Concepts of Regression
Regression is a statistical method used for modeling the relationship between a dependent variable (target) and one or more independent variables (predictors or features). The primary goal of regression analysis is to understand how changes in the independent variables affect the dependent variable.
Regression Equation:The foundation of regression analysis is the regression equation, which represents the relationship between the dependent variable (Y) and one or more independent variables (X₁, X₂, … Xₖ).
In simple linear regression, the equation is: Y = β₀ + β₁X + ε, where:
- Y is the dependent variable.
- X is the independent variable.
- β₀ and β₁ are the coefficients to be estimated (intercept and slope).
- ε represents the error term, which accounts for the unexplained variability in Y.
Coefficients (β₀ and β₁):Coefficients are values that the regression model estimates to quantify the relationship between the independent and dependent variables.
- β₀ (intercept): Represents the value of Y when X is 0.
- β₁ (slope): Represents the change in Y for a one-unit change in X.
Residuals:Residuals (or errors) are the differences between the observed values of the dependent variable (Y) and the predicted values (Ŷ) from the regression model.
- Residuals are calculated as: Residual = Y – Ŷ.
- Analyzing residuals helps assess the model’s fit and assumptions.
Goodness of Fit:Goodness of fit measures how well the regression model fits the data.
- One common measure is R-squared (R²), which quantifies the proportion of variance in Y that is explained by the independent variables. R² ranges from 0 to 1, with higher values indicating a better fit.
Cross-Validation:Cross-validation is a technique used to evaluate a model’s performance on unseen data.
- Common methods include k-fold cross-validation, where the dataset is divided into k subsets , and the model is trained and tested on different combinations of these folds to estimate its generalization performance
R-Squared Value:
R-squared (R²) is a statistical measure that is often used to evaluate the goodness of fit of a regression model. It quantifies the proportion of the variance in the dependent variable that is explained by the independent variables in the model. R² values range from 0 to 1, with higher values indicating a better fit.
Overfitting in Regression:Overfitting in regression occurs when the model is excessively complex and fits the training data too closely. It tries to capture not only the true underlying relationship between the predictors and the target variable but also noise, random fluctuations, and outliers present in the training data.
Consequences:
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- On the training data, an overfit model will exhibit a very high R-squared because it essentially “memorizes” the training data.
- On new, unseen data the model’s performance deteriorates significantly because it cannot generalize well beyond the training data. This results in a much lower R-squared, indicating that the model is not reliable for making predictions.
- For instance,The machine learning algorithm predicts university student academic performance and graduation outcomes based on factors such as family income, past academic performance, and academic qualifications of parents. However, the test data only includes candidates from a specific gender or ethnic group.
- In this case, the model may overfit to the specific gender or ethnic group present in the test data.
- It might learn patterns or biases that are not applicable to candidates from different gender or ethnic backgrounds.
- As a result, it struggles to make accurate predictions for candidates outside the narrow demographic represented in the test dataset. The solution can be like, training dataset should be more representative of the diversity of the university student population. Including data from a broader range of gender and ethnic backgrounds will help the model generalize and make fairer predictions for all students.