Observation Influence Diagnostic of a Data Assimilation System - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 90

Furthermore, the change in the estimate that occurs when the i th observation is

deleted is

S ii
.1 S ii / r i

y i y . i/

D (4.9)

i

y . i/
i

where is the LS estimate of y i obtained by leaving-out the i th observation

of the vector y and the th row of the matrix X . The method is useful to assess

the quality of the analysis by using the discarded observation, but impractical for
large systems. The formula shows that the impact of deleting

i

.y i ; x i / on y i can be

computed by knowing only the residual r i and the diagonal element S ii - the nearer

the self-sensitivity S ii is to one, the more impact on the estimate y i . A related result

concerns the so-called cross-validation (CV) score: that is, the LS objective function
obtained when each data point is in turn deleted ( Wahba 1990 , Theorem 4.2.1):

X X

.y i y i / 2
.1 S ii / 2

.y i y . i i / 2 D (4.10)

i D 1 i D 1

This theorem shows that the CV score can be computed by relying on the all-data
estimate separate LS

regressions on the leaving-out-one samples. Moreover, ( 4.9 ) shows how to compute
self-sensitivities by the leaving out one experiment.
The definitions of influence matrix ( 4.4 ) and self-sensitivity ( 4.5 ) are rather
general and can be applied also to non-LS and nonparametric statistics. In spline
regression, for example, the interpretation remains essentially the same as in
ordinary linear regression and most of the results, like the CV-theorem above, still
apply. In this context, Craven and Wahba ( 1979 ) proposed the generalized-CV score,
replacing in ( 4.10 )

y and the self-sensitivities, without actually performing m

. For further applications of influence
diagnostics beyond usual LS regression (and further references) see Ye ( 1998 )and
Shen et al. ( 2002 ). The notions related to the influence matrix that it has introduced
here will in the following section be derived in the context of a statistical analysis
scheme used for data assimilation in numerical weather prediction (NWP).

S ii by the mean tr . S /=q

4.3 Observational Influence and Self-Sensitivity

for a DA Scheme

4.3.1 Linear Statistical Estimation in Numerical Weather

Prediction

Data assimilation systems for NWP provide estimates of the atmospheric state x
by combining meteorological observations y with prior (or background) informa-
tion x b . A simple Bayesian Normal model provides the solution as the posterior
expectation for x ,given y and x b . The same solution can be achieved from a classical