## 3 Materials and Methods
### Data Processing and Sensor Configuration
Figure 2 presents the gyroscope data, reflecting angular changes during body movement, while Figure 3 displays the linear acceleration measured, which isolates the acceleration component excluding gravitational effects. Figure 4 shows the gravitational component of acceleration, highlighting the orientation of the smartphone during the activity. These visualizations serve as the basis for the feature extraction process discussed in the subsequent section. Statistical Feature Analysis for Activity Recognition  
Table 3 summarizes the statistical analysis of the feature vectors computed from each sliding window. The process began with raw data collection, followed by segmentation into windows corresponding to each activity. For each window, signal features were extracted and processed using statistical methods. This comprehensive feature extraction approach provides a detailed basis for the subsequent classification analysis and allows for a robust evaluation of model performance under varying sensor configurations. In this study, several key time-domain features were extracted to capture the essential characteristics of the sensor signals over time. The **mean** (\(\mu\)) provides the average value of the signal in each window, highlighting the overall level of activity:  
\[\mu=\frac{1}{n}\sum_{i=1}^{n}x_{i}, \tag{1}\]  
where \(n\) represents the number of data points in the window, and \(x_{i}\) denotes each individual sensor reading. Figure 4: Running Activity Gravity Sensor Data: gravitational acceleration values along the x, y, and z axes recorded between 80 and 85 s.  
Figure 3: Running Activity Linear Accelerometer Data: linear acceleration values along the x, y, and z axes recorded between 80 and 85 s. \begin{table}
\begin{tabular}{c l l} \hline \hline
**No.** & **Statistical Feature** & **Description** \\ \hline
1. & Mean & The average value of the signal over the window. \\
2. & Standard deviation & The standard deviation value over the window. \\
3. & Median absolute value & The median value over the window. \\
4. & Skewness & Describes the distribution’s \\
5. & Maximum & The maximum value of the signal over the window. \\
6. & Minimum & The minimum value of the signal over the window. \\
7. & Kurtosis & Describes the degree of \\  & & peakedness of the data. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Descriptive Statistical Features for Human Activity Recognition. The **standard deviation** (\(\sigma\)) quantifies the amount of variation in the signal, reflecting how much the values fluctuate around the mean:  
\[\sigma=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\mu)^{2}}, \tag{2}\]  
**Skewness** assesses the asymmetry of the signal distribution, indicating whether the values tend to be higher or lower than the mean. This is particularly useful for understanding the balance of activity levels:  
\[\text{Skewness}=\frac{n}{(n-1)(n-2)}\sum_{i=1}^{n}\biggl{(}\frac{x_{i}-\mu}{ \sigma}\biggr{)}^{3}, \tag{3}\]  
Additionally, the **maximum** (\(\max(x)\)) and **minimum** (\(\min(x)\)) values capture the range of the data, helping to identify the peak and lowest activity levels:  
\[\max(x)=\max\{x_{1},x_{2},\dots,x_{n}\}, \tag{4}\]  
\[\min(x)=\min\{x_{1},x_{2},\dots,x_{n}\}. \tag{5}\]  
Lastly, **kurtosis** measures the \"tailedness\" or sharpness of the signal distribution. This feature helps to identify the presence of outliers or extreme values that may affect activity classification:  
\[\text{Kurtosis}=\left(\frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum_{i=1}^{n}\biggl{(} \frac{x_{i}-\mu}{\sigma}\biggr{)}^{4}\right)-\frac{3(n-1)^{2}}{(n-2)(n-3)}. \tag{6}\]  
Together, these extracted features provide a rich representation of the sensor data and form the basis for further analysis in human activity recognition, allowing for robust classification models to be built based on the variability and distribution of the signal data [31].