Linear regression: fitting a straight line to data

You have measured how many cups of coffee the café sells at different temperatures: (15 °C, 20 cups), (20 °C, 35 cups), (25 °C, 50 cups). It looks like a line. If you find the line, you can predict sales at 30 °C.

A model here is simply sales = k · temperature + m. "Training the model" means: find k and m so the line fits the points as well as possible.

How well does a line fit? Calculate the error for each point: actual value − line's prediction. Square the errors (so positive and negative values do not cancel out, and large errors are penalised heavily) and take the average. This is called MSE, mean squared error, or loss.

With k = 3, m = −25 on the coffee points:

temp	actual	prediction	error	error²
15	20	20	0	0
20	35	35	0	0
25	50	50	0	0

Loss = 0 — perfect. With real data, it is never zero; you seek the minimum. How to find the best k and m automatically is the next node: gradient descent.