weighted-covariance — CRED case

CRED-inspired risk analytics pipeline assigns different importance to observations based on their reliability. You need to compute the weighted covariance so the downstream model captures the relationship between features using those observation weights.

Normalise the weights to sum to 1 ($p_i=w_i\mathrm{/}\sum wp\_i=w\_i/\backslash sum\; w$), compute the weighted mean $\mu ={\sum }_i{p}_i{x}_i\backslash mu=\backslash sum\_ip\_ix\_i$ and return $\mathrm{\Sigma }={\sum }_i{p}_i(x_i-\mu )(x_i-\mu {)}^{\mathrm{\top }}\backslash Sigma=\backslash sum\_ip\_i(x\_i-\backslash mu)(x\_i-\backslash mu)^\backslash top$. No small-sample bias correction is applied.

Implement solve(X, w).

Returns. Return a symmetric $d\times dd\backslash times\; d$ float NumPy matrix. The weights must be non-negative and not all zero.

Examples

Example 1

Input:

solve([[1, 2], [3, 4], [5, 1]], [1, 2, 1])

Output:

[[2.0, -0.5], [-0.5, 1.6875]]

Example 2

Input:

solve([[1.0, 0.0], [3.0, 2.0]], [1.0, 1.0])

Output:

[[1.0, 1.0], [1.0, 1.0]]