Covariance matrix

Background

The covariance matrix summarises how a set of variables vary together: entry $(i, j)$ is the covariance between feature $i$ and feature $j$, and the diagonal holds each feature's variance. It is the foundation of PCA, Gaussian models, whitening, and the Mahalanobis distance.

Problem statement

Implement calculate_covariance_matrix(vectors) where vectors[i] is the list of observations for feature $i$. Return the symmetric matrix of sample covariances:

$$\Sigma_{ij} = \frac{1}{n - 1}\sum_{k=1}^{n}\big(x_{ik} - \bar{x}_i\big)\big(x_{jk} - \bar{x}_j\big)$$

where $n$ is the number of observations per feature and $\bar{x}_i$ is feature $i$'s mean.

Input

• vectors — list[list[float]] of shape (n_features, n_observations): each inner list holds one feature's values across all observations.

Output

Returns a list[list[float]] of shape (n_features, n_features): the symmetric sample covariance matrix.

Examples

Example 1

Input:  [[1, 2, 3], [4, 5, 6]]
Output: [[1.0, 1.0], [1.0, 1.0]]

Explanation: feature 0 has mean 2 and feature 1 has mean 5. Each variance is $\frac{(-1)^2 + 0^2 + 1^2}{2} = 1$, and the covariance is $\frac{(-1)(-1) + 0 + (1)(1)}{2} = 1$.

Constraints

• Use the sample covariance: divide by $n - 1$, where $n$ is the number of observations.
• The matrix is symmetric: $\Sigma_{ij} = \Sigma_{ji}$.
• Every vectors[i] has the same length $n$.

Notes

• Mind the orientation: rows are features, columns are observations — the transpose of the usual (n_samples, n_features) design matrix (this matches np.cov with rowvar=True).
• Dividing by $n - 1$ (Bessel's correction) gives an unbiased estimate of the population covariance from a sample.