Observation Influence Diagnostic of a Data Assimilation System - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 89

and that

D @ y i
@y j

S ij

D @ y i
@y i

S ii (4.5)

for the off-diagonal ( i ¤ j ) and the diagonal ( i D j ) elements, respectively. Thus,

S ij is the rate of change of y j variations. The diagonal element

S ii , instead, measures the rate of change of the regression estimate

y i with respect to

y i with respect to

variations in the corresponding observation y i . For this reason the self-sensitivity (or

self-influence, or leverage) of the S ii , while

an off-diagonal element is a cross-sensitivity diagnostic between two data points.
Hoaglin and Welsch ( 1978 ) discuss some properties of the influence matrix. The
diagonal elements satisfy

i th data point is the i th diagonal element

0 S ii 1:::::::::i D 1;2;:::;m (4.6)

as S is a symmetric and idempotent projection matrix ( S D S 2 ). The covariance of
the error in the estimate y , and the covariance of the residual r D y y are related
to S by

/ D 2 S

var . y

/ D 2 .I m S

var . r / (4.7)

The trace of the influence matrix is

X

tr . S / D S ii D q D rank . S / (4.8)

i D 1

(in fact S has zeros). Thus, the trace is equal to

the number of parameters. The trace can be interpreted as the amount of information
extracted from the observations or degrees of freedom for signal ( Wahba et al. 1995 ).
The complementary trace, tr

m eigenvalues equals to 1 and m q

, on the other hand, is the degree of
freedom for noise , or simply the degree of freedom ( df ) of the error variance, widely
used for model checking (F test).
A zero self-sensitivity

. I S / D m tr . S /

S ii D 0 indicates that the i th observation has had no

influence at all in the fit, while indicates that an entire degree of freedom

(effectively one parameter) has been devoted to fitting just that data point. The aver-
age self-sensitivity value is

S ii D 1

S ii is considered 'large'
if its value is greater than three times the average ( Velleman and Welsch 1981 ).
By a symmetrical argument a self-sensitivity value that is less than one-third of the
average is considered 'small'.

q=m and an individual element