Linear regression is a fundamental statistical method that helps us understand and predict relationships between variables. It works by finding the best-fitting straight line through a set of data points, allowing us to model how one variable changes in response to another.
The linear regression model is expressed as y equals beta zero plus beta one times x plus epsilon. Beta zero is the intercept, representing the value of y when x equals zero. Beta one is the slope coefficient, showing how much y changes for each unit increase in x. The error term epsilon captures the random variation not explained by the linear relationship.
Ordinary Least Squares is the most common method for estimating the coefficients in linear regression. It works by finding the line that minimizes the sum of squared residuals. Residuals are the vertical distances between each data point and the fitted line. By squaring these distances and summing them up, OLS ensures that larger errors are penalized more heavily, leading to the best overall fit.
Evaluating a linear regression model involves several key metrics. R-squared measures the proportion of variance in the dependent variable explained by the model, ranging from zero to one, with higher values indicating better fit. Mean Squared Error quantifies the average squared difference between observed and predicted values. P-values test the statistical significance of coefficients, helping determine if relationships are meaningful or due to chance.
Once we have fitted our linear regression model, we can use it to make predictions for new data points. Simply plug in the new x value into our equation to get the predicted y value. Linear regression has wide applications across many fields, from sales forecasting and risk assessment to scientific modeling and quality control, making it one of the most valuable tools in data analysis.