The basic
quantity in the formalism is a local information (in space), based on the
observation of the states at one side (either left or right) of the position of
interest. From the internal statistics in the spatial pattern a local
conditional probability is formed, which will then define a local information
quantity as follows. In information terms one would expect that when a local
less common configuration is encountered at a certain position i in the sequence of cells, the conditional probability for that
configuration given the n-length symbol
sequence, for example, in the cells to the left of it will be relatively small.
This implies a high local information of that conditional probability,
, (1)
where zi denotes the symbol at position i,
see [Helvik et al, 2007]. A corresponding local
information 
conditioned on
the n cells to the right of the position i is similarly defined. Such a local information quantity has a
spatial average that can be written
. (2)
By using the
ergodicity theorem, one can conclude that in average the local information
equals the entropy s of the system,
. (3)
For details of
this derivation involving information theory for symbol sequences, see the
lecture notes [Lindgren, 2008]. A similar expression holds for the
corresponding right-sided quantity . We can then define a local information as an average
between the two,
, (4)
This means that
the local information Ii,n, as well
as the left- and right-handed versions, has a spatial average that equals the
entropy of the system. In the limit of an infinitely long conditioned sequence,
we define the local (left-sided) entropy sL(i) at position i,
, (5)
and the
right-sided counterpart is similarly defined.