Linear regression fits a line to examples so you can predict a continuous number. Imagine knowing the weights and prices of several diamonds, then estimating the price of another diamond from its weight. Here, carat is the input, or independent variable; price is the output, or dependent variable. section 8.11
Start with known carat–price pairs and plot them as points. The points need not fall on one line: the aim is to find a line that captures their overall trend. A line has the form y = b0 + b1 × x. Its slope, b1, describes how the prediction changes as carat changes; its intercept, b0, positions the line. Once those values are fitted, substitute a new carat value for x to predict price. section 8.11
A small learning implementation follows this sequence:
- Calculate the mean carat value and mean price. section 8.11
- Measure how each carat value and price differs from its mean. Use those differences to calculate the line’s slope, then use the means and slope to calculate its intercept. section 8.11
- Use the fitted line to make predictions for carat values it has not seen. section 8.11
def fit_regression_line(carats, prices):
n = len(carats)
mean_X = sum(carats) / n #A
mean_Y = sum(prices) / n #A
sum_x_squared = 0
for x_i in carats:
ans = (x_i - mean_X) ** 2
sum_x_squared += ans #B
sum_multiple = 0
for i in range(n):
ans = (carats[i] - mean_X) * (prices[i] - mean_Y)
sum_multiple += ans #C
b1 = sum_multiple / sum_x_squared #D
b0 = mean_Y - (b1 * mean_X) #E
min_x = min(carats)
max_x = max(carats)
y1 = b0 + b1 * min_x #F
y2 = b0 + b1 * max_x #F
return b0, b1, (min_x, y1), (max_x, y2)
This Python function makes the fitting steps visible: it computes means, the slope and intercept, then returns two points that can be used to draw the line. section 8.11
Check a diamond-price prediction
In one test example, a carat value scaled to 1,080 has an actual price of 2,869; the fitted line predicts 2,963. Comparing predictions with known prices in separate test data reveals errors that training data alone cannot show. An 80/20 training–test split is a common starting point, and R² is introduced as a way to measure the line’s performance. section 8.12
Choose regression when the answer is a continuous value, such as a price. If the answer is a category—such as whether a vehicle is a car or a truck—the task is classification instead. Algorithm choice also depends on the available features and their relationships. section 8.11 section 8.15
