  In this paper, we provide a new algorithm for the finite dimensional
approximation of the linear transfer Koopman and Perron-Frobenius operator from
time series data. We argue that existing approach for the finite dimensional
approximation of these transfer operators such as Dynamic Mode Decomposition
(DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two
important properties of these operators, namely positivity and Markov property.
The algorithm we propose in this paper preserve these two properties. We call
the proposed algorithm as naturally structured DMD since it retains the
inherent properties of these operators. Naturally structured DMD algorithm
leads to a better approximation of the steady-state dynamics of the system
regarding computing Koopman and Perron- Frobenius operator eigenfunctions and
eigenvalues. However preserving positivity properties is critical for capturing
the real transient dynamics of the system. This positivity of the transfer
operators and it's finite dimensional approximation also has an important
implication on the application of the transfer operator methods for controller
and estimator design for nonlinear systems from time series data.
