Fuzzy relations vs. Mamdani model (continued)

From: Pim van den Broek (pimvdb@cs.utwente.nl)
Date: Sun Dec 31 2000 - 04:36:01 MET

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    Hello,

    A couple of weeks ago there was a discussion in this group following a
    question of Andrej Albert on the difference between two models of
    approximate reasoning. Continuing this discussion I will consider a small
    example, and show that the choice of the model should be dependent on
    the interpretation of the rules, and moreover, that for some interpretations
    of the rules neither of the two models is appropriate.

    Suppose we have a set of rules of the form "IF A THEN B" where A is a set
    of symptoms and B is a set of diseases. The intention of the rules is to obtain
    a diagnosis of the disease(s) of a patient, given information about the symptoms
    this patient exhibits. We will only consider crisp sets here, taking the point of

    view that a model for approximate reasoning should produce reasonable results
    also when only crisp sets are considered.
    Given a rule "IF A THEN B" where A is a set of symptoms and B is a set of
    diseases and also given a set A' of symptoms, the resulting set B' of diseases
    will depend on the meaning of the rules. A rule might be applicable if the
    patient
    exhibits ANY of the symtoms, or if the patient exhibits ALL the symtoms.
    The result of the rule might mean that the patient suffers from ALL diseases from

    the set, or that the disease is ANY of the diseases of the set.
    This leads to four possible interpretations of a rule "IF A THEN B":

    Interpretation 1 : IF the patient exhibits all symptoms from A THEN the patient
    has
    all diseases of set B
    Interpretation 2 : IF the patient exhibits all symptoms from A THEN the disease
    of this patient belongs to set B
    Interpretation 3 : IF the patient exhibits anyl symptoms from A THEN the patient
    has
    all diseases of set B
    Interpretation 4 : IF the patient exhibits any symptoms from A THEN the disease
    of this patient belongs to set B

    Now consider the case where there are 3 symtoms and 3 diseases, and
    symptoms and diseases both are denoted by the numbers 1,2 and 3.

    Consider the rule : IF {1,2} THEN {1}
    We consider two input sets of symptoms: {1} and {3}.
    The following table contains for both input sets and all four interpretations of
    the
    rule the desired output set of diseases.

                 Int. 1 Int. 2 Int. 3 Int. 4
    Input {1} | {} {1,2,3} {1} {1}
    Input {2} | {} {1,2,3} {} {1,2,3}

    This table is computed as follows:
    The desired result is {1} ( the right hand-side of the rule), in case the rule
    is
    applicable, i.e. for input {1} and interpretations 3 and 4 of the rule.
    The desired result is {} if the rule is not applicable and the result denotes the

    diseases of the patient.
    The desired result is {1,2,3} if the rule is not applicable, and the result
    denotes the
    possible diseases of the patient.

    Now let us compare this table with the results obtained from both approximate
    reasoning models. Both models employ the formula

    B'(y) = sup_x min (A'(x), J(A(x),B(y)))

    In the Mamdani case, J is a t-norm, with the properties J(1,1)=1 and
    J(0,1)=J(1,0)=J(0,0)=0.
    In the second case (I call it the implication case), J is an implication
    operator, with
    the properties J(1,0)=0 and J(0,0)=J(0,1)=J(1,1)=1.

    The result of the computation is:

                 Mamdani Implication
    Input {1} | {1} {1}
    Input {2} | {} {1,2,3}

    So the Mamdani model produces the results which are desired in interpretation 3,
    and the implication model produces the results which are desired in
    interpretation 4.
    One easily verifies that this holds true for other rules and inputs also.

    So my conclusion for this example is : In case the interpretation of the rules is

    interpretation 3, use the Mamdani model. In case the interpretation of the rules
    is interpretation 4, use the implication model. In case the interpretation of the
    rules
    is interpretation 1 or interpretation 2, do not use any of these two models,
    since
    neither of them produces the desired results in case of crisp sets.

    My conclusion in general is: In a practical situation, a simple test using crisp
    sets can
    determine which of the two models of approximate reasoning is appropriate, or
    show
    that neither of them is appropriate.

    I look forward to receive your opinions!

    greetings,

    Pim van den Broek.

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    <!doctype html public "-//w3c//dtd html 4.0 transitional//en">
    <html>
    Hello,
    <p>A couple of weeks ago there was a discussion in this group following
    a
    <br>question of Andrej Albert on the difference between two models of
    <br>approximate reasoning. Continuing this discussion I will consider a
    small
    <br>example, and show that the choice of the model should be dependent
    on
    <br>the interpretation of the rules, and moreover, that for some interpretations
    <br>of the rules neither of the two models is appropriate.
    <p>Suppose we have a set of rules of the form "IF A THEN B" where A is
    a set
    <br>of symptoms and B is a set of diseases. The intention of the rules
    is to obtain
    <br>a diagnosis of the disease(s) of a patient, given information about
    the symptoms
    <br>this patient exhibits. We will only consider crisp sets here, taking
    the point of
    <br>view that a model for approximate reasoning should produce reasonable
    results
    <br>also when only crisp sets are considered.
    <br>Given a rule "IF A THEN B" where A is a set of symptoms and B is a
    set of
    <br>diseases and also given a set A' of symptoms, the resulting set B'
    of diseases
    <br>will depend on the meaning of the rules. A rule might be applicable
    if the patient
    <br>exhibits ANY of the symtoms, or if the patient exhibits ALL the symtoms.
    <br>The result of the rule might mean that the patient suffers from ALL
    diseases from
    <br>the set, or that the disease is ANY of the diseases of the set.
    <br>This leads to four possible interpretations of a rule "IF A THEN B":
    <p>Interpretation 1 :&nbsp; IF the patient exhibits all symptoms from A
    THEN the patient has
    <br>all diseases of set B
    <br>Interpretation 2 :&nbsp; IF the patient exhibits all symptoms from
    A THEN the disease
    <br>of this patient belongs to set B
    <br>Interpretation 3 :&nbsp; IF the patient exhibits anyl symptoms from
    A THEN the patient has
    <br>all diseases of set B
    <br>Interpretation 4 :&nbsp; IF the patient exhibits any symptoms from
    A THEN the disease
    <br>of this patient belongs to set B
    <p>Now consider the case where there are 3 symtoms and 3 diseases, and
    <br>symptoms and diseases both are denoted by the numbers 1,2 and 3.
    <p>Consider the rule : IF {1,2} THEN {1}
    <br>We consider two input sets of symptoms: {1} and {3}.
    <br>The following table contains for both input sets and all four interpretations
    of the
    <br>rule the desired output set of diseases.
    <p><tt>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    Int. 1&nbsp;&nbsp; Int. 2&nbsp;&nbsp;&nbsp; Int. 3&nbsp;&nbsp; Int. 4</tt>
    <br><tt>Input {1} |&nbsp; {}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {1,2,3}&nbsp;&nbsp;
    {1}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {1}</tt>
    <br><tt>Input {2} |&nbsp; {}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {1,2,3}&nbsp;&nbsp;
    {}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {1,2,3}</tt>
    <p>This table is computed as follows:
    <br>The desired result is {1}&nbsp; ( the right hand-side of the rule),
    in case the rule is
    <br>applicable, i.e. for input {1} and interpretations 3 and 4 of the rule.
    <br>The desired result is {} if the rule is not applicable and the result
    denotes the
    <br>diseases of the patient.
    <br>The desired result is {1,2,3} if the rule is not applicable, and the
    result denotes the
    <br>possible diseases of the patient.
    <p>Now let us compare this table with the results obtained from both approximate
    <br>reasoning models. Both models employ the formula
    <p><tt>B'(y) = sup_x min (A'(x), J(A(x),B(y)))</tt>
    <p>In the Mamdani case, J is a t-norm, with the properties J(1,1)=1 and
    <br>J(0,1)=J(1,0)=J(0,0)=0.
    <br>In the second case (I call it the implication case), J is an implication
    operator, with
    <br>the properties J(1,0)=0 and J(0,0)=J(0,1)=J(1,1)=1.
    <p>The result of the computation is:
    <p><tt>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    Mamdani&nbsp;&nbsp; Implication</tt>
    <br><tt>Input {1} |&nbsp; {1}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {1}</tt>
    <br><tt>Input {2} |&nbsp; {}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    {1,2,3}</tt>
    <p>So the Mamdani model produces the results which are desired in interpretation
    3,
    <br>and the implication model produces the results which are desired in
    interpretation 4.
    <br>One easily verifies that this holds true for other rules and inputs
    also.
    <p>So my conclusion for this example is : In case the interpretation of
    the rules is
    <br>interpretation 3, use the Mamdani model. In case the interpretation
    of the rules
    <br>is interpretation 4, use the implication model. In case the interpretation
    of the rules
    <br>is interpretation 1 or interpretation 2, do not use any of these two
    models, since
    <br>neither of them produces the desired results in case of crisp sets.
    <p>My conclusion in general is: In a practical situation, a simple test
    using crisp sets can
    <br>determine which of the two models of approximate reasoning is appropriate,
    or show
    <br>that neither of them is appropriate.
    <p>I look forward to receive your opinions!
    <p>greetings,
    <p>Pim van den Broek.
    <br>&nbsp;
    <br>&nbsp;</html>

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