Fitted values ($\hat{Y}$)

Fitted values are the responses predicted at the training covariates by a fitted regression model.

A linear smoother has fitted-value vector $\hat{Y}=HY$, where the smoother matrix $H$ depends on the covariates and tuning parameters but not on the response vector.

For a linear smoother $\hat{Y}=HY$, its effective degrees of freedom are $\mathrm{tr}(H)$. This measures sensitivity of fitted values to responses even when $H$ is not an orthogonal projection.