Wrong use of covariance to calculate within-class scatter matrix

[xiongtx]

In Chpt. 5, pg. 157 it says:

$$\Sigma_i = \frac{1}{n_i} \sum_{x \in D_i} (\mathbf{x} - \mathbf{m_i})(\mathbf{x} - \mathbf{m_i})^T$$

That should be $n_i - 1$, which is what np.cov uses to get an unbiased estimator.

Then in Chpt. 5, pg. 158 the within-class scatter matrix S_W is calculated as:

d = 13 # number of features
S_W = np.zeros((d, d))
for label,mv in zip(range(1, 4), mean_vecs):
    class_scatter = np.cov(X_train_std[y_train==label].T)
    S_W += class_scatter

The covariances are not being weighted by sample sizes. The correct implementation is:

for label,mv in zip(range(1, 4), mean_vecs):
    n_i = np.sum(y_train == label)
    class_scatter = (n_i - 1) * np.cov(X_train_std[y_train==label].T)
    S_W += class_scatter