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Optimal fire management of a threatened species
# -*- coding: utf-8 -*-
"""Optimal fire management of a threatened species
===============================================
This PyMDPtoolbox example is based on a paper [Possingham1997]_ preseneted by
Hugh Possingham and Geoff Tuck at the 1997 MODSIM conference. This version
only considers a single population, rather than the full two-patch spatially
structured model in the paper. The paper is freely available to read from the
link provided, so minimal details are given here.
.. [Possingham1997] Possingham H & Tuck G, 1997, ‘Application of stochastic
dynamic programming to optimal fire management of a spatially structured
threatened species’, *MODSIM 1997*, vol. 2, pp. 813–817. `Available online
<http://www.mssanz.org.au/MODSIM97/Vol%202/Possingham.pdf>`_.
"""
# Copyright (c) 2015 Steven A. W. Cordwell
#
# 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 <ORGANIZATION> 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
from mdptoolbox import mdp
# The number of population abundance classes
POPULATION_CLASSES = 7
# The number of years since a fire classes
FIRE_CLASSES = 13
# The number of states
STATES = POPULATION_CLASSES * FIRE_CLASSES
# The number of actions
ACTIONS = 2
ACTION_NOTHING = 0
ACTION_BURN = 1
def check_action(x):
"""Check that the action is in the valid range.
"""
if not (0 <= x < ACTIONS):
msg = "Invalid action '%s', it should be in {0, 1}." % str(x)
raise ValueError(msg)
def check_population_class(x):
"""Check that the population abundance class is in the valid range.
"""
if not (0 <= x < POPULATION_CLASSES):
msg = "Invalid population class '%s', it should be in {0, 1, …, %d}." \
% (str(x), POPULATION_CLASSES - 1)
raise ValueError(msg)
def check_fire_class(x):
"""Check that the time in years since last fire is in the valid range.
"""
if not (0 <= x < FIRE_CLASSES):
msg = "Invalid fire class '%s', it should be in {0, 1, …, %d}." % \
(str(x), FIRE_CLASSES - 1)
raise ValueError(msg)
def check_probability(x, name="probability"):
"""Check that a probability is between 0 and 1.
"""
if not (0 <= x <= 1):
msg = "Invalid %s '%s', it must be in [0, 1]." % (name, str(x))
raise ValueError(msg)
def get_habitat_suitability(years):
"""The habitat suitability of a patch relatve to the time since last fire.
The habitat quality is low immediately after a fire, rises rapidly until
five years after a fire, and declines once the habitat is mature. See
Figure 2 in Possingham and Tuck (1997) for more details.
Parameters
----------
years : int
The time in years since last fire.
Returns
-------
r : float
The habitat suitability.
"""
if years < 0:
msg = "Invalid years '%s', it should be positive." % str(years)
raise ValueError(msg)
if years <= 5:
return 0.2*years
elif 5 <= years <= 10:
return -0.1*years + 1.5
else:
return 0.5
def convert_state_to_index(population, fire):
"""Convert state parameters to transition probability matrix index.
Parameters
----------
population : int
The population abundance class of the threatened species.
fire : int
The time in years since last fire.
Returns
-------
index : int
The index into the transition probability matrix that corresponds to
the state parameters.
"""
check_population_class(population)
check_fire_class(fire)
return population*FIRE_CLASSES + fire
def convert_index_to_state(index):
"""Convert transition probability matrix index to state parameters.
Parameters
----------
index : int
The index into the transition probability matrix that corresponds to
the state parameters.
Returns
-------
population, fire : tuple of int
``population``, the population abundance class of the threatened
species. ``fire``, the time in years since last fire.
"""
if not (0 <= index < STATES):
msg = "Invalid index '%s', it should be in {0, 1, …, %d}." % \
(str(index), STATES - 1)
raise ValueError(msg)
population = index // FIRE_CLASSES
fire = index % FIRE_CLASSES
return (population, fire)
def transition_fire_state(F, a):
"""Transition the years since last fire based on the action taken.
Parameters
----------
F : int
The time in years since last fire.
a : int
The action undertaken.
Returns
-------
F : int
The time in years since last fire.
"""
## Efect of action on time in years since fire.
if a == ACTION_NOTHING:
# Increase the time since the patch has been burned by one year.
# The years since fire in patch is absorbed into the last class
if F < FIRE_CLASSES - 1:
F += 1
elif a == ACTION_BURN:
# When the patch is burned set the years since fire to 0.
F = 0
return F
def get_transition_probabilities(s, x, F, a):
"""Calculate the transition probabilities for the given state and action.
Parameters
----------
s : float
The class-independent probability of the population staying in its
current population abundance class.
x : int
The population abundance class of the threatened species.
F : int
The time in years since last fire.
a : int
The action undertaken.
Returns
-------
prob : array
The transition probabilities as a vector from state (``x``, ``F``) to
every other state given that action ``a`` is taken.
"""
# Check that input is in range
check_probability(s)
check_population_class(x)
check_fire_class(F)
check_action(a)
# a vector to store the transition probabilities
prob = np.zeros(STATES)
# the habitat suitability value
r = get_habitat_suitability(F)
F = transition_fire_state(F, a)
## Population transitions
if x == 0:
# population abundance class stays at 0 (extinct)
new_state = convert_state_to_index(0, F)
prob[new_state] = 1
elif x == POPULATION_CLASSES - 1:
# Population abundance class either stays at maximum or transitions
# down
transition_same = x
transition_down = x - 1
# If action 1 is taken, then the patch is burned so the population
# abundance moves down a class.
if a == ACTION_BURN:
transition_same -= 1
transition_down -= 1
# transition probability that abundance stays the same
new_state = convert_state_to_index(transition_same, F)
prob[new_state] = 1 - (1 - s)*(1 - r)
# transition probability that abundance goes down
new_state = convert_state_to_index(transition_down, F)
prob[new_state] = (1 - s)*(1 - r)
else:
# Population abundance class can stay the same, transition up, or
# transition down.
transition_same = x
transition_up = x + 1
transition_down = x - 1
# If action 1 is taken, then the patch is burned so the population
# abundance moves down a class.
if a == ACTION_BURN:
transition_same -= 1
transition_up -= 1
# Ensure that the abundance class doesn't go to -1
if transition_down > 0:
transition_down -= 1
# transition probability that abundance stays the same
new_state = convert_state_to_index(transition_same, F)
prob[new_state] = s
# transition probability that abundance goes up
new_state = convert_state_to_index(transition_up, F)
prob[new_state] = (1 - s)*r
# transition probability that abundance goes down
new_state = convert_state_to_index(transition_down, F)
# In the case when transition_down = 0 before the effect of an action
# is applied, then the final state is going to be the same as that for
# transition_same, so we need to add the probabilities together.
prob[new_state] += (1 - s)*(1 - r)
# Make sure that the probabilities sum to one
assert (prob.sum() - 1) < np.spacing(1)
return prob
def get_transition_and_reward_arrays(s):
"""Generate the fire management transition and reward matrices.
The output arrays from this function are valid input to the mdptoolbox.mdp
classes.
Let ``S`` = number of states, and ``A`` = number of actions.
Parameters
----------
s : float
The class-independent probability of the population staying in its
current population abundance class.
Returns
-------
out : tuple
``out[0]`` contains the transition probability matrices P and
``out[1]`` contains the reward vector R. P is an ``A`` × ``S`` × ``S``
numpy array and R is a numpy vector of length ``S``.
"""
check_probability(s)
# The transition probability array
transition = np.zeros((ACTIONS, STATES, STATES))
# The reward vector
reward = np.zeros(STATES)
# Loop over all states
for idx in range(STATES):
# Get the state index as inputs to our functions
x, F = convert_index_to_state(idx)
# The reward for being in this state is 1 if the population is extant
if x != 0:
reward[idx] = 1
# Loop over all actions
for a in range(ACTIONS):
# Assign the transition probabilities for this state, action pair
transition[a][idx] = get_transition_probabilities(s, x, F, a)
return (transition, reward)
def solve_mdp():
"""Solve the problem as a finite horizon Markov decision process.
The optimal policy at each stage is found using backwards induction.
Possingham and Tuck report strategies for a 50 year time horizon, so the
number of stages for the finite horizon algorithm is set to 50. There is no
discount factor reported, so we set it to 0.96 rather arbitrarily.
Returns
-------
sdp : mdptoolbox.mdp.FiniteHorizon
The PyMDPtoolbox object that represents a finite horizon MDP. The
optimal policy for each stage is accessed with mdp.policy, which is a
numpy array with 50 columns (one for each stage).
"""
transition, reward = get_transition_and_reward_arrays(0.5)
sdp = mdp.FiniteHorizon(transition, reward, 0.96, 50)
sdp.run()
return sdp
def print_policy(policy):
"""Print out a policy vector as a table to console
Let ``S`` = number of states.
The output is a table that has the population class as rows, and the years
since a fire as the columns. The items in the table are the optimal action
for that population class and years since fire combination.
Parameters
----------
p : array
``p`` is a numpy array of length ``S``.
"""
p = np.array(policy).reshape(POPULATION_CLASSES, FIRE_CLASSES)
print(" " + " ".join("%2d" % f for f in range(FIRE_CLASSES)))
print(" " + "---" * FIRE_CLASSES)
for x in range(POPULATION_CLASSES):
print(" %2d|" % x + " ".join("%2d" % p[x, f] for f in
range(FIRE_CLASSES)))
def simulate_transition(s, x, F, a):
"""Simulate a state transition.
Parameters
----------
s : float
The class-independent probability of the population staying in its
current population abundance class.
x : int
The population abundance class of the threatened species.
F : int
The time in years since last fire.
a : int
The action undertaken.
Returns
-------
x, F : int, int
The new abundance class, x, of the threatened species and the new years
last fire class, F.
"""
check_probability(s)
check_population_class(x)
check_fire_class(F)
check_action(a)
r = get_habitat_suitability(F)
F = transition_fire_state(F, a)
if x == POPULATION_CLASSES - 1:
# pass with probability 1 - (1 - s)*(1 - r)
if np.random.random() < (1 - s)*(1 - r):
x -= 1
elif 0 < x < POPULATION_CLASSES - 1:
# pass with probability s
if np.random.random() < 1 - s:
if np.random.random() < r: # with probability (1 - s)r
x += 1
else: # with probability (1 - s)(1 - r)
x -= 1
# Add the effect of a fire, making sure x doesn't go to -1
if a == ACTION_BURN and (x > 0):
x -= 1
return x, F
def _run_tests():
"""Run tests on the modules functions.
"""
assert get_habitat_suitability(0) == 0
assert get_habitat_suitability(2) == 0.4
assert get_habitat_suitability(5) == 1
assert get_habitat_suitability(8) == 0.7
assert get_habitat_suitability(10) == 0.5
assert get_habitat_suitability(15) == 0.5
state = convert_index_to_state(STATES - 1)
assert state == (POPULATION_CLASSES - 1, FIRE_CLASSES - 1)
state = convert_index_to_state(STATES - 2)
assert state == (POPULATION_CLASSES -1, FIRE_CLASSES - 2)
assert convert_index_to_state(0) == (0, 0)
for idx in range(STATES):
state1, state2 = convert_index_to_state(idx)
assert convert_state_to_index(state1, state2) == idx
print("Tests complete.")
if __name__ == "__main__":
import sys
try:
ARGV = sys.argv[1]
except IndexError:
ARGV = None
if ARGV == "test":
_run_tests()
else:
SDP = solve_mdp()
print_policy(SDP.policy[:, 0])
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