## 2 Materials and Methods
### Derivation of Smooth Functions  
Time-normalization was applied prior to any FDA procedures, as in previous research [18; 24]. The data were time-normalized to 100 data points. The primary procedure in FDA is to transform discrete data into a functional form. The _fda_ package in R (version 3.4.2) was used for the transformation of discrete data into a functional form and smoothing the functions and the code used is provided in Appendix A.1. R Code A1. The cyclical data were cut into a single cycle of AKS in order to analyze one complete cycle. The cyclical data were cut from the beginning of AKS (stance event) to the end of AKS (propelling kettlebell overhead event) using video recorded in the Vicon Nexus motion analysis system. Each cycle was transformed into a functional form. The data were assumed to have some noise or measurement error, which was assumed to be normally distributed. The data were expressed by the equation:  
\[y_{j}=x(t_{j})+\varepsilon_{j}, \tag{1}\]  
where \(y_{j}\) is the raw data, \(x(t_{j})\) is the signal, and \(\varepsilon_{j}\) is the noise or measurement error. A set of functional building blocks \(\phi_{k}\) with \(k=1,\,\ldots,\,K\) termed as basis functions were combined linearly [16]. The larger the number of \(K\), the closer these functions were to the exact interpolation of the data, whereas the smaller the number of \(K\), the smoother the data. A smaller number of \(K\) increases the residual difference between the smooth function and noisy data [1]. The smooth function \(x(t)\) was represented as a linear combination of basis functions:  
\[x(t)=\sum_{k=1}^{K}c_{k}\phi_{k}(t), \tag{2}\]  
where \(\phi_{k}(t)\) is the \(k^{\text{th}}\) basis function at time \(t\), \(c_{k}\) is the coefficient of the expansion, and \(K\) is the number of basis expansion [16]. There are several basis functions available--Fourier, B-spline, wavelet, exponential and power, polynomial, polygonal, step-function, constant (single), empirical, and designer bases functions--but the first two are the most commonly used [34]. The use of a basis function depends on the nature of the data. Basically, the Fourier basis function is used for periodic or cyclical functions, while the B-spline basis function is used for nonperiodic functions. The B-spline basis function was chosen since the cyclical data had been cut into individual cycles to portray the behavior of a complete cycle of AKS. The B-spline basis function is one of the most prominent spline functions besides M-spline, I-spline, and truncated power function [35]. It is numerically stable and flexible, which makes it the most frequently used basis function [36]. B-spline was a popular choice among previous researchers as the basis function [15; 18; 19; 22; 23; 24; 37]. There is no universal basis function that is suitable for all types of data. The spline basis function is a piecewise polynomial made up of divided internal observations to form subintervals with boundaries at points known as break points or breaks [16]. Knots are placed at breaks and are usually equally spaced. However, excessive knots form overfitted data, while fewer knots form underfitted data [38]. The order of a polynomial must be one point higher than its degree. By default, the order of spline is four; thus, the degree is three, which implies a cubic polynomial. Finding the first and second derivatives, velocity and acceleration, is the usual practice in biomechanics. The spline basis must be at least two orders higher than the highest-order derivative that is used [16]. Since order two is the highest order of the derivative of interest (acceleration), a minimum of order four of the spline basis needs to be used. The number of basis functions \(K\) is the summation of orders and the number of interior knots in the curve [16]. The interior knots refer to the knots at breaks, excluding the ones at the beginning and end of a function's domain. Since a large \(K\) (\(K=100\)) was used, thus, the number of equally spaced interior knots was 98.