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Simple Linear Regression

The model

Simple linear regression predicts a number from one feature:

ลท = wยทx + bลท = wยทx + b

  • ww is the slope (how much y changes per unit x)
  • bb is the intercept (y when x = 0)

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  flowchart LR
  x[Single feature x] -->|w, b| yhat[Prediction ลท]

false

Intuition

If w = 200w = 200 and xx is house size in sqft:

  • increasing size by 1 sqft increases predicted price by 200 (units of currency)

Scikit-learn example

Simple linear regression
import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
 
# X must be 2D in scikit-learn
X = np.array([500, 700, 800, 1000, 1200]).reshape(-1, 1)
y = np.array([100, 150, 170, 210, 240])
 
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
 
model = LinearRegression()
model.fit(X_train, y_train)
 
pred = model.predict(X_test)
print("w (slope):", model.coef_[0])
print("b (intercept):", model.intercept_)
print("MSE:", mean_squared_error(y_test, pred))
Simple linear regression
import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
 
# X must be 2D in scikit-learn
X = np.array([500, 700, 800, 1000, 1200]).reshape(-1, 1)
y = np.array([100, 150, 170, 210, 240])
 
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
 
model = LinearRegression()
model.fit(X_train, y_train)
 
pred = model.predict(X_test)
print("w (slope):", model.coef_[0])
print("b (intercept):", model.intercept_)
print("MSE:", mean_squared_error(y_test, pred))

Pitfalls

  • outliers can strongly affect the line
  • if the relationship is non-linear, the line underfits

Mini-checkpoint

Plot x vs y. Does it look roughly linear?

  • If yes, start here.
  • If no, consider polynomial regression or other models.

๐Ÿงช Try It Yourself

Exercise 1 โ€“ Train-Test Split

Exercise 2 โ€“ Fit a Linear Model

Exercise 3 โ€“ Evaluate with MSE

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