📊Principles of Data Science Unit 7 – Supervised Learning: Regression

What's Regression All About?

• Regression is a supervised learning technique used to predict continuous numerical values
• Aims to establish a relationship between independent variables (features) and a dependent variable (target)
• Fits a mathematical function to the training data to minimize the difference between predicted and actual values
• Can be used for forecasting, trend analysis, and understanding the impact of variables on an outcome
• Assumes a linear or non-linear relationship exists between the features and the target variable
  • Linear regression assumes a straight-line relationship
  • Non-linear regression captures more complex relationships (polynomial, exponential, etc.)
• Requires a labeled dataset with input features and corresponding target values for training
• Produces a trained model that can make predictions on new, unseen data points

Types of Regression Models

• Linear Regression: Assumes a linear relationship between features and the target variable
  • Simple Linear Regression: One independent variable and one dependent variable
  • Multiple Linear Regression: Multiple independent variables and one dependent variable
• Polynomial Regression: Models non-linear relationships by adding polynomial terms to the linear equation
• Ridge Regression: Linear regression with L2 regularization to handle multicollinearity and prevent overfitting
• Lasso Regression: Linear regression with L1 regularization for feature selection and model simplification
• Elastic Net Regression: Combines L1 and L2 regularization to balance between Lasso and Ridge regression
• Stepwise Regression: Iteratively adds or removes features based on their statistical significance
• Decision Tree Regression: Builds a tree-like model by splitting the data based on feature values

Key Concepts and Terminology

• Features: Independent variables used to predict the target variable (denoted as X)
• Target: Dependent variable that we aim to predict (denoted as y)
• Coefficients: Weights assigned to each feature in the regression equation (denoted as β)
• Intercept: The value of the target variable when all features are zero (denoted as β₀)
• Residuals: Differences between the predicted values and the actual values
• Overfitting: When a model learns the noise in the training data and fails to generalize well to new data
• Underfitting: When a model is too simple to capture the underlying patterns in the data
• Regularization: Techniques used to prevent overfitting by adding a penalty term to the loss function
  • L1 Regularization (Lasso): Adds the absolute values of coefficients to the loss function
  • L2 Regularization (Ridge): Adds the squared values of coefficients to the loss function

The Math Behind Regression

• Linear Regression Equation: $y = β₀ + β₁x₁ + β₂x₂ + ... + β_nx_n + ε$
  • $y$: Predicted target variable
  • $β₀$: Intercept
  • $β₁, β₂, ..., β_n$: Coefficients for each feature
  • $x₁, x₂, ..., x_n$: Feature values
  • $ε$: Error term (residuals)
• Ordinary Least Squares (OLS): Method used to estimate the coefficients by minimizing the sum of squared residuals
  • Objective: Minimize $\sum_{i=1}^{n} (y_i - \hat{y}_i)^2$, where $\hat{y}_i$ is the predicted value for the $i$-th observation
• Gradient Descent: Iterative optimization algorithm used to find the minimum of the cost function
  • Updates the coefficients in the direction of steepest descent to minimize the cost function
• Cost Function: Measures the difference between predicted and actual values (e.g., Mean Squared Error)
• Regularization Terms:
  • L1 (Lasso): $\lambda \sum_{j=1}^{p} |β_j|$
  • L2 (Ridge): $\lambda \sum_{j=1}^{p} β_j^2$
  • $\lambda$: Regularization parameter that controls the strength of regularization

Implementing Regression in Python

• Popular libraries: scikit-learn, statsmodels, TensorFlow, PyTorch
• Preprocessing steps:
  • Handling missing values (imputation or removal)
  • Encoding categorical variables (one-hot encoding, label encoding)
  • Scaling features (standardization, normalization)
• Splitting the data into training and testing sets using

  train_test_split

  from scikit-learn
• Creating and training the regression model:
  • Linear Regression:

    from sklearn.linear_model import LinearRegression

  • Ridge Regression:

    from sklearn.linear_model import Ridge

  • Lasso Regression:

    from sklearn.linear_model import Lasso

• Fitting the model to the training data using the

  fit()

  method
• Making predictions on the testing set using the

  predict()

  method
• Evaluating the model's performance using metrics like Mean Squared Error (MSE) or R-squared

Model Evaluation Techniques

• Train-Test Split: Dividing the dataset into separate training and testing sets
  • Training set: Used to train the model and learn the parameters
  • Testing set: Used to evaluate the model's performance on unseen data
• Cross-Validation: Technique to assess the model's performance and generalization ability
  • K-Fold Cross-Validation: Divides the data into K equal-sized folds, trains and evaluates the model K times
  • Leave-One-Out Cross-Validation (LOOCV): Uses each data point as a separate testing set
• Evaluation Metrics:
  • Mean Squared Error (MSE): Average of the squared differences between predicted and actual values
  • Root Mean Squared Error (RMSE): Square root of MSE, provides an interpretable metric in the same units as the target variable
  • Mean Absolute Error (MAE): Average of the absolute differences between predicted and actual values
  • R-squared (R²): Proportion of the variance in the target variable explained by the model
• Residual Analysis: Examining the differences between predicted and actual values to assess model assumptions and identify patterns or outliers

Real-World Applications

• House Price Prediction: Estimating the price of a house based on features like area, number of rooms, location, etc.
• Sales Forecasting: Predicting future sales based on historical data, seasonality, and other relevant factors
• Customer Lifetime Value Prediction: Estimating the total revenue a customer will generate over their lifetime
• Stock Price Prediction: Forecasting future stock prices based on historical data, market trends, and economic indicators
• Weather Forecasting: Predicting temperature, precipitation, or other weather variables based on atmospheric conditions
• Energy Consumption Prediction: Estimating energy usage based on factors like temperature, time of day, and building characteristics
• Medical Diagnosis: Predicting the likelihood of a disease based on patient symptoms, test results, and demographic information

Common Pitfalls and How to Avoid Them

• Multicollinearity: High correlation among independent variables, leading to unstable coefficients
  • Solution: Remove one of the correlated variables or use regularization techniques (Ridge, Lasso)
• Overfitting: Model fits the training data too closely, failing to generalize well to new data
  • Solution: Use regularization, cross-validation, or simplify the model
• Underfitting: Model is too simple to capture the underlying patterns in the data
  • Solution: Increase model complexity, add more relevant features, or use non-linear models
• Outliers: Data points that significantly deviate from the general trend, influencing the model's fit
  • Solution: Identify and handle outliers appropriately (remove, transform, or use robust regression methods)
• Non-linearity: Linear models may not capture non-linear relationships between features and the target variable
  • Solution: Use polynomial regression, decision trees, or other non-linear models
• Heteroscedasticity: Non-constant variance of the residuals across the range of predicted values
  • Solution: Use weighted least squares, transform the target variable, or consider non-linear models
• Autocorrelation: Correlation between the residuals in a time series or spatial data
  • Solution: Use time series models (e.g., ARIMA) or incorporate spatial dependencies