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NEURO-FUZZY INFERENCE SYSTEM APPLICATION FOR CREDIT RATING OF BANK

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                                                                                        D
                                       mahdiyeh_akhbari@yahoo.com




                                                                      SI
                                                                ><                      >
                                                   farimah@cc.iut.ac.ir
                                89/3/5 :                             87/7/28 :              >
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 89    / 92                  / @>            >          @                                                       2

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    3                            …@              @      >@       -                             @             >


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Ar




   >    >      (                             )         ( )@ > @ > > < > >
    @    @   >                                    >           .           >
                                                                      .> MDA ANFIS



1- Artificial neural networks.
2- adaptive network based fuzzy inference system or adaptive neuro-fuzzy inference system.
3- Multiple Discriminate Analysis.




                                                                                                                           www.SID.ir
    89                  / 92           / @>           >               @                                                              4

                            <@                            @                       >            @              >        >
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                                                                                                                                    .[11]
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                                 ive


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                                                                                                                                     B
>                                         >                           >               >@   @                               @
        >                          @          @                                   <                           @                  >
                                                      :                       >                @                               .[3 1<2]
                        .                                         <                                                   @ >



1- fuzzy adaptive network.




                                                                                                                                            www.SID.ir
     5                                     …@                @                   >@       -                            @                >


                      <               @                                                                                <@ >
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                                                 @ (                         @        >                    )                             -3
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    89            / 92           / @>                >                   @                                                          6

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                                        .                            >            (1974)                                        ٣

             < [16 <15 <13]                     <>                                @                                    @                >
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         ch



                                                                                                                       >
                         X1
         >
                                                                                                            Y1
    @>
Ar




                         X2                                                                                 Y2



                                                                             @



1- adaptive networks.
2- gradient descent.
3- chain rule.




                                                                                                                                            www.SID.ir
     7                                              …@               @       >@        -                             @               >


                                <                                                                       (1           )
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                                                                                       k                 L
    >>                  O ik                        (                )             (k,i)                             k > i
                           :                                    @            @>        @                                      Oi .
                                                                                                                               k

     Oi = Oi (O1 −1,...Ok −1, a , b, c,...),                                                                                         (1)
                                ive

      k    k k
                        n
                                                                                       @                                 c <b <a
                                                             k −1

 k
Oi                                                      .
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         ch



                  >                 @               > @ > > @> (1 ≤ p ≤ P )   p@ (٤@                                                       )
                                                                            :>       >

                      ∑ (Tm, p − OL ,p )2 ,
                      nL
                                                                                                                                     (2)
Ar




     Ep =                         m
                  m =1
                           m < OL ,p
                                m                                             >                p                          m <Tm,p




1- node function.
2- adaptive node.
3- fixed node.
4- energy function.




                                                                                                                                               www.SID.ir
89              / 92                / @>            >            @                                                        8

          @                                    .         @>           >           p@                                          >

                                                                                       .E =     ∑ Ep :
                                                                                                 P

                                                                                                p =1
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@           > @ >>                      p@          <            <        > < >                             @             >
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                                                                                      D
         ∂E p
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                                 L                                                                                   (3)
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           L

    :

         ∂E p
        ∂Oik p
                    =
                         @


                         ∑
                        n k +1

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                                                                 ,
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                                                                                                                     (4)
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                                               .>                > (4) (3) > >                  >
                                                                                                                    ∂O i , p
                                                                                                                       k

                            :       >               <                             @           @                 α
        ch



        ∂E p                    ∂E p ∂O*
         ∂α
                =       ∑       ∂O* ∂α
                                         ,                                                                          (5)
                    O * ∈S
                    .           α                                                           @
Ar




                                                                                                                S
                                                             :                α             E               @
        ∂E P ∂E p
          =∑       .                                                                                                (6)
        ∂α p =1 ∂α
                : >                            α                          @                                 <
                        ∂E
    ∆α = − η                                                                                                         (7)
                        ∂α




                                                                                                                                  www.SID.ir
     9                         …@                 @               >@        -                       @               >


                               :>                             >                             @       > > η
     η=                                                                                                             (8)
                      k
                               ,
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                                                                                    D
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                                                                                                      >
                                                                                                    .[15< 14]
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                                                                                                                     @
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ANFIS             >                         <[20]                                   <           @
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 >                                                > <>             >                          @                 ANFIS
                                               :                                                >       >
     if x is A1 and y is B1 , then f1 = p1x + q1y + r1                                                  1@
         ch



     if x is A 2 and y is B 2 , then f 2 = p2 x + q2 y + r2                                             2@
                                (           <4)               >           @
                                    :                    >>             ( <4)                   >
Ar




                                                                                                                ANFIS




1- step size.
2- batch learning.
3- off-line.
4- epoch.
5- sweep.
6- pattern learning.
7- on- line.




                                                                                                                                  www.SID.ir
 89          / 92        / @>        >        @                                                  10




                                                                      D
                                                          SI
                                     of
                           @     -                                        -1
                     ive

                                                                       (@> @ )                  @
         :                           (        )                       < @   >i
  O1, i = µ A i (x ), i = 1,2, or
                                                                                               (9)
      ch



  O1, i = µ Bi −2 (x ), i = 3,4.
            @ >                Ol,i (                             )       @              B A
@          >                                          @       >           >.              l@         > i
                                                                  .                >                 @
Ar




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                                                                                                           www.SID.ir
    11                             …@            @            >@         -                 @                        >


                                                                         (@        @       @           )        >@
                                                         > (@            >)                                    @
                                                                   :               >           @
    O2, i = w i = µ A i (x ) * µ Bi (x ), i = 1,2.                                                                 (10)
@        and           T-norm                    >                   <         @                                      >
@                             >.                     >     @                           >       <                          >
                                   . >                      @                              (           )




                                                                              D
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    O 3,i = w i =
                              wi
                            w1 + w 2
                                        :
                                     , i = 1,2.
                                                         >
                                                               SI(@           @
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                                                                                       @       )
                                                                                                                   (11)

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    O4, i = w i f i = w i ( p i x + q i y + ri ), for i = 1,2                                              (12)
wi                                                @                                @                   <ri qi <pi
                  .>         >                              @                                                  @
                                                                              ( >@ )                                @
     ch



>             @         @                      >                   >           @
             :              > @>            @ >> @                                @
                                            ∑ w i fi
Ar




    Z = O 5, i = ∑ w i f i = i                                                                                     (13)
                                            ∑ wi
                                                         , for i = 1,2.
                                             i




                                                                                                                              www.SID.ir
 89           / 92            / @>       >                @                                                           12

                              (ANFIS) ١                       @      –
          @                                      <                                <ANFIS
                                                      :              >              .   (r2 q2 <p2 <r1 <q1 <p1)
           w1             w2
    Z=             f1 +          f2
        w1 + w 2        w1 + w 2
    = w1f1 + w 2f 2                                                                                              (14)
    = ( w1x ) p1 + ( w1y)q1 + ( w1)r1 + ( w 2 x ) p2 + ( w 2 y)q 2 + ( w 2 )r2 .




                                                                                      D
       > .>               >                      >@ >
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>

      .[20]
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                               >
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                                                                     SI  @
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    Y=F(X1,X2,…,Xn)
                          @                                               @                < Xi          @            >
                                                                                                                           Y
      ch



                > >           @                                                        <             >
                                                              <"                               ":             :
      .                                                                  >        >                          @
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Ar




                                     @                               Y            >                 >.       >
                                                          .                                                  @
      i@                                         Pi                      <Y
              <(Yi=1) >                  >>               >                   >            > (xi)                         >


1- Adaptive network based fuzzy inference system or adaptive neuro-fuzzy inference system.




                                                                                                                               www.SID.ir
      13                            …@              @         >@       -                @         >


                     > (Yi=0) >                 >       >              i@                             1-Pi
                                            :                (15) @                                   <>
    E ( Yi / x i ) = 0 * P ( Yi = 0 / x i ) + 1 * P ( Yi = 1 / x i ) = P ( Yi = 1 / x i ) = Pi
                                                                                                      (15)
                                            Yi @                   @                        >>
            >                  <                                                    <       Pi i @
                                                                   .> @                       >
                                    >       <ANFIS                  >           @>                    >




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                                                                                                             www.SID.ir
    89            / 92                  / @>          >           @                                                           14

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                                                                       @>           @                                         -2-4
                              :>                         >             @                          <                            >




                                                                                                                                     www.SID.ir
    15                                   …@                 @             >@            -                       @                >

R1: If (x1 is Low and x2 is Low and x3 is Low, then ( Y1 = c1 + c1x1 + c1 x1 + c1 x 3 )
                                                            o     1      2        3
                                                                      2
R2: If (x1 is Low and x2 is Low and x3 is Medium, then ( Y 2 = c 2 + c1 x1 + c 2 x 2 + c2 x 3 )
                                                                 o             2        3
M
                                                                      27
R27:If(x 1is High and x2 is High and x3 is High then ( Y 27 = c 27 + c1 x1 + c 27 x 2 + c 27 x 3 )
                                                                               2          3
            ) x3 (                           ) x 2 <(                     ) x1 @>              @                <                    >
                                                                o



             (Medium )                         <(Low)                      > (                   >
    i@                                            <Yi .                       @                 @ @                              (High)
                            >            >>                                                                                      ><




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                @        @                   >k                                     <[20]             <>
                                                                                                                    :            >
                %10 <k                   <                  >                                  4 >                             -1
Ar




                                                                           .(               1,1                              > )
            <                                                                                  2 >                             -2
                                                       .(        0,9                                 > )                      %10 <k


1- initial step size.
2- step size decrease rate.
3- step size increase rate.




                                                                                                                                          www.SID.ir
    89              / 92                 / @>              >            @                                                              16

                                            <                                  >>                 2 4 <%10 >
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<                   @                   >                                      <              @                                    >
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                                                                               200 >                            RMSE
                                                           >                                 > >




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                                                                                        >
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                    27
                    ∑ w l Yl
            Y = l =1
            ˆ
                     27
                                                                                                                            (16)
                     ∑
                                 ive

                             wl
                     l =1
                                                                                       :>                          >               wl

            wl = ∏ µ                   (x i ) = µ          (x i ) * µ          (x i ) * µ         (x i )                       (17)
            ch



                                 Fil                 Fil                 Fil                Fil

            Fil .        >                           @              > Yl <                                                     >
             >.          >                      xi                  @ @         (@                                     )           @
                                       >@        @                      <     @                                            @
Ar




                                                 @             >      (17) @ > >                                           >           .
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        .                                                                 @   @                                    wl .                      >
                         @ > (@>                @                 )       @ @
>                         135 >                                 >               27                                 ><                  <
                                                                                                                                           .> >




                                                                                                                                                  www.SID.ir
         17                                  …@          @                 >@            -                            @                    >


          <        >     MATLAB                                      ANFIS                                                    @            @
    .>            > 27 >  <1,6GHz                                >       Pentium IV                                       >        @


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