## 3 Methods
### Feature Extraction
#### 3.2.6. Hjorth Parameters  
The Hjorth parameters constitute a trio of mathematical descriptors conceived by [PERSON] in the 1970s, designed for the quantitative analysis of time-series data characteristics.  
These parameters, along with other features such as heart rate variability and GSR responses, were instrumental in differentiating the five defined emotional states: excitement, happiness, anxiety, calmness, and sadness. The selection of these emotions was driven by their distinct physiological patterns, which were consistently observed across the VR experiences. In addition, they have received widespread application across various domains, particularly in the realm of physiological signal analysis, encompassing electrocardiogram (ECG) and galvanic skin response (GSR) signals.  
The principal Hjorth parameters--activity, mobility, and complexity--each unravel unique facets of signal dynamics, offering a multifaceted perspective on the underlying physiological processes.  
Activity, one of the Hjorth parameters, quantifies the signal's power and is determined by calculating the signal's variance. Analyzing the activity of the ECG and GSR signals allows researchers to evaluate the overall power or intensity of the physiological responses these signals represent. Variations in activity can shed light on the strength or magnitude  
Figure 5: The energy of ECG signals.  
of the physiological processes in question. The activity, being synonymous with variance, is mathematically defined as  
\[\text{Activity}=\sigma^{2}=E\Big{[}(x(t)-\mu)^{2}\Big{]} \tag{10}\]  
where \(E\) denotes the expected value, \(x(t)\) represents the signal at time \(t\), and \(\mu\) is the mean of the signal [46].  
Mobility is a Hjorth parameter indicative of a signal's dynamics, calculated as the square root of the ratio of the variance of the signal's first derivative to the variance of the signal itself. This parameter essentially captures the mean frequency or spectral breadth of the signal. Analyzing the mobility of ECG and GSR signals provides researchers with an understanding of dynamic shifts and fluctuations within physiological responses, aiding in the detection of frequency variations or spectral characteristics. The formula for mobility is given by  
\[Mobility=\sqrt{\frac{\text{Var}\Big{(}\frac{dx(t)}{dt}\Big{)}}{\text{Var}(x(t ))}} \tag{11}\]  
where \(\text{Var}\Big{(}\frac{dx(t)}{dt}\Big{)}\) is the variance of the first derivative of the signal and \(\text{Var}(x(t))\) is the variance of the signal itself [47].  
Complexity, another Hjorth parameter, provides a measure of the frequency changes within a signal. Calculated as the mobility of the signal's first derivative divided by the mobility of the signal itself, the complexity reflects the relative frequency changes or the rate of spectral modulation. By analyzing the complexity of physiological signals, such as ECG and GSR, researchers can uncover patterns and variations in frequency content. This sheds light on dynamic physiological processes and potential regulatory mechanisms. Put simply, complexity quantifies the changes in signal frequency, and its calculation is as follows:  
\[Complexity=\frac{\text{Mobility}\Big{(}\frac{dx(t)}{dt}\Big{)}}{\text{Mobility }(x(t))} \tag{12}\]  
where \(\text{Mobility}\Big{(}\frac{dx(t)}{dt}\Big{)}\) is the mobility of the signal's first derivative, and \(\text{Mobility}(x(t))\) is the mobility of the signal itself [48]. These measures provide valuable information on the dynamics of physiological responses captured by the ECG and GSR signals through quantitative assessments of signal properties.