated as the average of its specificity and sensitivity. The error rate (ER) was the complement of the NER and was calculated as ER = 1 − NER. Receiver operating characteristic (ROC) curves were then calculated separately for each class. The area under the ROC curve (AUC) was derived from the ROC curve.

Figure 1 shows a general flowchart of the proposed classification model in the present study.

Fig. 1
figure 1

Flowchart of the classification model

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2.5 Statistical analysis

In recent years, to improve the evaluation process of the performance of any ML model, statistical tests have been employed to evaluate which algorithm is superior to the other. Friedman statistical test was performed for analysis of PLS-DA classification results over three feature sets by comparing four different conditions. To provide a more comprehensive analysis of the data, we examined the differences in traditional CoP parameters and entropy measurements between the young and old groups across four different conditions: eyes open on a rigid surface, eyes closed on a rigid surface, eyes open on a compliant surface, and eyes closed on a compliant surface, by using an independent student t-test. The Shapiro–Wilk test was performed to evaluate the normality of the data. A test for homoscedasticity (Levene’s test) was also performed on all variables. For stationarity test, we performed a stationarity test for the data and all variables by using in SPSS for visual (graphs) examination. We checked the line graphs for any noticeable trends, seasonality, or patterns. For all statistical tests, IBM SPSS Statistics version 22 (IBM Corp., Armonk, NY, USA) was used. A significant level was considered p ≤ 0.05.

3 Results

3.1 Comparison results for PLS-DA

To understand the impact of age on postural stability and identify potential differences between young and old individuals, the present study explored the application of PLS-DA algorithms for classifying CoP data under various conditions, including eyes open and closed, on both rigid and compliant surfaces. The classification results obtained for old and young individuals under different conditions (eyes open and closed) on rigid and compliant surfaces were compared.

A hybrid approach was employed for feature extraction, in which relevant features were extracted from the CoP data to capture distinctive patterns and variations associated with different conditions. To characterize postural sway, time-domain variables, such as sway area and sway velocity, and nonlinear characteristics, including entropy measures, were used. Selected variables with scaled average values were considered optimal features for separating the two considered classes (old vs. young). Figure 2 shows the variable profiles for the young and old groups.

Fig. 2
figure 2

Averages of selected variables for old (red) and young (blue dot) classes

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Concerning both time-domain variables in Fig. 2a, they reflected the capability of each variable alone to separate the old and young classes. For sway velocity, the old group showed higher average values than the young group in all eye and surface conditions. While the old group showed a lower sway area than the young group when the eyes were open, it was higher than that in the young group when the eyes were closed on a compliant surface. Concerning SampEn and FuzzyEn measurements in the A-P and M-L directions, the old group had higher average values than the young group, apart from the entropy averages on the compliant surface in the M-L direction. Therefore, a good classification performance was expected when all variables were considered together in the PLS-DA model. Following PLS-DA, scores (describing the samples) were obtained. The score plot shows the distribution and separation of samples. Figure 3 shows the PLS-DA score plots for the young and old classes.

Fig. 3
figure 3

Score plots of LV2 vs. LV1 in the proposed PLS-DA model. It represented the young group class in a line with red color and the old group in a dotted line with blue color. The feature set from time domain alone, entropy measurements, and hybrid (both time-domain and entropy variables) are compared in a eyes open on rigid surface condition, b eyes closed on rigid surface condition, c eyes open on compliant surface condition, and d) eyes closed on rigid compliant surface condition

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In Fig. 3, the score plot of the selected LV2 vs. LV1s is provided to visualize the discrimination power of the PLS-DA model and identify young and old group classes with their patterns in the data. The LV1 axis represents the main source of variation and differentiation between old and young groups. The separation along this axis indicates the largest difference between the groups. LV2 contributes to the further separation of the samples within each group in the positive and negative directions. Therefore, we can observe the relationships between samples and extract meaningful insights related to selected variables (time domain alone, entropy alone, and hybrid feature set) in four eye and surface conditions.

The score plots show how well the PLS-DA model was able to separate two different classes in all conditions in terms of the between-group class separability of each feature set. While the time-domain parameters were found with a certain degree of overlap in Fig. 3a(i, ii, and iii), Fig. 3b(i), and Fig. 3d(ii), a greater degree of separation was observed in Fig. 3c(i, ii, and iii). As expected from the classification performance, in almost all Figures, the old group with hybrid parameters showed larger scores on the LV1 than the young group, except for Fig. 3d(ii and iii). Furthermore, LV2 showed that variables from the young groups had negative scores, and the variables from the old group had positive scores. These patterns can be interpreted as variables that have a positive effect on determining the young group and discriminating it from the old group. The old group was generally characterized by high values of all selected variables. The variable averages for the young and old group classes shown in Fig. 2 confirmed this result (similar to the exception in Fig. 2d(ii)).

3.2 Analysis of classification performances

Table 3 shows the overall accuracy rates in the training and cross-validation data that provide an understanding of how well the model performs in terms of correctly classifying old and young samples across old and young classes.

Table 3 Overall accuracy rates of the PLS-DA algorithm in training and cross-validation for
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In Table 3, a higher overall accuracy indicates a higher proportion of correct predictions, in which the model performed well in general. The best classification performance was found as 89% for training and 88% for cross-validation in the eyes open on the compliant surface. Under almost all conditions, the proposed PLS-DA model using the hybrid feature set exhibited a much higher overall accuracy.

While overall accuracy provides a general measure of the model performance, other evaluation metrics, including sensitivity, specificity, and precision, were also considered for evaluating the performance of the PLS-DA model using multiple metrics that provided a more comprehensive understanding of its strengths and weaknesses. Table 4 shows the specificity, sensitivity, and precision metrics that provide valuable information on the performance of the proposed PLS-DA model in training and cross-validation data, specifically in terms of its ability for each feature set under four different conditions. These metrics were calculated from the confusion matrix that captured how many young and old group samples were correctly or erroneously assigned to young and old group classes in the training and cross-validation.

Table 4 Classification performance results of the proposed PLS-DA model in training and cross-validation
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As shown in Table 4, higher specificity rates indicated a lower rate of false positives in the PLS-DA model, which was good at identifying young and old samples. A higher sensitivity indicated a lower false negative rate of the model that was good at capturing positive old and young samples and had a high recall rate. Higher precision indicated a lower rate of false positives, meaning that the model made fewer incorrect positive predictions. When all feature sets were compared under different conditions, it was noted that the hybrid feature set with eyes open on a compliant surface indicated the best classification performance for both the old and young groups. For the old group, while the results indicated 93% sensitivity, 94% specificity, and 93% precision in the training data, the results indicated 88% sensitivity, 93% specificity, and 91% precision in cross-validation data.

3.3 ROC curves

ROC curves were also used to compare the performance of the PLS-DA between the old and young groups regarding three feature sets and four sensory input conditions. A perfect classification method would yield a point in the upper left corner of the ROC space, representing maximum sensitivity and specificity, while a random classification would yield points along the diagonal line from the left bottom to the top right corner. Figure 4 and Fig. 5show the ROC curves for the older group and young group, respectively.

Fig. 4
figure 4

ROC curves for the old group class

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Fig. 5
figure 5

ROC curves for the young group class

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The ROC curves for both classes were acceptable and very good. The AUC value shows the overall performance of the model across all the conditions and feature sets. An AUC between 0.7 and 0.8 was considered fair, while an AUC between 0.8 and 0.9 was considered good. A model with an AUC above 0.9 was considered excellent, indicating a high degree of predictive power. In the present study, the overall AUC values between good and excellent scores indicated that the PLS-DA model was able to discriminate between the old and young group samples. For eyes open on a compliant surface, the AUC value revealed the highest performance, with a time domain of 0.96, entropy of 0.86, and hybrid feature sets of 0.96.

3.4 Statistical tests

Overall, our findings suggested that age-related differences in sway velocity, SampEN, FuzzyEn (both A-P and M-L directions), and sway area were observed in one out of the four conditions (See Appendix 1). However, for the sway area, no significant differences were detected in the eyes open and closed on rigid surface conditions. The results of the Shapiro–Wilk test showed that the data were not normally distributed (p ≤ 0.05). The result of homoscedasticity (Levene’s test) test was found p > 0.05 for Sway Velocity_X, Sway Velocity_Y, and SampEn_X. Variances across groups were assumed to be equal that was it suggested homoscedasticity. On the other hand, it was found p < 0.05 for SampEn_Y, FuzzyEn_X, and FuzzyEn_Y. It indicated heteroscedasticity. For stationarity test, visual (graphs) examination indicated that all variables seems stationary. Based on normality tests, a Friedman statistical test was performed as a nonparameter statistical tool for analyzing overall accuracy and AUC results obtained for the PLS-DA model. While there was a statistically significant difference in overall accuracy depending on measurements under four different conditions (χ2 (f) = 8,793; p = 0.032), there were no significant differences in AUCs (χ2 = 5,80; p = 0.122). Additionally, we compared three feature sets of the PLS-DA model across four different conditions within each group (see Appendix 2).

4 Discussion

The novel contribution of the present study was to investigate the classification performance of the PLS-DA model in classifying age-related postural sway changes in altered sensory test conditions, including eyes open/closed and standing on rigid and compliant surfaces. Three different feature sets (time domain, entropy measurements, and hybrid approaches including time-domain and entropy variables) were also compared for model performance describing the ability to maintain balance in different age groups. Rather than a time-domain and entropy feature set alone, a hybrid COP feature in eyes open and compliant surfaces revealed the best performance in discriminating older people from young people.

4.1 Comparison of the classification performance of the PLS-DA model

This study mainly focused on implementing a PLS-DA algorithm that handles high-dimensional and multilinear data by providing a solution to the complexities of postural control analysis rather than performance comparisons of different ML algorithms. It is noteworthy that while many traditional ML algorithms have contributed significantly to the field of postural control analysis, our study introduces PLS-DA as a promising alternative. PLS-DA has been also a widely used ML algorithm in various fields. According to our knowledge, this algorithm has not previously been applied to the classification of age-related COP features. This contribution highlights the innovative nature of this research, which not only advances the understanding of age-related postural control but also introduces how the PLS-DA algorithm can be a practical and effective methodology for clinicians and researchers alike. Additionally, this study suggests a small hybrid feature set that combines traditional statistical metrics with nonlinear dynamics. This feature selection approach can build a balance between model simplicity and predictive power, enhancing classification accuracy. The present study revealed that the high overall accuracy rates, such as 89% for training and 88% for cross-validation, indicated that the PLS-DA model performed exceptionally well in distinguishing between old and young groups across different sensory conditions. Notably, the PLS-DA model's performance was consistently better when using the small hybrid feature set, demonstrating the significance of combining time-domain and entropy measurements. This result suggests that considering both traditional postural sway analysis and entropy measurements can contribute to a more accurate classification. Indeed, there have been many contradictory results in the ML classification when comparing studies in the literature. Regarding classifier performance, many factors may affect the results, which may depend on the dataset, hyperparameter settings, and predictor variables [28]. Therefore, in this study, we compared the performance of various ML classification models using consistent procedures and the same dataset, especially in comparison to feature sets. However, no prior studies have used the PLS-DA model to classify CoP time series between young and older people. Based on the same dataset, the classification accuracies of the proposed PLS-DA using a smaller feature set outperformed other common ML models in the literature.

Giovanini et al. distinguished between different age groups using two different CoP time series with different durations. [29] They used the same data as the dataset [23]. However, unlike the present study, they computed very large feature sets, including temporal (mean distance, root mean square (RMS) distance, mean velocity, RMS velocity, standard deviation (SD) of velocity), spatial (sway path, length of COP path, excursion area, total mean velocity), spectral (mean frequency, median frequency, spectral power with 95%), and nonlinear features (SampEn of distance and velocity, multiscale sample entropy (MSE) of distance and velocity, scaling exponent of velocity, Hurst exponent of distance). They applied six common ML models with specific configurations and hyperparameters: SVMs, k-NN, NB MLP, RF, and decision tree (DT). They found that a sampling duration of 60 s provided more discriminative information than the other durations. The overall classification accuracy of all ML models was 61.3% [range = 57.0—64.9] for dataset 1 and 67.8% [range = 57.9—71.2] for dataset 2. The overall mean values of all ML models were found smaller for 30 s in duration which affected the CoP time series duration in different age groups. Reilly examined an effective feature selection procedure for more accurate and reliable classification results among older subjects with fall risk [30]. The same dataset was used for human balance evaluations [23]. This study extracted time-domain features (mean, SD, minimum, maximum, range and RMS path length, resultant distance, mean distance, RMS distance, total excursion, mean velocity, 95% confidence-circle area, mean rotational frequency of the COP, fractal dimension, and 95% confidence-circle of the fractal dimension) and frequency-domain features (total power of the frequency spectrum, mean frequency, median frequency, 95% power frequency, and peak frequency) were extracted from CoP. A total of sixty-seven features were identified using the Genetic Algorithm and ReliefF selection, and 18 features were obtained using the self-evaluating feature evaluation (SAFE) method. MLP, SVM, NB, and k-NN models were used to evaluate the performance of the selected time–frequency features. It was found that the mean classification accuracy of MLP was 80% [range = 61–96], SVM was 75% [range ge = 57–87], k-NN was 75% [range ge = 67–93], and NB was 77% [range = 67–93]. All the ML models had low reliability and high variability. Çetin and Bilgin differentiated between young and aged groups using the same dataset [31]. Feature set was extracted from traditional time-dependent variables including the mean standard deviation values of CoP and Force (F) change in the A-P and M-L directions. They used the DT, linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), logistic regression, SVM, and k-NN classifiers. They observed that force signals generally provided more successful results than COP signals. The SVM model exhibited the highest accuracy (81.67%) to separate the young and older groups. Ren et al. evaluated Mini-Balance Evaluation Systems Test scores and their subscores by using Artificial Intelligence [32]. They used also the same public dataset. They extracted and mapped thirty-one common features from the time and frequency domains of the CoP data for balance performance evaluation. They also developed twenty-five other feature sets extracted from the pixel-based CoP displacement diagram (PCPDD) for the CoP trajectory. New features were extracted from the time and frequency domains of the PCPDD. A total of 224 features from all experimental conditions were calculated. However, they performed regression analysis rather than classification for balance performance. They used five types of commonly used regressors: RF, MLP, RB network (RBFN), linear regression, and k-NN regression. They found that low mean absolute errors showed the reliability of AI techniques for assessing balance control subsystems.

Indeed, the advantages and disadvantages of many ML methods over each other are still a subject that has been widely discussed in the literature. In this context, some important considerations and the potential limitations were addressed for the PLS-DA model such as relying on certain assumptions about the data such as linearity between predictors and class labels, handling multicollinearity among predictor variables, using dimensionality reduction technique, being algorithmically unstable, about the generalizability [33]. However, it is important to note that the PLS-DA algorithm can be always inappropriate or ineffective. Additionally, the choice of classification method can rely on the specific context of the problem, the nature of the data, and the goals of the analysis. In this study, we attempted to use the PLS-DA model which can be a valuable ML algorithm, especially in situations where its strengths align with the characteristics of the CoP data and our enhancing feature set objectives. In the context of CoP data, it is common to have multiple features that may be correlated. PLS-DA can help extract latent variables that capture the shared information among these features, reducing multicollinearity and improving model stability. In cases where we used a small hybrid feature set, this reduction could be particularly valuable and provide interpretable results in terms of the contributions of latent variables and the importance of features. This could be advantageous when we wanted to understand which features were driving the differences between classes (e.g., young vs. old) in CoP data under different conditions (eyes and surface). Therefore, we considered the trade-offs between complexity and interpretability when selecting this PLS-DA classification algorithm.

4.2 Hybrid features in ML models

Recently, the quantification of balance has been investigated using CoP data recorded in quiet stance to discriminate older subjects from young, as well as fallers from nonfallers [33]. Quantitative age-related changes in the values of crucial measures could be used in diagnosing the deteriorations and diseases in aging and recommendations for treatments [35]. A large number of variables derived from the trajectory of the COP can be computed in either the A-P or M-L directions in time, frequency, time–frequency, and state-space models. This significantly augments the number of variables that can be considered, thereby resulting in statistical challenges associated with data dimensionality. Indeed, the number of features (also known as variables or input dimensions) in ML models can affect classification performance. A smaller feature set implies a lower-dimensional input space, which can reduce the complexity of the classification problem. With fewer features, the algorithm has fewer relationships to learn and less computational burden. A smaller feature set may contain only the most relevant and informative features for classification. The linear and nonlinear features of a CoP time series can capture the essential characteristics and patterns of age-related changes, leading to a more concise representation. Therefore, a smaller hybrid feature set included the linear and nonlinear components of postural sway for the ML classification. By contrast, a larger feature set can include more diverse and complex patterns. There may be a reduced risk of over-fitting, where the model becomes overly specialized for the training data and fails to generalize well to the test data (cross-validation). Small features may also help prevent the model from memorizing the noise or idiosyncrasies in the training set. Conversely, a larger feature set increases the risk of over-fitting, particularly if the dataset is small relative to the number of features. Regularization techniques and larger training datasets can help mitigate this issue. Currently, the development of wearable devices, long-term, and real-time data storage, and monitoring has become an essential issue. In this sense, the computational efficiency of ML models is an essential parameter for performance. Training models with a smaller feature set are generally faster and require fewer computational resources. With fewer dimensions, the algorithm can process data more efficiently, resulting in shorter training times and faster inference. For monitoring, the relationships between a reduced number of features and the classification outcome can be easier to comprehend and explain. In contrast, a larger feature set may increase the computational complexity of models that are harder to interpret and may require more advanced algorithms or distributed computing frameworks for efficient processing, making it challenging to understand the underlying decision-making process.

In the present study, feature selection methods were not used. The variables were selected according to those commonly used in the literature and were sufficiently described in detail to be reproducible. Sway area and velocity were considered as two common features calculated by the CoP as a time-domain feature set. For real-world use of postural sway data (biological dynamic data), we also used entropy measures that means order regularity or complexity. The idea of entropy was to reveal nonlinear dynamics of CoP time series so that data with repeating elements arise from more ordered systems, and would be reasonably characterized by a low value of entropy. In the literature, there have been a number of studies on dozens of entropy measures including major algorithms and their modified or derived versions [36]. For example, Shannon entropy, Rényi entropy, permutation entropy, approximate entropy, sample entropy, multiscale entropy, fuzzy entropy, dispersion entropy, slope entropy, bubble entropy, distribution entropy, phase entropy, spectral entropy, average entropy, entropy of entropy, differential entropy, bivariate entropy, and diffusion entropy [36,37,38,39]. However, in this study, entropy measures used infrequently were not considered. SampEn and FuzzyEn were performed to access CoP variability in order for objective results to be paid when driving conclusions based on the observation from a single measure, as well as their simplicity, widespread applicability, and advantages when used with careful consideration of previous studies as guidelines to select appropriate parameter values. Because it is well known that, the correct choice of the embedding parameters "m," tolerance range "r," and data length "N" is a important issue in entropy measurements. In fact, it is stated that SampEn is extremely sensitive to these characteristic quantities. Additionally, defining the brevity of physiological time series can be highly controversial. For example, a very short time series inherently will provide fewer data points for analysis, which can lead to increased variability in the calculated entropy measures. In a “very short time series,” there are fewer possible subsequences to match, resulting in a smaller number of template matches. This can lead to increased variability in the entropy estimates, making it challenging to obtain stable and reliable results. Therefore, when using datasets of less than 200 points for either SampEn or FuzzyEn, it has been suggested that special attention should be paid [40]. According to previous studies (in the literature), for clinical data, typically, “m” is to be set at 2, “r” to be set between 0.1 and 0.25 times the standard deviation of the data, and “N” as 1000 [25, 41,42,43,44]. However, there is still no consensus on determining these quantities, especially in short-term data [25, 45]. Although some studies introduced particular entropy algorithms to make them less sensitive input parameters [45, 46], there has yet to be a study that investigates this issue in human movement data specifically with gait data while the sensitivity of different entropy algorithms to changing parameters has been investigate. Therefore, we should consider that the choice of parameters used is critical for these types of datasets, in particular short data. In addition, it has not been agreed upon which algorithm performs consistently independent of parameter choice. Although SampEn has still disadvantages [40], in our study, we focused on the relative consistency of SampEn and FuzzyEn algorithms with commonly used parameters. Because we considered that if true differences exist, the selected algorithm should be able to discriminate between groups (young and old) and sensory input conditions (eyes and surface) and maintain relative consistency as the parameter is changed slightly.

As mentioned above, of course, dozens of different entropy measurements that can provide potentially complementary or highly correlated information in the CoP signal could also be applied to the ML algorithm. However, rather than using feature sets extracted from time-domain or nonlinear analysis alone, enriched hybrid features with reduced numbers were identified as relevant to balance assessment in the literature, as mentioned above. The hybrid feature set exhibited the best classification performance when comparing the traditional time-domain and entropy measurements alone. In recent years, wearable sensors and motion capture systems have become increasingly common for assessing postural stability. The unmet need for reliable approaches to extract physiologically meaningful information from static posturography is limited. Furthermore, the study of biomarkers of CoPs in aging is an emerging field, and research in this area is still ongoing. One study indicated that linear CoP measures alone were unable to differentiate older fallers from older nonfallers, but that the fallers had significantly larger SampEn values than nonfallers [47]. To avoid over-fitting, the cost of high-dimensional data, and lack of domain-specific relevance, a small feature set was chosen for its memory-efficient nature, which could be especially optimal for clinical practice, real-life applicability in daily life or home-based environments, and wearable systems. This does not require significant computational resources and time, unlike large and complex feature sets. The proposed hybrid feature set is easily interpretable and domain specific. This can reveal the interpretability of the resulting ML modes and an understanding of the underlying age-related factors influencing the CoP time series. Several previous studies have also reported ML models to predict fall risk using a hybrid feature set extracted from postural sway measurements. Sun et al. performed an RF classifier model to assess the accuracy and feature importance of various postural sway features to differentiate multiple sclerosis (MS) groups from healthy controls [48]. Twenty common sway measures including nonlinear (SampEn in A-P direction) and time-domain (sway range, sway path length, mean velocity in M-L direction, and sway area) features were extracted. They found that the RF classifier had a high classification accuracy of over 86% for differentiating low-risk MS and low-risk MS individuals from controls. Montesinus et al. investigated the ability of AmpEn and SampEn measures with different input parameters to discriminate age-related COP time series [47]. Their feature set included linear COP displacement measures (amplitude, SD, and mean velocity in the A-P and M-L directions, total length, total mean velocity, and area) and nonlinear measures (approximate entropy (ApEn) and SampEn). They used only statistical analyses. They found that SampEn could discriminate between groups and was more consistent than ApEn.

In the present study, the sway area in the older groups tended to be larger and velocity tended to be reduced when compared to the young groups, particularly on the compliant surface. Similarly, SampEnd and FuzzyEn were larger in older adults than in younger adults. This finding is consistent with previous studies that reported that decreased regularity (larger entropy values) might be reflective of poorer balance performance via reduced balance adaptability [49, 50]. Entropy measures such as SampEn and FuzzyEn provide potential information about the complexity and unpredictability of CoP signals. Recent nonlinear measurements have defined the application of the dynamic approach to the evaluation of CoP signals, which reveal the progression of motor behavior over time. These nonlinear measures allow for the regularity, adaptability, stability, and quantification of complexity by demonstrating the ability to capture the temporal direction of CoP displacement variations. Consequently, the analysis presented in this study proved that there was an increase in hybrid CoP measures in the older group compared to the young group. This could be useful in determining the differences in balance between young and older adults and addressing increased fall risk.

4.3 Effect of sensory ınputs

In this study, the hybrid features calculated from raw CoP data were essential for predictive model development. The differences in CoPs between young and older people suggest that balance and stability may decline with age. However, these differences can also be influenced by multiple factors, such as sensory input conditions, restrictive factors, and environmental factors. Maintaining balance and postural control is not controlled by a single system or set of reflexes. Both should be considered complex motor skills formed by the interaction of multiple sensory and motor processes. Sensory information from somatosensory, vestibular, and visual systems is integrated, and the relative weights placed on each of these input data depend on the objectives of the motor task and the environmental context [51]. The alteration in the sensory environment can lead to a re-weighting of the brain's relative reliance on each of the senses. For example, under normal conditions, healthy individuals who stand in a well-illuminated environment on a stable surface primarily benefit from the somatosensory system, as opposed to the visual and vestibular systems [52]. Conversely, when standing on an unstable surface, the importance of visual and vestibular information increases as reliance on surface inputs diminishes to maintain postural control. Because individuals reside in an ever-changing environment, the ability to recalibrate the weighting between these senses, which varies based on the sensory context, is crucial for ensuring stability. Age-related physical and sensory changes can disrupt the sensory recalibration abilities of older individuals and affect their postural stability [51]. Older adults may use different strategies to maintain balance. The main reason could be that older adults may rely more on feedback from their feet and other sensory systems to maintain balance compared to young adults who may rely more on feed-forward mechanisms.

This study aimed to understand the impact of concurrent sensory feedback on balance with aging. Looking at the literature, there have been various findings about the effect of concurrent sensory feedback [3]. In the present study, it can be noted that the PLS-DA model with hybrid features was more sensitive to the somatosensory feedback absence (unstable surface) than the visual feedback absence. In other words, it can be seen that the features revealed more distinguishing physiological information regarding the classification power of the model in eyes open to compliant surface conditions compared with other conditions. The older subjects were characterized by higher entropy values when compared to young people with eyes open and closed on rigid (stable) surfaces, but lower entropy values were obtained in eyes open on a compliant (unstable) surface. Similarly, time-domain variables, such as sway area and sway velocity in the A-P and M-L directions, presented higher values in the eyes-closed condition than in the eyes-open condition. These findings are consistent with the age-related changes in CoP reported in the literature [3, 53]. The absence of visual control in older and younger people also translated into lower AUC values for PLS-DA models on compliant surfaces but higher values for rigid surfaces. When comparing eyes open and closed, older groups may rely less on visual cues or make fewer adjustments to their CoP signals. The reason for these results can be explained by the decline in somatosensory functions of individuals over sixty years of age [54]. Therefore, older adults may represent a more regular pattern for CoP variability, as well as a larger body sway when compared to younger subjects [55]. Understanding the effects of somatosensory inputs on postural stability can help inform fall prevention strategies and interventions for older individuals, emphasizing the importance of balance training, muscle strengthening, and sensory integration to mitigate fall risk when encountering unstable surfaces.

4.4 Clinical significance

Maintaining balance control requires the proper functioning of the sensory systems, spinal cord, and central nervous system (CNS). These systems process signals and subsequently facilitate accurate muscle activity and structural responses in accordance with the directives of the CNS or spinal cord. Any limitations in the aforementioned structures involved in balance control can lead to an alteration in the manner in which individuals manage their stability, ultimately resulting in a detrimental impact on the overall balance control system. Therefore, the process of aging could have an impact on sensory integration across several parameters. For example, the process of aging might impact the central mechanisms that are responsible for combining sensory information from many sources. The intricate interplay between the somatosensory, visual, and vestibular systems in maintaining balance is a fundamental aspect of human locomotion, and disruptions in this interplay can have profound implications, particularly as individuals age. Additionally, aging might also impair the ability to accurately perceive sensory signals due to the degradation of the individual sensory systems [56].The posture problems might also occur as a result of reduced visual acuity, vibratory and proprioceptive loss, as well as decreased vestibular and cerebellar function. As a result, age-related alterations in peripheral sensory and cerebral functions lead to decreased reaction time and neuromuscular performance in older individuals [57]. These changes might undermine the ability to adapt to changes in posture and the environment, leading to inadequate performance in postural control.

The investigation on the influence of sensory inputs on postural stability in the study can hold great significance for clinical considerations. The absence of somatosensory feedback on unstable surfaces exhibited a greater sensitivity in the model, which is clinically relevant. These adjustments with the fact that older adults often rely more on proprioceptive inputs from the feet and lower extremities to maintain balance. Interventions that aim to improve proprioception and sensory integration, such as exercises that target foot and ankle proprioceptors, may play a crucial role in mitigating age-related postural challenges. For example, incorporating exercises that simulate uneven or compliant surfaces into rehabilitation programs may better prepare older adults for the diverse terrains they navigate in daily life. This personalized approach can expand with the broader shift in health care toward precision and individualized care. Additionally, falls are a significant health concern among older adults, often leading to severe consequences. Understanding the heightened sensitivity to somatosensory feedback in unstable conditions can allow clinicians to strategically focus on interventions that address this specific vulnerability. Targeting proprioceptive training and enhancing sensory awareness may be pivotal in fortifying individuals against the risk of falls, promoting independence, and ultimately improving their quality of life. By integration of wearable devices and advanced monitoring systems into clinical practice, real-time assessments of postural stability, guided by the model's sensitivity to sensory inputs, could offer dynamic and data-driven approaches to intervention planning. Regarding methodologic aspect, the hybrid characteristics proposed in this investigation offer a promising avenue for clinicians, providing a succinct yet potent instrument for evaluating age-related fluctuations in postural equilibrium. The findings of the study can emphasize the significance of a multidimensional approach to the analysis of postural control, integrating both conventional measurements and advanced computational models to obtain a comprehensive comprehension of balance in the aging population.

4.5 Limitations

This study has some limitations. The first limitation was that only the PLS-DA model was assessed. The performance of the model was not compared to any other common ML models such as SVMs, k-NN, and MLP. The performance of the proposed PLS-DA model depends not only on the features and dataset but also on the hyperparameters of the model because it can affect the performance, computational efficiency, interpretability, and robustness of the model. The second limitation was the selection of appropriate common hyperparameters. However, different hyperparameters can be tested to improve the classification performance and improve the generalization of new data. Entropy algorithms also depend on the input parameters that affect the results and their interpretations. The sensitivity of input parameters issue is not only unique to SampEn and FuzzyEn but also common concern in many entropy-based analyses. In the present study, common input parameters were selected for entropy measurements instead of testing different variations. These parameters were chosen based on prior research and best practices for analyzing for clinical data, “m” is set at 2, “r” is set between 0.1 and 0.25 times the standard deviation (SD) of the data. Similarly, for FuzzyEn calculation, other fuzzy membership functions, such as triangular or trapezoidal, could also be used. We did not consider a part of the sensitivity analysis, where we assess how different choice of membership function affects the results of the analysis. We aimed to indicate how a FuzzyEn with Gaussian membership function provides meaningful discriminates age-related patterns effectively for our specific research question. In the future study, we can aim to compare which input parameters in which entropy algorithm indicates relative consistency and which algorithm provides the best discrimination between groups. Finally, only the CoP data were considered. To avoid a large feature set, feature sets were extracted only from the time-domain and entropy measurements. However, this hybrid feature set could include any other relevant data, such as demographic information or clinical assessments. While this study provides important insights, there are some avenues for future research in this area. Further investigation should focus on validating the findings in a larger and more diverse population sample to enhance the generalizability of the results and examine the predictive capabilities of PLS-DA-based classification in tracking changes in fall risk and aging-related balance control over time.

5 Conclusıon

The present study highlights the robust classification performance of PLS-DA in effectively detecting age-related changes. By harnessing the synergistic potential of small hybrid features extracted from the analysis of CoP in the time-domain and entropy measurements, the accuracy and reliability of the classification process were significantly enhanced. These findings could not only underscore the robustness of PLS-DA but also provide important insights into further advancements in enhancing balance control in aging populations and personalized interventions for fall prevention strategies.