Intuition-Based Tutorial on OLS Linear Regression

1. Introduction: What Does Linear Regression Try to Achieve?

Linear regression is one of the most fundamental tools in statistics and machine learning. Its purpose is simple—but extremely powerful:

To model how multiple input variables influence a single output, using a linear equation.

This makes it useful in a huge range of fields—economics, finance, engineering, marketing, and even artificial intelligence. Despite its simplicity, linear regression contains ideas that eventually grow into neural networks and deep learning.

Most real-world data is messy. Points rarely fall on a perfect line or plane. Some students with many study hours still score poorly; some houses with high square footage sell for lower prices. Because of this randomness, linear regression must find a line or surface that gets as close as possible to all data points.

This leads to the idea of Ordinary Least Squares (OLS)—the engine that drives the entire method.

2. Multiple Linear Regression: Expanding the Equation

When you first encounter regression, you usually see it in the form:

$\hat{y} = mx + c$

But real-life problems rarely have just one input variable.

When we generalize to multiple variables, the equation becomes:

$\hat{y} = w_1 x_1 + w_2 x_2 + w_3 x_3 + \dots + w_n x_n + b$

Meaning of each part

x₁, x₂, x₃, … xₙ
The features or inputs. These might be age, salary, square footage, number of study hours, etc.
w₁, w₂, w₃, … wₙ
The weights or coefficients.
Each weight tells us how strongly an input affects the output:
- A large positive weight means that as the variable increases, the output increases strongly.
- A negative weight means the variable pulls the output downward.
- A small weight means the variable has little influence.
b (bias/intercept/offset)
The baseline prediction.
It represents what the model predicts when every input is zero.
ŷ
The predicted output.

Why this equation is linear

Because each term involves a weight multiplied directly by an input.
There are no squares, roots, exponentials, or interactions.
The model forms a flat surface (line/plane/hyperplane) in the input space.

3. The Role of the Intercept (Bias): Why It Is Essential

The intercept b is one of the most misunderstood but crucial parts of regression.

The instructor in the source calls it the offset or bias.
In machine learning, the term bias is standard, especially in neural networks.

Why do we need the intercept?

Imagine a model that predicts house prices using only:

$\hat{y} = w_1 \times (\text{square footage})$

If square footage = 0, this model predicts the price = 0.
But in reality, even an empty plot or destroyed house still has value based on:

land value
location
taxes
economic factors

Therefore:

We need a baseline prediction before any input contributes.

That baseline is the intercept.

Geometrically

The intercept shifts the entire line or plane up or down, allowing the model to sit properly in the cloud of data points.

If you force the model to pass through the origin (by removing the intercept), it will almost always:

tilt unnaturally
fit the data poorly
produce large errors

✔ In neural networks

Every neuron contains:

$\hat{y} = \mathbf{w} \cdot \mathbf{x} + b$

Without the bias, the neuron cannot shift its activation function and becomes far less expressive.

This simple concept begins in linear regression but becomes a foundational idea for deep learning.

4. The Core of OLS: Minimizing the Total Error

OLS chooses weights and bias values by trying to minimize the sum of squared errors.

For each data point:

$\displaystyle \text{error}_i = y_i - \hat{y}_i$

OLS squares the error:

$\displaystyle (y_i - \hat{y}_i)^2$

And adds all squared errors across all m data points:

$\displaystyle \text{Total Error} = \sum_{i=1}^{m} (y_i - \hat{y}_i)^2$

The goal is:

Choose w₁, w₂, …, wₙ, b such that the total squared error is as small as possible.

✔ Why square the errors?

Positive and negative errors cannot cancel out
Large mistakes are punished strongly
The resulting function is smooth and differentiable
This makes it easy to compute the optimal solution using calculus

✔ What OLS gives you

OLS provides a unique set of coefficients that:

best reflect the trend in the data
create the flattest possible surface that minimizes total deviation
ensure the model is unbiased and performs well on average

5. Geometric Interpretation: From Lines to Hyperplanes

One of the most illuminating ways to understand regression is through geometry.

If you have:

Number of inputs	The model fits a…
1 variable	Line
2 variables	Plane
3 variables	3D hyperplane
n variables	n-dimensional hyperplane

Even though humans cannot visualize more than three dimensions, the math extends naturally.

✔ What is regression looking for geometrically?

It is searching for the closest possible flat surface that runs near all data points in the dataset.

Every data point has a vertical distance from the hyperplane.
OLS tries to minimize the sum of these squared vertical distances.

✔ Why vertical distances?

Because vertical differences represent errors in predicting Y, not errors in X.

6. What the Coefficients Really Mean (Deep Intuition)

✔ Weight = “How much does this input matter?”

A large weight means the input has a strong impact on the prediction.
A small weight means the variable is less important.
Zero weight means the input has no relationship with the output.

✔ Sign of the weight

Positive: Increasing the input increases the output
Negative: Increasing the input decreases the output

✔ Example

If the model is:

$\displaystyle \hat{y} = 5x_1 - 2x_2 + 10$

Then:

For every 1-unit increase in x₁, output increases by 5 units
For every 1-unit increase in x₂, output decreases by 2 units
Even when all inputs are 0, output begins at 10

This kind of interpretation makes regression extremely valuable for understanding relationships between variables.

7. Regression as a Single Neuron: The Link to Neural Networks

Linear regression is much more than a statistical tool—it is the simplest form of a neural network.

A neuron computes:

$\displaystyle z = w_1 x_1 + w_2 x_2 + \dots + w_n x_n + b$

Then an activation function (like ReLU or sigmoid) is applied:

$\displaystyle \hat{y} = \sigma(z)$

If you remove the activation function, the neuron becomes:

$\displaystyle \hat{y} = w_1 x_1 + w_2 x_2 + \dots + w_n x_n + b$

Which is exactly the linear regression equation.

Why this matters

Understanding regression builds intuition for:

how neural networks combine inputs
how weights represent learned relationships
how bias shifts decisions
how errors are minimized during training

Neural networks use the same concept but stack many neurons together to learn complex, nonlinear patterns.

Regression is the foundation.