## 2 Materials and Methods
### Inertial Sensor  
The inertial sensor used to collect data during the TUG test included a triaxial accelerometer sensor board (Freescale RD3152 MMA7260Q) with a sampling rate of 45 readings/s, an internal battery, and an SD storage card placed inside a plastic box. The sensor was small and lightweight and could be worn by the test subjects without causing any discomfort. The location of the strapped sensor approximated the center of mass location with respect to the belt; it was placed between the 3 rd and 5 th lumbar vertebrae (L3-L5) [22], as shown in Figure 1. The sensor weighed approximately 26.5 g, and the length, width, and height were 232, 7, and 21 mm, respectively, placing no substantial burden on the subject. The accelerometer collected data from the X (vertical, V; up: +, down: -), Y (mediolateral, ML; right: +, left: -), and \(Z\) (anterior-posterior, AP; forward: +, backward: -) axes. We used MSE to calculate the complexity of the three axes as features. While calculating the sample entropy for each coarse-grained series, the MSE calculation process involved the following sub-processes [27]: (1) coarse granulation, (2) sample entropy, and (3) the complexity index. The \"coarse-graining\" process is based on the scale factor of the segment windows, in which the average of the data values for each segment window is calculated, forming a new time series. Each element \(y_{j}{}^{(\tau)}\) of the new coarse-grained time series is calculated according to  
\[y_{i}{}^{(x)}\ =\ \frac{1}{\tau}\sum_{i-(j-1)x+1}^{jx}f_{i},\ 1\leq j\leq\frac{N}{\tau} \tag{1}\]  
where \(\tau\) is the scale factor, \(N\) is the size of the original dataset, and \(x_{i}\) is a data point in the original time series. For a scale factor of 1, the coarse-grained time series is simply the original time series; a scale factor of 2 involves the averaging of a pair of consecutive points from the original time series, which becomes the time series from which to create a scale factor of 3, and so on. When calculating the first and remaining time series, the duration between them is determined and the maximum value is taken, as shown in Figure 2.  
The sample entropy (SampEn) was calculated for each coarse-grained time series in order to obtain entropy measurement values at each scale [42]. SampEn at each time scale \(\tau\) was expressed as the negative of the natural logarithm of the conditional probability \(C^{m}(r)\), where the two sequences are compared within a similarity tolerance \(r\) for consecutive points \(m\) and \(m+1\). The formula for SampEn is  
\[\text{SampEn}(N,m,r)\ =\ -\ln\frac{C^{m+1}\left(r\right)}{C^{m}(r)} \tag{2}\]  
where \(N\) is the total number of data points, \(m\) is the number of consecutive data points, and \(r\) is the tolerance for accepting the match, which was chosen to be between 10% and 20% of the sample standard deviation \(\sigma\) of the time series. In this study, the parameters were set as follows: \(m=2\) and \(r=0.15\sigma\).  
The complexity index is a function of sample entropy, which in turn is a function of the specification factor and is calculated using Equation (3). The sum of the sample entropy (SampEn) at each scale is the complexity index (CI), as shown in Figure 3.  
\[\text{Complexity Index}\ =\ \sum_{i=1}^{N}\text{\emph{Sample Entropy}}(i) \tag{3}\]  
Figure 1: Schematic of the wearable sensor and triaxial.  
Figure 2: Schematic of Phase 2 (sample entropy) of the multiscale entropy (MSE) analysis.