The DFFITS above only describe how a single observation affects the
prediction of itself. Naturally one could also consider how
observation
$i$ affects the prediction of the $j$'th observation by
looking at $\hat{y}_{j(i)}$.
If one were to carry this through and try to analyze how an
observation affects the prediction of all other observations, this
would lead to a somewhat intractable $n\times n$ matrix.

Cook's distance is a single measure describing how an individual
observation affects all predictions, thus summarizing the information
into an $n$-vector.
$$
D_i = \frac{\sum_j \left ( \hat{y}_j - \hat{y}_{j(i)} \right )^2 }{pMSE}
=\frac{1}{ps^2} || \hat{\mathbf{y}} - \hat{\mathbf{y}}_{(i)}||^2
$$

Given deletion formulas, it is not too hard to see that
$$
D_i =\frac{\hat{e}_i^2}{ps^2}\frac{h_{ii}}{(1-h_{ii})^2}
$$
and it is seen that this measure is large when either the residual
$\hat{e}_i$ is large or the influence measure $h_{ii}$ from the hat matrix
is large.