## 2 Materials and Methods
### Clustering Analysis  
Clustering algorithms are essential in unsupervised machine learning, enabling the grouping of data points based on patterns or inherent structures without predefined labels [19]. This study used clustering to identify subgroups within the dataset that shared similar physical and neuroimaging characteristics. In principle, all clustering methods differ, with unique approaches and assumptions. Yet, most clustering techniques can also be tuned to different datasets with the relevant preprocessing and careful adjustments of parameters. Clustering analysis can be divided into four main approaches:  
* Hierarchical clustering;
* Density-based clustering;
* Probabilistic clustering;
* Partitioning clustering. Hierarchical clustering forms one nested hierarchy of clusters using agglomerative (bottom-up) or divisive (top-down) strategies, enabling data analysis at many different granularity levels. Such an approach makes the DBSCAN effective for datasets with non-linear cluster boundaries, though its performance can degrade with varying cluster densities [20]. Density-based methods such as the DBSCAN (Density-Based Spatial Clustering of Applications with Noise) group points based on dense regions in the feature space, identifying clusters of arbitrary shapes while isolating noise as outliers. This makes the DBSCAN effective for datasets with non-linear cluster boundaries, though its performance can degrade with varying cluster densities [21]. Model-based clustering assumes that data are generated from a mixture of probability distributions, often Gaussian, and seeks to optimize these models to represent the data best. This approach accommodates overlapping clusters but relies on accurate model assumptions [22]. Probabilistic methods assume that data points are generated from a mixture of probability distributions, typically Gaussian. The clustering process involves estimating distribution parameters to represent the dataset accurately. The Expectation-Maximization (EM) algorithm is an example of a probabilistic approach. It alternates between two steps: the Expectation (E) step, where data points are probabilistically assigned to clusters based on current model estimates, and the Maximization (M) step, where the distribution parameters are updated to maximize the likelihood of the observed data. Partitioning methods such as the k-means direct the data into a fixed number of clusters by minimizing the within-cluster sum of squares; the spatial compactness of the resulting clusters ensures that the partition is suitable. Due to its effectiveness and simplicity, the k-means is commonly used in applications dealing with large continuous numerical datasets. However, the algorithm operates on the assumption of spherical clusters of an equal size. It is susceptible to the initial placement of centroids, a limitation that can be alleviated through advanced initialization methods such as k-means++ [23]. Despite these limitations, the scalability and interpretability of the k-means make it a popular choice in many domains, including medical and health-related research. K-means clustering was selected for this study due to its ability to efficiently handle the continuous features: the body mass index (BMI) and skeletal muscle index (SMI). The algorithm's reliance on minimizing within-cluster variance aligned group participants based on similar physical characteristics. The k-means++ initialization method enhanced the algorithm's robustness by reducing the sensitivity to random centroid initialization, ensuring more consistent convergence. The k-means algorithm operates by iteratively partitioning the dataset \(X=\{x_{1},x_{2},\ldots,x_{n}\}\), comprising \(n\) data points in \(k\) clusters \(C=\{C_{1},C_{2},\ldots,C_{k}\}\).