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Zahra Rajabi
pymdptoolbox
Commits
ad59e63d
Commit
ad59e63d
authored
Mar 12, 2014
by
Steven Cordwell
Browse files
rewrite firemdp example
parent
4f9df03f
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src/examples/firemdp.py
src/examples/firemdp.py
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src/examples/firemdp.py
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ad59e63d
# * coding: utf8 *
"""
Created on Sun Mar 9 17:20:30 2014
"""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
<http://www.mssanz.org.au/MODSIM97/Vol%202/Possingham.pdf>`_.
@author: Steven A W Cordwell
"""
# Copyright (c) 2014 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.
from
mdptoolbox
import
mdp
import
random
import
numpy
as
np
NUMBER_OF_POPULATION_CLASSES
=
7
NUMBER_OF_FIRE_CLASSES
=
13
STATES
=
NUMBER_OF_POPULATION_CLASSES
*
NUMBER_OF_FIRE_CLASSES
# 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
):
...
...
@@ 33,11 +75,11 @@ def convertStateToIndex(population, fire):
the state parameters.
"""
assert
0
<=
population
<
NUMBER_OF_
POPULATION_CLASSES
,
"'population' "
\
"
must be in
(0, 1...%s)"
%
str
(
NUMBER_OF_
POPULATION_CLASSES

1
)
assert
0
<=
fire
<
NUMBER_OF_
FIRE_CLASSES
,
"'fire' must be in "
\
"(0, 1...%s) "
%
str
(
NUMBER_OF_
FIRE_CLASSES

1
)
return
(
population
*
NUMBER_OF_
FIRE_CLASSES
+
fire
)
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.
...
...
@@ 55,12 +97,12 @@ def convertIndexToState(index):
species. ``fire``, the time in years since last fire.
"""
assert
index
<
NUMBER_OF_POPULATION_CLASSES
*
NUMBER_OF_FIRE_CLASS
ES
population
=
index
//
NUMBER_OF_
FIRE_CLASSES
fire
=
index
%
NUMBER_OF_
FIRE_CLASSES
assert
index
<
STAT
ES
population
=
index
//
FIRE_CLASSES
fire
=
index
%
FIRE_CLASSES
return
(
population
,
fire
)
def
getHabitatSuitability
(
time
):
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
...
...
@@ 69,8 +111,8 @@ def getHabitatSuitability(time):
Parameters

time
: int
Time in y
ears since last fire.
years
: int
Y
ears since last fire.
Returns

...
...
@@ 78,151 +120,259 @@ def getHabitatSuitability(time):
The habitat suitability.
"""
assert
time
>=
0
,
"'
time
' must be a positive number"
if
time
<=
5
:
return
(
0.2
*
time
)
elif
5
<=
time
<=
10
:
return
(

0.1
*
time
+
1.5
)
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
getTransitionAndRewardMatrices
(
s
=
0.5
):
P
=
np
.
zeros
((
ACTIONS
,
STATES
,
STATES
))
R
=
np
.
zeros
((
STATES
,))
for
idx
in
range
(
STATES
):
x
,
F
=
convertIndexToState
(
idx
)
if
x
!=
0
:
R
[
idx
]
=
1
for
a
in
range
(
ACTIONS
):
P
[
a
][
idx
]
=
getTransitionProbabilities
(
x
,
F
,
s
,
a
)
return
(
P
,
R
)
def
getTransitionProbabilities
(
x
,
F
,
s
,
action
):
tp
=
np
.
zeros
((
STATES
,))
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
ction
==
0
:
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
<
NUMBER_OF_
FIRE_CLASSES

1
:
if
F
<
FIRE_CLASSES

1
:
F
+=
1
elif
a
ction
==
1
:
# When the patch is burned set the
time
since fire to 0.
elif
a
==
1
:
# When the patch is burned set the
years
since fire to 0.
F
=
0
# Population transitions
if
x
==
0
:
# population abundance class stays at 0 (extinct)
# Demographic model probabilities
idx
=
convertStateToIndex
(
0
,
F
)
tp
[
idx
]
=
1
elif
x
==
NUMBER_OF_POPULATION_CLASSES

1
:
# 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
transition_a
=
x
transition_b
=
x

1
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
ction
==
1
:
transition_a
=
1
transition_b
=
1
if
a
==
1
:
x_1
=
1
x_2
=
1
# Demographic model probabilities
idx
=
convertStateToIndex
(
transition_a
,
F
)
tp
[
idx
]
=
1

(
1

s
)
*
(
1

r
)
idx
=
convertStateToIndex
(
transition_b
,
F
)
tp
[
idx
]
=
(
1

s
)
*
(
1

r
)
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.
transition_a
=
x
transition_b
=
x
+
1
transition_c
=
x

1
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
action
==
1
:
transition_a
=
1
transition_b
=
1
if
transition_c
>
0
:
transition_c
=
1
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
idx
=
convertStateToIndex
(
transition_a
,
F
)
tp
[
idx
]
=
s
idx
=
convertStateToIndex
(
transition_b
,
F
)
tp
[
idx
]
=
(
1

s
)
*
r
idx
=
convertStateToIndex
(
transition_c
,
F
)
tp
[
idx
]
+=
(
1

s
)
*
(
1

r
)
return
(
tp
)
def
simulateTransition
(
state
,
s
,
r
,
fire
):
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 classindependent 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

state
: int
x
: int
The current abundance class of the threatened species.
s : float
The stateindependent probability of the population staying in its
current
state
.
current
abundance class
.
r : float
The probability the population moves up one
state, assuming it is not
staying in its current state. ``r`` depends on ``F``, the
time in years
since the last fire.
fire : bool
ean
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

state
: int
x
: int
The new abundance class of the threatened species.
"""
assert
0
<=
state
<=
6
,
"'state' must be in (0, 1...6)"
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"
state
=
int
(
state
)
if
state
==
0
:
x
=
int
(
x
)
if
x
==
0
:
pass
elif
state
==
6
:
elif
x
==
POPULATION_CLASSES

1
:
if
random
.
random
()
<=
1

(
1

s
)
*
(
1

r
):
pass
else
:
# with probability (1  s)(1  r)
state
=
5
x

=
1
else
:
if
random
.
random
()
<=
s
:
pass
else
:
else
:
if
random
.
random
()
<=
r
:
# with probability (1  s)r
state
+=
1
x
+=
1
else
:
# with probability (1  s)(1  r)
state
=
1
# Add the effect of a fire
if
fire
:
state
=
1
# Make sure state didn't transition to 1
if
state
<
0
:
state
=
0
return
(
state
)
def
solveMDP
():
P
,
R
=
getTransitionAndRewardMatrices
()
sdp
=
mdp
.
FiniteHorizon
(
P
,
R
,
0.9
,
50
)
sdp
.
run
()
return
(
sdp
)
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
tests
():
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
(
90
)
==
(
NUMBER_OF_
POPULATION_CLASSES

1
,
NUMBER_OF_
FIRE_CLASSES

1
)
assert
convertIndexToState
(
89
)
==
(
NUMBER_OF_
POPULATION_CLASSES

1
,
NUMBER_OF_
FIRE_CLASSES

2
)
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__"
:
tests
()
import
sys
try
:
argv
=
sys
.
argv
[
1
]
except
IndexError
:
argv
=
None
if
argv
==
"test"
:
_runTests
()
else
:
sdp
=
solveMDP
()
printPolicy
(
sdp
.
policy
[:,
0
])
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