Geoscience Reference
In-Depth Information
H.X/
D
X
x
2
p.x/
p.x/:
log
(19.30)
Where
p
is a pdf. One can also define joint entropy
H.X;Y/
D
X
x
2
X
p.x;y/
log
p.x;y/
(19.31)
y
2
Y
p.x/
and relative entropy (e.g., Kullback-Leibler distance) between probabilities
q.x/
and
D.p
k
q/
D
X
x
2
log
p.x/
p.x/
q.x/
:
(19.32)
The mutual information
is defined as the relative entropy between the joint
distribution and the product distribution
I.X
I
Y/
I.X
I
Y/
D
X
x
2
X
log
p.x;y/
p.x;y/
p.x/p.y/
;
(19.33)
y
2
Y
and it represents a reduction of uncertainty due to information sharing between vari-
ables. The mutual information is non-negative and becomes zero for independent
variables. One can also see that
I.X
I
X/
D
H.X/
(19.34)
since there is no new information that can reduce uncertainty.
References
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systems. SIAM J Sci Comput 26:411-447
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131:634-642
Auligne T, Lorenc A, Michel Y, Montmerle T, Jones A, Hu M, Dudhia J (2011) Toward a new
cloud analysis and prediction system. Bull Am Meteorol Soc 92:207-210
Axelsson O (1994) Iterative solution methods. Cambridge University Press, Cambridge, p 668
Axelsson O, Barker VA (1984) Finite-element solution of boundary-layer problems. Theory and
computations. Academic Press, Orlando, p 432
Bauer P, Geer A, Lopez P, Salmond D (2010) Direct 4D-Var assimilation of all- sky radiances.
Part I: implementation. Q J Roy Meteorol Soc 136A:1868-1885
Bauer P, Lopez P, Salmond D, Benedetti A, Saarinen S, Moreau E (2006) Implementation of
1D
4D-Var assimilation of precipitation-affected microwave radiances at ECMWF. II: 4D-Var.
Q J Roy Meteorol Soc 132:2307-2332
Bauer P, Ohring G, Kummerow C, Auligne T (2011) Assimilating satellite observations of clouds
and precipitation into NWP models. Bull Am Meteorol Soc 92:ES25-ES28
C
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