from abc import ABCMeta, abstractmethod
import numpy as np
from scipy import integrate
[docs]class BaseDiscretizer(ABCMeta):
"""
Base class for the discretizer classes in pgmpy. The discretizer
classes are used to discretize a continuous random variable
distribution into discrete probability masses.
Parameters
----------
factor: A ContinuousNode or a ContinuousFactor object
the continuous node or factor representing the distribution
to be discretized.
low, high: float
the range over which the function will be discretized.
cardinality: int
the number of states required in the discretized output.
Examples
--------
>>> from scipy.stats import norm
>>> from pgmpy.factors.continuous import ContinuousNode
>>> normal = ContinuousNode(norm(0, 1).pdf)
>>> from pgmpy.discretize import BaseDiscretizer
>>> class ChildDiscretizer(BaseDiscretizer):
... def get_discrete_values(self):
... pass
>>> discretizer = ChildDiscretizer(normal, -3, 3, 10)
>>> discretizer.factor
<pgmpy.factors.continuous.ContinuousNode.ContinuousNode object at 0x04C98190>
>>> discretizer.cardinality
10
>>> discretizer.get_labels()
['x=-3.0', 'x=-2.4', 'x=-1.8', 'x=-1.2', 'x=-0.6', 'x=0.0', 'x=0.6', 'x=1.2', 'x=1.8', 'x=2.4']
"""
def __init__(self, factor, low, high, cardinality):
self.factor = factor
self.low = low
self.high = high
self.cardinality = cardinality
[docs] @abstractmethod
def get_discrete_values(self):
"""
This method implements the algorithm to discretize the given
continuous distribution.
It must be implemented by all the subclasses of BaseDiscretizer.
Returns
-------
A list of discrete values or a DiscreteFactor object.
"""
pass
[docs] def get_labels(self):
"""
Returns a list of strings representing the values about
which the discretization method calculates the probability
masses.
Default value is the points -
[low, low+step, low+2*step, ......... , high-step]
unless the method is overridden by a subclass.
Examples
--------
>>> from pgmpy.factors import ContinuousNode
>>> from pgmpy.discretize import BaseDiscretizer
>>> class ChildDiscretizer(BaseDiscretizer):
... def get_discrete_values(self):
... pass
>>> from scipy.stats import norm
>>> node = ContinuousNode(norm(0).pdf)
>>> child = ChildDiscretizer(node, -5, 5, 20)
>>> chld.get_labels()
['x=-5.0', 'x=-4.5', 'x=-4.0', 'x=-3.5', 'x=-3.0', 'x=-2.5',
'x=-2.0', 'x=-1.5', 'x=-1.0', 'x=-0.5', 'x=0.0', 'x=0.5', 'x=1.0',
'x=1.5', 'x=2.0', 'x=2.5', 'x=3.0', 'x=3.5', 'x=4.0', 'x=4.5']
"""
step = (self.high - self.low) / self.cardinality
labels = [
f"x={str(i)}" for i in np.round(np.arange(self.low, self.high, step), 3)
]
return labels
[docs]class RoundingDiscretizer(BaseDiscretizer):
"""
This class uses the rounding method for discretizing the
given continuous distribution.
For the rounding method,
The probability mass is,
cdf(x+step/2)-cdf(x), for x = low
cdf(x+step/2)-cdf(x-step/2), for low < x <= high
where, cdf is the cumulative density function of the distribution
and step = (high-low)/cardinality.
Examples
--------
>>> import numpy as np
>>> from pgmpy.factors.continuous import ContinuousNode
>>> from pgmpy.factors.continuous import RoundingDiscretizer
>>> std_normal_pdf = lambda x : np.exp(-x*x/2) / (np.sqrt(2*np.pi))
>>> std_normal = ContinuousNode(std_normal_pdf)
>>> std_normal.discretize(RoundingDiscretizer, low=-3, high=3,
... cardinality=12)
[0.001629865203424451, 0.009244709419989363, 0.027834684208773178,
0.065590616803038182, 0.120977578710013, 0.17466632194020804,
0.19741265136584729, 0.17466632194020937, 0.12097757871001302,
0.065590616803036905, 0.027834684208772664, 0.0092447094199902269]
"""
[docs] def get_discrete_values(self):
step = (self.high - self.low) / self.cardinality
# for x=[low]
discrete_values = [
self.factor.cdf(self.low + step / 2) - self.factor.cdf(self.low)
]
# for x=[low+step, low+2*step, ........., high-step]
points = np.linspace(self.low + step, self.high - step, self.cardinality - 1)
discrete_values.extend(
[
self.factor.cdf(i + step / 2) - self.factor.cdf(i - step / 2)
for i in points
]
)
return discrete_values
[docs]class UnbiasedDiscretizer(BaseDiscretizer):
"""
This class uses the unbiased method for discretizing the
given continuous distribution.
The unbiased method for discretization is the matching of the
first moment method. It involves calculating the first order
limited moment of the distribution which is done by the _lim_moment
method.
For this method,
The probability mass is,
(E(x) - E(x + step))/step + 1 - cdf(x), for x = low
(2 * E(x) - E(x - step) - E(x + step))/step, for low < x < high
(E(x) - E(x - step))/step - 1 + cdf(x), for x = high
where, E(x) is the first limiting moment of the distribution
about the point x, cdf is the cumulative density function
and step = (high-low)/cardinality.
References
----------
Klugman, S. A., Panjer, H. H. and Willmot, G. E.,
Loss Models, From Data to Decisions, Fourth Edition,
Wiley, section 9.6.5.2 (Method of local moment matching) and
exercise 9.41.
Examples
--------
>>> import numpy as np
>>> from pgmpy.factors import ContinuousNode
>>> from pgmpy.factors.continuous import UnbiasedDiscretizer
# exponential distribution with rate = 2
>>> exp_pdf = lambda x: 2*np.exp(-2*x) if x>=0 else 0
>>> exp_node = ContinuousNode(exp_pdf)
>>> exp_node.discretize(UnbiasedDiscretizer, low=0, high=5, cardinality=10)
[0.39627368905806137, 0.4049838434034298, 0.13331784003148325,
0.043887287876647259, 0.014447413395300212, 0.0047559685431339703,
0.0015656350182896128, 0.00051540201980112557, 0.00016965346326140994,
3.7867260839208328e-05]
"""
[docs] def get_discrete_values(self):
lev = self._lim_moment
step = (self.high - self.low) / (self.cardinality - 1)
# for x=[low]
discrete_values = [
(lev(self.low) - lev(self.low + step)) / step
+ 1
- self.factor.cdf(self.low)
]
# for x=[low+step, low+2*step, ........., high-step]
points = np.linspace(self.low + step, self.high - step, self.cardinality - 2)
discrete_values.extend(
[(2 * lev(i) - lev(i - step) - lev(i + step)) / step for i in points]
)
# for x=[high]
discrete_values.append(
(lev(self.high) - lev(self.high - step)) / step
- 1
+ self.factor.cdf(self.high)
)
return discrete_values
def _lim_moment(self, u, order=1):
"""
This method calculates the kth order limiting moment of
the distribution. It is given by -
E(u) = Integral (-inf to u) [ (x^k)*pdf(x) dx ] + (u^k)(1-cdf(u))
where, pdf is the probability density function and cdf is the
cumulative density function of the distribution.
Reference
---------
Klugman, S. A., Panjer, H. H. and Willmot, G. E.,
Loss Models, From Data to Decisions, Fourth Edition,
Wiley, definition 3.5 and equation 3.8.
Parameters
----------
u: float
The point at which the moment is to be calculated.
order: int
The order of the moment, default is first order.
"""
def fun(x):
return np.power(x, order) * self.factor.pdf(x)
return integrate.quad(fun, -np.inf, u)[0] + np.power(u, order) * (
1 - self.factor.cdf(u)
)
[docs] def get_labels(self):
labels = list(
f"x={str(i)}"
for i in np.round(np.linspace(self.low, self.high, self.cardinality), 3)
)
return labels
