# -*- coding: utf-8 -*-
"""Markov Decision Process (MDP) Toolbox: ``example`` module
=========================================================
The ``example`` module provides functions to generate valid MDP transition and
reward matrices.
Available functions
-------------------
forest
A simple forest management example
rand
A random example
"""
# Copyright (c) 2011-2014 Steven A. W. Cordwell
# Copyright (c) 2009 INRA
#
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are met:
#
# * Redistributions of source code must retain the above copyright notice,
# this list of conditions and the following disclaimer.
# * Redistributions in binary form must reproduce the above copyright notice,
# this list of conditions and the following disclaimer in the documentation
# and/or other materials provided with the distribution.
# * Neither the name of the nor the names of its contributors
# may be used to endorse or promote products derived from this software
# without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
# POSSIBILITY OF SUCH DAMAGE.
import numpy as _np
import scipy.sparse as _sp
def forest(S=3, r1=4, r2=2, p=0.1, is_sparse=False):
"""Generate a MDP example based on a simple forest management scenario.
This function is used to generate a transition probability
(``A`` × ``S`` × ``S``) array ``P`` and a reward (``S`` × ``A``) matrix
``R`` that model the following problem. A forest is managed by two actions:
'Wait' and 'Cut'. An action is decided each year with first the objective
to maintain an old forest for wildlife and second to make money selling cut
wood. Each year there is a probability ``p`` that a fire burns the forest.
Here is how the problem is modelled.
Let {0, 1 . . . ``S``-1 } be the states of the forest, with ``S``-1 being
the oldest. Let 'Wait' be action 0 and 'Cut' be action 1.
After a fire, the forest is in the youngest state, that is state 0.
The transition matrix ``P`` of the problem can then be defined as follows::
| p 1-p 0.......0 |
| . 0 1-p 0....0 |
P[0,:,:] = | . . 0 . |
| . . . |
| . . 1-p |
| p 0 0....0 1-p |
| 1 0..........0 |
| . . . |
P[1,:,:] = | . . . |
| . . . |
| . . . |
| 1 0..........0 |
The reward matrix R is defined as follows::
| 0 |
| . |
R[:,0] = | . |
| . |
| 0 |
| r1 |
| 0 |
| 1 |
R[:,1] = | . |
| . |
| 1 |
| r2 |
Parameters
---------
S : int, optional
The number of states, which should be an integer greater than 1.
Default: 3.
r1 : float, optional
The reward when the forest is in its oldest state and action 'Wait' is
performed. Default: 4.
r2 : float, optional
The reward when the forest is in its oldest state and action 'Cut' is
performed. Default: 2.
p : float, optional
The probability of wild fire occurence, in the range ]0, 1[. Default:
0.1.
is_sparse : bool, optional
If True, then the probability transition matrices will be returned in
sparse format, otherwise they will be in dense format. Default: False.
Returns
-------
out : tuple
``out[0]`` contains the transition probability matrix P and ``out[1]``
contains the reward matrix R. If ``is_sparse=False`` then P is a numpy
array with a shape of ``(A, S, S)`` and R is a numpy array with a shape
of ``(S, A)``. If ``is_sparse=True`` then P is a tuple of length ``A``
where each ``P[a]`` is a scipy sparse CSR format matrix of shape
``(S, S)``; R remains the same as in the case of ``is_sparse=False``.
Examples
--------
>>> import mdptoolbox.example
>>> P, R = mdptoolbox.example.forest()
>>> P
array([[[ 0.1, 0.9, 0. ],
[ 0.1, 0. , 0.9],
[ 0.1, 0. , 0.9]],
[[ 1. , 0. , 0. ],
[ 1. , 0. , 0. ],
[ 1. , 0. , 0. ]]])
>>> R
array([[ 0., 0.],
[ 0., 1.],
[ 4., 2.]])
>>> Psp, Rsp = mdptoolbox.example.forest(is_sparse=True)
>>> len(Psp)
2
>>> Psp[0]
<3x3 sparse matrix of type ''
with 6 stored elements in Compressed Sparse Row format>
>>> Psp[1]
<3x3 sparse matrix of type ''
with 3 stored elements in Compressed Sparse Row format>
>>> Rsp
array([[ 0., 0.],
[ 0., 1.],
[ 4., 2.]])
>>> (Psp[0].todense() == P[0]).all()
True
>>> (Rsp == R).all()
True
"""
assert S > 1, "The number of states S must be greater than 1."
assert (r1 > 0) and (r2 > 0), "The rewards must be non-negative."
assert 0 <= p <= 1, "The probability p must be in [0; 1]."
# Definition of Transition matrix
if is_sparse:
P = []
rows = list(range(S)) * 2
cols = [0] * S + list(range(1, S)) + [S - 1]
vals = [p] * S + [1-p] * S
P.append(_sp.coo_matrix((vals, (rows, cols)), shape=(S,S)).tocsr())
rows = list(range(S))
cols = [0] * S
vals = [1] * S
P.append(_sp.coo_matrix((vals, (rows, cols)), shape=(S,S)).tocsr())
else:
P = _np.zeros((2, S, S))
P[0, :, :] = (1 - p) * _np.diag(_np.ones(S - 1), 1)
P[0, :, 0] = p
P[0, S - 1, S - 1] = (1 - p)
P[1, :, :] = _np.zeros((S, S))
P[1, :, 0] = 1
# Definition of Reward matrix
R = _np.zeros((S, 2))
R[S - 1, 0] = r1
R[:, 1] = _np.ones(S)
R[0, 1] = 0
R[S - 1, 1] = r2
# we want to return the generated transition and reward matrices
return(P, R)
def rand(S, A, is_sparse=False, mask=None):
"""Generate a random Markov Decision Process.
Parameters
----------
S : int
Number of states (> 1)
A : int
Number of actions (> 1)
is_sparse : bool, optional
False to have matrices in dense format, True to have sparse matrices.
Default: False.
mask : array, optional
Array with 0 and 1 (0 indicates a place for a zero probability), shape
can be ``(S, S)`` or ``(A, S, S)``. Default: random.
Returns
-------
out : tuple
``out[0]`` contains the transition probability matrix P and ``out[1]``
contains the reward matrix R. If ``is_sparse=False`` then P is a numpy
array with a shape of ``(A, S, S)`` and R is a numpy array with a shape
of ``(S, A)``. If ``is_sparse=True`` then P and R are tuples of length
``A``, where each ``P[a]`` is a scipy sparse CSR format matrix of shape
``(S, S)`` and each ``R[a]`` is a scipy sparse csr format matrix of
shape ``(S, 1)``.
Examples
--------
>>> import numpy, mdptoolbox.example
>>> numpy.random.seed(0) # Needed to get the output below
>>> P, R = mdptoolbox.example.rand(4, 3)
>>> P
array([[[ 0.21977283, 0.14889403, 0.30343592, 0.32789723],
[ 1. , 0. , 0. , 0. ],
[ 0. , 0.43718772, 0.54480359, 0.01800869],
[ 0.39766289, 0.39997167, 0.12547318, 0.07689227]],
[[ 1. , 0. , 0. , 0. ],
[ 0.32261337, 0.15483812, 0.32271303, 0.19983549],
[ 0.33816885, 0.2766999 , 0.12960299, 0.25552826],
[ 0.41299411, 0. , 0.58369957, 0.00330633]],
[[ 0.32343037, 0.15178596, 0.28733094, 0.23745272],
[ 0.36348538, 0.24483321, 0.16114188, 0.23053953],
[ 1. , 0. , 0. , 0. ],
[ 0. , 0. , 1. , 0. ]]])
>>> R
array([[[-0.23311696, 0.58345008, 0.05778984, 0.13608912],
[-0.07704128, 0. , -0. , 0. ],
[ 0. , 0.22419145, 0.23386799, 0.88749616],
[-0.3691433 , -0.27257846, 0.14039354, -0.12279697]],
[[-0.77924972, 0. , -0. , -0. ],
[ 0.47852716, -0.92162442, -0.43438607, -0.75960688],
[-0.81211898, 0.15189299, 0.8585924 , -0.3628621 ],
[ 0.35563307, -0. , 0.47038804, 0.92437709]],
[[-0.4051261 , 0.62759564, -0.20698852, 0.76220639],
[-0.9616136 , -0.39685037, 0.32034707, -0.41984479],
[-0.13716313, 0. , -0. , -0. ],
[ 0. , -0. , 0.55810204, 0. ]]])
>>> numpy.random.seed(0) # Needed to get the output below
>>> Psp, Rsp = mdptoolbox.example.rand(100, 5, is_sparse=True)
>>> len(Psp), len(Rsp)
(5, 5)
>>> Psp[0]
<100x100 sparse matrix of type ''
with 3296 stored elements in Compressed Sparse Row format>
>>> Rsp[0]
<100x100 sparse matrix of type ''
with 3296 stored elements in Compressed Sparse Row format>
>>> # The number of non-zero elements (nnz) in P and R are equal
>>> Psp[1].nnz == Rsp[1].nnz
True
"""
# making sure the states and actions are more than one
assert S > 1, "The number of states S must be greater than 1."
assert A > 1, "The number of actions A must be greater than 1."
# if the user hasn't specified a mask, then we will make a random one now
if mask is not None:
# the mask needs to be SxS or AxSxS
try:
assert mask.shape in ((S, S), (A, S, S)), "'mask' must have " \
"dimensions S×S or A×S×S."
except AttributeError:
raise TypeError("'mask' must be a numpy array or matrix.")
# generate the transition and reward matrices based on S, A and mask
if is_sparse:
# definition of transition matrix : square stochastic matrix
P = [None] * A
# definition of reward matrix (values between -1 and +1)
R = [None] * A
for a in range(A):
# it may be more efficient to implement this by constructing lists
# of rows, columns and values then creating a coo_matrix, but this
# works for now
PP = _sp.dok_matrix((S, S))
RR = _sp.dok_matrix((S, S))
for s in range(S):
if mask is None:
m = _np.random.random(S)
m[m <= 2/3.0] = 0
m[m > 2/3.0] = 1
elif mask.shape == (A, S, S):
m = mask[a][s] # mask[a, s, :]
else:
m = mask[s]
n = int(m.sum()) # m[s, :]
if n == 0:
m[_np.random.randint(0, S)] = 1
n = 1
# find the columns of the vector that have non-zero elements
nz = m.nonzero()
if len(nz) == 1:
cols = nz[0]
else:
cols = nz[1]
vals = _np.random.random(n)
vals = vals / vals.sum()
reward = 2*_np.random.random(n) - _np.ones(n)
PP[s, cols] = vals
RR[s, cols] = reward
# PP.tocsr() takes the same amount of time as PP.tocoo().tocsr()
# so constructing PP and RR as coo_matrix in the first place is
# probably "better"
P[a] = PP.tocsr()
R[a] = RR.tocsr()
else:
# definition of transition matrix : square stochastic matrix
P = _np.zeros((A, S, S))
# definition of reward matrix (values between -1 and +1)
R = _np.zeros((A, S, S))
for a in range(A):
for s in range(S):
# create our own random mask if there is no user supplied one
if mask is None:
m = _np.random.random(S)
r = _np.random.random()
m[m <= r] = 0
m[m > r] = 1
elif mask.shape == (A, S, S):
m = mask[a][s] # mask[a, s, :]
else:
m = mask[s]
# Make sure that there is atleast one transition in each state
if m.sum() == 0:
m[_np.random.randint(0, S)] = 1
P[a][s] = m * _np.random.random(S)
P[a][s] = P[a][s] / P[a][s].sum()
R[a][s] = (m * (2 * _np.random.random(S) -
_np.ones(S, dtype=int)))
# we want to return the generated transition and reward matrices
return(P, R)