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Fuzzy set example

Step 1 -- Define Fuzzy Sets

Let's define six fuzzy sets 

low energy
high energy

low protein
high protein

high feed
low feed

lets measure feed, energy and protein on a 0 to 10 scale
(this is just to keep the example simple any scale
could have been used)


for these sets assume the following
membership function for low
;[s]
3:0,1;19,2;47,1;344,-1;
3:0,13,9,Times,0,12,0,0,0;2,16,12,Times,0,14,0,0,0;1,25,18,Times,1,24,0,0,0;
:[font = input; preserveAspect; startGroup]
$DefaultFont = {"Chicago" , 12}
low[x_]:= 1- (1/(1+Exp[-((x-5)/1.0)]))

Plot[low[x],{x,0,10},AxesLabel -> {"Value","Membership"},GridLines -> Automatic]
:[font = output; output; inactive; preserveAspect]
{"Chicago", 12}
;[o]
{Chicago, 12}
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[(6)] .59524 .01472 0 2 Msboxa
[(8)] .78571 .01472 0 2 Msboxa
[(10)] .97619 .01472 0 2 Msboxa
[(Value)] 1.025 .01472 -1 0 Msboxa
[(0.2)] .01131 .13323 1 0 Msboxa
[(0.4)] .01131 .25174 1 0 Msboxa
[(0.6)] .01131 .37026 1 0 Msboxa
[(0.8)] .01131 .48877 1 0 Msboxa
[(1)] .01131 .60728 1 0 Msboxa
[(Membership)] .02381 .61803 0 -4 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
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.02381 .55988 m
.02756 .55988 L
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.02756 .58358 L
s
P
[(Membership)] .02381 .61803 0 -4 Mshowa
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clip
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Graphics["<<>>"]
;[o]
-Graphics-
:[font = text; inactive; preserveAspect]
for these sets assume the following 
membership function for high
;[s]
1:0,1;66,-1;
2:0,13,9,Times,0,12,0,0,0;1,19,14,Times,0,18,0,0,0;
:[font = input; preserveAspect; startGroup]

high[x_]:= 1/(1+Exp[-((x-5)/1.0)])

Plot[high[x],{x,0,10},AxesLabel -> {"Value","Membership"},GridLines -> Automatic]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Chicago findfont 12  scalefont  setfont
% Scaling calculations
0.0238095 0.0952381 0.0147151 0.59257 [
[(2)] .21429 .01472 0 2 Msboxa
[(4)] .40476 .01472 0 2 Msboxa
[(6)] .59524 .01472 0 2 Msboxa
[(8)] .78571 .01472 0 2 Msboxa
[(10)] .97619 .01472 0 2 Msboxa
[(Value)] 1.025 .01472 -1 0 Msboxa
[(0.2)] .01131 .13323 1 0 Msboxa
[(0.4)] .01131 .25174 1 0 Msboxa
[(0.6)] .01131 .37026 1 0 Msboxa
[(0.8)] .01131 .48877 1 0 Msboxa
[(1)] .01131 .60728 1 0 Msboxa
[(Membership)] .02381 .61803 0 -4 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
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.001 w
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.001 w
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p
.001 w
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.001 w
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P
p
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s
P
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Graphics["<<>>"]
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What does this mean? 

If the energy level is 6 then this observation is scored on the low
energy fuzzy set as 
about a .4 and on the high energy fuzzy set as about  a .6. 

Thus this observation is not classed as either high or low energy but is
associated to varying degrees with both concepts. This is the key to
fuzzy sets.

What can we do with this?

Step 2 Relating Observations to Fuzzy Sets

Lets take an example with 10 observations of energy and protein.


;[s]
5:0,3;355,1;356,2;399,1;465,0;467,-1;
4:1,13,9,Times,0,12,0,0,0;2,19,14,Times,0,18,0,0,0;1,25,18,Times,0,24,0,0,0;1,16,12,Times,0,14,0,0,0;
:[font = input; preserveAspect; startGroup]
energy = {0.0,0.0,0.0,0.0,3.2,6.1,10.0,3.2,3.2,10.0}

:[font = output; output; inactive; preserveAspect; endGroup]
{0., 0., 0., 0., 3.2, 6.1, 10., 3.2, 3.2, 10.}
;[o]
{0., 0., 0., 0., 3.2, 6.1, 10., 3.2, 3.2, 10.}
:[font = input; preserveAspect; startGroup]
protein = {0.0,3.2,6.1,10.0,0.0,0.0,0.0,3.1,3.3,10.0}
:[font = output; output; inactive; preserveAspect; endGroup]
{0., 3.2, 6.1, 10., 0., 0., 0., 3.1, 3.3, 10.}
;[o]
{0., 3.2, 6.1, 10., 0., 0., 0., 3.1, 3.3, 10.}
:[font = text; inactive; preserveAspect]
The following evaluate the observations in terms of their membership in the
four fuzzy sets we have defined.   


;[s]
3:0,0;58,1;68,0;114,-1;
2:2,13,9,Times,0,12,0,0,0;1,16,12,Times,1,14,0,0,0;
:[font = input; preserveAspect; startGroup]
lowenergy  = low[energy]
:[font = output; output; inactive; preserveAspect; endGroup]
{0.9933071490757151445, 0.9933071490757151445, 
 
  0.9933071490757151445, 0.9933071490757151445, 
 
  0.8581489350995122104, 0.2497398944048823958, 
 
  0.006692850924284855546, 0.8581489350995122104, 
 
  0.8581489350995122104, 0.006692850924284855546}
;[o]
{0.993307, 0.993307, 0.993307, 0.993307, 0.858149, 0.24974, 
 
  0.00669285, 0.858149, 0.858149, 0.00669285}
:[font = input; preserveAspect; startGroup]
lowprotein = low[protein]
:[font = output; output; inactive; preserveAspect; endGroup]
{0.9933071490757151445, 0.8581489350995122104, 
 
  0.2497398944048823958, 0.006692850924284855546, 
 
  0.9933071490757151445, 0.9933071490757151445, 
 
  0.9933071490757151445, 0.8698915256370021479, 
 
  0.8455347349164652957, 0.006692850924284855546}
;[o]
{0.993307, 0.858149, 0.24974, 0.00669285, 0.993307, 0.993307, 
 
  0.993307, 0.869892, 0.845535, 0.00669285}
:[font = input; preserveAspect; startGroup]
highenergy = high[energy]

:[font = output; output; inactive; preserveAspect; endGroup]
{0.006692850924284855559, 0.006692850924284855559, 
 
  0.006692850924284855559, 0.006692850924284855559, 
 
  0.1418510649004877896, 0.7502601055951176042, 
 
  0.9933071490757151445, 0.1418510649004877896, 
 
  0.1418510649004877896, 0.9933071490757151445}
;[o]
{0.00669285, 0.00669285, 0.00669285, 0.00669285, 0.141851, 0.75026, 
 
  0.993307, 0.141851, 0.141851, 0.993307}
:[font = input; preserveAspect; startGroup]
highprotein = high[protein]
:[font = output; output; inactive; preserveAspect; endGroup]
{0.006692850924284855559, 0.1418510649004877896, 
 
  0.7502601055951176042, 0.9933071490757151445, 
 
  0.006692850924284855559, 0.006692850924284855559, 
 
  0.006692850924284855559, 0.1301084743629978521, 
 
  0.1544652650835347043, 0.9933071490757151445}
;[o]
{0.00669285, 0.141851, 0.75026, 0.993307, 0.00669285, 0.00669285, 
 
  0.00669285, 0.130108, 0.154465, 0.993307}
:[font = text; inactive; preserveAspect]

Observation 3 for example has an enegy value
of 0 and a protein value of 6.1
:[font = input; preserveAspect; startGroup]

lowenergy[[3]]
lowprotein[[3]]
highenergy[[3]]
highprotein[[3]]
:[font = output; output; inactive; preserveAspect]
0.9933071490757151445
;[o]
0.993307
:[font = output; output; inactive; preserveAspect]
0.2497398944048823958
;[o]
0.24974
:[font = output; output; inactive; preserveAspect]
0.006692850924284855559
;[o]
0.00669285
:[font = output; output; inactive; preserveAspect; endGroup]
0.7502601055951176042
;[o]
0.75026
:[font = text; inactive; preserveAspect]


Step 3 Define Decision Rules

For example an expert might define the following common sense rules:

Rule 1 if protein is low and energy is low then feed is high
Rule 2 if protein is low and energy is high then feed is low
Rule 3 if protein is high and energy is low then feed is high
Rule 4 if protein is high and energy is high then feed is low

Step 4 Inference

Now we want to relate the observations to the rules.

First we are interested in  to what degree  an observation has membership in 
the fuzzy set associated with both  lowenergy and lowprotein (rule 1) and the other combinations associated with the remaining rules. The following memberships need to be evaluated.

lowenergy and lowprotein
lowenergy and highprotein
highenergy and lowprotein
highenergy and highprotein

These and operations can be considered as fuzzy sets with their membership
functions defined as the minimum of the values of the memberships on 
the component sets.


;[s]
5:0,0;2,1;30,0;349,1;365,0;954,-1;
2:3,13,9,Times,0,12,0,0,0;2,25,18,Times,1,24,0,0,0;
:[font = input; preserveAspect]
MinOpt := Min[FzSet1[[j]],FzSet2[[j]]]

FzSet1 = lowenergy;
FzSet2 = lowprotein;
:[font = input; preserveAspect; startGroup]
lowelowp =Table[MinOpt,{j,1,10}]
:[font = output; output; inactive; preserveAspect; endGroup]
{0.9933071490757151445, 0.8581489350995122104, 
 
  0.2497398944048823958, 0.006692850924284855546, 
 
  0.8581489350995122104, 0.2497398944048823958, 
 
  0.006692850924284855546, 0.8581489350995122104, 
 
  0.8455347349164652957, 0.006692850924284855546}
;[o]
{0.993307, 0.858149, 0.24974, 0.00669285, 0.858149, 0.24974, 
 
  0.00669285, 0.858149, 0.845535, 0.00669285}
:[font = input; preserveAspect; startGroup]
FzSet1 = lowenergy;
FzSet2 = highprotein;
lowehighp =Table[MinOpt,{j,1,10}]
:[font = output; output; inactive; preserveAspect; endGroup]
{0.006692850924284855559, 0.1418510649004877896, 
 
  0.7502601055951176042, 0.9933071490757151445, 
 
  0.006692850924284855559, 0.006692850924284855559, 
 
  0.006692850924284855546, 0.1301084743629978521, 
 
  0.1544652650835347043, 0.006692850924284855546}
;[o]
{0.00669285, 0.141851, 0.75026, 0.993307, 0.00669285, 0.00669285, 
 
  0.00669285, 0.130108, 0.154465, 0.00669285}
:[font = input; preserveAspect; startGroup]
FzSet1 = highenergy;
FzSet2 = lowprotein;
highelowp =Table[MinOpt,{j,1,10}]
:[font = output; output; inactive; preserveAspect; endGroup]
{0.006692850924284855559, 0.006692850924284855559, 
 
  0.006692850924284855559, 0.006692850924284855546, 
 
  0.1418510649004877896, 0.7502601055951176042, 
 
  0.9933071490757151445, 0.1418510649004877896, 
 
  0.1418510649004877896, 0.006692850924284855546}
;[o]
{0.00669285, 0.00669285, 0.00669285, 0.00669285, 0.141851, 0.75026, 
 
  0.993307, 0.141851, 0.141851, 0.00669285}
:[font = input; preserveAspect; startGroup]
FzSet1 = highenergy;
FzSet2 = highprotein;
highehighp =Table[MinOpt,{j,1,10}]
:[font = output; output; inactive; preserveAspect; endGroup]
{0.006692850924284855559, 0.006692850924284855559, 
 
  0.006692850924284855559, 0.006692850924284855559, 
 
  0.006692850924284855559, 0.006692850924284855559, 
 
  0.006692850924284855559, 0.1301084743629978521, 
 
  0.1418510649004877896, 0.9933071490757151445}
;[o]
{0.00669285, 0.00669285, 0.00669285, 0.00669285, 0.00669285, 
 
  0.00669285, 0.00669285, 0.130108, 0.141851, 0.993307}
:[font = text; inactive; preserveAspect]
Further these fuzzy sets can be used to define the outcomes of rules.
In this case the score of the rules premise is applied to
the rules conclusion.

For the rules defined step 3  each observation is scored on each rule. For example observation 3
has the following scores on these rules. These scores reflect the maximum level of membership for the observation on the fuzzy set defined for the output. (High feed and Low feed)

Rule 1 if protein is low and energy is low then feed is high
Rule 2 if protein is low and energy is high then feed is low
Rule 3 if protein is high and energy is low then feed is high
Rule 4 if protein is high and energy is high then feed is low
;[s]
3:0,0;39,1;68,0;675,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]



lowelowp[[3]]
lowehighp[[3]]
highelowp[[3]]
highehighp[[3]]
:[font = output; output; inactive; preserveAspect]
0.2497398944048823958
;[o]
0.24974
:[font = output; output; inactive; preserveAspect]
0.7502601055951176042
;[o]
0.75026
:[font = output; output; inactive; preserveAspect]
0.006692850924284855559
;[o]
0.00669285
:[font = output; output; inactive; preserveAspect; endGroup]
0.006692850924284855559
;[o]
0.00669285
:[font = text; inactive; preserveAspect]






The above output is associated with both the high feed and low feed rules.

What now?


For example observation 3 would result in the following
:[font = input; preserveAspect; startGroup]
minhigh[x_,y_]:=Min[y,high[x]]
minlow[x_,y_]:=Min[y,low[x]]
y = lowelowp[[3]]
R1 = Plot[minhigh[x,y],{x,0,10},PlotRange -> {0,1.0},AxesLabel -> {"Feed Level","Score Rule 1"},GridLines -> Automatic]
:[font = output; output; inactive; preserveAspect]
0.2497398944048823958
;[o]
0.24974
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[(0.2)] .01131 .12361 1 0 Msboxa
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[(0.6)] .01131 .37082 1 0 Msboxa
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y = lowehighp[[3]]
R2 = Plot[minhigh[x,y],{x,0,10},PlotRange -> {0,1.0},AxesLabel -> {"Feed Level","Score Rule 2"},GridLines -> Automatic]
:[font = output; output; inactive; preserveAspect]
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Step 5 Composition

The key is to define the fuzzy set that is inferred by the rule for a specific
observation. One way to do this is take the maximum of the score for the membership
function defined for the outcomes.










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y1= lowelowp[[3]]
y2 = lowehighp[[3]]
y3 = highelowp[[3]]
y4 = highehighp[[3]]

fud[x_]:= Max[minhigh[x,y1],minhigh[x,y2],minlow[x,y3],minlow[x,y4]]
Plot[fud[x],{x,0,10},PlotRange -> {0,1.0},AxesLabel -> {"Feed Level","Composite Score"},GridLines -> Automatic]



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Show[R1,R2,R3,R4,AxesLabel -> {"Feed Level","Score on Rules"}]
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Step 6 Defuzzification

To determine how much feed to use we need to defuzzify the rules. To do
this we first combine the results from all rules into a new fuzzy set and
then transform this composited fuzzy set to a crisp number.


;[s]
3:0,2;22,1;23,0;232,-1;
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cent = Table[fud[x],{x,0,10,.1}]

cum = FoldList[Plus,0,cent]
centat = Last[cum]/2


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;[o]
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21.72316320954368409
;[o]
21.7232
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Troid := If[cum[[j]] <= centat && cum[[j+1]] >= centat,j,0 ]
Droid = Table[Troid,{j,1,100}]
Centroid = Max[Droid]/10.
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7.3
;[o]
7.3
:[font = text; inactive; preserveAspect]


But what about observation 10


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y1= lowelowp[[10]]
y2 = lowehighp[[10]]
y3 = highelowp[[10]]
y4 = highehighp[[10]]

fud[x_]:= Max[minhigh[x,y1],minlow[x,y2],minhigh[x,y3],minlow[x,y4]]
Plot[fud[x],{x,0,10},PlotRange -> {0,1.0},AxesLabel -> {"Feed Level","Composite Score"},GridLines -> Automatic]



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0.00669285
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0.9933071490757151445
;[o]
0.993307
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Graphics["<<>>"]
;[o]
-Graphics-
:[font = input; preserveAspect; startGroup]
cent = Table[fud[x],{x,0,10,.1}]
cum = FoldList[Plus,0,cent]
centat = Last[cum]/2
Troid := If[cum[[j]] <= centat && cum[[j+1]] >= centat,j,0 ]
Droid = Table[Troid,{j,1,100}]
Centroid = Max[Droid]/10.

:[font = output; output; inactive; preserveAspect]
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y1= lowelowp[[5]]
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fud[x_]:= Max[minhigh[x,y1],minlow[x,y2],minhigh[x,y3],minlow[x,y4]]
Plot[fud[x],{x,0,10},PlotRange -> {0,1.0},AxesLabel -> {"Feed Level","Composite Score"},GridLines -> Automatic]


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cent = Table[fud[x],{x,0,10,.1}]
cum = FoldList[Plus,0,cent]
centat = Last[cum]/2
Troid := If[cum[[j]] <= centat && cum[[j+1]] >= centat,j,0 ]
Droid = Table[Troid,{j,1,100}]
Centroid = Max[Droid]/10.

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7.4
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7.4
^*)