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Fuzzy Mathematics with FuzzPy (Part 2)

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In the first part of this post, we explored the foundations of fuzzy sets and fuzzy graphs, and discussed how you can use FuzzPy, a general fuzzy mathematics library for python, to work with these types. Today, we will expand a bit on those foundations by learning about fuzzy numbers, and creating visualizations with FuzzPy.

Fuzzy Numbers

Think of a fuzzy number as a number whose actual value is uncertain. We do not have its value, but we have a set of estimations, each with an associated degree of likelihood, or grade. If you remember the definition of a fuzzy set, you will remember that the structure of a fuzzy set is very similar to what I just described, so it only makes sense to use a fuzzy set to represent what is known about a fuzzy number.

Effectively, a fuzzy number is a fuzzy subset of the real line \mathbb{R}, where the value of each member of the set is an estimated value of the fuzzy number, and the \mu-value of the member is its grade.

triangular_fuzzy_number

Triangular Fuzzy Number

One can represent the estimated values as well as their associated \mu-values in the form of a Cartesian coordinate system, with the Y-axis representing \mu-values, and the X-axis representing estimated values.
We can then see that certain patterns emerge in the shape of the visualization, and these patterns are used to distinguish between the different types of fuzzy numbers:

  • Polygonal sets are represented by an arbitrary set of segments on the plane.
  • Trapezoidal sets contain two kernels (estimations whose \mu-value is exactly 1.0), and two X-intercepts.
  • Triangular sets contain exactly one kernel, and two X-intercepts
  • Gaussian sets have \mu-values determined by a normal distribution, thus providing a smoother distribution.

There are a few additional types as well, but these are the four basic types, and are all supported out of the box by FuzzPy.

Because the points of interest in each type are different, instanciating a new FuzzyNumber in FuzzPy is done differently depending on the type. Let’s instantiate objects for each of these types using FuzzPy, and we’ll play a bit with these objects in a moment. Make sure you have FuzzPy installed first (easy_install -U fuzzpy) and save the following code in fuzzynums.py:

import fuzz
 
# Instantiating a PolynomialFuzzyNumber is done by providing
# a list of (value, mu) tuple to the constructor:
polygonal = [fuzz.PolygonalFuzzyNumber([(0.0, 0.0), \
    (0.25, 1.0), (1.0, 0.5), (1.3, 1.0), (1.8, 0.0)])]
 
polygonal += [fuzz.PolygonalFuzzyNumber([(0.0, 0.0), \
    (1.0, 0.5), (3.0, 1.0), (4.0, 1.0), (6.0, 0.0)])]
 
# A trapezoidal needs to be given a tuple containing the values
# of the kernels (mu = 1.0) and a tuple containing the values
# of the support (mu = 0.0)
trapezoidal = [fuzz.TrapezoidalFuzzyNumber((1.0, 2.0), \
    (0.3, 3.0))]
 
trapezoidal += [fuzz.TrapezoidalFuzzyNumber((1.0, 6.0), \
    (0.5, 10.0))]
 
# A triangular set needs to be provided the value of the
# kernel, and a tuple containing the values of the support.
triangular = [fuzz.TriangularFuzzyNumber(1.0, (0.0, 3.0))]
triangular += [fuzz.TriangularFuzzyNumber(4.0, (0.0, 4.5))]
 
# Gaussian fuzzy sets simply need the mean, and standard
# derivation for the estimated value.
gaussian = [fuzz.GaussianFuzzyNumber(1.0, 0.2)]
gaussian += [fuzz.GaussianFuzzyNumber(12.0, 1.0)]

We can now fire up the python interpreter and work with our objects:

>>> # Polygonal Union:
... print polygonal[0] | polygonal[1]
PolygonalFuzzyNumber: kernel [(0.25, 0.25), (1.3, 1.3), \
(3.0, 4.0)], support [(0.0, 6.0)]
 
>>> # Polygonal Intersection:
... print polygonal[0] & polygonal[1]
PolygonalFuzzyNumber: kernel [], support [(0.0, 6.0)]
 
>>> # Trapezoidal Addition:
... print trapezoidal[0] + trapezoidal[1]
TrapezoidalFuzzyNumber: kernel (2.0, 8.0), support (0.8000000\
0000000004, 13.0)
 
>>> # Gaussian Addition
... print gaussian[0] + gaussian[1]
GaussianFuzzyNumber: kernel (13.0, 13.0), support (4.85663149\
07018668, 21.143368509298135)

The class inheritance chain for FuzzPy’s FuzzyNumber derived classes looks like this:

fuzzy_number_class_inheritance.png

Fuzzy Number Class Inheritance

Visualizations

FuzzPy ships with a small visualization framework that is able to create visualizations for almost all FuzzPy types, and encode them in a variety of formats. In most cases, creating and storing a visualization is a trivial operation:

from fuzzynums import *
from fuzz.visualization import VisManager
 
gaussian = fuzz.GaussianFuzzyNumber(1.0, 0.2)
plugin = VisManager.create_backend(gaussian)
(vis_format, vis_data) = plugin.visualize()
 
with (open("visualization.%s" % vis_format, "wb")) as fp:
    fp.write(vis_data)

In this example, we passed our FuzzyNumber instance to the VisManager.create_backend() method, which is a plugin factory method, and it returned the first plugin it could find that can be used to create a visualization of our FuzzyNumber.

The plugin’s visualize() method was called, and returned a tuple containing the file format of the visualization, and its payload. We then created a file using the returned format as extension, and stored the payload in that file.

You can also force the VisManager object to use a certain plugin if you do not want to rely on the default plugin picked for the type of object to visualize:

plugin = VisManager.create_backend(gaussian, 'num_gnuplot')

Similarly, a plugin’s visualize() method can accept arguments to customize the final output. For example, the following code would tell the plugin to use the ‘gif’ image format:

(vis_format, vis_data) = plugin.visualize(format="gif")

You will need to have Graphviz installed for graph visualizations, and Gnuplot for fuzzy number visualizations.

Now that you know how to use all the FuzzPy types and how to create visualizations for your data, you can start using FuzzPy in your own project, or get involved in the development process. Check out the project’s GitHub page for more information.

fuzzy_digraph

Fuzzy Digraph Visualization

polygonal_fuzzy_number

Polygonal Fuzzy Number

Fuzzy Mathematics with FuzzPy (Part 1)

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FuzzPy is a python library that exposes specialized datatypes to deal with a wide range of fuzzy number types, fuzzy sets, and fuzzy graphs. Binary operations against each type, such as set unions and intersections, can be performed using some of python’s native binary operators (|, &, ==), or specialized methods if you wish to deviate from the default functions for these computations.

FuzzPy also provides several helper methods such as kernel and neighbors isolation, alpha-cuts, cardinality testing, shortest-path computation, and minimum spanning trees. A powerful visualization framework also allows you to quickly create and save visualizations for your data.

In this post, we will examine fuzzy sets and fuzzy graphs, and see how one can use FuzzPy to work with these types, providing examples for each. In the next post of this series, we will examine the different types of fuzzy numbers, generate visualizations, and explore advanced concepts such as automatic fuzzification, and importing graph data from fuzzy adjacency matrices.

Fuzzy Sets

A fuzzy set is simply a set in which each element has an associated degree of membership into the set. This membership value is called \mu-value of the element. A \mu-value of zero indicates that the element does not belong to the set, while any value between zero (exclusive) and one (inclusive), indicates that the element is, to a certain degree, a set member.

Creating a fuzzy set with FuzzPy is just a matter of creating a new FuzzySet object, then either calling the add(FuzzyElement(value, mu)) method for each element to import, or using the update(elements) method, providing it with a list of FuzzyElement(value, \mu-value) tuples for bulk import.

from fuzz import FuzzySet, FuzzyElement
 
# Create two lists of FuzzyElements
elements_a = [(1, 0.3), (2, 0.5), (3, 0.7), (4, 0.9)]
elements_b = [(3, 0.8), (6, 0.6), (1, 0.4), (8, 0.2)]
 
elements_a = [FuzzyElement(x[0], x[1]) for x in elements_a]
elements_b = [FuzzyElement(x[0], x[1]) for x in elements_b]
 
# Create two fuzzy sets
set_a = FuzzySet()
set_b = FuzzySet()
 
set_a.update(elements_a)
set_b.update(elements_b)

We’ve just created a couple fuzzy sets with four elements each. Notice that two of the elements are common to both set_a and set_b (albeit with different \mu-values). We can now perform several operations against these sample sets. Save the code above in a file called example.py and fire up the python interpreter to test fuzzy set functionality against our sets:

>>> from example import *
>>> # Set Difference
...
>>> set_a - set_b
FuzzySet([FuzzyElement(2, 0.500000), FuzzyElement(4, 0.900000)])
>>> set_b - set_a
FuzzySet([FuzzyElement(8, 0.200000), FuzzyElement(6, 0.600000)])
# Set Union
...
>>> set_a | set_b
FuzzySet([FuzzyElement(1, 0.400000), FuzzyElement(2, 0.500000), FuzzyElement(3, 0.800000), FuzzyElement(4, 0.900000), FuzzyElement(6, 0.600000), FuzzyElement(8, 0.200000)])
# Set Intersection
...
>>> set_a & set_b
FuzzySet([FuzzyElement(1, 0.300000), FuzzyElement(3, 0.700000)])

Fuzzy graphs

In a fuzzy graph or digraph, each vertice and each edge can be associated with a \mu-value. The \mu-value of a vertice indicates the vertice’s degree of membership into the graph, while an edge or arc’s \mu-value indicates the degree of connectivity between the head and tail vertices.

The creation of fuzzy graphs is also pretty straightforward: create a set or fuzzy set (see above), then feed that set to the fuzz.Graph constructor, along with the directed boolean value to specify whether you’d like to create a graph or a digraph. You can then use the fuzz.Graph.connect(head, tail, mu) method to add edges to the graph:

import fuzz
 
# Create two sets of vertices
set_a = set([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
set_b = set([11, 12, 13, 14, 15, 16, 17, 18, 19, 20])
 
# Create our graph objects
graph_a = fuzz.FuzzyGraph(viter=set_a, directed=False)
graph_b = fuzz.FuzzyGraph(viter=set_b, directed=True)
 
# Create some arcs in graph_a
graph_a.connect(1, 3, 0.5)
graph_a.connect(3, 6, 0.06)
graph_a.connect(2, 8, 0.2)
graph_a.connect(4, 1, 0.9)
 
# And some in graph_b
graph_b.connect(11, 19, 0.9)
graph_b.connect(14, 12, 0.3)
graph_b.connect(15, 14, 0.5)

Save the code above in a file named graphs.py and fire up your python interpreter so we can experiment with graph operations:

>>> from graphs import *
>>> # Retrieve the mu value of the edge between 1 and 3
... 
>>> graph_a.mu(3, 1)
0.5
>>> # Retrieve an alpha-cut of graph_b against mu-value of 0.5
... 
>>> graph_b.alpha(0.5)
Graph(viter = set([11, 12, 13, 14, 15, 16, 17, 18, 19, 20]), eiter = set([(15, 14), (11, 19)]), directed = True)
>>> # Is graph_a a subgraph of graph_b? (no)
... graph_a.issubgraph(graph_b)
False
>>> # Detect adjacent (connected) vertices
...
>>> graph_a.adjacent(1, 2)
False
>>> graph_a.adjacent(1, 3)
True
>>> # Find all the neighbours of the vertex with value 3
... 
>>> graph_a.neighbors(3)
set([1, 6])
>>> # Find the shortest path to all vertices using 6 as origin
... 
>>> graph_a.dijkstra(6)
{1: 3, 2: None, 3: 6, 4: 1, 5: None, 6: None, 7: None, 8: None, 9: None, 10: None}
>>> # Find the minimum spanning tree of graph_a
... 
>>> graph_a.minimum_spanning_tree()
Graph(viter = set([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]), eiter = set([(2, 8), (1, 3), (4, 1), (3, 6)]), directed = False)

There are a few more operations available for both crisp and fuzzy graphs/digraphs. You should consult the API doc for a comprehensive list.

You should also consult the examples provided as part of the module on GitHub to see FuzzPy in action.

Algorithms in Python: Binary exponentiation

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The typical approach to exponentiation of a base b by an exponent n is to repeatedly multiply the base by itself, as such: b^n = \prod_{i=1}^{n} b

This is easy to compute for small values but quickly chokes with very large values.

A faster approach involves converting the exponent into base 2, then multiplying the running total and squaring the base each time a 1-bit is encountered. The python implementation looks like this:

def binary_exponent(base, exponent):
    """\
    Binary Exponentiation
 
    Instead of computing the exponentiation in the traditional way,
    convert the exponent to its reverse binary representation.
 
    Each time a 1-bit is encountered, we multiply the running total by
    the base, then square the base.
    """
    # Convert n to base 2, then reverse it.
    exponent = bin(exponent)[2:][::-1]
 
    result = 1
    for i in range(0, len(exponent)):
        if exponent[i] == '1':
            result *= base
        base *= base
    return result

Similarly, the same approach can be taken to perform modular exponentiation against very large exponents:

def modular_exponent(base, exponent, mod):
    """\
    Modular exponentiation through binary decomposition.
 
    We use the same technique as for the binary exponentiation above in
    order to find the modulo of our very large exponent and an arbitrary
    integer mod.
    """
    exponent = bin(exponent)[2:][::-1]
 
    x = 1
    power = base % mod
    for i in range(0, len(exponent)):
        if exponent[i] == '1':
            x = (x * power) % mod
        power = (power ** 2) % mod
    return x

The optimization is especially visible for the modulo operation, since we are working on smaller numbers with each iteration, whereas for simple exponentiation, the CPython interpreter can already optimize the operation.
The speedup for modular exponentiation on my system was in the order of 72x the speed of a simple (a^n) \mod{m} whereas the the computation of the exponentiation took about the same time in both cases.

Algorithms in Python: Binary Operations

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Today I’d like to demonstrate a simple implementation of Kenneth Rosen‘s binary addition and multiplication algorithms as outlined in "Discrete Mathematics and its Applications". Both algorithms are very simple and work the same way we’ve all learned to do decimal additions and multiplications by hand in grade-school.

As usual, I’ve also provided a unit-test suite for each algorithm.

Please note: This is only intended to demonstrate the algorithms. If you are actually trying to do some real work with binary numbers in Python, note that the language itself provides a highly optimized implementation of binary numbers and related operations. Here are a few examples:

# Decimal to binary conversion:
>>> bin(15)
'0b1111'
 
# Binary to decimal conversion:
>>> str(0b1111)
'15'
 
# Binary addition:
>>> bin(0b1111 + 0b10011)
'0b100010'
 
# Binary multiplication:
>>> bin(0b1111 * 0b10011)
'0b100011101'

Binary Addition

We will use strings in this example to represent sequences of bits. The algorithm operates right-to-left, maintaining a carry each time it adds two 1-valued bits.

Code:

"""Simple binary operation algorithm."""
 
import math
 
# a and b should be string representation of the binary components
# You could easily extend this to overload the + operator for binaries,
# but this is already built into python.
def binary_addition(bin_a, bin_b):
    """Performs a binary addition against two provided binary numbers"""
 
    # Pad the components so they are of equal length
    max_length = max(len(bin_a), len(bin_b))
    bin_a, bin_b = bin_a.zfill(max_length), bin_b.zfill(max_length)
 
 
    carry = 0
    result = ''
 
    # Note that we work from right to left
    for i in range(0, max_length)[::-1]:
        tmp = int(math.floor((int(bin_a[i]) + int(bin_b[i]) + carry)/2))
        res = str(int(bin_a[i]) + int(bin_b[i]) + carry - 2*tmp)
        result += res
        carry = tmp
 
    result = (result + str(carry))[::-1]
    try:
        return result[result.index('1'):]
    except ValueError, ex:
        return '0'

Unit Test:

"""Unit test for binary_addition.py"""
 
import unittest
from binary_addition import binary_addition
 
class TestBinaryAddition(unittest.TestCase):
 
    """Comparing python binary addition against our own"""
 
    def test_add_same_length(self):
 
        """Adding 2 binary numbers of the same length"""
 
        bin_a = '001001101'
        bin_b = '011011010'
 
        self._compare_additions(bin_a, bin_b)
 
    def test_add_b_larger(self):
 
        """ B has more digits than A """
 
        bin_a = '01000101'
        bin_b = '1110010110101011101101011101000101'
 
        self._compare_additions(bin_a, bin_b)
 
    def test_add_b_smaller(self):
 
        """B has less digits than A"""
 
 
        bin_a = '1110001001001001001101101011000101'
        bin_b = '0110010'
 
        self._compare_additions(bin_a, bin_b)
 
    def _compare_additions(self, bin_a, bin_b):
 
        """Compares the python implementation against ours"""
 
        bin_add = binary_addition(bin_a, bin_b)
        py_add = bin(eval("0b%s" % bin_a) + eval("0b%s" % bin_b))
 
        algo_res = bin_add[bin_add.index('1'):]
        py_res = py_add[py_add.index('1'):]
 
        self.assertEqual(py_res, algo_res)
 
 
 
if '__main__' == __name__:
    unittest.main()

Binary Multiplication

We are still using strings to represent bit sequences. We are still working from right to left, shifting to the left every number from the first number that needs to be multiplied by 1 in the second number. We then append the result to a list that we sum at the end, using the previous algorithm.

Code:

"""Simple Binary multiplication algorithm"""
 
 
import sys
import os.path
 
from binary_addition import binary_addition
 
 
def binary_multiplication(bin_a, bin_b):
    """Multiplies two binary numbers by using python lists to
    easily shift digits around"""
 
    temp_result = []
    result = "0"
 
    # Remove any unnecessary padding
    bin_a = bin_a[bin_a.index('1'):]
    bin_b = bin_b[bin_b.index('1'):]
 
    for i in range(0, len(bin_b))[::-1]:
        if bin_b[i] == '1':
            temp_result.append("%s%s" % (bin_a, "0" * (len(bin_b)-i-1)))
        else:
            temp_result.append("0")
 
 
    for val in temp_result:
        result = binary_addition(str(result), str(val))
 
    return result

Unit Test:

"""Unit test for binary_addition.py"""
 
import unittest
from binary_multiplication import binary_multiplication
 
class TestBinaryAddition(unittest.TestCase):
 
    """Comparing python binary multiplication against our own"""
 
    def test_add_same_length(self):
 
        """Multiplying 2 binary numbers of the same length"""
 
        bin_a = '001001101'
        bin_b = '011011010'
 
        self._compare_multiplications(bin_a, bin_b)
 
    def test_simple(self):
 
        """Multiplying 2 simple binary numbers"""
        bin_a = '110'
        bin_b = '101'
 
        self._compare_multiplications(bin_a, bin_b)
 
    def test_add_b_larger(self):
 
        """B has more digits than A"""
 
        bin_a = '010'
        bin_b = '111001011000101'
 
        self._compare_multiplications(bin_a, bin_b)
 
    def test_add_b_smaller(self):
 
        """ B has less digits than A """
 
        bin_a = '111001011000101'
        bin_b = '010'
 
        self._compare_multiplications(bin_a, bin_b)
 
    def _compare_multiplications(self, bin_a, bin_b):
 
        """Compares the python implementation against ours"""
 
        bin_mult = binary_multiplication(bin_a, bin_b)
        py_mult = bin(eval("0b%s" % bin_a) * eval("0b%s" % bin_b))
 
        algo_res = bin_mult[bin_mult.index('1'):]
        py_res = py_mult[py_mult.index('1'):]
 
        self.assertEqual(py_res, algo_res)
 
 
 
if '__main__' == __name__:
    unittest.main()

Algorithms in Python: Base Expansion

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I felt it would be helpful to folks interested in Python and studying algorithms, to review some commonly studied algorithms in Computer Science by providing a small description and a Python implementation of each algorithm.

This week, we’ll cover an introductory algorithm for converting from one numeral base to another. Here is the python code:

"""Simple implementation of a base expansion algorithm"""
 
import math
 
def base_expand(base, val):
    """This simple function performs a base-expansion from decimal
    using moduli and a translation table. The translation table is
    a clear limitation here, in that it implies the maximum base
    is 36."""
 
    if (base < 2) or (base > 36):
        raise BaseOutOfBoundsError(base)
 
    trans_table = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ'
    res = ''
 
    while val != 0:
        res += trans_table[(int(val % base))]
        val = math.floor(val / base)
    return res[::-1]
 
class BaseOutOfBoundsError(Exception):
    """Base must be between 2 and 36"""
    def __init__(self, val):
        self.val = val
    def __str__(self):
        print "\nInvalid base: %s. Base must be (x | x > 1; x < 37)" % \
            self.val

Here is a unit-test for base_expand.py:

"""Unit tests for base_expansion.py"""
 
import unittest
from base_expansion import base_expand, BaseOutOfBoundsError
 
# Unit tests for base_expansion.py
class TestVals(unittest.TestCase):
    """Test suite"""
 
    def test_known_values(self):
        """Testing against known values"""
 
        vals = [{'val': 8, 'base': 2, 'expect': '1000'},
            {'val': 915652, 'base': 16, 'expect': 'DF8C4'},
            {'val': 256, 'base': 10, 'expect': '256'},
            {'val': 88189, 'base': 8, 'expect': '254175'}]
 
        for val in vals:
            self.assertEqual(val['expect'], \
                base_expand(val['base'], val['val']))
 
    def test_invalid_base(self):
        """ Testing invalid bases """
 
        bases = [1, 37, 50, 100]
        for base in bases:
            self.assertRaises(BaseOutOfBoundsError, base_expand, \
                base, 1)
 
 
 
if '__main__' == __name__:
    unittest.main()