Stationary

This is a python library for computing stationary distributions of finite Markov process, with a particular emphasis on finite population dynamics.

The library can approximate solutions for arbitrary finite Markov processes and compute exact stationary distributions for reversible Markov processes on discretized simplices. The approximate solutions are very accurate (see below). One need only supply a list of weighted edges:

edges = [(source_state, target_state, transition_probability), ...]
s = stationary_distribution(edges)

The states of the process can be any mutable or hashable python object: integers, strings, etc. The function stationary_distribution accepts a few parameters, including a logspace=True option for transition probabilities that are very small and need to be handle in log space. To save memory for large state spaces, the library uses a sparse matrix implementation of the graph associated to the Markov process. A relatively modern computer should be able to fit a few million states into memory.

Also included are functions to generate transition probabilities for the Moran process with mutation and various generalizations, including Fermi processes, dynamics on graphs, dynamics in populations of varying sizes, and the Wright-Fisher process.

For example, the following image (a stationary distribution for a rock-paper-scissors dynamic on a population of size 560) was created with this library and a ternary plotting library:

For very large state spaces, the stationary distribution calculation can be offloaded to an included C++ implementation (faster and smaller memory footprint).

Calculation of Stationary Distributions

The library computes stationary distributions in a variety of ways. Transition probabilities are represented by sparse matrices or by functions on the product of the state space. The latter is useful when the transition matrix is too large to fit into memory, such as the Wright-Fisher process for a population of size N with n-types, which requires

floating point values to specify the transition matrix.

The stationary distribution calculation function stationary.stationary_distribution takes either a list of weighted edges (as above) or a function specifying the transitions and the collection of states of the process. You can specify the following:

• Transitions or a function that computes transitions
• Compute in log-space with logspace=True, useful (necessary) for processes with very small probabilities
• Compute the stationary distribution exactly or approximately with exact=True (default is false). If False, the library computes large powers of the transition matrix times an initial state. If exact=True, the library attempts to use the following formula:

This formula only works for reversible processes on the simplex -- a particular encoding of states and paths is assumed.

The library can also compute exact solutions for the neutral fitness landscape for the Moran process.

Examples

Let's walk through a detailed example. For the classical Moran process, we have a population of two types, A and B. Type B has relative fitness r versus the fitness of type A (which is 1). It is well-known that the fixation probability of type A is

It is also known that the stationary distribution of the Moran process with mutation converges to the following distribution when the mutation rate goes to zero:

where the stationary distribution is over the population states [(0, N), (1, N-1), ..., (N, 0)].

In fixation_examples.py we compare the ratio of fixation probabilities with the stationary distribution for small values of mu, mu=10^-24, producing the following plot:

In the top plot there is no visual distinction between the two values. The lower plot has the difference in the two calculations, showing that the error is very small.

There are a few convenience functions that make such plots easy. To compute the stationary distribution of the Moran process is just a few lines of code:

    from stationary import convenience
    r = 2
    game_matrix = [[1, 1], [r, r]]
    N = 100
    mu = 1./ N
    # compute the transitions, stationary distribution, and entropy rate
    edges, s, er = convenience.moran(N, game_matrix, mu, exact=True, logspace=True)
    print s[(0, N)], s[(N, 0)]
    >>> 0.247107738567 4.63894759631e-29

More Examples

There are many examples in the test suite and some more complex examples in the examples subdirectory.

C++ Implementation

For larger state spaces a more efficient C++ implementation is provided. Compile with the following command:

g++ -std=c++11 -O3 stationary.cpp

You can still use the python functions to develop the process (e.g. the transition probabilies) and to plot. See export_to_cpp for an example.

Unit Tests

The library contains a number of tests to ensure that the calculations are accurate, including comparisons to closed forms when available.

To run the suite of unit tests, use the command

nosetests -s tests

Note that there are many tests and some take a considerable amount of time (several minutes in some cases).