# -*- coding: utf-8 -*-
"""Optimal fire management for a single population 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 two populations considered
in the original 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
`_.
"""
# Copyright (c) 2014 Steven A. W. Cordwell
#
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#
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# modification, are permitted provided that the following conditions are met:
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# * Neither the name of the nor the names of its contributors
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from mdptoolbox import mdp
import random
import numpy as np
# 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
def convertStateToIndex(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.
"""
assert 0 <= population < POPULATION_CLASSES, "'population' must be in " \
"(0, 1...%s)" % str(POPULATION_CLASSES - 1)
assert 0 <= fire < FIRE_CLASSES, "'fire' must be in " \
"(0, 1...%s) " % str(FIRE_CLASSES - 1)
return(population * FIRE_CLASSES + fire)
def convertIndexToState(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.
"""
assert index < STATES
population = index // FIRE_CLASSES
fire = index % FIRE_CLASSES
return(population, fire)
def getHabitatSuitability(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
Years since last fire.
Returns
-------
r : float
The habitat suitability.
"""
assert years >= 0, "'years' must be a positive number"
if years <= 5:
return(0.2 * years)
elif 5 <= years <= 10:
return(-0.1 * years + 1.5)
else:
return(0.5)
def getTransitionProbabilities(s, x, F, a):
"""Calculate the transition probabilities for the given state and action.
Parameters
----------
s : float
The probability of a population remaining in its current abundance
class
x : int
The population abundance class
F : int
The number of years since a fire
a : int
The action to be performed
Returns
-------
prob : array
The transition probabilities as a vector from state (x, F) to every
other state given action ``a`` is performed.
"""
assert 0 <= x < POPULATION_CLASSES
assert 0 <= F < FIRE_CLASSES
assert 0 <= s <= 1
assert 0 <= a < ACTIONS
prob = np.zeros((STATES,))
r = getHabitatSuitability(F)
# Efect of action on time in years since fire.
if a == 0:
# 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 == 1:
# When the patch is burned set the years since fire to 0.
F = 0
# Population transitions
if x == 0:
# Demographic model probabilities
# population abundance class stays at 0 (extinct)
new_state = convertStateToIndex(0, F)
prob[new_state] = 1
elif x == POPULATION_CLASSES - 1:
# Population abundance class either stays at maximum or transitions
# down
x_1 = x
x_2 = x - 1
# Effect of action on the state
# If action 1 is taken, then the patch is burned so the population
# abundance moves down a class.
if a == 1:
x_1 -= 1
x_2 -= 1
# Demographic model probabilities
new_state = convertStateToIndex(x_1, F)
prob[new_state] = 1 - (1 - s) * (1 - r) # abundance stays the same
new_state = convertStateToIndex(x_2, F)
prob[new_state] = (1 - s) * (1 - r) # abundance goes down
else:
# Population abundance class can stay the same, transition up, or
# transition down.
x_1 = x
x_2 = x + 1
x_3 = x - 1
# Effect of action on the state
# If action 1 is taken, then the patch is burned so the population
# abundance moves down a class.
if a == 1:
x_1 -= 1
x_2 -= 1
# Ensure that the abundance class doesn't go to -1
if x_3 > 0:
x_3 -= 1
# Demographic model probabilities
new_state = convertStateToIndex(x_1, F)
prob[new_state] = s # abundance stays the same
new_state = convertStateToIndex(x_2, F)
prob[new_state] = (1 - s) * r # abundance goes up
new_state = convertStateToIndex(x_3, F)
# In the case when x_3 = 0 before the effect of an action is applied,
# then the final state is going to be the same as that for x_1, so we
# need to add the probabilities together.
prob[new_state] += (1 - s) * (1 - r) # abundance goes down
return(prob)
def getTransitionAndRewardArrays(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``.
"""
assert 0 <= s <= 1, "'s' must be between 0 and 1"
# The transition probability array
P = np.zeros((ACTIONS, STATES, STATES))
# The reward vector
R = np.zeros(STATES)
# Loop over all states
for idx in range(STATES):
# Get the state index as inputs to our functions
x, F = convertIndexToState(idx)
# The reward for being in this state is 1 if the population is extant
if x != 0:
R[idx] = 1
# Loop over all actions
for a in range(ACTIONS):
# Assign the transition probabilities for this state, action pair
P[a][idx] = getTransitionProbabilities(x, F, s, a)
return(P, R)
def solveMDP():
"""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
-------
mdp : 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).
"""
P, R = getTransitionAndRewardArrays(0.5)
sdp = mdp.FiniteHorizon(P, R, 0.96, 50)
sdp.run()
return(sdp)
def printPolicy(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)
range_F = range(FIRE_CLASSES)
print(" " + " ".join("%2d" % f for f in range_F))
print(" " + "---" * FIRE_CLASSES)
for x in range(POPULATION_CLASSES):
print(" %2d|" % x + " ".join("%2d" % p[x, f] for f in range_F))
def simulateTransition(x, s, r, fire):
"""Simulate a state transition.
Parameters
----------
x : int
The current abundance class of the threatened species.
s : float
The state-independent probability of the population staying in its
current abundance class.
r : float
The probability the population moves up one abundance class, assuming
it is not staying in its current state. ``r`` depends on ``F``, the
time in years since the last fire.
fire : bool
True if there has been a fire in the current year, otherwise False.
Returns
-------
x : int
The new abundance class of the threatened species.
"""
assert 0 <= x < POPULATION_CLASSES, "'x' must be in " \
"{0, 1...%s}" % POPULATION_CLASSES - 1
assert 0 <= s <= 1, "'s' must be in [0; 1]"
assert 0 <= r <= 1, "'r' must be in [0; 1]"
assert fire in (True, False), "'fire' must be a boolean value"
x = int(x)
if x == 0:
pass
elif x == POPULATION_CLASSES - 1:
if random.random() <= 1 - (1 - s) * (1 - r):
pass
else: # with probability (1 - s)(1 - r)
x -= 1
else:
if random.random() <= s:
pass
else:
if 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 fire and (x > 0):
x -= 1
return(x)
def _runTests():
#Run tests on the modules functions.
assert getHabitatSuitability(0) == 0
assert getHabitatSuitability(2) == 0.4
assert getHabitatSuitability(5) == 1
assert getHabitatSuitability(8) == 0.7
assert getHabitatSuitability(10) == 0.5
assert getHabitatSuitability(15) == 0.5
assert convertIndexToState(STATES-1) == (POPULATION_CLASSES - 1,
FIRE_CLASSES - 1)
assert convertIndexToState(STATES-2) == (POPULATION_CLASSES -1,
FIRE_CLASSES - 2)
assert convertIndexToState(0) == (0, 0)
for idx in range(STATES):
s1, s2 = convertIndexToState(idx)
assert convertStateToIndex(s1, s2) == idx
print("Tests complete.")
if __name__ == "__main__":
import sys
try:
argv = sys.argv[1]
except IndexError:
argv = None
if argv == "test":
_runTests()
else:
sdp = solveMDP()
printPolicy(sdp.policy[:, 0])