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Multivariate Regression Techniques & Handling Outliers

1. Multivariate Regression Techniques

Multivariate Regression is used when there are multiple dependent (target) variables predicted using multiple independent variables. It extends simple and multiple regression to handle multiple response variables simultaneously.

Types of Multivariate Regression Models:

1. Multivariate Multiple Linear Regression (MMLR):

   • Predicts multiple dependent variables using multiple independent variables.

   • Equation:

     $$Y_1, Y_2, ..., Y_n = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_p X_p + \epsilon$$

   • Used in econometrics, healthcare, and finance.

2. Principal Component Regression (PCR):

   • Applies PCA to reduce dimensionality before performing regression.

   • Used when independent variables are highly correlated.

3. Partial Least Squares Regression (PLSR):

   • Similar to PCR but optimizes for predicting response variables.

   • Useful when predictors are highly collinear.

4. Ridge & Lasso Regression:

   • Ridge: Adds L2 regularization to handle multicollinearity.

   • Lasso: Adds L1 regularization, performing feature selection.

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2. Handling Outliers in Multivariate Regression

Outliers can distort regression models, reducing accuracy. Techniques to handle them include:

1. Detecting Outliers:

   • Mahalanobis Distance: Measures how far a point is from the multivariate mean.

   • Cook’s Distance: Identifies influential points affecting regression.

   • Boxplots & Z-Scores: Helps find extreme values.

2. Handling Outliers:

   • Winsorization: Capping extreme values to a certain percentile.

   • Transformation: Applying log or Box-Cox transformation to reduce skewness.

   • Robust Regression: Using Huber regression or RANSAC, which reduce outlier influence.

   • Removing Outliers: If outliers are due to errors, they can be removed.

Conclusion

Multivariate regression models capture complex relationships between multiple dependent and independent variables. Handling outliers properly is essential for ensuring model accuracy and robustness.