Linear regression — sum of squared residuals (SSR)

Background

The sum of squared residuals (SSR) is the loss linear regression minimises. For a line $\hat{y}_i = m x_i + c$ it is

$$\text{SSR} = J(m, c) = \sum_{i=1}^{n} \bigl(y_i - (m x_i + c)\bigr)^2$$

Squaring serves two purposes: it stops positive and negative residuals cancelling, and it punishes a big miss far more than a small one (an error twice as large adds four times as much). A single far-off point can dominate the total — exactly the effect the lesson's draggable SSR explorer shows.

Problem statement

Implement ssr(x, y, m, c) that returns the scalar SSR of the line $y = m x + c$ on the data (x, y).

Input

• x — array-like of inputs.
• y — array-like of true targets, the same length.
• m — float, the slope.
• c — float, the intercept.

Output

Returns a Python float: $\sum_i (y_i - (m x_i + c))^2$.

Examples

Example 1 — a perfect fit scores zero

Input:  x = [2, 4, 5], y = [2, 8, 11], m = 3, c = -4
Output: 0.0

Explanation: the line $y = 3x - 4$ passes through every point, so every residual is 0.

Example 2 — the lesson's bad starting guess

Input:  x = [51], y = [149], m = 1, c = 1
Output: 9409.0

Explanation: prediction $= 52$, residual $= 149 - 52 = 97$, and $97^2 = 9409$.

Constraints

• Build the prediction $m x + c$, take the residual $y - \hat{y}$, square, and sum.
• Return a plain float (wrap the NumPy scalar in float()).

Notes

• This is the height of the "mountain" in the lesson's gradient-descent analogy. Gradient descent's whole job is to walk this number downhill — which is exactly the next problem.