Linear regression

Linear Regression: Understanding the Fundamentals

Linear regression is a fundamental technique in machine learning. It's a powerful tool that allows us to capture the relationship between variables by fitting a straight line to the data points. The beauty of linear regression lies in its simplicity and interpretability. We can directly observe how changes in the independent variables impact the dependent variable. It serves as a crucial starting point for understanding the principles of modeling and prediction.

While linear regression may seem basic compared to more advanced models, it plays a pivotal role in building our understanding of more complex techniques, such as deep learning. By grasping the foundations of linear regression, we gain insight into the principles of optimization and minimizing errors. We estimate the coefficients that align the line with the data points, ensuring the best possible fit.

Linear regression can be optimized using a technique called gradient descent. The goal of gradient descent is to find the optimal values for the coefficients of the linear regression model that minimize the prediction errors. The process involves iteratively adjusting the coefficients in the direction of the steepest descent of the loss function.

By delving into the world of gradient descent, we unlock the power to fine-tune the parameters of the linear regression model and improve its performance. We calculate the gradient of the loss function with respect to the coefficients and update the coefficients accordingly, gradually approaching the optimal values that minimize the errors.

The beauty of gradient descent lies in its ability to navigate the high-dimensional space of coefficients efficiently. It allows us to iteratively refine our model, making it more accurate and capturing the underlying relationships in the data.

Now let's step into the details step by step:

Step 1: The Data

Here we can see the data points. Try playing with them :

Our ultimate objective is to find a line that accurately represents the underlying relationship between the variables. This line is what we refer to as the "best fit" or "optimal fit" line. By identifying this line, we can make predictions and understand how changes in one variable correspond to changes in the other.

Finding the line means finding the best values for the parameters of the line, namely the slope (w) and the y-intercept (b). These parameters define the equation of the line, which is represented as y = wx + b. Namely, to fit the line to the data means that we want to adjust the values of w and b in such a way that the line closely aligns with the data points.

In the next visualization, try playing with the sliders that control parameters w and b and try to find the best fit line—the line that aligns best with the data points:

However, trying to find these parameters by hand can be extremely time-consuming and inefficient, especially when dealing with large datasets or complex models with a lot of parameters. The number of possible parameter combinations can be enormous, making it impractical to manually explore all the potential options.

We want an algorithm that can explore the high-dimensional parameter space, iteratively optimize the model, and uncover the best parameter values that minimize the prediction errors, enabling us to capture the underlying patterns in the data and improve the overall performance of the model.

But first, we need to define what prediction error is.

Prediction error refers to the discrepancy between the predicted values of the model and the actual observed values in the data. It quantifies how well the model's predictions align with the ground truth. In the context of linear regression, prediction error is typically measured using a loss function, such as mean squared error (MSE). The MSE calculates the average of the squared differences between the predicted y-values and the true y-values for each data point.

The formula for MSE is as follows:

$$ MSE = \frac{1}{2n} \sum_{i=1}^{n}(y_i - \hat{y}_i)^2 $$

where \(n\) represents the number of data points or observations, \(y_i\) denotes the true value of the target variable for the \(i\)th data point, and \(\hat{y}_i\) represents the predicted value of the target variable for the \(i\)th data point.

The MSE formula calculates the squared difference between the predicted and true values for each data point, sums them up, and then divides by the total number of data points n. This provides an average measure of the prediction error, with higher values indicating larger errors and poorer model performance.

Minimizing the MSE during the training process of linear regression involves adjusting the model's parameters (slope and intercept) to find the values that yield the smallest MSE. This optimization aims to find the line that best fits the data, minimizing the overall squared differences between the predicted and true values.

By minimizing the MSE, we strive to improve the accuracy and precision of our linear regression model, enhancing its ability to capture the underlying relationship between the variables and make more accurate predictions.

In the next visualization, for each parameter that we choose, we can see the loss for the line they define. Here the error is literally sums of square ares. Try to find the parameters that will minimize the loss:

Now that we understand the loss function, let's change our perspective a bit and talk about the parameter space.
Each point in the parameter space is a configuration of parameters for the model with the form (w, b).

Try playing with it a bit by dragging the red dot in the parameter space and observe how the line changes:

How can we view the loss function in that space? Simply put, for every configuration of (w, b), we will use the z-axis to represent the loss. Try moving around the parameter point to see that and get a feel for how the loss function looks like. Afterward, press the button "show loss function" and see the whole loss. Observe that in points where this loss is low, the line is best.

I hope you saw that in places where the loss function is low, the line is better. So the only question left is how do we get to these parameters? In real life, we don't have this visualization where we can find it with the eyes. We should use an algorithm that will take the data and return good parameters, parameters that minimize the loss.
We will do it in the next post... thanks