Fundamentals, data analitics linear regression and bias

🧩 What Is X and What Is X_b? 🔹 X

X is your feature matrix — it contains the input values you want the model to learn from.

Example: If you have one feature (say, number of pages in a book), X looks like this:

X = [[100], [200], [300], [400]]

It’s a column of data — every row is one training example.

💡 Shape: (m, n)

m = number of training examples (rows)

n = number of features (columns)

So in this case, m = 4, n = 1.

🔹 X_b

Before using the Normal Equation, we must add a column of 1s to X.

Why? Because our formula for linear regression is:

𝑦

𝜃 0 + 𝜃 1 𝑥 y=θ 0 ​

+θ 1 ​

x

That θ₀ (the intercept or bias term) multiplies by 1 every time. To make it fit the same matrix format, we add a column of 1s to X.

X_b = [[1, 100], [1, 200], [1, 300], [1, 400]]

Now the first column (all 1s) corresponds to θ₀, and the second column corresponds to θ₁.

💡 Shape of X_b: (m, n+1)

🧠 What Is the Bias Term (θ₀ or intercept_)?

The bias term (also called intercept) is the value of y when all features are 0. It basically shifts the regression line up or down so it fits the data better.

Example:

If your model is

𝑦

4 + 3 𝑥 y=4+3x

then:

θ₀ (bias) = 4 → this means when x = 0, y starts at 4

θ₁ (weight) = 3 → this means for every 1 increase in x, y goes up by 3

So, in your Scikit-Learn output:

lin_reg.intercept_, lin_reg.coef_

you got:

(array([4.21509616]), array([[2.77011339]]))

That means:

Bias/intercept = 4.215

Slope/weight = 2.770

So your final model is:

𝑦

= 4.215 + 2.770 𝑥 y

​

=4.215+2.770x 🔁 And What Does the “T” Mean in X.T?

X.T means the transpose of X.

Transpose means switch rows and columns.

If X looks like this:

1 2 3 4 5 6

Then X.T looks like this:

1 4 2 5 3 6

This is used because in matrix algebra, to multiply correctly, dimensions must align. So the Normal Equation uses X.T to flip the matrix for proper multiplication:

𝜃

( 𝑋 𝑇 𝑋 ) − 1 𝑋 𝑇 𝑦 θ=(X T X) −1 X T y 🧮 Putting It All Together Symbol Meaning Example X Feature matrix Pages in each book X_b Feature matrix with column of 1s Adds bias term y Target values Prices of books θ₀ Intercept (bias) Where the line crosses y-axis θ₁…θₙ Weights/slopes How much each feature affects y X.T Transpose of X Flips rows ↔ columns for math compatibility 🧠 Intuitive Summary

👉 The bias term lets your line start at the right height. 👉 The weights control how steep the line is. 👉 The Normal Equation finds the best bias and weight combination so the line is as close as possible to all data points.

Would you like me to add a small visual (graph) showing how the bias moves the line up/down and how the slope changes its angle? It would make the concept of bias and weights much clearer visually.