Pearson correlation coefficient

Background

The Pearson correlation coefficient $r$ measures the strength and direction of the linear relationship between two variables, normalised to $[-1, 1]$: $+1$ is a perfect increasing line, $-1$ a perfect decreasing line, and $0$ means no linear relationship. It is the covariance divided by the product of the two standard deviations.

Problem statement

Implement pearson_correlation(x, y) returning the Pearson correlation between two equal-length vectors:

$$r = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i (x_i - \bar{x})^2}\;\sqrt{\sum_i (y_i - \bar{y})^2}}$$

Input

• x — np.ndarray of shape (n,).
• y — np.ndarray of shape (n,).

Output

Returns a float in $[-1, 1]$.

Examples

Example 1

Input:  x = [1, 2, 3, 4], y = [2, 4, 6, 8]
Output: 1.0

Explanation: $y = 2x$ is a perfect increasing linear relationship, so $r = 1$.

Constraints

• Centre both vectors by their means before forming the products.
• Divide by the product of the centred-vector norms; the scale factors cancel, so sample-vs-population normalisation does not matter.
• Return a float; the result lies in $[-1, 1]$.

Notes

• Pearson $r$ captures only linear dependence — variables can be strongly related (e.g. $y = x^2$) yet have $r \approx 0$.
• $r$ is the cosine of the angle between the mean-centred vectors, which is why it is invariant to shifting and positive scaling of either variable.