PyDTMC 6.6.0

A framework for discrete-time Markov chains analysis.

PyDTMC is a full-featured, lightweight library for discrete-time Markov chains analysis. It provides classes and functions for creating, manipulating, simulating and visualizing markovian stochastic processes.

Status:
Info:
PyPI:
Conda:
Donation:

Requirements

The Python environment must include the following packages:

• Matplotlib
• NetworkX
• NumPy
• SciPy

The package Sphinx is required for building the package documentation. The package pytest is required for performing unit tests. For a better user experience, it's recommended to install Graphviz and pydot before using the plot_graph function.

Installation & Upgrade

PyPI:

$ pip install PyDTMC
$ pip install --upgrade PyDTMC

Git:

$ pip install https://github.com/TommasoBelluzzo/PyDTMC/tarball/master
$ pip install --upgrade https://github.com/TommasoBelluzzo/PyDTMC/tarball/master

$ pip install git+https://github.com/TommasoBelluzzo/PyDTMC.git#egg=PyDTMC
$ pip install --upgrade git+https://github.com/TommasoBelluzzo/PyDTMC.git#egg=PyDTMC

Conda:

$ conda install -c conda-forge pydtmc
$ conda update -c conda-forge pydtmc

$ conda install -c tommasobelluzzo pydtmc
$ conda update -c tommasobelluzzo pydtmc

Usage

The core element of the library is the MarkovChain class, which can be instantiated as follows:

>>> p = [[0.2, 0.7, 0.0, 0.1], [0.0, 0.6, 0.3, 0.1], [0.0, 0.0, 1.0, 0.0], [0.5, 0.0, 0.5, 0.0]]
>>> mc = MarkovChain(p, ['A', 'B', 'C', 'D'])
>>> print(mc)

DISCRETE-TIME MARKOV CHAIN
 SIZE:           4
 RANK:           4
 CLASSES:        2
  > RECURRENT:   1
  > TRANSIENT:   1
 ERGODIC:        NO
  > APERIODIC:   YES
  > IRREDUCIBLE: NO
 ABSORBING:      YES
 REGULAR:        NO
 REVERSIBLE:     NO

Below a few examples of MarkovChain properties:

>>> print(mc.is_ergodic)
False

>>> print(mc.recurrent_states)
['C']

>>> print(mc.transient_states)
['A', 'B', 'D']

>>> print(mc.steady_states)
[array([0.0, 0.0, 1.0, 0.0])]

>>> print(mc.is_absorbing)
True

>>> print(mc.fundamental_matrix)
[[1.50943396, 2.64150943, 0.41509434]
 [0.18867925, 2.83018868, 0.30188679]
 [0.75471698, 1.32075472, 1.20754717]]
 
>>> print(mc.kemeny_constant)
5.547169811320755

>>> print(mc.entropy_rate)
0.0

Below a few examples of MarkovChain methods:

>>> print(mc.absorption_probabilities())
[1.0 1.0 1.0]

>>> print(mc.expected_rewards(10, [2, -3, 8, -7]))
[-2.76071635, -12.01665113, 23.23460025, -8.45723276]

>>> print(mc.expected_transitions(2))
[[0.085, 0.2975, 0.0000, 0.0425]
 [0.000, 0.3450, 0.1725, 0.0575]
 [0.000, 0.0000, 0.7000, 0.0000]
 [0.150, 0.0000, 0.1500, 0.0000]]

>>> print(mc.first_passage_probabilities(5, 3))
[[0.5000, 0.0000, 0.5000, 0.0000]
 [0.0000, 0.3500, 0.0000, 0.0500]
 [0.0000, 0.0700, 0.1300, 0.0450]
 [0.0000, 0.0315, 0.1065, 0.0300]
 [0.0000, 0.0098, 0.0761, 0.0186]]
 
>>> print(mc.hitting_probabilities([0, 1]))
[1.0, 1.0, 0.0, 0.5]
 
>>> print(mc.mean_absorption_times())
[4.56603774, 3.32075472, 3.28301887]

>>> print(mc.mean_number_visits())
[[0.50943396, 2.64150943, inf, 0.41509434]
 [0.18867925, 1.83018868, inf, 0.30188679]
 [0.00000000, 0.00000000, inf, 0.00000000]
 [0.75471698, 1.32075472, inf, 0.20754717]]
 
>>> print(mc.walk(10, seed=32))
['D', 'A', 'B', 'B', 'C', 'C', 'C', 'C', 'C', 'C', 'C']

>>> walk = ["A"]
>>> for i in range(1, 11):
...     current_state = walk[-1]
...     next_state = mc.next_state(current_state, seed=32)
...     print(f'{i:02} {current_state} -> {next_state}')
...     walk.append(next_state)
 1) A -> B
 2) B -> C
 3) C -> C
 4) C -> C
 5) C -> C
 6) C -> C
 7) C -> C
 8) C -> C
 9) C -> C
10) C -> C

Plotting functions can provide a visual representation of MarkovChain instances; in order to display the output of plots immediately, the interactive mode of Matplotlib must be turned on:

>>> plot_eigenvalues(mc)
>>> plot_graph(mc)
>>> plot_redistributions(mc, 10, plot_type='heatmap', dpi=300)
>>> plot_redistributions(mc, 10, plot_type='projection', dpi=300)
>>> plot_walk(mc, 10, plot_type='histogram', dpi=300)
>>> plot_walk(mc, 10, plot_type='sequence', dpi=300)
>>> plot_walk(mc, 10, plot_type='transitions', dpi=300)