Sensititivy Analysis in the Degenerate Case for Perron-Eigenvalues
We use a paper from N. Singhal and V. Pande as reference and start
from there. We proceed in analogy to appendix B. We extend the Sensitivity
Analysis for the case, that an eigenvalue is degenerate. Since we
are interested in the computation of the stationary distribution of
a system, which has been sampled by short trajectories. This can lead
to transition matrices, which do not have one unique Perron-Eigenvector,
but a few, which correspond to completely seperated subspaces. In
particular this can happen, when the trajectory is started in state,
that are known to exist, but have not been observed by previous trajectories.
We will concentrate only on the case, where the eigenvector subspace
to the eigenvalue one is degenerated. In this particular case a few
step even simplify.
To compute the stationary distribution we need to compute the left
eigenvalues of

to the eigenvalue one or of

to the eigenvalue zero, which is equivalent to find a basis for the
kernel or the nullspace. To compute left eigenvalues, with an algorithmus
written for right eigenvalues, one needs to transpose the matrix in
question and proceed as usual.
We start with an eigenvector equation, where

is the Matrix of right eigenvectors (in columns) to the eigenvalue
one and the

is a zeromatrix of the same size. In analogy

is the matrix of left eigenvectors
(in rows).
Since we are looking at statistical matrices, we know, that the right
PerronEigenvectors are constant on a seperated subspace, so if we
choose the right eigenvectors to be orthogonal, these are zero everywhere
execept on the indices referring to a state in a certain subspace.
These indices are constant and we will choose these to be equal to
one.
So now we have split the matrix in several subspaces indicated by
the characteristic vectors

. The left eigenvectors indicate
now the stationary distribution inside one of these subspaces. To
proceed, we need to choose a basis of left eigenvectors, so that these
are also confined to the subspaces. We express this by
Finally we choose the

, so that the stationary distribution
in one subspace is given by

. This is equal to
and is different to (B4), where the euklidian norm is chosen. We assume
explicitly for the sensitivity, that the subspaces will be kept and
not, that the degeneracy is destroyed. Meaning, that the

are constant
This leads to a form for equation (B6)
For computational reasons we only need to compute the entries of the
variation of

to

, when

are in the same
subspace, otherwise the value is zero. At least I think, I have to
check that. At least I assume, when the degeneracy is maintained.