Fuzzy Modeling Library (FMLib)

Fuzzy Modeling Library (FMLib)
fun1-2-1681: Modeling of the Three-Dimensional Surface F1

Description

Name: fun1-2-1681
Type: Laboratory problem

Number of input variables: 2
Number of training examples: 1681

Domain of the input variable 1: [-5, 5]
Domain of the input variable 2: [-5, 5]
Range of the output variable: [0, 50]

The aim in this problem is to model the three-dimensional surface generated by the mathematical function F1 shown below.

In this problem, seven linguistic terms are usually considered for each variable in linguistic fuzzy modeling.

Data Sets

A training data set uniformly distributed in the two-dimensional definition space has been obtained experimentally. In this way, a set with 1,681 values has been generated for the function F1 taking 41 values for each one of the two input variables considered to be uniformly distributed in their intervals.

On the other hand, the test data is obtained generating the input variable values at random in the concrete universes of discourse for each one of them, and computing the associated output variable value. Two test data sets with 168 (9.1%) and 420 (20%) examples have been generated.

Training data set (1681 examples): fun1-2-1681.tra
Test data set 1 (168 examples): fun1-2-1681-1.tst
Test data set 2 (420 examples): fun1-2-1681-2.tst

Results

Linguistic Fuzzy Modeling
Method Type	Reference	Method	No. Rules	No. Labels	Training	Test	Comments
Learning only the rule base (highest interpretability)
	[CCH01]	Wang & Mendel	49	21	2.048137	2.287129	Test data set 2
	[CCH01]	COR	49	21	1.605482	1.175941	Test data set 2
Learning/tuning also the data base
Learning/tuning also the data base	[CH97]	MOGUL-D	62	21	0.335800	0.262500	--

Precise Fuzzy Modeling
Method Type	Reference	Method	No. Rules	No. Labels	Training	Test	Comments
Approximate FRBSs
	[CH97]	MOGUL-A1	76	228	1.462900	0.951800	--
	[HLV98]	MOGUL-GLP	64	192	---	0.768021	--
TSK-type FRBSs
TSK-type FRBSs	[CH99]	MOGUL-TSK	49	21	0.006921	0.007498	--

References

The application was originally proposed in:

[CH97]

O. Cordón, F. Herrera, A three-stage evolutionary process for learning descriptive and approximate fuzzy logic controller knowledge bases from examples, International Journal of Approximate Reasoning 17:4 (1997) 369-407.

The data has been also used in the following papers:

[HLV98]	F. Herrera, M. Lozano, J.L. Verdegay, A learning process for fuzzy control rules using genetic algorithms, Fuzzy Sets and Systems 100 (1998) 143-158.
[CH99]	O. Cordón, F. Herrera, A two-stage evolutionary process for designing TSK fuzzy rule-based systems, IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics 29:6 (1999) 703-715.
[CCH01]	J. Casillas, O. Cordón, F. Herrera, COR: A methodology to improve ad hoc data-driven linguistic rule learning methods by inducing cooperation among rules, IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics, 2001. To appear.