Implementing Multiple Linear Regression from Scratch
Introduction
Welcome to our exciting second class in the Regression and Gradient Descent series! In the previous lesson, we covered Simple Linear Regression. Now, we're transitioning toward Multiple Linear Regression, a powerful tool for examining the relationship between a dependent variable and several independent variables.
Consider a case where we need to predict house prices, which undoubtedly depend on multiple factors, such as location, size, and the number of rooms. Multiple Linear Regression accounts for these simultaneous predictors. In today's lesson, you'll learn how to implement this concept in C++!
Multiple Linear Regression - The Concept
Linear Algebra Behind: Dataset Representation
Linear Algebra Behind: Making a Prediction
Linear Algebra Behind: Math Solution
Implementing Multiple Linear Regression from Scratch
Model's Performance Evaluation
Lesson Summary and Practice
Congratulations on mastering Multiple Linear Regression! You've effectively bridged the gap from concept to implementation, designing a regression model in C++ from scratch.
Prepare for the upcoming lesson to delve more deeply into Regression Analysis. Meanwhile, make sure to practice and refine your newly acquired skills!
Each row represents the m features for a single data point. Notice how we include a column of 1's the represent the intercept (also called bias) of each equation.
For each row (equation), there is a corresponding y value. So y looks like:
y=y1y2⋮yn
The normal equation results in a vector:
β0β1⋮βm
Now, for any set of features x1 through xm, we can predict the y^ value as:
y^=(1⋅β0)+(β1⋅x1)+(β2⋅x2)+...+(βm⋅xm)
To calculate all the predictions at once, we take the dot product of X and β
To implement Multiple Linear Regression, we'll leverage some Linear Algebra concepts. Using the Normal Equation, we can calculate the coefficients for our regression equation:
β=(XTX)−1XTy
Where X is a matrix of features and y is a vector of the target variable values. Like Simple Linear Regression, residuals (the differences between actual and predicted values) play a significant role. The smaller these residuals, the better the model fits.
Let's roll up our sleeves and start coding! We'll primarily rely on Eigen to handle numerical operations and matrices.
After completing our model, we need to evaluate its performance. We use the coefficient of determination (R2 score) for this purpose. It indicates how well our model fits the data. The formula is:
R2=1−SStotalSSresiduals
Here, SSresiduals is the residual sum of squares, and SStotal is the total sum of squares:
SSresiduals=i=1∑n(yi−y^i)2
where yi are the observed values and y^i are the predicted values from the regression model.
SStotal=i=1∑n(yi−yˉ)2
where yi are the observed values and yˉ is the mean of the observed data.
A higher R2 value (closer to 1) indicates a better model fit.
