The Math Behind Machine Learning: Linear Regression Explained
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Linear regression is a fundamental machine learning algorithm that finds the best straight line through a set of data points. Given input and output variables, it discovers the linear relationship that best predicts the output based on the input. This red line represents our model's prediction, showing how the algorithm learns to map inputs to outputs.
The linear model is expressed as y equals beta zero plus beta one times x. Beta zero is the intercept, which is where the line crosses the y-axis. Beta one is the slope, representing how much y changes for each unit increase in x. These two parameters completely define our linear relationship and are what we need to learn from the data.
To find the best line, we need a way to measure how good our predictions are. The cost function, specifically Mean Squared Error, calculates the average of the squared differences between actual and predicted values. These orange lines show the errors for each data point. Our goal is to find the line parameters that minimize this total error.
Gradient descent is an optimization algorithm that finds the minimum of the cost function. Starting from an initial parameter value, it calculates the gradient and moves in the opposite direction. The red dot shows the current parameter value, and the green line shows the gradient direction. By repeatedly taking steps proportional to the negative gradient, we eventually reach the global minimum where the cost is lowest.
The Normal Equation offers an alternative to gradient descent by providing a direct analytical solution. Using matrix operations, it calculates the optimal parameters in one step without any iterations. The design matrix X contains our input features, and the formula gives us the exact best-fit line. While this method is precise and doesn't require tuning learning rates, it becomes computationally expensive for very large datasets due to matrix inversion operations.