## 3 Problem Statement and Mathematical Formalization
### Mathematical Formalization  
We formulate the problem of this study as follows:  
* Let \(U\) be the set of users, \(I_{u}(t)\) represent the set of interactions for a given user \(u\) at time \(t\). * The UI adaptation function, which determines the UI for a given user \(u\) at time \(t\), is defined as \[UI(u,t)=f(P_{u},I_{u}(t),H_{u}),\] (2) where \(P_{u}\) represents the user's preferences, \(I_{u}(t)\) captures real-time user interaction and \(H_{u}\) denotes the user's interaction history. - To improve personalization, users are grouped based on interaction patterns using K-Means++ clustering. - Let \(G=\{G_{1},G_{2},\ldots G_{L}\}\) denote the set of user clusters. - Each user \(u\) belongs to a cluster \(G_{i}\), where: \[G_{i}=argmin\ d(X_{u},C_{k}),\] (3) where \(d(X_{u},C_{k})\) is the distance metric between the user's features and cluster centroid \(C_{k}\). This clustering mechanism enables the creation of adaptive UI groups, allowing for more targeted and personalized UI adaptations. - The effectiveness of a dynamic UI adapation strategy is evalauted using optimization function: \[max\in\sum_{u\in U}\alpha\cdot E_{u}+\beta\cdot UX_{u}+\gamma\cdot S_{u}-\delta \cdot D_{u},\] (4) where \(\circ\)\(E_{u}\) measures the level of user interaction with the system. It quantifies user actions (click frequency, page views, session duration, search queries, checkout behavior) that indicate engagement. \[E_{u}=w_{1}\cdot c_{u}+w_{2}\cdot p_{u}+w_{3}\cdot s_{u}+w_{4}\cdot q_{u}+w_{5 }\cdot ch_{u}.\] (5)  
The coefficients \(w_{i}\) represent the weights assigned to various user engagement actions. Theses weights reflect the relative importance of each action in contributing to business objectives. Actions with a higher impact on outcomes such as \"purchase\" or \"checkout\" are assigned greater weights (e.g., \(w_{1}=10\), \(w_{2}=8\)) than actions like \"view\" or \"scroll\" (e.g., \(w_{3}=6\), \(w_{4}=4\)). This prioritization aligns with key business metrics such as revenue generation and purchase intent. To ensure adaptability, the weights are dynamically updated over time based on user behavior and real-time feedback. The adjustment follows the update formula:  
\[w_{i}(t)=w_{i}(t-1)+\alpha\cdot\sigma_{i}, \tag{6}\]  
where  
* \(w_{i}(t)\) is the updated weight for the \(i^{\text{th}}\) action at time t.
* \(w_{i}(t-1)\) is the previous weight at time \(t-1\). * \(\alpha\) is a learning rate controlling the adjustment speed. * \(\sigma_{i}\) represent the influence of new data or feedback on the \(i^{\text{th}}\) action. \(UX_{u}\) represent the user experience score which evaluates UI fluidity, responsiveness, and ease of use. It is defined by sum of retention rate \(R_{u}\), response time \(R_{u}\) and error rate \(R_{u}\) at time \(t\) that computed as: \[UX_{u}=\frac{1}{T}\sum_{t=1}^{T}\lambda_{1}\cdot R_{u}(t)+\lambda_{2}\cdot RT_{u }(t)+\lambda_{1}\cdot ER_{u}(t),\] (7)  
\(\lambda_{i}\) are the weighting factors to \(i^{\text{th}}\) aspect. The coefficients \(\lambda_{i}\) represent the weighting factors for the \(i^{\text{th}}\) aspect of user experience, such as ease of use, visual design, and content satisfaction. These weights are derived by combining user survey responses with interaction log data. Users are asked to rate each UX aspect on a Likert scale (e.g., 1 to 5), and these ratings are aggregated across the user base. To determine which aspects, contribute most significantly to the overall UX score, we apply Principal Component Analysis (PCA). PCA helps identify the most influential dimensions by analyzing variance across the collected ratings and interactions. This approach ensures that aspects most strongly correlated with user satisfaction receive higher weight, making the UX evaluation both data-driven and user-centric. \(S_{u}\) represent user satisfaction and capture subjective user feedback through surveys, ratings, or implicit behavior analysis.