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 , where the value of each member of the set is an estimated value of the fuzzy number, and the -value of the member is its grade.

Triangular Fuzzy Number

One can represent the estimated values as well as their associated -values in the form of a Cartesian coordinate system, with the Y-axis representing -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 -value is exactly 1.0), and two X-intercepts.
• Triangular sets contain exactly one kernel, and two X-intercepts
• Gaussian sets have -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

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 Visualization

Polygonal Fuzzy Number

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1. Fuzzy Mathematics with FuzzPy (Part 1)
2. Algorithms in Python: Binary exponentiation
3. Algorithms in Python: Binary Operations

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 -value of the element. A -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, -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 -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 -value. The -value of a vertice indicates the vertice’s degree of membership into the graph, while an edge or arc’s -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.

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1. Fuzzy Mathematics with FuzzPy (Part 2)
2. Algorithms in Python: Binary exponentiation
3. Algorithms in Python: Binary Operations

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 whereas the the computation of the exponentiation took about the same time in both cases.

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1. Algorithms in Python: Binary Operations
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3. Fuzzy Mathematics with FuzzPy (Part 1)

I didn’t get around to sit down and come up with a write-up about the experience as i’ve been working on other projects every evening this week, but I finally have a minute to do so now.

As far as the convention event itself went, there is definitely a lot of good and a few notable logistical oversights that are worth mentioning, but i’m not really interested in addressing either on here, as I’m really more interested in sharing my impression of the technical panels I’ve attended.

I have to say, I’m very excited about every single one of the panels I’ve been able to attend this year. They were all highly informative, and the speakers very motivated and passionate about their material. Here’s a short summary of the events I was especially excited about:

• Neural Networks
  This panel was held by Dr. Stanley C. Mortel. It explained the basic concepts behind the idea of building and training neural computation networks. It was a very abstract fly-by course, which I feel is a very appropriate way to introduce this type of material. There was a second part of this panel available the next day but we weren’t fortunate enough to attend it.
• Reading by Will Wheaton
  Just kidding
• Beginning PyGame Programming
  This tutorial by Craig Maloney was my first real introduction to PyGame. Craig had a nice little demo environment all set up and ready for the presentation. He flew pretty quickly through many of the concepts behind PyGame while writing a Pong demo. Although he went through the material pretty quickly, I’m very interested in learning more about the platform as a result, so it’s safe to say the tutorial was a success as far as I’m concerned.
• Open Hardware with Arduino
  The speaker for this talk was W Craig Trader, and I have to say Craig was not only extremely knowledgeable about the Arduino (and obviously several other) platform, but also pretty excited about it, and since microcontrolers isn’t something I’ve ever bothered to learn anything about, I really didn’t expect to get so excited about the talk. The Arduino platform appears to be very accessible technically and financially, and also pretty powerful. Craig did an amazing job showing us the strengths and weaknesses of the platform and getting our whole group pretty excited to play with it!
• High Performance PHP
  This talk held by Rasmus Lerdorf, the creator of PHP and an infrastructure architect at Yahoo! Inc., took us through several performance optimization techniques for PHP apps, although many of the concepts featured in the talk could easily be applied to any apache-based app. It was very refreshing to finally see someone as experienced and well-rounded as him go through the tribulations of identifying and addressing performance bottlenecks in PHP apps. I was very interested in both the individual techniques highlighted during the talk, as well as the problem-solving process of a highly experienced software engineer, so that talk was the highlight of the con for me.

Unfortunately, due to very unfortunate logistical shortcomings, our group was unable to attend a lot of the panels we were looking forward to check out, but despite this, the con was a resounding success in every aspect you can think of. I’ve met some really cool and interesting people, learned a lot of very exciting stuff that will be guiding some of my personal research for months to come, learned some technical concepts that will directly impact my work performance, and had way too much fun.

The next stop on our list will probably be PyOhio. We had a chance to chat with Catherine Devlin, a fellow Pythonista & Oracle geek from IntelliTech Systems, who told us about it, and Aaron and I are looking into putting together a talk proposal for the con, if we can come up with it before the deadline, and we already have some pretty interesting ideas, so I’m looking forward to it. I’d also really love to attend PyCon 2010 in Atlanta!

If you’re interested in attending a convention where Linux and FOSS enthusiasts get a chance to have fun with the Sci-Fi crowd for a weekend of fun, PenguiCon is definitely for you!!

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