Linear regression is a popular supervised machine learning algorithm for predicting a continuous target variable (the dependent variable) based on one or more predictor variables (independent variables). The primary goal of linear regression is to find the best-fit linear relationship between the predictors and the target variable.
Here’s how linear regression works, including the objective function and gradient descent:
Objective Function (Cost Function): The objective in linear regression is to find the parameters (coefficients) of the linear model that minimize the difference between the predicted values and the actual values of the target variable. This is typically done using a cost function. The most common cost function for linear regression is the Mean Squared Error (MSE) or the Sum of Squared Residuals (SSR), which is defined as:
MSE = (1 / 2m) * Σ(yi — ŷi)²
● MSE: Mean Squared Error, the cost to be minimized.
● m: The number of data points in the dataset.
● yi: The actual value of the target variable for the ith data point.
● ŷi: The predicted value of the target variable for the ith data point.
Gradient Descent: Gradient Descent is an optimization algorithm that minimizes the cost function. It iteratively updates the model parameters to find the values that minimize the cost. For linear regression, the model parameters are the coefficients (weights) of the linear equation.
The gradient descent algorithm starts with some initial values for the coefficients and iteratively updates them. The update rule for linear regression is as follows: θj = θj — α * (∂MSE/∂θj)
● θj: The jth coefficient (weight) of the linear regression model.
● α (alpha): The learning rate, a hyperparameter that controls the step size in the parameter space.
● ∂MSE/∂θj: The partial derivative of the MSE concerning θj, which represents the gradient of the cost function about that parameter.
The algorithm continues to update the coefficients until convergence is reached, which occurs when the changes in the cost function become very small or after a fixed number of iterations. Here’s a high-level overview of the steps in gradient descent for linear regression:
1– Initialize the coefficients θj.
2– Compute the predicted values ŷi using the current coefficients.
3– Compute the gradient (∂MSE/∂θj) for each coefficient.
4– Update each coefficient θj using the update rule.
5– Repeat steps 2–4 until convergence is achieved or a predetermined number of iterations is reached.
The choice of the learning rate (α) is crucial in gradient descent, as it can affect the algorithm’s convergence. If α is too small, the algorithm may converge slowly, and if it’s too large, it may overshoot the minimum. Proper tuning of the learning rate is often necessary for effective gradient descent.
● Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It assumes a linear relationship between the variables and aims to find the best-fitting line (or hyperplane in multiple dimensions) to make predictions.
● Linear regression assumes linearity, independence of errors, homoscedasticity (constant variance of errors), and normally distributed errors.
● Simple linear regression involves one independent variable, while multiple linear regression involves more than one independent variable.
● R-squared measures the proportion of the variance in the dependent variable that can be explained by the independent variables. It ranges from 0 to 1, where higher values indicate a better fit of the model to the data.
● The coefficients represent the change in the dependent variable for a one-unit change in the corresponding independent variable while holding all other variables constant.
● Correlation measures the strength and direction of a linear relationship between two variables, but it doesn’t imply causation. Regression, however, models the effect of one or more independent variables on a dependent variable.
● Some common methods include R-squared, adjusted R-squared, F-statistic, and p-values for the coefficients.
● The F-statistic is used to test the overall significance of the regression model. It compares the model’s fit with independent variables to a null model with no independent variables.
● Common problems include multicollinearity, heteroscedasticity, and outliers. Solutions may involve removing or transforming variables, using robust standard errors, or applying regularization techniques.
● Regularization techniques like Ridge and Lasso regression prevent overfitting by adding a penalty term to the loss function. Ridge adds L2 regularization, while Lasso adds L1 regularization. They are applicable when dealing with high-dimensional data and multicollinearity.
● The bias-variance trade-off refers to the trade-off between a model’s ability to fit the data well (low bias) and generalize to new data (low variance). Increasing model complexity typically reduces bias but increases variance, and vice versa.