## 2 Modeling Procedure
### Secondary Data Modeling  
A model was created through secondary data analysis of a randomized controlled trial conducted in the United Kingdom, which suggested that smartphones are effective at motivating individuals with limited intrinsic motivation for exercise to engage in physical activity [16]. There were three groups of 55 male adult participants each. Data from participant number 115 were used for the semi-physical identification because he had the most reliable set of data for the study period with the least amount of missing information. The structure of the intervention is based on a simplified version of the dynamical systems model of the Social Cognitive Theory (SCT) presented by [PERSON] et al. [15], which is illustrated as a fluid analogy in Figure 1. The only output considered is the behavior (\(\eta_{4}\)), which is measured through the number of daily steps taken by the individual. This \"subsystem\" is used to describe the algorithm of physical activity intervention related to social norms to encourage physical activity behavior. The simplified model included inputs (\(\xi_{1}\)), (\(\xi_{2}\)), (\(\xi_{5}\)) and (\(\xi_{7}\)), while ignoring inputs (\(\xi_{3}\)) (Persuasion), (\(\xi_{4}\)) (Internal Cues), and (\(\xi_{6}\)) (Intrapersonal States) as they were irrelevant to the study. These inputs will be assumed to be equal to zero. The meaning of some inputs is:  
* Skills training (\(\xi_{1}\)). These activities help to increase (or decrease) the self-management skills of the individual. * Observed behavior of others (\(\xi_{2}\)). This input refers to vicarious learning or social learning. This is a type of learning that occurs when an individual can acquire knowledge and skills by observing others and learning from the outcomes of their actions. * Perceived barriers (\(\xi_{5}\)). This input refers to external conditions that affect behavior. For instance, unfavorable weather conditions, limited availability of exercise facilities, or insufficiently secure walking paths can all hinder physical activity levels. This input refers to conditions that are perceived by the participants. * Environmental context (\(\xi_{7}\)). This input refers to situational factors which directly influence the behavioral outcomes. This study takes into account the date of measurement, which is identified as either a weekend or a weekday. This approach used principles from semi-physical system identification and relied on the widely-accepted prediction-error identification methods (PEM) [17]. Semi-physical identification combines physical knowledge and experimental data to create a mathematical model of a physical system (which was created based on the data obtained from male subjects in Great Britain and was previously presented in an earlier study published in 2016) [16]. It involves using simplified physical models or analogies to represent a complex system (human behavior can be considered a complex system, composed of interconnected components that interact nonlinearly) and then fitting experimental data to these models to estimate system parameters. This approach can be used to predict the system's behavior under different conditions. The estimation process was carried out in MATLAB(r) using the idgrey and greyset commands from the System Identification Toolbox [18]. The idgrey model is used to represent a system as a state-space model with identifiable coefficients. The greyset function is employed to estimate the unknown parameters of the idgrey model. Results obtained  
Figure 1: Fluid analogy for the simplified version of the SCT model [9]. are depicted in Figure 2. By comparing the estimated behavior (\(\eta_{4}\)) based on the number of steps obtained from semi-physical identification using simulated data with the real experimental data from male adult participant number 115, a fitting of 30.5% was achieved with nine iterations, which serves as a good starting point.