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nrc.fuzzy
Class FuzzySet

java.lang.Object
  |
  +--nrc.fuzzy.FuzzySet
All Implemented Interfaces:
java.lang.Cloneable, java.io.Serializable
Direct Known Subclasses:
LFuzzySet, LRFuzzySet, RFuzzySet
public class FuzzySet
extends java.lang.Object
implements java.lang.Cloneable, java.io.Serializable

A fuzzy set is a mapping of a set of real numbers onto a membership value in the range [0, 1]. In this fuzzy package a fuzzy set is represented by a set of pairs u/x, where u is the membership value for the real number x. We can represent the set of values as { u1/x1 u2/x2 ... un/xn }. The x values in the set are in increasing order (x1 < x2 < ... < xn). Values prior to x1 have the same membership value as x1 and values after xn have the same membership value as xn. Values between xi and x(i+1) are determined by the value that lies on the straight line between the 2 consecutive points. In effect we are representing a graph with straight lines joining the points in the fuzzy set with horizontal lines connecting to the first and last points. Consider the fuzzy set { 0.0/0.3 1.0/0.5 0.0/0.7 }. As shown in the diagram below, this is a triangular shaped fuzzy set. This is a very compact representation of a fuzzy set. A fuzzy set with a single point, for example { 0.5/25 }, is represents a single horizontal line (in the example a fuzzy set with membership value of 0.5 for all x values). Note that this is not a single point! A few examples are shown below.

A Visual Guide To FuzzySets
FuzzySet Visual Representation
Simple triangular FuzzySet:
  • this FuzzySet is represented by the 3 points 
    { 0.0/0.3 1.0/0.5 0.0/0.7 }
    It is a convex FuzzySet.
Slightly more complex FuzzySET:
  • this FuzzySet is represented by the 5 points 
    { 0.0/0.1 1.0/0.3 0.65/0.4 1.0/0.5 0.0/0.8 }
    It is a non-convex FuzzySet.


To simplify the creation of FuzzySets a hierarchy of subclasses have been defined. The FuzzySets that can be built by this set of subclasses represents a fairly complete set of common shapes. The user is able to generate further subclasses to meet specific needs. The hierarchy is shown below.

NOTE: Although the range of allowed membership values is [0, 1], it is possible to form FuzzySets with membership values > 1.0. There are instances where this is desirable. For example in some approaches it is preferred to collect the outputs of fuzzy rules by doing a 'fuzzySum' of the resultant FuzzyValues (global contribution of rules). The fuzzySum operation can result in FuzzySets with membership values > 1.0. When a FuzzySet is created with a FuzzySet constructor (that uses an array of Doubles or SetPoints) the values are restricted to being between 0 and 1. However, when the fuzzySum operation or the methods appendSetPoint and insertSetPoint are used to add points to the FuzzySet, membership values > 1.0 are allowed.

See Also:
LFuzzySet, RFuzzySet, LRFuzzySet, TriangleFuzzySet, TrapezoidFuzzySet, SingletonFuzzySet, RectangleFuzzySet, PIFuzzySet, SFuzzySet, LeftLinearFuzzySet, ZFuzzySet, RightLinearFuzzySet, FuzzyVariable, FuzzyValue, Serialized Form

Nested Class Summary
protected  class FuzzySet.UITools
          The UITools class was created to support the Fuzzy Union, Intersection and Sum methods.
 
Field Summary
protected static java.lang.Boolean BFALSE
          BFALSE and BTRUE are static Boolean constants used to avoid generating instances of Booleans over and over.
protected static java.lang.Boolean BTRUE
           
protected static boolean COLLINEAR_OR_PARALLEL
          A constant used to flag a colinear or parallel result from the lineSegmentIntersection method in class UITools.
protected static int INCREMENT
          A constant specifying the amount by which the FuzzySet will be expanded each time an additional point is added and more space is required to accomodate that point.
protected static int INTERSECTION
          A constant used to flag the Intersection of FuzzySets operation.
protected static int MAX_INITIAL_POINTS
          A constant specifying the maximum number of initial points in the FuzzySet.
protected static int MAXMIN
          A constant used to flag the maximumOfIntersection operation.
protected static boolean NO_INTERSECTION
          A constant used to flag a no intersection result from the lineSegmentIntersection method in class UITools.
protected  int numPoints
          The numPoints variable keeps track of the number of points contained in set, the array of SetPoints.
protected  nrc.fuzzy.SetPoint[] set
          This is the heart of the FuzzySet, an array of SetPoints which represent the FuzzySet.
protected  boolean simplified
          The simplified variable is a boolean used to indicate whether or not the FuzzySet has been 'simplied' (revove extra points in the set -- see the simplifiySet method)
protected static int SUM
          A constant used to flag the Sum of FuzzySets operation.
protected  FuzzySet.UITools tool
          A Union/Intersection tool object which provides methods required only for Union/Intersection method functionality, and also contains variables to keep track of the state of the Union/Intersection method.
protected static int toStringPrecision
          A static variable which determines how many decimal places will be printed by the toString method of FuzzySet.
protected static int UNION
          A constant used to flag the Union of FuzzySets operation.
 
Constructor Summary
FuzzySet()
          Constructs an empty FuzzySet with a default maximum number of elements allocated for the set array.
FuzzySet(double[] x, double[] y, int numPoints)
          Constructs a FuzzySet containing the set of points represented by the array of x values and the array of y values passed.
FuzzySet(nrc.fuzzy.FuzzySet newSet)
          Constructs a new FuzzySet which is essentially a copy of the FuzzySet passed as a parameter.
FuzzySet(int initMaxSize)
          Constructs an empty FuzzySet with a specified number of elements allocated for the set array.
FuzzySet(nrc.fuzzy.SetPoint[] setPoints, int numPoints)
          Constructs a FuzzySet containing the set of points represented by the array of SetPoints passed.
 
Method Summary
 void appendSetPoint(double X, double Y)
          Appends the SetPoint parameter to the end of the SetPoint array which constitutes the FuzzySet.
 void appendSetPoint(nrc.fuzzy.SetPoint a)
          Appends the SetPoint parameter to the end of the SetPoint array which constitutes the FuzzySet.
 int checkXValueOrder()
          Checks to assure that all the x values of the FuzzySet are in ascending order.
 int checkYValueRange()
          Checks to assure that all the y values of the FuzzySet are within the strict range of 0.0 to 1.0 inclusive, or [0.0, 1.0].
protected  nrc.fuzzy.FuzzySet concat(nrc.fuzzy.FuzzySet otherSet)
          Concatenates the FuzzySet passed as a parameter to this instance of FuzzySet.
protected  nrc.fuzzy.FuzzySet concat(nrc.fuzzy.FuzzySet a, nrc.fuzzy.FuzzySet b)
          Concatenates the two FuzzySets passed as a parameters.
 void confineToXBounds(double lowXBound, double highXBound)
          Confines the FuzzySet to the x boundaries specified, usually the Universe of Discourse.
 nrc.fuzzy.FuzzySet copyFuzzySet()
          Returns a copy of this instance of FuzzySet.
 boolean equals(nrc.fuzzy.FuzzySet otherSet)
          Compare this FuzzySet with another for equality.
protected  double findMaxYOverlapValue(nrc.fuzzy.SetPoint a, nrc.fuzzy.SetPoint b, nrc.fuzzy.SetPoint c, nrc.fuzzy.SetPoint d)
          Returns the maximum y value of the overlapping portion of two vertical line segments that run in opposite directions.
 nrc.fuzzy.FuzzySet fuzzyComplement()
          Takes the compliment of this FuzzySet.
protected static boolean fuzzyEqual(double value1, double value2)
          Tests two double values for equality with a fine tolerance.
 nrc.fuzzy.FuzzySet fuzzyIntersection(nrc.fuzzy.FuzzySet otherSet)
          Returns the intersection of this FuzzySet with the FuzzySet argument.
static nrc.fuzzy.FuzzySet fuzzyIntersection(nrc.fuzzy.FuzzySet setA, nrc.fuzzy.FuzzySet setB)
          Returns the intersection set of the two FuzzySet arguments.
protected  java.lang.Object fuzzyIntersectionUnion(nrc.fuzzy.FuzzySet otherSet, int op)
          Returns the intersection set, the union set, the sum set or the maximum of intersection of this FuzzySet with the FuzzySet argument, depending on the operation specified by the op argument.
 nrc.fuzzy.FuzzySet fuzzyNormalize()
          Normalize this FuzzySet.
 nrc.fuzzy.FuzzySet fuzzyScale(double yvalue)
          Scale this FuzzySet.
 nrc.fuzzy.FuzzySet fuzzySum(nrc.fuzzy.FuzzySet otherSet)
          Returns the sum of this FuzzySet with the FuzzySet argument.
static nrc.fuzzy.FuzzySet fuzzySum(nrc.fuzzy.FuzzySet setA, nrc.fuzzy.FuzzySet setB)
          Returns the sum of the two FuzzySet arguments.
 nrc.fuzzy.FuzzySet fuzzyUnion(nrc.fuzzy.FuzzySet otherSet)
          Returns the union of this FuzzySet with the FuzzySet argument.
static nrc.fuzzy.FuzzySet fuzzyUnion(nrc.fuzzy.FuzzySet setA, nrc.fuzzy.FuzzySet setB)
          Returns the union set of the two FuzzySet arguments.
 nrc.fuzzy.IntervalVector getAlphaCut(boolean cutType, double cut, double minUOD, double maxUOD)
          Returns the alpha cut of this FuzzySet at the membership level specified by the cut argument.
 double getMaxY()
          Returns the maximum membership value of the FuzzySet.
 double getMembership(double X)
          Returns the membership value of the FuzzySet at the specified x value.
 double getMinY()
          Returns the minimum membership value of the FuzzySet.
 nrc.fuzzy.SetPoint getPoint(int i)
          Returns the SetPoint at the specified index in the FuzzySet.
protected  nrc.fuzzy.IntervalVector getSupport(double minUOD, double maxUOD)
          Returns the support of this FuzzySet.
protected  FuzzySet.UITools getUITools()
          Get the UITools object associated with the FuzzySet.
 double getX(int i)
          Returns the x value of the SetPoint at the specified index in the FuzzySet.
 double getXforMembership(double m)
          Returns the 1st X value with the specified membership value in the FuzzySet This is done by interpolation.
 double getY(int i)
          Returns the y value of the SetPoint at the specified index in the FuzzySet.
protected  boolean goingDown(double y1, double y2)
          Tests whether the y value parameters are in descending order.
protected  boolean goingUp(double y1, double y2)
          Tests whether the y value parameters are in ascending order.
 nrc.fuzzy.FuzzySet horizontalIntersection(double y)
          Returns the horizontal intersection of the FuzzySet at the membership level specified by the y parameter.
protected  nrc.fuzzy.FuzzySet horizontalIntersectionUnion(double y, int op)
          Returns the horizontal intersection or union of the FuzzySet at the membership level specified by the y parameter.
 nrc.fuzzy.FuzzySet horizontalUnion(double y)
          Returns the horizontal union of the FuzzySet at the membership level specified by the y parameter.
protected  void incrementArrayLength()
          Add space to the set array holding the SetPoints
protected  void incrementArrayLength(int newSize)
          Add space to the set array holding the SetPoints
 void insertSetPoint(double X, double Y)
          Inserts the double parameters X and Y into the SetPoint array which constitutes the FuzzySet.
 void insertSetPoint(nrc.fuzzy.SetPoint a)
          Inserts the SetPoint parameter into the SetPoint array which constitutes the FuzzySet.
 boolean isConvex()
          Returns true if this FuzzySet is convex.
 boolean isEmpty()
          Returns true if this FuzzySet is empty; in other words, if this FuzzySet does not contain any points.
 boolean isNormal()
          Returns true if this FuzzySet is normal.
 boolean isSimplified()
          Returns true if this FuzzySet has been simplified; in other words, if this FuzzySet does not contain any unnecessary points (see simplifySet method).
 double maximumDefuzzify(double xMin, double xMax)
          Finds the mean of maxima of a fuzzy set as the defuzzification value.
 double maximumOfIntersection(nrc.fuzzy.FuzzySet otherSet)
          Returns the maximum membership value on the intersection of this FuzzySet with the FuzzySet argument.
static double maximumOfIntersection(nrc.fuzzy.FuzzySet setA, nrc.fuzzy.FuzzySet setB)
          Returns the maximum membership value on the intersection of the two FuzzySet arguments.
 double momentDefuzzify(double xMin, double xMax)
          Moment defuzzification defuzzifies a fuzzy set returning a floating point (double value) that represents the fuzzy set.
 boolean noIntersectionTest(nrc.fuzzy.FuzzySet b)
          Returns a tentative indication of whether or not the FuzzySets intersect.
 boolean nonIntersectionTest(nrc.fuzzy.FuzzySet otherSet)
          Tests whether or not two FuzzySets intersect.
 java.lang.String plotFuzzySet(java.lang.String plotChar, double lowX, double highX)
          Plots the fuzzy set in an ascii format producing a String that can be displayed The string must be displayed using a non-proportional spaced font.
static java.lang.String plotFuzzySets(java.lang.String plotChars, double lowX, double highX, nrc.fuzzy.FuzzySet[] fSets)
          Plots the fuzzy sets in an ascii format producing a String that can be displayed The string must be displayed using a non-proportional spaced font such as courier.
 boolean removeSetPoint(double X, double Y)
          Find the SetPoint with x,y values in the FuzzySet and remove it if found.
 boolean removeSetPoint(nrc.fuzzy.SetPoint a)
          Find the SetPoint in the FuzzySet and remove it if found.
protected  boolean sameDirection(double y1, double y2, double y3)
          Tests whether the y values are in strictly ascending or descending order in the order they are passed as parameters.
protected  boolean sameSign(double number1, double number2)
          Tests whether the two double value arguments have the same sign.
static void setToStringPrecision(int numDecimalPlaces)
          Sets the precision, in terms of the number of decimal places that will be printed for each double value, of the toString method.
protected  void shiftArrayLeft(int index)
          shifts all the SetPoint array elements which have an index value greater than or equal to the parameter index left one position.
protected  void shiftArrayRight(int index)
          shifts all the SetPoint array elements which have an index value greater than or equal to the parameter index right one position.
 void simplifySet()
          Simplifies the array of points by removing any extraneous points from the FuzzySet.
 int size()
          Returns the size, or the number of points, in the FuzzySet.
protected  void testArrayLength()
          Test to see if set array of SetPoints is full and if so increment space in array.
 java.lang.String toString()
          Returns a string representation of the FuzzySet.
protected  void trimToSize()
          Remove excess space that might be allocated in the array holding the fuzzy set points.
 double weightedAverageDefuzzify(double xMin, double xMax)
          Finds the weighted average of the values of a fuzzy set as the defuzzification value.
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait
 

Field Detail

UNION

protected static final int UNION
A constant used to flag the Union of FuzzySets operation.

See Also:
Constant Field Values

INTERSECTION

protected static final int INTERSECTION
A constant used to flag the Intersection of FuzzySets operation.

See Also:
Constant Field Values

SUM

protected static final int SUM
A constant used to flag the Sum of FuzzySets operation.

See Also:
Constant Field Values

MAXMIN

protected static final int MAXMIN
A constant used to flag the maximumOfIntersection operation.

See Also:
Constant Field Values

COLLINEAR_OR_PARALLEL

protected static final boolean COLLINEAR_OR_PARALLEL
A constant used to flag a colinear or parallel result from the lineSegmentIntersection method in class UITools.

See Also:
Constant Field Values

NO_INTERSECTION

protected static final boolean NO_INTERSECTION
A constant used to flag a no intersection result from the lineSegmentIntersection method in class UITools.

See Also:
Constant Field Values

MAX_INITIAL_POINTS

protected static final int MAX_INITIAL_POINTS
A constant specifying the maximum number of initial points in the FuzzySet.

See Also:
Constant Field Values

INCREMENT

protected static final int INCREMENT
A constant specifying the amount by which the FuzzySet will be expanded each time an additional point is added and more space is required to accomodate that point.

See Also:
Constant Field Values

toStringPrecision

protected static int toStringPrecision
A static variable which determines how many decimal places will be printed by the toString method of FuzzySet. This precision can be set using the setToStringPrecision method.

tool

protected transient FuzzySet.UITools tool
A Union/Intersection tool object which provides methods required only for Union/Intersection method functionality, and also contains variables to keep track of the state of the Union/Intersection method.

set

protected nrc.fuzzy.SetPoint[] set
This is the heart of the FuzzySet, an array of SetPoints which represent the FuzzySet.

numPoints

protected int numPoints
The numPoints variable keeps track of the number of points contained in set, the array of SetPoints.

BFALSE

protected static final java.lang.Boolean BFALSE
BFALSE and BTRUE are static Boolean constants used to avoid generating instances of Booleans over and over.

BTRUE

protected static final java.lang.Boolean BTRUE

simplified

protected boolean simplified
The simplified variable is a boolean used to indicate whether or not the FuzzySet has been 'simplied' (revove extra points in the set -- see the simplifiySet method)

Constructor Detail

FuzzySet

public FuzzySet()
Constructs an empty FuzzySet with a default maximum number of elements allocated for the set array. The array will have no SetPoints in it and the size of the FuzzySet will be 0.

FuzzySet

public FuzzySet(int initMaxSize)
Constructs an empty FuzzySet with a specified number of elements allocated for the set array. The array will have no SetPoints in it and the size of the FuzzySet will be 0.

Parameters:
initMaxSize - the maximum initial size of the set array.

FuzzySet

public FuzzySet(double[] x,
                double[] y,
                int numPoints)
         throws XValuesOutOfOrderException,
                YValueOutOfRangeException
Constructs a FuzzySet containing the set of points represented by the array of x values and the array of y values passed. The points of the set will be examined to make sure that the y values (membership values) lie between 0.0 and 1.0. To create FuzzySets with membership values > 1.0 use appendSetPoint or insertSetPoint. The x values are checked to make sure they are in proper increasing order.

Parameters:
x - the double array which represents the x values of the points which are to make up the FuzzySet
y - the double array which represents the y values of the points which are to make up the FuzzySet
numPoints - the int that denotes the number of points contained in the x/y arrays
Throws:
XValuesOutOfOrderException - if the x values in the x[] double array are not in strictly ascending order.
YValueOutOfRangeException - if the y values in the y[] double array are not in the strict range [0.0, 1.0].

FuzzySet

public FuzzySet(nrc.fuzzy.SetPoint[] setPoints,
                int numPoints)
         throws XValuesOutOfOrderException,
                YValueOutOfRangeException
Constructs a FuzzySet containing the set of points represented by the array of SetPoints passed. The points of the set will be examined to make sure that the y values (membership values) lie between 0.0 and 1.0. To create FuzzySets with membership values > 1.0 use appendSetPoint or insertSetPoint. The x values are checked to make sure they are in proper increasing order.

Parameters:
setPoints - the SetPoint array containing the points which are to constitute the FuzzySet
numPoints - the number of points in the SetPoint array
Throws:
XValuesOutOfOrderException - if the x values in the SetPoints in the SetPoint array are not in strictly ascending order.
YValueOutOfRangeException - if the y values in the SetPoint array are not in the strict range [0.0, 1.0].

FuzzySet

public FuzzySet(nrc.fuzzy.FuzzySet newSet)
Constructs a new FuzzySet which is essentially a copy of the FuzzySet passed as a parameter.

Parameters:
newSet - the FuzzySet which will be copied to create the new FuzzySet
Method Detail

equals

public boolean equals(nrc.fuzzy.FuzzySet otherSet)
Compare this FuzzySet with another for equality. The sets are equal only if they have the same x and y values (within an adjustable tolerance -- see Parameters.FUZZY_TOLERANCE). Note that it is assumed that the sets have been passed through the simplifySet method to reduce the number of points to the minimum required to represent the set, otherwise sets that are equivalent may not be considered equal.

Returns:
true if they are equal otherwise return false.
See Also:
Parameters

fuzzyNormalize

public nrc.fuzzy.FuzzySet fuzzyNormalize()
Normalize this FuzzySet. Normalization involves identifying the point in the FuzzyValue with the highest membership value and multiplying all the membership values in the FuzzyValue by a scale factor such that this highest point then has a membership value of 1.0.

Returns:
a newly constructed FuzzySet containing the normalization of this FuzzySet

fuzzyScale

public nrc.fuzzy.FuzzySet fuzzyScale(double yvalue)
Scale this FuzzySet. Returns the fuzzy set which is scaled by: 
    y/maxYValueOfSet

Effectively this adjusts the set so that the maximum membership value of the set has the value y and all other membership values are scaled accordingly. If the membership values are all less than y then a copy of the FuzzySet is returned (ie. it scales values down but never up).

Returns:
a newly constructed FuzzySet containing the scaled version of this FuzzySet

fuzzyComplement

public nrc.fuzzy.FuzzySet fuzzyComplement()
Takes the compliment of this FuzzySet. More specifically, it takes the compliment of the membership (y) values of the SetPoints of the FuzzySet.

Mathematically (NOT), u(x) = 1 - u(x), or y = 1 - y.

This assumes that all of the membership values of the fuzzy set are <= 1.0 and >= 0.0. If values lie outside this then values >1.0 will be set to 0.0 and those < 0.0 will be set to 1.0

Returns:
a newly constructed FuzzySet containing the complement of this FuzzySet

getUITools

protected FuzzySet.UITools getUITools()
Get the UITools object associated with the FuzzySet. If there is none allocate one.

Returns:
the UITools object associated with the FuzzySet

maximumOfIntersection

public double maximumOfIntersection(nrc.fuzzy.FuzzySet otherSet)
Returns the maximum membership value on the intersection of this FuzzySet with the FuzzySet argument. Consider the two diagrams below. On the right appears a diagram of two sets, one set in black with green SetPoints, and the other set in grey with orange SetPoints. The maximum y value of the intersection is denoted by a red dot on the graph on the right.

Two Sets Two Sets showing Maximum of Intersection

Parameters:
otherSet - the FuzzySet which is intersected with this FuzzySet to determine the maximum y value of intersection
Returns:
the double value which represents the maximum y value on the intersection of the two FuzzySets

maximumOfIntersection

public static double maximumOfIntersection(nrc.fuzzy.FuzzySet setA,
                                           nrc.fuzzy.FuzzySet setB)
Returns the maximum membership value on the intersection of the two FuzzySet arguments. For a visual depiction of the max min of intersection, please see the documentation for the method maximumOfIntersection(FuzzySet otherSet).

Parameters:
setA - one of the two FuzzySets
setB - one of the two FuzzySets
Returns:
the double value which represents the maximum y value on the intersection of the two FuzzySets

fuzzyIntersection

public nrc.fuzzy.FuzzySet fuzzyIntersection(nrc.fuzzy.FuzzySet otherSet)
Returns the intersection of this FuzzySet with the FuzzySet argument. The visual representation of the intersection of two example FuzzySets is depicted below. One set is black with green SetPoints, and the other set is grey with orange SetPoints. The diagram on the left is of the two FuzzySets, and the diagram on the right is of the two sets overlaid by the intersection set in red. Intersection is synonymous with the logical operator AND.

Two Sets Two Sets showing Intersection Set

Parameters:
otherSet - the FuzzySet to be intersected with this FuzzySet.
Returns:
a new FuzzySet object that represents the intersection set of the two FuzzySets

fuzzyIntersection

public static nrc.fuzzy.FuzzySet fuzzyIntersection(nrc.fuzzy.FuzzySet setA,
                                                   nrc.fuzzy.FuzzySet setB)
Returns the intersection set of the two FuzzySet arguments. For a visual depiction of the intersection set and more information, please see the documentation for the method fuzzyIntersection(FuzzySet otherSet).

Parameters:
setA - one of the two FuzzySets
setB - one of the two FuzzySets
Returns:
the FuzzySet intersection set of the two FuzzySets

fuzzyUnion

public nrc.fuzzy.FuzzySet fuzzyUnion(nrc.fuzzy.FuzzySet otherSet)
Returns the union of this FuzzySet with the FuzzySet argument. The visual representation of the union of two example FuzzySets is depicted below. One set is black with green SetPoints, and the other set is grey with orange SetPoints. The diagram on the left is of the two FuzzySets, and the diagram on the right is of the two sets overlaid by the union set in red. Union is synonymous with the logical operator OR.

Two Sets Two Sets showing Union Set

Parameters:
otherSet - the FuzzySet to be unionized with this FuzzySet.
Returns:
a new FuzzySet object that represents the union set of the two FuzzySets

fuzzyUnion

public static nrc.fuzzy.FuzzySet fuzzyUnion(nrc.fuzzy.FuzzySet setA,
                                            nrc.fuzzy.FuzzySet setB)
Returns the union set of the two FuzzySet arguments. For a visual depiction of the union set and more information, please see the documentation for the method fuzzyUnion(FuzzySet otherSet).

Parameters:
setA - one of the two FuzzySets
setB - one of the two FuzzySets
Returns:
the FuzzySet union set of the two FuzzySets

fuzzySum

public nrc.fuzzy.FuzzySet fuzzySum(nrc.fuzzy.FuzzySet otherSet)
Returns the sum of this FuzzySet with the FuzzySet argument. The sum of 2 fuzzy sets is a set where the membership (y) value at every x position is the sum of the memberships values of the two sets at the corresponding x values.

NOTE WELL: The sum can lead to FuzzySets with membership values greater than 1.0. This is sometimes used to collect output of fuzzy rules (global contribution) rather than doing the union of the outputs.

Parameters:
otherSet - the FuzzySet to be summed with this FuzzySet.
Returns:
a new FuzzySet object that represents the sum of the two FuzzySets

fuzzySum

public static nrc.fuzzy.FuzzySet fuzzySum(nrc.fuzzy.FuzzySet setA,
                                          nrc.fuzzy.FuzzySet setB)
Returns the sum of the two FuzzySet arguments. The sum of 2 fuzzy sets is a set where the membership (y) value at every x position is the sum of the memberships values of the two sets at the corresponding x values.

NOTE WELL: The sum can lead to FuzzySets with membership values greater than 1.0. This is sometimes used to collect output of fuzzy rules (global contribution) rather than doing the union of the outputs.

Parameters:
setA - one of the two FuzzySets
setB - one of the two FuzzySets
Returns:
the FuzzySet sum of the two FuzzySets

fuzzyIntersectionUnion

protected java.lang.Object fuzzyIntersectionUnion(nrc.fuzzy.FuzzySet otherSet,
                                                  int op)
Returns the intersection set, the union set, the sum set or the maximum of intersection of this FuzzySet with the FuzzySet argument, depending on the operation specified by the op argument. The value of the op argument can be one of the following FuzzySet constants: UNION, INTERSECTION, SUM, or MAXMIN.

Parameters:
otherSet - the other FuzzySet with which to perform the specified operation with this FuzzySet
op - the int which specifies whether intersection, union, sum or max min of intersection is to be performed
Returns:
an Object which can either be an instanceof Double or FuzzySet. If the Object is an instanceof Double, then that Double object contains the value of the maximum of intersection. If the Object is an instanceof FuzzySet, then that FuzzySet object contains either an intersection, union or sum set; any method that calls this method is expecting one or the other.

copyFuzzySet

public nrc.fuzzy.FuzzySet copyFuzzySet()
                                throws java.lang.CloneNotSupportedException
Returns a copy of this instance of FuzzySet.

Returns:
a copy of this FuzzySet
Throws:
java.lang.CloneNotSupportedException - if the object to be cloned does not implement the Cloneable Interface.

simplifySet

public void simplifySet()
Simplifies the array of points by removing any extraneous points from the FuzzySet. If the following cases occur, the method takes the described actions:

checkXValueOrder

public int checkXValueOrder()
Checks to assure that all the x values of the FuzzySet are in ascending order. Returns -1 if they are ordered properly and the index of the first of the values that is out of order if they are not.

Returns:
Returns -1 if all x values are ordered properly and the index of the first of the values that is out of order if they are not
Throws:
XValuesOutOfOrderException - if the x values of the FuzzySet are not in strictly ascending order.

checkYValueRange

public int checkYValueRange()
Checks to assure that all the y values of the FuzzySet are within the strict range of 0.0 to 1.0 inclusive, or [0.0, 1.0]. Returns -1 if they are in the correct range and the index of the first value that is out of range if they are not.

Returns:
Returns -1 if all membership values are in the range [0, 1] and the index of the first value that is out of range if they are not.
Throws:
YValueOutOfRangeException - if the y values in the FuzzySet are not all within the range [0.0, 1.0]

getMembership

public double getMembership(double X)
Returns the membership value of the FuzzySet at the specified x value. This is done by interpolation. Note that if there are several membership values at the x value (e.g. a vertical line at this x value) then the maximum membership value is returned).

Parameters:
X - the x value at which to find the membership value of the FuzzySet
Returns:
the membership value of the FuzzySet at the specified x value or -1.0 if no points in the set

getXforMembership

public double getXforMembership(double m)
                         throws NoXValueForMembershipException
Returns the 1st X value with the specified membership value in the FuzzySet This is done by interpolation. Note that if there are multiple x values with this membership value then the only the 1st X value is returned.
Note that this is most often used to get the X value corresponding to a membership value in a FuzzySet that is strictly increasing from 0.0 to 1.0 or strictly decreasing from 1.0 to 0.0 (e.g. an SFuzzySet or a ZFuzzySet).

Returns:
the 1st X value of the FuzzySet with the specified membership value or exception if the set does not have this membership value
NoXValueForMembershipException

confineToXBounds

public void confineToXBounds(double lowXBound,
                             double highXBound)
                      throws XValuesOutOfOrderException
Confines the FuzzySet to the x boundaries specified, usually the Universe of Discourse. This is achieved by truncating the FuzzySet at the x boundaries. Consider the example below:

The FuzzySet on the left is a depiction of the set before being confined to the UOD (Universe of Discourse), and the FuzzySet to the right is the same set after being confined. Essentially, the set to the right is the same FuzzySet; however, all points outside the UOD now have a membership value of zero.

Note that this is infact equivalent to doing an intersection between the given FuzzySet and the RectangleFuzzySet(lowXBound, highXBound), but this is considerably more efficient.

Parameters:
lowXBound - the low x value at which to bound the set
highXBound - the high x value at which to bound the set
Throws:
XValuesOutOfOrderException - if low and high bounds out of order

size

public int size()
Returns the size, or the number of points, in the FuzzySet.

Returns:
the integer which represents the number of points in the FuzzySet.

getX

public double getX(int i)
Returns the x value of the SetPoint at the specified index in the FuzzySet.

Parameters:
i - the integer index of the SetPoint whose x value is to be returned
Returns:
the double x value of the SetPoint at the specified index in the FuzzySet
Throws:
ArrayOutOfBoundsException - if the parameter i, the specified index, is an illegal index

getY

public double getY(int i)
Returns the y value of the SetPoint at the specified index in the FuzzySet.

Parameters:
i - the integer index of the SetPoint whose y value is to be returned
Returns:
the double y value of the SetPoint at the specified index in the FuzzySet
Throws:
ArrayOutOfBoundsException - if the parameter i, the specified index, is an illegal index

getMinY

public double getMinY()
Returns the minimum membership value of the FuzzySet.

Returns:
the double value representing the minimum membership value of the FuzzySet. If the FuzzySet is empty returns 0.0.

getMaxY

public double getMaxY()
Returns the maximum membership value of the FuzzySet.

Returns:
the double value representing the maximum membership value of the FuzzySet. If the FuzzySet is empty returns 0.0.

getPoint

public nrc.fuzzy.SetPoint getPoint(int i)
Returns the SetPoint at the specified index in the FuzzySet.

Parameters:
i - the index of the SetPoint to return
Returns:
the SetPoint in the FuzzySet at the specified index
Throws:
java.lang.ArrayIndexOutOfBoundsException - if the parameter numPoints is greater than the length of the SetPoint array, or the number of SetPoints contained in the array.

setToStringPrecision

public static void setToStringPrecision(int numDecimalPlaces)
Sets the precision, in terms of the number of decimal places that will be printed for each double value, of the toString method.

Parameters:
numDecimalPlaces - the desired number of decimal places for each double value printed by the toString method

toString

public java.lang.String toString()
Returns a string representation of the FuzzySet. For example, if the FuzzySet consisted of the following points, where the format is (x, u(x)), where u(x) is the membership value or the y value:
then the string representation takes the form: u(x)/x.

Overrides:
toString in class java.lang.Object
Returns:
the String representation of the FuzzySet

fuzzyEqual

protected static boolean fuzzyEqual(double value1,
                                    double value2)
Tests two double values for equality with a fine tolerance.

Parameters:
value1 - the first double value
value2 - the second double value
Returns:
true if the values are equal within a certain tolerance.

getSupport

protected nrc.fuzzy.IntervalVector getSupport(double minUOD,
                                              double maxUOD)
Returns the support of this FuzzySet. The support set of a FuzzySet is the set of its elements that have a membership function other than the trivial membership of zero.

Returns:
an IntervalVector object which represents the support set of this FuzzyValue, or null if the FuzzySet has a uniform membership value of zero.

getAlphaCut

public nrc.fuzzy.IntervalVector getAlphaCut(boolean cutType,
                                            double cut,
                                            double minUOD,
                                            double maxUOD)
Returns the alpha cut of this FuzzySet at the membership level specified by the cut argument. Formally, a distinction is made betweeen two types of alpha cuts, the strong and the weak alpha cut.

The WEAK alpha cut of a FuzzySet is the set of all elements in the Universe of Discourse for which the membership function of the FuzzySet is greater than or equal to the cut argument.

The STRONG alpha cut of a FuzzySet is the set of all elements in the Universe of Discourse for which the membership function of the FuzzySet is greater than the cut argument. Note that a STRONG alpha cut at 0.0 is equivalent to the Support of the fuzzy set)

Parameters:
cutType - either Paramters.WEAK or Parameters.STRONG (defaults to WEAK if any other value)
cut - the double membership value at which the alpha cut is taken. The resulting support set is the set of all elements for which the membership value is greater than or equal to the cut value. (normally a value between 0.0 and 1.0)
minUOD - the minimum x value to be considered
maxUOD - the maximum x value to be considered. Note if minUOD > maxUOD the values will be switched.
Returns:
the IntervalVector which represents the alpha cut of this FuzzyValue at the double value of the cut argument, or null if the alpha cut at the cut level results in zero intervals.

isSimplified

public boolean isSimplified()
Returns true if this FuzzySet has been simplified; in other words, if this FuzzySet does not contain any unnecessary points (see simplifySet method).

Returns:
true if this FuzzySet is empty

isEmpty

public boolean isEmpty()
Returns true if this FuzzySet is empty; in other words, if this FuzzySet does not contain any points.

Returns:
true if this FuzzySet is empty

isNormal

public boolean isNormal()
Returns true if this FuzzySet is normal. The definition of normal is that there is at least one point in the universe of discourse where the membership function reaches unity (in terms of this fuzzy package, unity is equal to 1.0). Note that since we do allow membership values to be > 1.0 to accomodate special uses of the FuzzySets, there must be no values > 1.0 for it to be normal.

Returns:
true if this FuzzySet is normal

isConvex

public boolean isConvex()
Returns true if this FuzzySet is convex. The definition of convex is that the membership function of a convex FuzzySet does not go "up-and-down" more than once.

Returns:
true if this FuzzySet is convex

sameSign

protected boolean sameSign(double number1,
                           double number2)
Tests whether the two double value arguments have the same sign.

Parameters:
number1 - one of the two double values to compare
number2 - one of the two double values to compare
Returns:
true if the two double values have the same sign, ie. they are either both negative or both positive. It is important to note that if one of the values is zero, the method will always return true, as this method regards zero as being both (or neither, depending on how you look at it) negative and positive.

sameDirection

protected boolean sameDirection(double y1,
                                double y2,
                                double y3)
Tests whether the y values are in strictly ascending or descending order in the order they are passed as parameters. Please see below for a visual depiction of the situation.

Parameters:
y1 - the first y value
y2 - the second y value
y3 - the third y value
Returns:
true if the y values are in strictly ascending or descending order, as shown in the diagram above

goingUp

protected boolean goingUp(double y1,
                          double y2)
Tests whether the y value parameters are in ascending order. This means that the first y value is less than the second y value.

Parameters:
y1 - the first y value
y2 - the second y value
Returns:
true if the first y value is less than the second y value

goingDown

protected boolean goingDown(double y1,
                            double y2)
Tests whether the y value parameters are in descending order. This means that the first y value is greater than the second y value.

Parameters:
y1 - the first y value
y2 - the second y value
Returns:
true if the first y value is greater than the second y value

horizontalUnion

public nrc.fuzzy.FuzzySet horizontalUnion(double y)
Returns the horizontal union of the FuzzySet at the membership level specified by the y parameter. Please see the diagram below for a visual depiction. If y is < 0 then y is set to 0.

Parameters:
y - the y value with which the set is being unioned
Returns:
a new FuzzySet object representing the horizontal union for the FuzzySet with the y value argument

horizontalIntersection

public nrc.fuzzy.FuzzySet horizontalIntersection(double y)
Returns the horizontal intersection of the FuzzySet at the membership level specified by the y parameter. Please see the diagram below for a visual depiction. If y is < 0 then y is set to 0.

Parameters:
y - the y value with which the set is being intersected
Returns:
a new FuzzySet object representing the horizontal intersection for the FuzzySet with the y value argument

horizontalIntersectionUnion

protected nrc.fuzzy.FuzzySet horizontalIntersectionUnion(double y,
                                                         int op)
Returns the horizontal intersection or union of the FuzzySet at the membership level specified by the y parameter. Whether a horizontal intersection is performed, or a horizontal union, is determined by the op parameter. If y is < 0 then y is set to 0.

Parameters:
y - the y value with which the set is being intersected or unionized
op - the integer which specifies the operation to perform. The only possible values for this parameter are the constants INTERSECTION and UNION.
Returns:
a new FuzzySet object that represents the horizontal union or horizontal intersection, depending on what was requested

noIntersectionTest

public boolean noIntersectionTest(nrc.fuzzy.FuzzySet b)
Returns a tentative indication of whether or not the FuzzySets intersect. If this method returns true, meaning that to its knowledge the sets don't intersect, this can be taken for gospel. However, if the method returns false, it is indicative that an intersection of the FuzzySets is probable, but not certain. This is done as a fast test for no intersection.

Parameters:
b - a FuzzySet to be tested against this FuzzySet no intersection
Returns:
true if the FuzzySets definitely do not intersect; false if there is a possibility that they do

nonIntersectionTest

public boolean nonIntersectionTest(nrc.fuzzy.FuzzySet otherSet)
Tests whether or not two FuzzySets intersect. This intersection test is complete and thorough unlike the noIntersectionTest method. Without dwelling on what we would intuitively know as intersection of two FuzzySets, the more subtle differences between intersection and nonintersection are depicted below.

No Overlap, No Intersection Overlap, therefore Intersection

Parameters:
otherSet - the other FuzzySet which might possibly intersect with this FuzzySet
Returns:
true if the FuzzySets do not intersect, false if they do intersect

findMaxYOverlapValue

protected double findMaxYOverlapValue(nrc.fuzzy.SetPoint a,
                                      nrc.fuzzy.SetPoint b,
                                      nrc.fuzzy.SetPoint c,
                                      nrc.fuzzy.SetPoint d)
Returns the maximum y value of the overlapping portion of two vertical line segments that run in opposite directions. If the line segments are in the same direction (in other words, both ascending vertically or descending vertically) or they do not overlap the return value is -1.0.

Parameters:
a - the first point of the first line segment
b - the second point of the first line segment
c - the first point of the second line segment
d - the second point of the second ine segment
Returns:
  • -1.0 if the line segments are in the same direction, in other words, one is descending and the other is ascending. This is also the return value if the line segments do not overlap.
  • the maximum y value of the two line segments.

insertSetPoint

public void insertSetPoint(nrc.fuzzy.SetPoint a)
Inserts the SetPoint parameter into the SetPoint array which constitutes the FuzzySet. The new SetPoint is placed in the FuzzySet so as to keep the x values of the set in consistently increasing order.

For example, let us pretend that the existing FuzzySet looks like this:

0/1 1/2 1/4 0.5/4 0/5 (y/x... or u(x)/x format)

If the point 0.25/1.5 was inserted by this method, the result would be: 0/1 0.25/1.5 1/2 1/4 0.5/4 0/5. If the point 0.75/4 was inserted, the result would be: 0/1 1/2 1/4 0.5/4 0.75/4 0/5. Therefore, it is important to note that if a point is inserted and it's x value is the same as that of an existing point, it will be inserted into the FuzzySet at a position after the existing point with the same x value.

Parameters:
a - the SetPoint to be inserted into the FuzzySet

insertSetPoint

public void insertSetPoint(double X,
                           double Y)
Inserts the double parameters X and Y into the SetPoint array which constitutes the FuzzySet. The new SetPoint consisting of the passed X and Y values is placed in the FuzzySet so as to keep the x values of the set in consistently increasing order.

For example, let us pretend that the existing FuzzySet looks like this:

0/1 1/2 1/4 0.5/4 0/5 (y/x... or u(x)/x format)

If the point 0.25/1.5 was inserted by this method, the result would be: 0/1 0.25/1.5 1/2 1/4 0.5/4 0/5. If the point 0.75/4 was inserted, the result would be: 0/1 1/2 1/4 0.5/4 0.75/4 0/5. Therefore, it is important to note that if a point is inserted and it's x value is the same as that of an existing point, it will be inserted into the FuzzySet at a position after the existing point with the same x value.
Also note that any membership values must be >= 0. If not they will be set to 0.

Parameters:
X - the double x value to be inserted in the FuzzySet
Y - the double y value to be inserted in the FuzzySet

appendSetPoint

public void appendSetPoint(nrc.fuzzy.SetPoint a)
Appends the SetPoint parameter to the end of the SetPoint array which constitutes the FuzzySet. The new SetPoint is placed in the FuzzySet so as to keep the x values of the set in consistently increasing order. If the x value of the SetPoint being added is < the last x value in the array it will work backwards to place it in proper x order.

For example, let us pretend that the existing FuzzySet looks like this:

0/1 1/2 1/4 0.5/4 0/5 (y/x ... or u(x)/x format)

If the point 0.25/5.5 was appended by this method, the result would be: 0/1 1/2 1/4 0.5/4 0/5 0.25/5.5. If the point 0.75/4 was inserted, the result would be: 0/1 1/2 1/4 0.5/4 0.75/4 0/5. Therefore, it is important to note that if a point is appended and it's x value is the less than that of the last point, it will be placed into the FuzzySet at a position after all points with the x values less than or equal to it's x value.
Also note that any membership values must be >= 0. If not they will be set to 0.

Parameters:
a - the SetPoint to be inserted into the FuzzySet

appendSetPoint

public void appendSetPoint(double X,
                           double Y)
Appends the SetPoint parameter to the end of the SetPoint array which constitutes the FuzzySet. The new SetPoint is placed in the FuzzySet so as to keep the x values of the set in consistently increasing order. If the x value of the SetPoint being added is < the last x value in the array it will work backwards to place it in proper x order.

For example, let us pretend that the existing FuzzySet looks like this:

0/1 1/2 1/4 0.5/4 0/5 (y/x ... or u(x)/x format)

If the point 0.25/5.5 was appended by this method, the result would be: 0/1 1/2 1/4 0.5/4 0/5 0.25/5.5. If the point 0.75/4 was inserted, the result would be: 0/1 1/2 1/4 0.5/4 0.75/4 0/5. Therefore, it is important to note that if a point is appended and it's x value is the less than that of the last point, it will be placed into the FuzzySet at a position after all points with the x values less than or equal to it's x value.
Also note that any membership values must be >= 0. If not they will be set to 0.

Parameters:
X - the double x value to be inserted in the FuzzySet
Y - the double y value to be inserted in the FuzzySet

removeSetPoint

public boolean removeSetPoint(nrc.fuzzy.SetPoint a)
Find the SetPoint in the FuzzySet and remove it if found.

Parameters:
a - the SetPoint
Returns:
a boolean value with the following significance
  • true: the point was found and removed
  • false: the point did not exist in the FuzzySet and no action was taken

removeSetPoint

public boolean removeSetPoint(double X,
                              double Y)
Find the SetPoint with x,y values in the FuzzySet and remove it if found.

Parameters:
X - the double value
Y - the double value
Returns:
a boolean value with the following significance
  • true: the point was found and removed
  • false: the point did not exist in the FuzzySet and no action was taken

concat

protected nrc.fuzzy.FuzzySet concat(nrc.fuzzy.FuzzySet otherSet)
Concatenates the FuzzySet passed as a parameter to this instance of FuzzySet. This method does not affect the FuzzySet calling it or the FuzzySet passed to it.

Parameters:
otherSet - the FuzzySet to concatenate to this instance of FuzzySet
Returns:
a newly constructed FuzzySet which is the concatenation of this instance with the passed FuzzySet.

concat

protected nrc.fuzzy.FuzzySet concat(nrc.fuzzy.FuzzySet a,
                                    nrc.fuzzy.FuzzySet b)
Concatenates the two FuzzySets passed as a parameters. This method does not affect or alter the FuzzySets passed to it.

Parameters:
a - the first of the FuzzySets to be concatenated
b - the second of the FuzzySets to be concatenated
Returns:
a newly constructed FuzzySet which is the concatenation of the two FuzzySet parameters.

trimToSize

protected void trimToSize()
Remove excess space that might be allocated in the array holding the fuzzy set points.

testArrayLength

protected void testArrayLength()
Test to see if set array of SetPoints is full and if so increment space in array.

incrementArrayLength

protected void incrementArrayLength()
Add space to the set array holding the SetPoints

incrementArrayLength

protected void incrementArrayLength(int newSize)
Add space to the set array holding the SetPoints

shiftArrayLeft

protected void shiftArrayLeft(int index)
shifts all the SetPoint array elements which have an index value greater than or equal to the parameter index left one position. In addition to performing the shift, this method also decrements the count of the elements n the array (UNLESS it is EMPTY!!).

Parameters:
index - the index value of the first element in the array which is to be shifted. This element and all further sequential elements in the array are shifted down one index value.

shiftArrayRight

protected void shiftArrayRight(int index)
shifts all the SetPoint array elements which have an index value greater than or equal to the parameter index right one position. In addition to performing the shift, this method also increments the count of the elements in the array (UNLESS it is EMPTY!!).

Important Note: any method that calls this method to make space for a new element must necessarily call the testArrayLength method first to ensure that there is enough room in the array for the shifting to occur.

Parameters:
index - the index value of the first element in the array which is to be shifted. This element and all further sequential elements in the array are shifted up one index value.

momentDefuzzify

public double momentDefuzzify(double xMin,
                              double xMax)
                       throws InvalidDefuzzifyException,
                              XValuesOutOfOrderException
Moment defuzzification defuzzifies a fuzzy set returning a floating point (double value) that represents the fuzzy set. It calculates the first moment of area of a fuzzy set about the y axis. The set is subdivided into different shapes by partitioning vertically at each point in the set, resulting in rectangles, triangles, and trapezoids. The centre of gravity (moment) and area of each subdivision is calculated using the appropriate formulas for each shape. The first moment of area of the whole set is then: 
      sum of ( moment(i) * area(i) )          <-- top
      ------------------------------
          sum of (area(i))                    <-- bottom
If the total area is 0 then throw an exception since the moment is not defined.

Parameters:
xMin - minimum x value to use when performing defuzzification
xMax - maximum x value to use when performing defuzzification
Returns:
the floating value that is the first moment of the fuzzy set
Throws:
InvalidDefuzzifyException - there is no valid moment for the fuzzy set (generally the area is 0 under the fuzzy set graph)
XValuesOutOfOrderException - occurs when the xMin parameter is greater than or equal to the xMax parameter.

maximumDefuzzify

public double maximumDefuzzify(double xMin,
                               double xMax)
                        throws XValuesOutOfOrderException,
                               InvalidDefuzzifyException
Finds the mean of maxima of a fuzzy set as the defuzzification value. NOTE: This doesn't always work well because there can be x ranges where the y value is constant at the max value and other places where the max is only reached for a single x value. When this happens the single value gets too much of a say in the defuzzified value. So use moment defuzzify in cases like this. 
             /------\                   /\
            /        \                 /  \
         --/          \---------------/    \-------------
                 ^       ^
                 |       |
                 |       | gives this as the mean of maximum
                 | this is more reasonable???
Centre of gravity (moment) defuzzify is likely more useful most of the time.

Parameters:
xMin - minimum x value to use when performing defuzzification
xMax - maximum x value to use when performing defuzzification
Returns:
the floating value that is the first average of the maximum values of the fuzzy set
Throws:
InvalidDefuzzifyException - there are no points in the FuzzySet
XValuesOutOfOrderException - occurs when the xMin parameter is greater than or equal to the xMax parameter.

weightedAverageDefuzzify

public double weightedAverageDefuzzify(double xMin,
                                       double xMax)
                                throws XValuesOutOfOrderException,
                                       InvalidDefuzzifyException
Finds the weighted average of the values of a fuzzy set as the defuzzification value. This is slightly different than the maximumDefuzzify since the maximumDefuzzify uses only points that have the same membership value (the one that is the maximum in the set of points). The weightedAverageDefuzzify uses all of the points with non-zero membership values and calculates the average of the x values using the membership values as weights in this average. NOTE: This doesn't always work well because there can be x ranges where the y value is constant at the max value and other places where the max is only reached for a single x value. When this happens the single value gets too much of a say in the defuzzified value. So use moment defuzzify in cases like this. 
             /------\                   /\
            /        \                 /  \
         --/          \---------------/    \-------------
                 ^       ^
                 |       |
                 |       | gives this as the weighted average
                 | this is more reasonable???
However, it is especially effective if the output has a number of singleton points in it. This might be, for example, when one has only singleton values describing the output fuzzy values (Takagi-Sugeno-Kang zero order type rules). In this case one gets a result that would be expected and one that would result in an exception with the momentDefuzzify (area is 0) and a poor result with the maximumDefuzzify (since only the values with the max membership are used - 20 in the example below). 
    
      1.0                 |
                          |
      0.5        |        |
                 |        |        |
         --------|--------|--------|--------
         0      10        20       30      40
                        ^
                        |
                        | gives 18.57 as the weighted average   
 
      (10*0.5 + 20*1 + 30*0.25) / (0.5 + 1 + 0.25) = 18.57
Centre of gravity (moment) defuzzify is likely more useful most of the time.

Parameters:
xMin - minimum x value to use when performing defuzzification
xMax - maximum x value to use when performing defuzzification
Returns:
the floating value that is the weighted average of the fuzzy set
Throws:
InvalidDefuzzifyException - there are no points in the FuzzySet
XValuesOutOfOrderException - occurs when the MinUOD is greater than or equal to the MaxUOD parameter (this should never happen).

plotFuzzySet

public java.lang.String plotFuzzySet(java.lang.String plotChar,
                                     double lowX,
                                     double highX)
Plots the fuzzy set in an ascii format producing a String that can be displayed The string must be displayed using a non-proportional spaced font.

plotFuzzySets

public static java.lang.String plotFuzzySets(java.lang.String plotChars,
                                             double lowX,
                                             double highX,
                                             nrc.fuzzy.FuzzySet[] fSets)
Plots the fuzzy sets in an ascii format producing a String that can be displayed The string must be displayed using a non-proportional spaced font such as courier.

Parameters:
fSets - array of fuzzy sets to plot the single graph
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