## 3 Proposed Research Methodology
### Models
#### 3.3.1 Logistic Regression  
Logistic regression (LR) is a specialized form of generalized linear model designed for scenarios where the outcome variable is binary, indicative of'success' or 'failure', and is associated with probabilities. Linear regression techniques are not suitable for such cases because probabilities are constrained to the interval (0, 1). [PERSON] [56] details how  
\begin{table}
\begin{tabular}{c c} \hline
**Pre-Game Predictors** & **In-Game Predictors** \\ \hline _Average difference between teams in:_ & Whether Team 1 eliminated an opponent first \\ Total games played & Whether Team 1 secured the first objective of the game \\ Rank & _Average difference between teams in:_ \\ Experience on selected character & Difference in average gold acquired at 10 min \\ Total experience across all characters & Difference in average gold acquired at 15 min \\ Number of characters played across all games & Difference in average damage taken by 10 min \\ Mean experience across all characters & Difference in average damage taken by 15 min \\ Current win/loss streak & Difference in average damage dealt by 10 min \\  & Difference in average damage dealt by 15 min \\ \hline \end{tabular}
\end{table}
Table 3: The final engineered feature set comprises seven pre-game predictors, eight in-game predictors, and a binary label.  
the mathematical framework of the LR overcomes this limitation by focusing on the log odds of the binary outcome variable. For a binary outcome \(Y\) with a'success' outcome \(Y_{s}\) occurring with probability \(p\), the odds of \(Y_{s}\) are \(\frac{p}{1-p}\). Odds span \((0,\,\infty)\) for \(p\) approaching 1 and 0. The mapping from probabilities to odds is nonlinear, with probabilities less than 0.5 corresponding to odds between 0 and 1 and probabilities greater than 0.5 corresponding to odds greater than 1. This asymmetry is resolved by linearizing the relationship between explanatory variables and the binary response variable by taking the natural logarithm of the odds, \(\log\Bigl{(}\frac{p}{1-p}\Bigr{)}\). This transformation expands the span of values of odds to \((-\infty,+\infty)\), thereby enabling linear modeling of the relationship between predictors and the response variable through the logit link function (Equation (1)).  
\[\log\Bigl{(}\frac{p}{1-p}\Bigr{)}=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+ \cdots\beta_{n}x_{n} \tag{1}\]  
The transformation of probabilities to log odds, which maps the raw binary response values to \(-\infty\) or \(+\infty\), renders traditional least squares fitting methods inapplicable. To address this, logistic regression employs maximum likelihood estimation to optimize the fit. This method works by associating the observed response values with the expected values predicted by the model for a chosen initial set of parameters, denoted as \(\beta_{0}\ldots\beta_{n}\), optimizing fit by minimizing log-loss via Gradient Descent [57]. The response variable can be mapped back to probability via the logistic function (Equation (2)).  
\[y=\frac{1}{1+e^{-(\beta_{0}+\beta_{1}x_{1}+\cdots\,\beta_{n}x_{n}\,)}} \tag{2}\]  
Regression stands out for its simplicity, adaptability to both categorical and numeric predictors, and ease with which it can be extended through regularization techniques to prevent overfitting. Its high interpretability is a significant advantage, as it offers explicit coefficients for each predictor, indicating the direction and magnitude of their effects on the outcome. However, it operates under the assumption of a linear relationship between the predictors and the log odds of the response variable, which can lead to biased estimates in more complex problem spaces where this linearity does not hold. Additionally, in situations where the feature space exceeds the number of observations, logistic regression is susceptible to overfitting, highlighting its limitations in handling high-dimensional data without adequate regularization or dimensionality reduction.