I need a mathematical equation of the form z= f(x, y) where x's domain is (0,1), y's domain is (0,4), and z must be [0 , 1]. Are there any methodology.
To be more accurate, x is the weight of a fuzzy system, y is the output of this fuzzy system, and z is the final weighted output of this fuzzy system, which will be entered in another process.
Would a trigonometric function work?
For example, z = sin(pi/2*(x/2+y/8)) ?
Thank you very much Professor Daniel.
I do not know about this equation, but the main point is that the z value must behave as x and y. I means if x1 > x2 and y1>y2 then must z1>z2 in the same proportion.
Hi Shaker,
Looks like you are looking for a monotonic increasing curvature. I think you can come up with many equations. On top of Dr. Poitras' suggestion, here is mine, an elliptic paraboloid:
z = 0.5*x2 + 0.5*(y/4)2
BY the way, what is the input to the fuzzy system?
Hi Professor Chak,
Thank you.
The value of x is the medical weight of the fuzzy system, and the y value is it risk level.
We have another fuzzy system which collect all of the first layer's systems to take the final decision.
the second layer system expects that all of its inputs to be percentage of risk [0, 1].
As a result, we need a mechanism to convert the two numbers at each system in to one representative number to enter into the second layer system.
For your suggested equation, yes. its results may be in [0,1], lets try
x=0.3 and y= 4 (as extreme value) then z=0.5. the value of z can not represent the values of x and y.
I hope that you got the idea.
Thanks again.
@r Shaker,
@ Followers
The question is not simple as it appears.
Notice that the domain of z =f(x,y) is an open rectangle
R={(x,y), 0
Dear Professor Kaddoura,
Thank you.
Ok. Let's assume that z is in (0,1). But it is important for z to be monotonic increasing as x and y increase.
Thanks.
Hi Shaker,
Thanks for your reply. You may work out some functions based on Prof. Kaddoura's suggestion.
If you are unfamiliar with various math functions, you may pick something else that suits your understanding and needs. You will have greater flexibility to design a system in the intermediate layer that fits your requirements by using a two-input single-output fuzzy system or a neural network.
For example, with the following input data set, you can freely assign the values of z that best represent the values of x and y, according to your medical knowledge. No one knows better than you do. If you know exactly what you are doing, you can design the intermediate fuzzy system from scratch. Else, you can use ANFIS (adaptive neuro-fuzzy inference system) to train the given data to predict the values of z.
(x, y)
0.00, 0
0.00, 1
0.00, 2
0.00, 3
0.00, 4
0.25, 0
0.25, 1
0.25, 2
0.25, 3
0.25, 4
0.50, 0
0.50, 1
0.50, 2
0.50, 3
0.50, 4
0.50, 5
0.75, 0
0.75, 1
0.75, 2
0.75, 3
0.75, 4
1.00, 0
1.00, 1
1.00, 2
1.00, 3
1.00, 4
https://www.mathworks.com/help/fuzzy/neurofuzzydesigner-app.html
Dear Professor Chak,
It seems a good idea. I am a computer scientist not physician, and all the time it is difficult to find a medical expert.
Any way, I can try even I will complicate the process because I have 7 fuzzy systems connected to each other. I was thinking in a math equation that convert these two numbers into one number just in the fly.
Thank you very much.
Regards.
Professor Kaddoura
Thank you.
for z=f(x,y)=ax+(((1-a)/4))y
if x=0.4 and y= 3.7, then z= 0.6625.
if x=0.7 and y=3.7 then z=0.8125.
if x= 0.9 and y= 3.7 then z= 0.9125
if x= 0.1 and y= 3.7 then z=0.5125.
if x=0.1 and y= 1 then z= 0.175
It looks good for me.
I will check it again and come to you.
Thank you so much.
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