Open AccessArticle

An Approach for Detecting Parkinson’s Disease by Integrating Optimal Feature Selection Strategies with Dense Multiscale Sample Entropy

Minh Tai Pham Nguyen

Minh Khue Phan Tran

Tadashi Nakano

³,

Thi Hong Tran

³ and

Quoc Duy Nam Nguyen

^3,*

Faculty of Advanced Program, Ho Chi Minh City Open University, Ho Chi Minh City 700000, Vietnam

Faculty of Information Technology, Ho Chi Minh City Open University, Ho Chi Minh City 700000, Vietnam

Department of Core Informatics, Graduate School of Informatics, Osaka Metropolitan University, Osaka 558-8585, Japan

Author to whom correspondence should be addressed.

Information 2025, 16(1), 1; https://doi.org/10.3390/info16010001

Submission received: 5 November 2024 / Revised: 9 December 2024 / Accepted: 17 December 2024 / Published: 24 December 2024

(This article belongs to the Special Issue Feature Papers in Artificial Intelligence 2024)

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Graphical abstract
"> Figure 1
(a) Gender and label distribution across each class in severity classification task; (b) gender and label distribution across each class in PD classification task. "> Figure 2
The methodology procedure has four stages: preprocessing, feature extraction, feature selection, and classification. "> Figure 3
The process of signal division using the time-slicing window method and outlier removal through the quartile approach and histogram analysis. "> Figure 4
Boxplot analysis of model consistency for PD and severity classification tasks. "> Figure 5
(a) Adjusted p-value matrix of paired T-tests comparing accuracies between classifier–feature selection method pairs (M-SamEn); (b) adjusted p-value matrix of paired t-tests comparing accuracies between classifier–feature selection method pairs (DM-SamEn). "> Figure 6
(a) Correlation matrix of the feature set extracted using the M-SamEn method; (b) correlation matrix of the feature set extracted using the DM-SamEn method. "> Figure 7
(a) Distribution of selected features by signal source (original and computed signals from Equations (<a href="#FD1-information-16-00001" class="html-disp-formula">1</a>)–(<a href="#FD3-information-16-00001" class="html-disp-formula">3</a>) after the feature selection stage; (b) distribution of selected features by feature extraction method after the feature selection stage. "> Figure 8
Comparison of feature count across feature selection methods (*: p-value < 0.05; **: p-value < 0.01). ">

Versions Notes

Abstract

Parkinson’s disease (PD) is a neurological disorder that severely affects motor function, especially gait, requiring accurate diagnosis and assessment instruments. This study presents Dense Multiscale Sample Entropy (DM-SamEn) as an innovative method for diminishing feature dimensions while maintaining the uniqueness of signal features. DM-SamEn employs a weighting mechanism that considers the dynamic properties of the signal, thereby reducing redundancy and improving the distinctiveness of features extracted from vertical ground reaction force (VGRF) signals in patients with Parkinson’s disease. Subsequent to the extraction process, correlation-based feature selection (CFS) and sequential backward selection (SBS) refine feature sets, improving algorithmic accuracy. To validate the feature extraction and selection stage, three classifiers—Adaptive Weighted K-Nearest Neighbors (AW-KNN), Radial Basis Function Support Vector Machine (RBF-SVM), and Multilayer Perceptron (MLP)—were employed to evaluate classification efficacy and ascertain optimal performance across selection strategies, including CFS, SBS, and the hybrid SBS-CFS approach. K-fold cross-validation was employed to provide improved evaluation of model performance by assessing the model on various data subsets, thereby mitigating the risk of overfitting and augmenting the robustness of the results. As a result, the model demonstrated a significant ability to differentiate between PD patients and healthy controls, with classification accuracy reported as ACC [CI 95%: 97.82–98.5%] for disease identification and ACC [CI 95%: 96.3–97.3%] for severity assessment. Optimal performance was primarily achieved through feature sets chosen using SBS and the integrated SBS-CFS methods. The findings highlight the model’s potential as an effective instrument for diagnosing PD and assessing its severity, contributing to advancements in clinical management of the condition.

Keywords:

Parkinson’s Disease detection; dense multiscale sample entropy; feature selection strategy; feature selection process; machine learning

Graphical Abstract

1. Introduction

Neurodegenerative diseases (NDDs) include a variety of diseases characterized by a gradual decline in nervous system structure and function. Parkinson’s disease (PD), a significant movement disorder, affects primarily older people, with an estimated 4 to 6.5 million affected individuals worldwide, including approximately 1 million instances in the United States, accounting for almost 1% of those over 60 [1]. Annually, the United States documents 60,000 new diagnoses of Parkinson’s disease. PD is diagnosed with tremors, rigidity, poor balance, and slow movement, which can cause major problems with walking, a higher risk of falling, and less independence. Although most cases seem sporadic, roughly 15% of PD occurrences exhibit familial patterns, with around 10% associated with certain genetic variations [2].

Gait analysis is important for finding biomechanical problems related to Parkinson’s disease (PD). Furthermore, metrics such as vertical ground reaction force (VGRF) signals, stride lengths, and swing timings can be beneficial in early diagnosis. Such evaluations are essential for the implementation of prompt therapies to potentially slow the progression of PD. Interpreting these measurements typically requires specialized knowledge, which presents difficulties for less experienced physicians. This problem indicates the urgency of algorithms that directly employ raw VGRF data to improve clinical evaluations, standardize diagnostic procedures, and increase accessibility for physicians and engineers [3].

Early research has focused on developing effective methods for the diagnosis and assessment of the severity of Parkinson’s disease. For instance, Aşurolu et al. attained an accuracy of 99.5%, a sensitivity of 98.7%, and a specificity of 99.5% by applying a CNN combined with a locally weighted random forest [4]. Another study by Zhao et al. and El Maachi et al. used CNN-based models to tell the difference between people with Parkinson’s disease and healthy people, and both studies obtained precision rates of more than 90% [5,6]. In addition, in another work [7], Fast Fourier Transform (FFT) was used in conjunction with an Artificial Neural Network (ANN) to classify the severity, resulting in an accuracy of 97%. Another approach proposes a deep learning architecture, “NDDNet”, aimed at resolving this issue, achieving an average accuracy of 96.75% in identifying three categories of neurodegenerative diseases [8].

The Dense Multiscale Sample Entropy (DM-SamEn) method builds upon the conventional Multiscale Sample Entropy (M-SamEn) technique by mitigating its fundamental limitations, especially the issues of feature redundancy and collinearity across scales. In M-SamEn, the entropy values are computed at various scales, frequently demonstrating a high correlation in this study, potentially resulting in less biased and redundant feature representations. Additionally, DM-SamEn addresses this issue by implementing a weighted aggregation mechanism that employs the exponential function of signal values as scale-specific weights. This guarantees that scales with enhanced diagnostic relevance have a more substantial impact on the cumulative entropy value, leading to a more unique feature set. By emphasizing scales with significant variations, DM-SamEn minimizes feature redundancy, enhances interpretability, and strengthens the robustness of the extracted features.

In response to the above problems, this study proposes a three-point strategy:

-: Utilize the DM-SamEn method to reduce feature dimensionality while ensuring feature uniqueness and robustness.
-: Utilize two feature selection methods, sequential backward selection (SBS) and correlation-based feature selection (CFS), to identify the most significant features from the initial set. A hybrid SBS-CFS approach is also employed.
-: Validate the feature extraction (DM-SamEn) and selection (CFS and SBS) stages with three classifiers: adaptive weighted K-nearest neighbors (AW-KNN), radial basis function support vector machine (RBF-SVM), and multilayer perceptron (MLP). These classifiers evaluate this classification performance, which also determines the optimal results in the various feature selection strategies.

2. Material

This study utilized an extensive dataset from the PhysioNet platform [9], mostly derived from three significant investigations: those by Yogev et al. (Ga), Hausdorff et al. (Ju), and Frenkel-Toledo et al. (Si) [10,11,12,13]. Table 1 encapsulates the dataset, comprising 208 records of vertical ground reaction force (VGRF) signals that illustrate gait patterns in two distinct groups. The initial group consists of 93 individuals diagnosed with idiopathic Parkinson’s disease (PD) (mean age: 66.3 years; 63% male), whereas the control group comprises 73 healthy individuals with comparable demographics (mean age: 63.65 years; 55% male). Participants walked at a self-determined velocity on a flat surface for roughly two min, while vertical ground reaction force (VGRF) data were recorded using eight sensors positioned beneath each foot insole, measuring force in Newtons over time. All 16 sensor outputs were digitized at a frequency of 100 Hz, along with supplementary signals representing the combined outputs of the eight sensors per foot.

The study’s database comprises demographic information, serving two primary purposes: differentiating the Control (Co) group from the Parkinson’s Disease (PD) group while rating PD severity via the Hoehn and Yahr (HY) scale (notations 2; 2.5; 3) [14]. A gender propensity was noted, with males exhibiting a higher likelihood of developing Parkinson’s disease (59% for classification tests and 62% for severity tasks). Age influences the beginning of Parkinson’s disease, occurring between 63.65 and 70.8 years, while gait speed diminishes with severity, ranging from 0.79 to 1.03 m/s. A minor imbalance was present in the classification data (Co 44%, PD 56%), whereas a significant imbalance was observed in severity, with “2” as the predominant class at 59%, followed by “2.5” at 30%, and “3” as the least prevalent at 11% (Figure 1).

3. Methodology

Figure 2 depicts a four-step approach that includes preprocessing, feature extraction, feature selection, and classification stages. The preprocessing phase comprises time slicing, outlier elimination, and signal manipulation to enhance signal quality and augment dataset size while reducing noise and artifacts that may hinder classification. We utilize both conventional features and dense multiscale sample entropy (DM-SamEn) features in the feature extraction process to efficiently minimize feature dimensionality. The feature selection step encompasses correlation-based feature selection (CFS), sequential backward selection (SBS), and the hybrid SBS-CFS strategy to retain the most essential features. Adaptive weighted K-Nearest Neighbors (AW-KNN), radial basis function support vector machine (RBF-SVM), and multilayer perceptron (MLP) classifiers assess classification efficacy in the concluding classification phase.

3.1. Preprocessing Stage

“Mean foot” signals (MS) were introduced by aggregating left (LS) and right (RS) foot signals of raw vertical ground reaction force (VGRF) signals for feature extraction. These signals are represented as (MS, LS, RS) for each sample, denoted as X. Initially captured at 100 Hz for almost two minutes, these X signals were divided into non-overlapping 10-s segments to augment sample diversity for training. However, this segmentation produced certain anomalous subsets; hence, we employed the outlier signal detection approach in Theorem 1 to exclude outliers (Figure 3), thereby refining the dataset for further phases.

Theorem 1.

Outlier Detection Based on Second-order Differences

Let

{S_{i}}_{i = 1}^{n}

represent a set of n signals, where each signal (

S_{i}

) is a sequence of real-valued observations (

S_{i} = {s_{i 1}, s_{i 2}, \dots, s_{i m}}

) of length m. The second-order difference of a signal (

S_{i}

) is as follows:

Δ^{2} S_{i} [j] = s_{i, j + 2} - 2 s_{i, j + 1} + s_{i j}, j = 1, \dots, m - 2 .

Then, the **variability score** of the signal (

S_{i}

) is computed as follows:

Score (S_{i}) = \sum_{j = 1}^{m - 2} |Δ^{2} S_{i} [j]| .

Let

{x_{i}}_{i = 1}^{n}

be the variability score values of all signals, where

x_{i} = Score (S_{i})

. The distribution of

{x_{i}}_{i = 1}^{n}

defines the lower quartile (

Q_{1}

), the upper quartile (

Q_{3}

), and the interquartile range (

IQR = Q_{3} - Q_{1}

A signal (

S_{i}

) is classified as an outlier if

x_{i} < Q_{1} - k \cdot IQR or x_{i} > Q_{3} + k \cdot IQR,

where

k > 0

is a scaling factor (typically

k = 1.5

Following preprocessing, the dataset was expanded, with X represented as subsets (sX). Due to the limited number of subjects in this study, each sX signal was regarded as an independent sample. Furthermore, noise was incorporated into the sX signals (Theorem 2) to reduce the risk of data leakage, which might reduce the reliability of the trained models and the overall validity of the study results.

Theorem 2.

Frequency-Specific Noise Addition for Data Leakage Prevention

Let

s = {s_{1}, s_{2}, \dots, s_{n}}

be a discrete signal of length n.

Let

S

be the discrete Fourier transform (DFT) of signal

s

S = FFT (s),

To prevent data leakage by introducing controlled perturbations with a noise-level parameter of

α \in [0, 1]

noise = α \cdot (max (s) - min (s)) \cdot R,

where

R = {R_{1}, R_{2}, \dots, R_{n}}

is a vector of independent random variables drawn from a standard normal distribution, i.e.,

R_{i} \sim N (0, 1)

for

i = 1, \dots, n

The noisy frequency-domain representation is then given by

S_{noisy} = S + noise .

Finally, the noisy signal (

s_{noisy}

) is obtained by applying the inverse Fourier transform to

S_{noisy}

s_{noisy} = IFFT (S_{noisy}),

This procedure introduces noise in the frequency domain to obscure patterns that could provoke data leakage, while maintaining essential signal properties within the defined noise level (α).

Subsequently, the sX signals were transformed by utilizing the first derivative, second derivative, and cumulative sum transformations, as delineated in Equations (1)–(3), to extract more profound information from the original signals. Each sX signal consists of four derived signals, yielding a total of 12 signals per subset. This approach is influenced by Nam et al.’s approach [15] that augments the feature set, equipping classifiers with greater data depth for higher performance during the training, validation, and testing stages.

x^{'} (t) = \frac{d x (t)}{d t}

(1)

x^{″} (t) = \frac{d^{2} x}{d t^{2}}

(2)

y [n] = \sum_{k = 0}^{n} x [k]

(3)

3.2. Feature Extraction Stage

3.2.1. Conventional Features

Before feature extraction, signals are standardized by rescaling their range to the [0,1] interval, as seen in Equation (4). Subsequent to this normalization process, conventional features, including mean and standard deviation, are determined for each signal, as illustrated in Equations (5) and (6).

x_{norm} = \frac{x - min (x)}{max (x) - min (x)}

(4)

μ = \frac{1}{N} \sum_{i = 1}^{N} x [i]

(5)

σ = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(x [i] - μ)}^{2}}

(6)

3.2.2. Dense Multiscale Sample Entropy (DM-SamEn) Features

Dense Multiscale sample entropy (DM-SamEn) comprises three primary terms: sample entropy, multiscale analysis, and dense. This approach predominantly utilizes sample entropy to assess the predictability and complexity of time-series data at several scales, finally aggregating these entropy values into an overall representative value for the input series.

Sample entropy (SamEn) is a statistical instrument utilized for evaluating the complexity or irregularity of time-series data [16]. SamEn mitigates the influences of sequence length and sample size biases, establishing a dependable method for comparing entropy across datasets of varying lengths. Elevated entropy values indicate a more complex dataset, whilst diminished values signify a simpler dataset. This method is very effective for analyzing physiological data to identify situations via fluctuations in complexity [17]. The SamEn calculations are contingent upon the embedding dimension (m), tolerance (r), and signal length (N), with ideal values for m and r established at 3 and 0.15, respectively, in this work. Theorem 3 graphically demonstrates the procedure of the sample entropy method.

Theorem 3.

Sample Entropy (SamEn)

For a time series (U =

{u_{1}, u_{2}, u_{3}, . . ., u_{N}}

1.: Generate X:
X = ${x_{1}, x_{2}, x_{3}, . . ., x_{N - m + 1}}$ , $x_{i}$ = ${u_{i}, x_{i + 1}, . . ., x_{i + m - 1}}$ , $i \leq N - m + 1$ ;
2.: Construct $C_{n}^{m}$ :

$C_{n}^{m} = \sum_{j = 1}^{N - m + 1} \frac{C_{i j}^{m}}{N - m + 1}, i \leq N - m + 1$

with

$C_{i j}^{m} = \{\begin{matrix} 1 & if d | x_{i}, x_{j} | \leq r \\ 0 & if d | x_{i}, x_{j} | > r \end{matrix}, d | x_{i}, x_{j} | = m a x | x_{i} - x_{j} |;$
3.: Define $ϕ^{m} (r, N)$ and $ϕ^{m + 1} (r, N)$ :

$Φ^{m} (r, N) = \frac{1}{N - m + 1} \sum_{i = 1}^{N - m + 1} (C_{i}^{m}), Φ^{m + 1} (r, N) = \frac{1}{N - m} \sum_{i = 1}^{N - m} (C_{i}^{m + 1});$
4.: Compute SamEn:

$S a m E n (m, r, N) = - ln (\frac{Φ^{m + 1} (r, N)}{Φ^{m} (r, N)})$

Multiscale sample entropy (M-SamEn) extends SamEn by analyzing the complexity of time series over multiple scales [18]. This provides a comprehensive understanding of signal complexity across an extensive array of time periods. M-SamEn provides a coarser analysis by incrementally downsampling the original time series to provide multiple temporal scales, in contrast to SamEn, which assesses complexity at a singular scale. SamEn is computed at each scale, providing insight into the signal’s structural dynamics across several time intervals. M-SamEn can identify nuanced, scale-dependent fluctuations in signal complexity by measuring entropy levels across various scales [19]. In this investigation, M-SamEn was utilized with scales ranging from 1 to 6, with Equation (7) elucidating the downsampling procedure for the original time series.

y_{j}^{s} = \frac{1}{s} \sum_{i = k_{j}}^{j s} x_{i}, k_{j} = (j - 1) s + 1, 1 \leq j \leq \frac{N}{s}

(7)

In this study, the term “dense” denotes an aggregated value that encapsulates all M-SamEn scale values [20]. Aan analysis of M-SamEn values obtained from the identical time-series data demonstrated robust correlation coefficients (>0.7), suggesting a possible collinearity issue that results in model instability and overfitting by intensifying the effect of linked predictors, thereby hiding the distinct effects of individual variables [21]. High collinearity complicates the interpretation of model coefficients, weakens prediction reliability, and skews the individual contributions of associated variables. Hence, to resolve these challenges and improve classifier stability, DM-SamEn is introduced as an aggregated form for M-SamEn values (Equation (8)). In addition, the DM-SamEn value can serve as an indicator of the overall predictability and complexity of the input time series across various temporal scales.

D M - S a m E n = \frac{\sum_{N}^{i = 1} {exp}^{x_{i}} * S a m E n_{i}}{\sum_{N}^{i = 1} {exp}^{x_{i}}}

(8)

3.3. Feature Selection Stage

Post-feature extraction, high correlations (>0.7) among features still remained in the feature set. In order to resolve this issue, we propose three strategies for feature selection: (1) Correlation-Based Feature Selection (CFS), (2) Sequential Backward Selection (SBS), and (3) a hybrid approach incorporating SBS and CFS.

3.3.1. Correlation-Based Feature Selection (CFS)

The correlation-based feature selection (CFS) method, developed by K. Michalak et al. in 2010, aims to improve the effectiveness of feature selection while maintaining classification accuracy [22]. CFS is based on the principle that an optimal feature subset should consist of features that are strongly correlated with the class label while demonstrating low correlations among themselves, minimizing redundancy and enhancing model interpretability [23]. By incorporating a threshold t, CFS enhances the feature selection process to ensure that the chosen features offer distinct information while avoiding collinearity. This threshold-based method facilitates the management of model complexity and enhances stability by prioritizing non-redundant features that have significant predictive value. This study implemented a threshold t of 0.7, as outlined in Theorem 4.

CFS was implemented as the primary filter in strategy 1 to identify key features from the initial set and as a secondary filter following sequential backward selection (SBS) in strategy 3.

Theorem 4.

Correlation-based Feature Selection (CFS)

Input: X-Input Features Spaces ( $m \times n$ ), t-Threshold
Output: Selected-Feature-List of indices
Procedure:

1

Initialize Selected-Feature as an empty list.

2

For each pair of features

(f_{i}, f_{j})

in X (where

i < j

)

a

Compute the absolute correlation coefficient between

f_{i}

and

f_{j}

)

b

If the absolute correlation coefficient is less than or equal to the threshold t

–: Add $f_{i}$ and $f_{j}$ to Selected-Feature

3

Remove any duplicate entries from Selected-Feature (keeping only unique feature indices)

Return Selected-Feature

3.3.2. Sequential Backward Selection (SBS)

Sequential backward selection (SBS) is an optimization technique designed to enhance the feature space and boost the performance of machine learning models via dimensionality reduction. SBS operates by systematically removing the least significant features from the entire feature set, with the objective of maintaining or improving classification accuracy [24]. The method begins with the full set of features and, at each iteration, eliminates the feature whose removal results in the smallest decrease (or greatest increase) in model performance. This method aims to minimize redundancy and computational cost by identifying a subset of features that maintains the predictive capability of the original set. SBS is particularly advantageous for handling high-dimensional data, as reducing the number of features helps mitigate overfitting and improve model stability [25]. The SBS algorithm is outlined in Theorem 5 as a pseudocode.

In strategies 2 and 3, the SBS serves as the primary filter for identifying significant features. In strategy 3, SBS operates through a sequential feature elimination process that continues as long as model performance is stable. Conversely, CFS relies on a correlation threshold (t), which may unintentionally remove elements essential for model effectiveness. Therefore, SBS operates as the primary filter, whereas CFS acts as the secondary filter.

Theorem 5.

Sequential Backward Selection (SBS)

Input:
X-Input Features Spaces ( $m \times n$ ), y-vector ( $n \times 1$ )
model-classifier, k-number of features to select (stopping criterion)
Output:Selected-Feature-List of indices
Procedure:

1

Initialize Selected-Feature as an empty list.

2

While the number of Selected-Feature > k:

a

Set min-performance = ∞ and candidate-feature = None

b

For each f in Selected-Feature:

i

Temporarily remove f from Selected-Feature

ii

Train the model using X[:, Selected-Feature] and y

iii

Evaluate the model performance

iv

If model-performance < min-performance:

–: set min-performance = model-performance
–: add f in candidate-feature

c

Remove candidate-feature from Selected-Feature

Return Selected-Feature

3.4. Classification Stage

This study employed three classification models to train the feature sets identified in the preceding step: radial basis function support vector machine (RBF-SVM), adaptive weighted K-nearest neighbors (AW-KNN), and multilayer perceptron (MLP). The selection of these models was guided by three primary objectives. The primary aim is to identify the model that exhibits optimal performance when trained with the selected features. The second aim is to evaluate the alignment of the observed training results with the initial research predictions. Finally, to illustrate the effectiveness of the feature extraction and selection processes, it is reasonable to utilize simpler classifiers or models with moderate training capabilities.

3.4.1. Adaptive Weighted K-Nearest Neighbors (AW-KNN)

Adaptive weighted k-nearest neighbors (AW-KNN) improves upon the primary K-nearest neighbors (KNN) algorithm. This approach strengthens classification accuracy by assigning weights to each neighbor according to their distance from the query point. The weighted method employed by AW-KNN facilitates a ranking of nearer neighbors, reducing the impact of more distant points that could introduce noise or ambiguity [26]. AW-KNN differs from standard KNN by dynamically adjusting weights for neighbors, enhancing its effectiveness in datasets characterized by overlapping classes or non-uniform distributions. The algorithm computes the distance from the query point (x) to each neighboring point (

x_{i}

) among the chosen k [27]. The algorithm allocates increased influence to nearer neighbors, with each neighbor’s weight inversely correlated to its distance from the query point. This study utilized k = 10 for Manhattan distance (Equation (9)) and squared inverse weighting (Equation (10)) for the calculations of distance and weight, respectively.

d (x, x_{i}) = \sum_{j = 1}^{n} | x_{j} - x_{i, j} |

(9)

w_{i} = \frac{1}{d {(x, x_{i})}^{2} + ϵ}

(10)

3.4.2. Radial Basis Function Support Vector Machine (RBF-SVM)

The radial basis function support vector machine (RBF-SVM) extends support vector machine (SVM) by utilizing the radial basis function (RBF) kernel (Equation (11)), commonly referred to as the Gaussian kernel, to manage non-linear data through transformation into a higher-dimensional feature space [28]. The RBF kernel is well-suited for datasets exhibiting non-linear relationships, as it transforms the input space into a feature space that facilitates linear separation [29]. RBF-SVM effectively classifies complex patterns by identifying a hyperplane that optimally separates classes within the transformed space (Equation (12)).

RBF-SVM is fundamentally designed for binary classification; however, for the severity problem in this study, which encompasses three classes, the “one-vs-all” (OvA) approach is implemented [30]. The OvA method is employed to adapt binary classifiers for multi-class classification by training an individual classifier for each class, framing it as a binary problem of differentiating that class from the others. The OvA approach generates k separate binary classifiers for a dataset including k classes. Each classifier is trained with one class designated as the positive class (

+ 1

) and all other classes as the negative class (

- 1

), resulting in a distinct decision boundary for differentiating that one class from the others. This approach is direct and successful, particularly when combined with a strong kernel such as RBF, which produces optimal decision boundaries for intricate datasets.

K (x, x^{'}) = exp (- γ ∥ x - x^{'} ∥^{2})

(11)

min_{w, b, ξ} \frac{1}{2} {∥ w ∥}^{2} + C \sum_{i = 1}^{N} ξ_{i}

(12)

3.4.3. Multilayer Perceptron (MLP)

A Multilayer Perceptron (MLP) is a type of artificial neural network (ANN) including several sequential layers of neurons, which include an input layer, one or more hidden layers, and an output layer, with each layer being fully interconnected to the subsequent layer. Multilayer perceptrons (MLPs) are extensively employed for supervised learning tasks, including classification and regression, due to their ability to represent intricate, non-linear relationships of data [31]. The use of non-linear activation functions in the hidden layers allows MLPs to proficiently extract non-linear features, facilitating complex data representation and the mapping of data to higher-dimensional space. The training procedure for a multilayer perceptron employs the backpropagation algorithm, which refines a loss function by modifying weights according to gradients derived from error propagation (Equation (13)). This study utilized the Sigmoid function (Equation (14)) and the Mean Squared Error (MSE) loss function (Equation (15)), employing a network topology consisting of three hidden layers, each containing 128 nodes.

w \leftarrow w - η \frac{\partial L}{\partial w}

(13)

σ (x) = \frac{1}{1 + e^{- x}}

(14)

L = - \frac{1}{N} \sum_{i = 1}^{N} [y_{i} log ({\hat{y}}_{i}) + (1 - y_{i}) log (1 - {\hat{y}}_{i})]

(15)

3.4.4. Model Validation

The stability of the training process was demonstrated via two main methods and one supplementary sub-strategy. The main strategies examine critical aspects affecting results and are discussed in following sections, while the sub-strategy aims to reinforce the study’s efficiency, emphasize its limitations, and drive future research directions.

First, the dataset was divided into two segments: a training/validation set comprising 80% and a test set comprising 20%. The test set was consistent during the training phase, whereas the training/validation set was divided, with 80% allocated for training and 20% for validation. The training process was conducted 1000 times, utilizing randomized selection of training and validation samples from the subset to ensure robustness. All trained models underwent evaluation with a fixed test set, and the results are presented as a box plot in Figure 4, demonstrating the stability of model performance across iterations.

Second, the dataset was evaluated using the k-fold cross-validation (k-foldCV) strategy with k = 10. In this approach, the dataset was divided into 10 folds, and the model was trained and tested 10 times, each time using a different fold as the test set, while the remaining 9 folds formed the training set. This ensured the complete separation of every data point for training and testing, thereby providing a comprehensive assessment of the model’s performance. The results of the k-fold cross-validation provide a comprehensive estimate of model performance and reduce the risk of overfitting by testing the model across varied data splits.

Finally, the sub-strategy required Leave-One-Subject-Out Cross-Validation (LOSOCV). This method divided the dataset of 166 subjects into 166 parts, executing training and testing of the model 166 times. In each iteration, data from a single subject served as the test set, while the remaining 165 subjects formed the training set, preventing any data leakage from the test subject into the training process. In contrast to the other two procedures, LOSOCV evaluates each subject independently, allowing it to be particularly resilient to overfitting. However, it differs from current methods and the existing study outlined in Section 4.2.4, which interpret signals post noise addition as independent units. In LOSOCV, the limited dataset of 166 patients could inherently limit expected model accuracy to approximately 75–80%, which is considered acceptable. This strategy stresses a realistic evaluation by ensuring that signals from test subjects do not affect training, emphasizing generalizability above exaggerated results.

4. Results and Discussion

The post-training results and the pertinent outcomes are both illustrated and discussed in this section.

4.1. Results

4.1.1. PD Classification Task Results

Table 2 indicates that all three training models attained high accuracy levels, surpassing 85%, validating the research methodology and fulfilling the objectives of the PD recognition tests. When comparing different ways to extract features, sets that used M-SamEn did slightly better than those that used DM-SamEn, with a 2~2.5% advantage over similar feature selection methods. From the viewpoint of feature selection, models exhibited optimal performance with the SBS technique, followed by the SBS-CFS hybrid method and the CFS method. Regarding classifier performance, RBF-SVM attained the highest results, somewhat exceeding AW-KNN, with a significant performance disparity noted between RBF-SVM and MLP. The maximum attained model accuracy was 98.38% (DM-SamEn, SBS, and RBF-SVM), succeeded by 98.25% (M-SamEn, SBS, and RBF-SVM), whereas the minimum recorded accuracy was 85.16% (DM-SamEn, SBS-CFS, and MLP).

Figure 4 presents stable mean accuracy results, consistently exceeding 85%, with a few outliers observed. The method–model pair that achieved the highest mean accuracy in the PD classification task (DM-SamEn) is (SBS, AW-KNN), followed by (SBS-CFS, AW-KNN), and the method–model pair that achieved the lowest mean accuracy is (CFS, RBF-SVM). Table 3 shows the results of subject-level validation utilizing the LOSOCV strategy, with all models reaching an accuracy over 77%. The maximum recorded accuracy was 85.67% (CFS, RBF-SVM), succeeded by 85.47% (SBS-CFS, RBF-SVM), while the lowest achieved accuracy was 77.65% (CFS, MLP). The results in Table 3 reveal greater standard deviations than those in Table 2, which is linked to the restricted training data in the subject-level validation strategy.

4.1.2. Severity Classification Task Results

Table 4 indicates that the performance of all three models in the PD severity recognition task surpasses 85%. Consistent with the results in Table 2, feature sets employing M-SamEn marginally surpassed those utilizing DM-SamEn across identical feature selection methods. In terms of feature selection, the models exhibited optimal performance with the SBS technique, followed by the SBS-CFS hybrid and, finally, the CFS method. Among the classifiers, RBF-SVM yielded the most favorable results, marginally exceeding AW-KNN by less than 0.5%; however, a substantial performance disparity persisted between RBF-SVM and MLP, indicating MLP’s probable unsuitability for this process. The maximum recorded accuracy was 96.89% (M-SamEn, SBS, and RBF-SVM), followed by 96.80% (DM-SamEn, SBS, and RBF-SVM), while the minimum accuracy was 85.75% (DM-SamEn, SBS-CFS, and MLP).

In the severity classification task shown in Figure 4, the average accuracy results demonstrate high stability, with all values higher than 85%, similar to those observed in the PD classification task. The PD classification task results show that SBS (AW-KNN) is the method–model pair with the best average accuracy, followed by SBS-CFS (AW-KNN). The task records the lowest average accuracy for (CFS, RBF-SVM) and (CFS, MLP). In the subject-level validation shown in Table 5, the minimum recorded performance was 77.23% (SBS-CFS, MLP), while the maximum reached approximately 86.52% (SBS, RBF-SVM). In comparison to Table 4, the standard deviations in Table 5 are substantially elevated, revealing that the size of the LOSOCV input data strongly impacts the results. The performance variations between the signal-level validation (Table 2 and Table 4) and the subject-level validation (Table 3 and Table 5) are roughly 10~15%, emphasizing the effect of the validation strategy on model results.

4.1.3. Throughput Performance and CO₂ Emission Results

Table 6 illustrates the throughput performance of the classification models across different tasks, feature extraction techniques, and feature selection techniques, supported by Table 7, which provides power consumption results in terms of CO₂ emissions. The studies were performed using a device equipped with 128 GB RAM, an Intel(R) Core(TM) i9-10980XE CPU working at 3.00 GHz, and a Ubuntu 22.04 operating system.

Initially, when comparing the two feature extraction methods, models employing the M-SamEn feature set showed reduced throughput compared to those utilizing DM-SamEn. The disparity is due to the total number of features—M-SamEn produces 96 features, while DM-SamEn decreases this value to 36, improving computing efficiency. Moreover, the smaller feature set of DM-SamEn leads to reduced CO₂ emissions relative to M-SamEn, emphasizing the computational and ecological benefits.

Secondly, among the feature selection techniques, models employing SBS typically demonstrate poorer throughput relative to those applying CFS and SBS-CFS. SBS maintains a greater quantity of features, eliminating only those that substantially affect model performance, preserving elevated computational complexity. Inversely, both CFS and SBS-CFS emphasize the elimination of highly correlated features, resulting in decreased processing time. SBS-CFS generally attains better throughput compared to CFS. The similarity can be seen in the CO₂ emission results, with SBS exhibiting the greatest emissions, followed by SBS-CFS, and CFS showing the lowest emissions.

Finally, among the classification models, the MLP demonstrated the highest throughput performance, reaching 2.05 ×

10^{5}

obj/s, while concurrently obtaining the lowest CO₂ emissions among the three models. In the PD classification task, RBF-SVM showed an insignificant advantage over AW-KNN in terms of speed, while AW-KNN showed a slightly higher throughput in the severity classification task. The difference can be linked to the quantity of labels—PD classification includes two labels, whereas severity classification comprises three. The increased label issues impact SVM more substantially than KNN due to the former’s dependence on complex decision boundaries, but KNN’s lazy learning technique is less impacted.

4.1.4. Paired t-Test Results

The accuracy results derived from training and validation in Section 3.4.4 were further examined using a paired t-test, with p-values adjusted via the Benjamini–Hochberg procedure for each combination of classification models and feature selection methods. Figure 5 displays the p-value matrix for various pairings. The analysis found that the majority of p-values are <0.05 for both M-SamEn and DM-SamEn, suggesting significant differences in the mean accuracy values among the classification model–selection method combinations. This demonstrates that each combination has unique performance traits. However, several p-values surpass 0.05, indicating the absence of statistically significant differences for these combinations.

According to Table 2 and Table 4, high p-values tend to indicate a crucial accuracy barrier beyond which further increases are challenging to overcome. The criterion for the PD classification task is between 98% and 98.5%, but for the severity classification task, the threshold is between 96% and 97%. These thresholds may signify performance saturation, wherein the models and feature selection techniques achieve their optimum accuracy under the given constraints.

4.2. Discussion

The results (Section 4.1) demonstrate that the proposed procedure (Figure 2), which integrates DM-SamEn for feature extraction with CFS, SBS, and the hybrid SBS-CFS for feature selection and utilizes AW-KNN, RBF-SVM, and MLP classifiers, attained high classification accuracy (>85%) and exhibited robust model stability. Further analysis is necessary to determine whether DM-SamEn features derived from signals generated via Equations (1)–(3) primarily contribute to model training or if feature selection methods identify DM-SamEn features as the predominant factors for optimal model performance. Furthermore, it is crucial to assess the quantity of feature reduction necessary to sustain the model’s average accuracy relative to the original feature set.

4.2.1. Analysis of Multicollinearity and Redundancy Problems in Feature Sets

Multicollinearity is an important issue considered in this study due to the fundamental properties of the feature extraction techniques, M-SamEn and the proposed DM-SamEn. The biggest challenge is the clinical interpretability of these features: entropy values quantify fluctuation patterns in the VGRF signal and are used as features for classification tasks. M-SamEn produces multiple entropy values across multiple scales, resulting in feature redundancy and elevated correlation coefficients among features. Multicollinearity can lead to inadequate post-trained models, since excessive feature dependency can cause overfitting, indicated by unnaturally boosted performance metrics. In order to address this issue, DM-SamEn consolidates entropy values into a single representative value, thereby drastically reducing redundancy. Figure 6 depicts the correlation matrix for the feature sets derived from M-SamEn (96 features) and DM-SamEn (36 features). The color gradient, ranging from white to dark blue, depicts correlation values ranging from 0 to 1. Despite a reduction in feature size using DM-SamEn, a significant quantity of highly correlated features (>0.7) exists.

Three feature selection strategies—CFS, SBS, and SBS-CFS—were implemented to further reduce multicollinearity and enhance performance. These strategies aim to minimize the quantity of correlated features while preserving crucial predictive information. This strategy enhances the reliability of the trained models, increases inference throughput, and reduces model size. The condensed models are beneficial for implementation on portable devices, allowing for potential trials in clinical contexts.

4.2.2. Analysis of Dominant Feature Types in Selected Feature Sets

This study assessed the impact of secondary signals generated in Section 3.1 on the classification results for both PD and severity classification tasks, concentrating on the feature sets derived from the selected pairs of high-performing classification models and selection methods. Figure 7a demonstrates that secondary signals derived from Equations (1)–(3) comprised 65% and 57% of the features employed in PD and severity classification tasks, respectively. This major contribution emphasizes the significance of the preprocessing stage, as the crucial factors are derived from these secondary signals, highlighting their essential role in improving the training process and overall classification performance.

The subsequent analysis analyzes features obtained from the DM-SamEn method and evaluates their impact on the selection of high-performing classification model pairs. Figure 7b illustrates that DM-SamEn’s features include an extensive percentage, representing 55% of features in PD classification and 59% in severity classification tasks. The findings suggest that the entropy values generated by the DM-SamEn approach provide the essential information required to enhance classification accuracy and boost classifier performance.

4.2.3. Analysis of Feature Selection Efficiency

Figure 8 illustrates four groups that indicate the number of features retained following the feature selection stage for each method. Each group comprises four columns: the blue and gray columns denote the PD classification task utilizing the M-SamEn and DM-SamEn methods, respectively, whereas the orange and yellow columns signify the same for the severity classification task.

In the “Original” group, feature extraction with M-SamEn yielded 96 features, whereas DM-SamEn diminished this to 36 features. As indicated in Section 4.1, the training results reveal a minor variance in performance between models trained using M-SamEn and DM-SamEn, with DM-SamEn yielding results that are merely 2–2.5% poorer. Within the “CFS” group, M-SamEn shrank the feature-set size from 96 to 64 and 59 for the PD and severity classification tasks, respectively, while DM-SamEn reduced the feature count from 36 to 20 for both tasks. Despite a reduction in features, the performance of the “CFS” models remains elevated, as shown in Table 2 and Table 4, but there is a performance decline beyond 2% when utilizing DM-SamEn instead of M-SamEn.

In the “SBS”, the quantity of features is not noticeably diminished over the selection stage. Applying M-SamEn yields a reduction in feature size from 96 to 90 and 87 for the PD and severity classification tasks, respectively. With DM-SamEn, the features decreased from 36 to 31 and 24 for the two tasks. The result indicate that models from the “SBS” group exhibit superior performance while retaining most of the features. Finally, the “SBS-CFS” group presents marginally poorer results compared to the “SBS” group. This group competently integrated the advantages of both techniques—sequential elimination from SBS and elimination of highly correlated features from CFS—thereby reducing irrelevant and duplicated features while ensuring accurate results. In M-SamEn, the feature-set size was diminished from 96 to 56 for the PD classification task and to 64 for the severity classification task. For DM-SamEn, the feature-set size was reduced from 36 to 17 and 14 for the PD and severity classification tasks, respectively. The results obviously illustrate a trade-off between the size of the feature set provided to the classifiers and the performance of the models post training.

4.2.4. Performance Comparison with Existing Studies

This section evaluates the performance reported in the current study in relation to existing research. Table 8 summarizes that other studies report high accuracy rates, exceeding 93%, consistent with our findings. The maximum accuracy attained in this study was 98.38% for PD classification and 96.80% for severity classification. Most previous studies utilize more sophisticated classification models, indicating increased computational power and energy usage. Our approach, utilizing the DM-SamEn method for feature extraction and the SBS-CFS method for feature selection, effectively identifies key features while minimizing computational demands. The methodology employed in this study attains a competitive accuracy through the use of simpler models, thereby minimizing the requirements for substantial computational power and energy consumption.

4.2.5. Limitations of the Study

Despite the impressive results of this study, some limits must be recognized, as they offer significant guidance for future research goals. This study utilized a reliable and publicly accessible dataset as the primary data source. This reliance presents issues, including data imbalance and dependence on a singular dataset, which may restrict the generalizability of the results. The dataset may poorly represent varied populations of Parkinson’s disease (PD) patients or adjustments to data collection strategies. External dataset validation is essential to strengthen the reliability of the results.

Secondly, this study acknowledged the problem of data imbalance but resisted using augmentation techniques like the Synthetic Minority Oversampling Technique (SMOTE) due to concerns about the higher risks of data leakage that come with the generation of artificial data. Augmentation algorithms may not consistently provide the most efficient option for rebalancing datasets in this case. Moreover, the restricted dataset size made subject-level validation impractical, even when the post-training performance surpassed 77%, as indicated in Table 3 and Table 5. A noise injection approach was presented in Theorem 2 to prevent data leakage. However, this approach requires further evaluation with larger and more diverse datasets to validate its effectiveness.

Thirdly, the study lacked real-world validation and prospective testing, which would have offered a deeper evaluation of the robustness and utility of the proposed study in clinical contexts. Collaboration with clinical Parkinson’s disease research groups and access to real-world datasets are essential to performing external validation. In addition, despite the DM-SamEn approach exhibiting technical efficiency, the entropy values it produces presently lack clinical significance for the early diagnosis of Parkinson’s disease. Future studies should focus on optimizing the feature set to improve its clinical significance and applicability in early detection.

Ultimately, due to the inherent noisiness of VGRF signals, this approach requires significant improvements to minimize noise effects and adapt hardware constraints, particularly in devices with restricted computational capabilities, such as IoT devices.

5. Conclusions

This study outlines a method for classifying Parkinson’s disease and its severity by analyzing vertical ground reaction force signals from patients. We proposed DM-SamEn as an alternative feature extraction method to standard M-SamEn, which effectively shrinks the feature-set size while maintaining training performance, thereby reducing computational demands. We enhanced model reliability and training efficiency by evaluating three feature selection methods: CFS, SBS, and a hybrid SBS-CFS approach. These addressed multicollinearity within the feature set. Ultimately, three classifiers—AW-KNN, RBF-SVM, and MLP—were employed, demonstrating the effectiveness of the feature extraction and selection methods while avoiding the necessity for overly complex models. Despite these advancements, this study admits several limitations, including the lack of external validation in clinical situations, constrained clinical interpretability of features developed by the DM-SamEn method, data imbalance, limited dataset availability, and challenges with adapting the methodology for resource-limited hardware such as IoT devices.

Author Contributions

Methodology and original draft preparation, M.T.P.N.; original draft preparation and supervision, M.K.P.T.; review and editing and supervision, T.N.; funding acquisition, T.H.T.; methodology, formal analysis, original draft preparation, and supervision, Q.D.N.N.All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the Japan Science and Technology Agency (JST) under Strategic Basic Research Programs Precursory Research for Embryonic Science and Technology (PRESTO) under Grant JPMJPR20M6.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement:

The dataset used in the study can be found at https://physionet.org/content/gaitpdb/1.0.0/ accessed on 16 December 2024.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. (a) Gender and label distribution across each class in severity classification task; (b) gender and label distribution across each class in PD classification task.

Figure 2. The methodology procedure has four stages: preprocessing, feature extraction, feature selection, and classification.

Figure 3. The process of signal division using the time-slicing window method and outlier removal through the quartile approach and histogram analysis.

Figure 4. Boxplot analysis of model consistency for PD and severity classification tasks.

Figure 5. (a) Adjusted p-value matrix of paired T-tests comparing accuracies between classifier–feature selection method pairs (M-SamEn); (b) adjusted p-value matrix of paired t-tests comparing accuracies between classifier–feature selection method pairs (DM-SamEn).

Figure 6. (a) Correlation matrix of the feature set extracted using the M-SamEn method; (b) correlation matrix of the feature set extracted using the DM-SamEn method.

Figure 7. (a) Distribution of selected features by signal source (original and computed signals from Equations (1)–(3) after the feature selection stage; (b) distribution of selected features by feature extraction method after the feature selection stage.

Figure 8. Comparison of feature count across feature selection methods (*: p-value < 0.05; **: p-value < 0.01).

Table 1. Demographic information of the PD dataset.

Task	PD Classification		Severity Classification
Label	Co	PD	2	2.5	3
Age	63.65	66.3	64.21	68.78	70.8
Height (m)	1.68	1.67	1.67	1.67	1.64
Weight (kg)	72.534	72.172	73.182	72.289	66.3
Gait Speed (m/s)	1.24	1.03	1.08	1	0.79

Table 2. PD classification task performance metrics (10-FoldCV).

Model	FS	Label		M-SamEn					DM-SamEn
Model	FS	Label		Pre	Sen	Spec	F1	Acc	Pre	Sen	Spec	F1	Acc
AW-KNN	CFS	PD	Mean	98.28	98.41	96.16	98.34	97.72	94.46	96.32	87.47	95.38	93.56
		PD	SD	0.41	0.69	0.81	0.36	0.46	0.80	1.01	1.40	0.66	0.85
		CO	Mean	96.48	96.16	98.41	96.31	97.72	91.75	87.47	96.32	89.36	93.56
		CO	SD	1.32	0.81	0.7	0.66	0.46	2.27	1.4	1.01	1.33	0.85
	SBS	PD	Mean	98.69	98.76	97.04	98.72	98.24	98.04	98.36	95.60	98.19	97.51
		PD	SD	0.31	0.75	0.75	0.39	0.5	0.36	0.84	0.72	0.44	0.58
		CO	Mean	97.30	97.04	98.76	91.16	98.24	96.34	95.60	98.36	95.95	97.51
		CO	SD	1.4	0.75	0.75	0.75	0.5	1.73	0.72	0.84	0.86	0.58
	SBS-CFS	PD	Mean	98.31	98.41	96.19	98.36	97.74	94.34	97.24	89.58	89.36	94.11
		PD	SD	0.30	0.66	0.69	0.37	0.48	0.72	1.04	1.27	0.57	0.74
		CO	Mean	96.48	96.19	98.41	96.33	97.74	93.65	87.98	92.61	95.04	94.11
		CO	SD	1.12	0.69	0.66	0.70	0.48	2.25	1.27	1.04	1.12	0.74
RBF-SVM	CFS	PD	Mean	93.65	87.98	92.61	98.56	98.00	94.96	96.19	88.7	95.57	93.87
		PD	SD	0.50	0.53	1.2	0.43	0.6	0.89	0.64	1.58	0.67	0.84
		CO	Mean	97.47	95.96	98.92	96.70	98.00	91.27	88.7	96.19	89.97	93.87
		CO	SD	1.27	1.12	0.53	1.06	0.6	1.12	1.58	0.64	1.13	0.84
	SBS	PD	Mean	98.27	99.21	96.11	98.73	98.25	98.59	99.07	96.81	98.83	98.38
		PD	SD	0.49	0.45	1.04	0.26	0.35	0.46	0.56	0.98	0.2	0.26
		CO	Mean	98.18	96.11	99.21	97.12	98.25	97.93	96.81	99.07	97.35	98.38
		CO	SD	0.98	1.04	0.45	0.57	0.35	1.16	0.98	0.56	0.43	0.26
	SBS-CFS	PD	Mean	98.43	99.00	96.43	98.71	98.21	96.05	97.26	91.18	96.64	95.36
		PD	SD	0.42	0.44	1.04	0.28	0.42	0.98	0.78	1.87	0.82	1.06
		CO	Mean	97.66	96.43	99.00	97.04	98.21	93.81	91.18	97.29	92.41	95.36
		CO	SD	1.08	1.04	0.44	0.80	0.42	1.56	1.87	0.78	1.56	1.06
MLP	CFS	PD	Mean	95.35	95.34	89.58	95.34	93.58	88.53	89.83	74.14	89.16	84.98
		PD	SD	0.78	0.90	1.95	0.52	0.65	1.49	1.32	2.25	1.23	1.38
		CO	Mean	89.67	89.58	95.34	89.59	93.58	76.68	74.14	89.83	75.34	84.98
		CO	SD	1.47	1.95	0.90	1.09	0.65	1.63	2.25	1.32	1.36	1.38
	SBS	PD	Mean	95.78	95.87	90.68	95.82	94.24	93.70	94.42	86.07	94.08	91.83
		PD	SD	0.87	0.88	1.49	0.57	0.7	0.94	1.15	1.69	0.70	0.84
		CO	Mean	90.82	90.68	95.87	90.72	94.24	87.84	86.68	94.46	86.75	91.83
		CO	SD	1.71	1.49	0.88	0.87	0.7	2.09	1.69	1.15	1.05	0.84
	SBS-CFS	PD	Mean	94.90	94.97	88.73	94.93	93.02	88.30	90.42	73.24	89.38	95.16
		PD	SD	0.93	0.71	1.50	0.71	0.88	1.10	1.09	2.83	0.63	0.88
		CO	Mean	88.75	88.73	94.97	88.72	93.02	77.27	73.25	90.40	75.21	95.16
		CO	SD	1.49	1.51	0.71	1.18	0.88	2.88	2.82	1.09	2.24	0.88

Table 3. PD classification task performance metrics (LOSOCV).

Model	FS	Label		M-SamEn					DM-SamEn
Model	FS	Label		Pre	Sen	Spec	F1	Acc	Pre	Sen	Spec	F1	Acc
AW-KNN	CFS	PD	Mean	88.78	83.76	86.62	83.81	85.19	82.04	81.98	79.69	80.43	80.83
		PD	SD	2.24	3.31	3.04	2.60	2.22	2.36	2.96	2.94	2.35	2.19
		CO	Mean	87.33	86.62	83.76	84.67	85.19	83.72	79.69	81.98	79.99	80.83
		CO	SD	2.38	3.04	3.31	2.22	2.22	2.45	2.94	2.96	2.41	2.19
	SBS	PD	Mean	87.71	84.39	85.67	83.93	85.03	87.43	83.84	85.35	83.27	83.27
		PD	SD	2.35	3.22	2.94	2.53	2.19	2.37	3.4	3.02	2.67	2.27
		CO	Mean	87.73	85.67	84.39	84.57	85.03	87.56	85.35	83.84	84.18	83.27
		CO	SD	2.38	2.94	3.22	2.38	2.19	2.49	3.02	3.4	2.42	2.27
	SBS-CFS	PD	Mean	88.30	83.93	86.59	84.44	85.26	84.94	83.37	82.26	81.94	82.81
		PD	SD	2.36	3.2	2.85	2.41	2.17	2.36	3.21	3.17	2.55	2.22
		CO	Mean	87.03	86.59	83.93	85.01	85.26	86.02	82.26	83.37	81.82	82.81
		CO	SD	2.33	2.85	3.2	2.32	2.17	2.43	3.17	3.21	2.54	2.22
RBF-SVM	CFS	PD	Mean	84.87	91.01	80.33	86.36	85.67	77.61	86.6	72.15	80.72	79.37
		PD	SD	2.37	2.55	3.35	2.15	2.13	2.24	2.5	3.2	2.07	2.14
		CO	Mean	92.24	80.33	91.01	83.35	85.67	85.66	72.15	86.60	76.48	79.37
		CO	SD	2.05	3.35	2.55	2.13	2.13	2.43	3.2	2.5	2,64	2.14
	SBS	PD	Mean	84.36	91.06	78.97	86.07	85.01	83.42	90.99	78.09	85.26	84.54
		PD	SD	2.52	2.53	3.75	2.25	2.32	2.37	2.72	3.41	2.28	2.19
		CO	Mean	91.74	78.97	91.06	81.88	85.01	92.04	78.08	90.99	81.96	84.54
		CO	SD	2.19	3.75	2.53	3.03	2.32	2.21	3.41	2.72	2.19	2.19
	SBS-CFS	PD	Mean	84.6	91.2	79.74	86.43	85.47	80.40	88.02	74.56	82.40	81.29
		PD	SD	2.42	2.42	3.44	2.12	2.19	2.42	2.71	3.57	2.20	2.24
		CO	Mean	91.58	79.74	91.20	82.99	85.47	88.71	74.56	88.02	78.46	81.29
		CO	SD	2.23	3.44	2.42	2.72	2.19	2.42	3.57	2.71	2.83	2.24
MLP	CFS	PD	Mean	81.9	86.68	77.27	82.59	81.98	78.34	81.16	74.13	77.59	77.65
		PD	SD	2.37	2.63	3.24	2.13	2.08	2.44	3.17	3.3	2.45	2.24
		CO	Mean	87.37	77.27	86.68	79.79	81.98	82.5	74.13	81.16	75.67	77.65
		CO	SD	2.2	3.24	2.63	2.52	2.08	2.55	3.3	3.17	2.62	2.24
	SBS	PD	Mean	83.28	87.74	78.92	84.04	83.33	82.89	86.37	78.71	82.97	82.54
		PD	SD	2.4	2.45	3.31	2.07	2.07	2.42	2.93	3.24	2.24	2.09
		CO	Mean	88.30	78.92	87.74	81.07	83.33	88.03	78.71	86.37	80.61	82.54
		CO	SD	2.13	3.31	2.45	2.60	2.07	2.16	3.24	2.93	2.49	2.09
	SBS-CFS	PD	Mean	82.12	88.08	77.59	83.63	82.84	81.66	82.54	77.95	79.81	80.24
		PD	SD	2.2	2.41	3.24	1.96	1.93	2.48	3.26	3.37	2.49	2.23
		CO	Mean	88.85	77.59	88.08	80.31	82.84	85.20	77.95	82.54	78.86	80.24
		CO	SD	1.94	3.24	2.41	2.51	1.93	2.49	3.37	3.26	3.42	2.23

Table 4. Severity Classification Task Performance Metrics (10-FoldCV).

Model	FS	Label		M-SamEn					Dm-SamEn
Model	FS	Label		Pre	Sen	Spec	F1	Acc	Pre	Sen	Spec	F1	Acc
AW-KNN	CFS	2	Mean	96.71	96.81	96.51	96.15	96.15	94.35	94.76	94.15	94.56	93.02
		2	SD	0.95	0.84	1.07	0.54	0.44	1.08	1.47	1.02	1.03	1.03
		2.5	Mean	96.50	97.16	97.91	96.80	96.15	92.48	94.47	95.35	93.44	93.02
		2.5	SD	1.19	0.97	0.70	0.41	0.44	1.86	1.30	1.23	1.24	1.03
		3	Mean	92.40	89.79	99.04	90.92	96.15	87.79	80.15	98.64	83.75	93.02
		3	SD	3.06	3.54	0.43	1.76	0.44	3.12	3.58	0.32	2.94	1.03
	SBS	2	Mean	97.12	97.29	96.96	97.20	96.72	97.06	96.53	96.89	96.79	96.23
		2	SD	0.46	0.73	0.56	0.34	0.66	0.65	1.08	0.79	0.6	0.65
		2.5	Mean	97.50	97.56	98.46	97.51	96.72	96.40	97.25	97.82	96.81	96.23
		2.5	SD	1.26	1.01	0.78	0.67	0.66	1.35	0.95	0.75	0.52	0.65
		3	Mean	92.73	91.12	99.04	91.77	96.72	92.10	91.55	98.98	91.69	96.23
		3	SD	3.13	3.16	0.39	1.66	0.66	2.58	3.68	0.47	1.88	0.65
	SBS-CFS	2	Mean	97.20	96.97	97.10	97.08	96.48	94.86	95.04	94.64	94.94	94.02
		2	SD	0.84	0.81	0.83	0.55	0.53	1.23	1.26	1.28	1.03	0.83
		2.5	Mean	96.85	97.54	98.07	97.18	96.48	94.50	95.09	96.63	94.79	94.02
		2.5	SD	1.27	0.73	0.79	0.64	0.53	1.05	1.04	0.67	0.91	0.83
		3	Mean	91.71	90.55	98.98	91.18	96.48	88.06	85.42	98.55	86.65	94.02
		3	SD	2.79	3.42	0.39	1.52	0.53	2.04	3.14	0.29	2.04	0.83
RBF-SVM	CFS	2	Mean	96.76	97.45	96.58	97.15	96.72	93.56	94.72	93.29	94.12	92.84
		2	SD	0.95	0.96	0.97	0.71	0.67	1.61	1.39	1.43	1.24	1.18
		2.5	Mean	97.82	97.36	98.71	97.59	96.72	93.00	93.86	95.82	93.40	92.84
		2.5	SD	0.79	0.89	0.46	0.59	0.67	1.54	1.72	0.76	1.18	1.18
		3	Mean	92.73	90.98	99.11	91.68	96.72	88.24	80.79	98.61	84.23	92.84
		3	SD	3.06	3.88	0.35	2.12	0.67	2.44	3.5	0.39	1.95	1.18
	SBS	2	Mean	96.94	97.90	96.85	97.25	96.86	96.99	97.88	96.96	97.30	96.80
		2	SD	0.67	0.85	0.73	0.49	0.43	0.90	0.73	0.81	0.68	0.82
		2.5	Mean	98.10	97.62	98.86	97.84	96.86	97.94	97.58	98.78	97.75	96.80
		2.5	SD	0.78	1.05	0.47	0.57	0.43	0.66	1.01	0.38	0.68	0.82
		3	Mean	93.97	89.69	98.26	91.58	96.86	92.09	90.92	98.89	91.37	96.80
		3	SD	2.98	4.53	0.38	2.36	0.43	2.65	4.12	0.42	2.39	0.82
	SBS-CFS	2	Mean	96.76	97.61	96.59	97.18	96.86	94.44	95.99	94.14	95.20	94.18
		2	SD	0.95	1.12	0.94	0.78	0.77	1.02	1.60	0.89	1.13	1.1
		2.5	Mean	97.91	97.64	98.76	97.78	96.86	95.61	94.54	97.43	95.06	94.18
		2.5	SD	0.79	0.97	0.52	0.56	0.77	1.88	1.41	1.04	1.48	1.1
		3	Mean	93.90	90.73	99.23	91.94	96.86	87.78	84.41	98.48	85.86	94.18
		3	SD	3.85	4.13	0.44	2.42	0.77	2.36	3.32	0.345	1.65	1.1
MLP	CFS	2	Mean	92.91	93.49	92.37	93.18	91.26	89.90	88.31	90.18	89.76	86.78
		2	SD	1.24	1.28	1.56	0.74	0.56	1.82	2.45	1.93	1.36	1.65
		2.5	Mean	91.19	90.28	94.70	90.70	91.26	87.12	88.11	89.05	85.26	86.78
		2.5	SD	1.07	1.73	0.67	0.94	0.56	3.89	2.07	2.52	2.27	1.65
		3	Mean	84.06	84.19	98.18	83.93	91.26	75.98	69.37	97.32	72.03	86.78
		3	SD	5.45	5.11	0.76	4.27	0.56	6.81	6.27	0.572	5.88	1.65
	SBS	2	Mean	93.28	93.98	92.98	93.59	91.78	92.93	93.08	92.65	92.99	90.77
		2	SD	1.42	1.35	1.39	1.06	0.88	1.35	1.08	1.38	0.92	0.71
		2.5	Mean	91.73	91.93	95.01	91.81	91.78	89.51	91.29	93.52	90.32	90.77
		2.5	SD	1.32	1.33	0.85	0.81	0.88	2.32	1.92	1.32	0.87	0.71
		3	Mean	84.89	82.03	98.13	82.83	91.78	84.71	79.32	98.18	81.78	90.77
		3	SD	4.77	4.51	0.50	2.76	0.88	4.36	4.58	0.67	2.55	0.71
	SBS-CFS	2	Mean	92.05	92.66	91.67	92.31	90.19	89.70	88.15	88.59	88.97	95.75
		2	SD	1.40	1.63	1.40	1.34	1.22	1.83	1.98	1.71	1.59	1.36
		2.5	Mean	89.59	90.24	93.64	89.91	90.19	83.35	85.53	89.49	84.61	95.75
		2.5	SD	2.39	1.89	1.42	1.15	1.22	2.15	1.63	1.40	1.62	1.36
		3	Mean	83.16	79.05	98.12	80.74	90.19	75.91	74.64	97.67	75.31	95.75
		3	SD	4.56	5.67	0.512	4.06	1.22	6.96	6.17	0.69	5.20	1.36

Table 5. Severity Classification Task Performance Metrics (LOSOCV).

Model	FS	Label		M-SamEn					Dm-SamEn
Model	FS	Label		Pre	Sen	Spec	F1	Acc	Pre	Sen	Spec	F1	Acc
AW-KNN	CFS	2	Mean	87.74	81.8	92.38	82.89	85.90	84.66	78.55	91.56	79.87	79.49
		2	SD	3.27	4.45	2.25	3.09	2.66	3.74	4.91	2.25	3.78	2.88
		2.5	Mean	87.11	81.87	93.66	82.26	85.90	79.18	81.89	87.57	78.16	79.49
		2.5	SD	3.4	5.32	1.71	4.2	2.66	3.76	4.92	2.52	3.95	2.88
		3	Mean	87.55	91.01	91.30	87.48	85.90	83.19	78.04	90.10	77.71	79.49
		3	SD	3.19	3.41	2.63	2.90	2.66	3.8	5.54	2.61	4.29	2.88
	SBS	2	Mean	88.21	82.30	92.13	82.43	85.37	86.30	82.29	91.46	81.82	83.77
		2	SD	3.55	4.58	2.71	3.69	2.85	3.67	4.64	2.54	3.76	3.05
		2.5	Mean	88.10	82.75	94.16	85.02	85.37	86.49	82.18	92.33	81.73	83.77
		2.5	SD	3.53	5.3	1.74	2.85	2.85	3.37	5.11	2.13	4.09	3.05
		3	Mean	87.59	91.07	91.76	85.37	85.37	86.6	86.8	91.85	85.04	83.77
		3	SD	3.17	3.98	2.36	3.28	2.85	3.77	4.97	2.39	3.77	3.05
	SBS-CFS	2	Mean	87.13	82.41	92.38	83.25	85.27	85.40	80.23	91.88	80.95	81.52
		2	SD	3.81	4.66	2.48	3.46	2.83	3.5	4.44	2.18	3.55	2.97
		2.5	Mean	88.75	82.77	94.49	83.64	85.27	80.51	81.8	88.77	79.22	81.52
		2.5	SD	3.23	5.19	1.67	4.21	2.83	3.67	4.88	2.52	3.93	2.97
		3	Mean	87.28	90.63	91.03	86.82	85.27	85.89	82.54	91.62	81.36	81.52
		3	SD	3.24	3.8	2.68	3.02	2.83	3.26	4.94	2.19	3.82	2.97
RBF-SVM	CFS	2	Mean	85.58	89.67	89.40	85.72	86.26	80.67	83.63	88.66	81.14	81.35
		2	SD	3.78	3.46	3.30	3.25	2.83	3.84	4.47	2.43	3.35	2.76
		2.5	Mean	87.03	85.45	92.83	84.08	86.26	81.99	82.81	88.91	80.46	81.35
		2.5	SD	3.44	5.04	2.01	4.12	2.83	3.68	4.35	2.69	3.52	2.76
		3	Mean	94.80	83.66	97.16	86.37	86.26	88.65	77.6	94.44	80.14	81.35
		3	SD	2.2	5.53	1.38	4.19	2.83	3.07	5.27	1.63	4.08	2.76
	SBS	2	Mean	85.81	88.79	90.02	86.61	86.52	83.64	89.36	88.70	85.66	86.23
		2	SD	3.88	3.58	2.99	3.33	2.77	4.15	3.81	3.08	3.12	2.85
		2.5	Mean	87.19	85.44	92.63	84.26	86.52	86.56	84.95	92.70	83.74	86.23
		2.5	SD	3.55	4.85	2.14	3.9	2.77	3.37	4.99	1.9	4.03	2.85
		3	Mean	95.02	85.31	97.12	86.78	86.52	96.05	84.36	97.93	87.72	86.23
		3	SD	2.3	5.34	1.37	4.2	2.77	1.75	5.27	0.92	3.88	2.85
	SBS-CFS	2	Mean	85.6	90.44	89.42	86.19	86.40	80.82	83.31	88.33	80.48	81.31
		2	SD	3.73	3.3	3.25	3.09	2.76	4.26	5.01	2.72	3.91	3.07
		2.5	Mean	85.85	84.02	92.79	83.68	86.40	81.46	81.31	88.77	79.08	81.31
		2.5	SD	3.83	5.27	1.94	4.17	2.76	3.88	4.81	2.71	3.91	3.07
		3	Mean	95.26	84.74	97.38	87.34	86.40	90.67	79.29	94.86	92.18	81.31
		3	SD	1.98	5.38	1.17	4.05	2.76	3.11	5.48	2.03	4.36	3.07
MLP	CFS	2	Mean	82.87	83.77	89.34	81.73	81.67	78.83	80.84	87.29	77.63	77.47
		2	SD	3.55	3.8	2.76	3.29	2.7	3.7	4.61	2.46	3.61	2.81
		2.5	Mean	78.81	82.24	87.42	78.65	81.67	76.65	76.72	85.66	74.19	77.47
		2.5	SD	3.48	3.48	2.43	3.61	2.7	3.85	4.62	2.81	3.51	2.81
		3	Mean	92.51	79.01	95.74	81.9	81.67	86.48	74.86	93.26	77.40	77.47
		3	SD	2.44	5.24	1.66	4.06	2.7	3.3	5.25	1.76	4.08	2.81
	SBS	2	Mean	80.75	81.74	88.16	79.36	80.37	78.61	79.83	87.89	78.53	79.03
		2	SD	3.97	4.34	2.9	3.72	2.91	4.15	4.6	2.77	3.79	2.84
		2.5	Mean	78.73	80.83	87.63	77.52	80.37	76.59	79.52	85.77	79.03	79.03
		2.5	SD	4.01	5.15	2.55	4.26	2.91	3.95	4.82	2.71	3.83	2.84
		3	Mean	90.58	78.53	94.75	81.44	80.37	89.8	77.73	94.87	80.78	79.03
		3	SD	3.15	5.03	1.96	3.89	2.91	3.13	5.53	1.72	4.2	2.84
	SBS-CFS	2	Mean	81.54	83.93	88.15	80.86	81.03	77.56	80.43	85.52	76.84	77.23
		2	SD	3.62	3.97	2.89	3.32	2.86	4.25	4.69	3.25	3.97	3.48
		2.5	Mean	80.03	82.06	88.19	78.92	81.03	77.06	75.37	87.12	73.47	77.23
		2.5	SD	3.73	4.89	2.65	4.04	2.86	3.99	5.46	2.49	4.26	3.48
		3	Mean	91.26	77.10	95.19	80.02	81.03	86.76	75.89	93.20	78.76	77.23
		3	SD	2.68	4.26	1.67	4.15	2.86	3.76	5.04	2.26	4.20	3.48

Table 6. Inference throughput of classifiers and feature selection methods in PD and severity classification tasks.

Model	FS	Inference Throughput ( $10^{5}$ obj/s)
		PD Classification		Severity Classification
		M-SamEn	DM-SamEn	M-SamEn	DM-SamEn
	CFS	0.24	0.55	0.22	0.52
AW-KNN	SBS	0.18	0.41	0.18	0.43
	SBS-CFS	0.24	0.61	0.24	0.57
	CFS	0.4	0.51	0.16	0.34
RBF-SVM	SBS	0.34	0.51	0.15	0.31
	SBS-CFS	0.37	0.63	0.16	0.59
	CFS	1.34	1.99	0.92	1.7
MLP	SBS	1.04	1.7	0.75	1.33
	SBS-CFS	1.27	2.05	0.99	1.76

Table 7. CO₂ emissions of classifiers and feature selection methods in PD and severity classification tasks.

Model	FS	CO₂ Emissions ( $10^{- 8}$ kg/obj)
		PD Classification				Severity Classification
		M-SamEn		DM-SamEn		M-SamEn		DM-SamEn
		Train	Test	Train	Test	Train	Test	Train	Test
	CFS	0.23	0.194	0.083	0.081	0.132	0.136	0.056	0.062
AW-KNN	SBS	0.298	0.251	0.12	0.11	0.169	0.171	0.07	0.077
	SBS-CFS	0.211	0.182	0.073	0.07	0.123	0.129	0.049	0.056
	CFS	0.915	0.117	1.22	0.08	1.027	0.1889	1.06	0.088
RBF-SVM	SBS	0.937	0.133	0.811	0.085	1.071	0.201	0.789	0.095
	SBS-CFS	0.872	0.115	1.04	0.068	1.001	0.18	0.87	0.055
	CFS	0.321	0.038	0.31	0.023	0.28	0.035	0.254	0.02
MLP	SBS	0.348	0.047	0.31	0.027	0.171	0.043	0.253	0.026
	SBS-CFS	0.317	0.038	0.293	0.022	0.272	0.034	0.257	0.019

Table 8. Performance comparison with existing research methods.

Existing Research	Algorithm		Task	Accuracy
[32]	Adam-LSTM		PD Classification	98.60%
[32]	Adam-LSTM		Severity Classification	96.6%
[33]	PSR, SEE, DQSD, VMD, and SVM		PD Classification	98.92%
[33]	PSR, SEE, DQSD, VMD, and SVM		Severity Classification	93.37%
[34]	Parallel 2D-DCNN		PD Classification	95.5%
[34]	Parallel 2D-DCNN		Severity Classification	95.75%
[35]	Transformer 1D-STE		PD Classification	95.2%
[35]	Transformer 1D-STE		Severity Classification	N/A
[36]	PSR, EMD, and NN		PD Classification	98.8%
[36]	PSR, EMD, and NN		Severity Classification	N/A
[37]	RFE, and RF		PD Classification	94.28%
[37]	RFE, and RF		Severity Classification	N/A
[38]	PARAFAC, and TD		PD Classification	97%
[38]	PARAFAC, and TD		Severity Classification	N/A
Proposed Algorithm	AW-KNN, RBF-SVM, MLP		PD Classification	98.38%
Proposed Algorithm	AW-KNN, RBF-SVM, MLP		Severity Classification	96.80%
DQSD: Factor Signal Decomposition VMD: Variational Mode Decomposition PSR: Phase Space Reconstruction SEE: Shannon Energy Envelope 1D-STE: 1D-Spatial Transformer Encoder		EMD: Empirical Mode Decomposition RFE: Recursive Feature Elimination RF: Random Forest TD: Tucker Decomposition PARAFAC: Parallel Factor Analysis

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Nguyen, M.T.P.; Tran, M.K.P.; Nakano, T.; Tran, T.H.; Nguyen, Q.D.N. An Approach for Detecting Parkinson’s Disease by Integrating Optimal Feature Selection Strategies with Dense Multiscale Sample Entropy. Information 2025, 16, 1. https://doi.org/10.3390/info16010001

AMA Style

Nguyen MTP, Tran MKP, Nakano T, Tran TH, Nguyen QDN. An Approach for Detecting Parkinson’s Disease by Integrating Optimal Feature Selection Strategies with Dense Multiscale Sample Entropy. Information. 2025; 16(1):1. https://doi.org/10.3390/info16010001

Chicago/Turabian Style

Nguyen, Minh Tai Pham, Minh Khue Phan Tran, Tadashi Nakano, Thi Hong Tran, and Quoc Duy Nam Nguyen. 2025. "An Approach for Detecting Parkinson’s Disease by Integrating Optimal Feature Selection Strategies with Dense Multiscale Sample Entropy" Information 16, no. 1: 1. https://doi.org/10.3390/info16010001

APA Style

Nguyen, M. T. P., Tran, M. K. P., Nakano, T., Tran, T. H., & Nguyen, Q. D. N. (2025). An Approach for Detecting Parkinson’s Disease by Integrating Optimal Feature Selection Strategies with Dense Multiscale Sample Entropy. Information, 16(1), 1. https://doi.org/10.3390/info16010001

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