From Intuition to Equation

1 · Observe

Say it in words first

An agent might say: 'each extra square metre adds about ₹80,000, and there is a small fixed adjustment'. That sentence is already a linear regression. The equation just gives its two ingredients names.

2 · Express

The equation

Different (w, b) pairs draw different lines. To compare them we need one number that says how far a line is from the sales. The usual choice is the mean squared error (MSE): square each miss, then average.

3 · Visualize

Three candidate rules

Same eight flats as before. We try three rules and compute the mean squared error for each.

The first rule is badly off. The second is close. The third, the fitted line rounded to two decimals, is the best of the three. The best possible line, with unrounded numbers, has an MSE of 21.0.

4 · Discover

Reading w and b

w = 0.806 lakh per sq m, which is ₹80,600 for each extra square metre. b = −6.82 lakh is the line's value at area 0, a place no real flat lives. The intercept is a bookkeeping number that makes the line sit at the right height, not a price anyone pays.

5 · Apply

Two worked examples

Example 1, a prediction. For a 75 sq m flat: 0.806 × 75 + (−6.82) = 53.63 lakh.

Example 2, changing the rule. Raise w from 0.806 to 0.9 and keep b: a 100 sq m flat moves from 73.8 to 83.2 lakh. A change of 0.094 in w moves the prediction by 0.094 × 100 = 9.4 lakh, which is why w must be chosen carefully.

6 · Where it goes wrong

Three ways it goes wrong

7 · In Python

The equation in code

Here is how to implement it, step by step. The code below does exactly these steps, and every line is editable.

1. Write the rule as price = w × area + b.
2. Choose a pair (w, b), for example 0.8 and −8.
3. Predict every flat's price with that pair.
4. Square each miss and average them to get the MSE.
5. Try other pairs. The pair with the smallest MSE is the better line.

import numpy as np

area = np.array([40, 50, 60, 70, 80, 90, 100, 110], float)
price = np.array([30, 34, 36, 52, 55, 58, 80, 84], float)
w, b = np.polyfit(area, price, 1)

x = 75
print("by hand: w * x + b =", round(w, 4), "*", x, "+", round(b, 3), "=", round(w * x + b, 2))
print("each extra sq m adds", round(w * 100, 1), "thousand rupees (w =", round(w, 3), "lakh per sq m)")
print("a flat of zero area is predicted at", round(b, 2), "lakh -> the intercept is a baseline, not a real flat")
for guess_w, guess_b in [(0.5, 0), (0.8, -8), (round(w, 2), round(b, 1))]:
    err = np.mean((price - (guess_w * area + guess_b)) ** 2)
    print("rule", guess_w, "x area +", guess_b, "-> mean squared error", round(err, 1))

by hand: w * x + b = 0.806 * 75 + -6.821 = 53.63
each extra sq m adds 80.6 thousand rupees (w = 0.806 lakh per sq m)
a flat of zero area is predicted at -6.82 lakh -> the intercept is a baseline, not a real flat
rule 0.5 x area + 0 -> mean squared error 330.1
rule 0.8 x area + -8 -> mean squared error 23.6
rule 0.81 x area + -6.8 -> mean squared error 21.1