18,September, 2023

$In\; todays\; class\; professor\; discussed\; about\; given\; equation,$

y=b0​+b1​X1​+b2​X2​+b3​X3​+E(error)

In this equation:

• $y$ represents the dependent variable or the target you are trying to predict.
• X1,X2,X1​,X2​, and X3​ are independent variables or predictors.
• b0,b1,b2,b0​,b1​,b2​, and b3​ are the coefficients of the respective predictors.
• $E$ represents the error term, which accounts for the variability in $y$ that cannot be explained by the predictors.

1. Linear Relationship: This equation still assumes a linear relationship between the dependent variable ($y$) and the independent variables (X1,X2,X1​,X2​, and X3​). Each coefficient (b1,b2,b1​,b2​, and b3​) represents the change in $y$ for a one-unit change in the corresponding predictor, assuming all other predictors remain constant.
2. Overfitting: The risk of overfitting still applies in multiple linear regression, particularly if you have a high number of predictors relative to your sample size. Including too many predictors without enough data can lead to overfitting, just like in polynomial regression.
3. Model Evaluation: To assess the performance of this multiple linear regression model, you can use techniques such as R-squared (coefficient of determination), p-values for the coefficients, and residual analysis to ensure the model’s validity.
4. Regularization: In cases where you have many predictors or suspect multicollinearity (correlation between predictors), you may consider using regularization techniques like Ridge or Lasso regression to prevent overfitting and improve model generalization.
5. Interpretation: Interpretation of coefficients (b1,b2,b1​,b2​, and b3b3​) remains the same as in simple linear regression. Each coefficient tells you the effect of a one-unit change in the corresponding predictor on the dependent variable, holding other predictors constant.
6. Assumptions: Like in simple linear regression, multiple linear regression assumes that the errors ($E$) are normally distributed, have constant variance (homoscedasticity), and are independent of each other (no autocorrelation).