adapting partial least squares into fuzzy regression - Agrostat 2010

ADAPTING PARTIAL LEAST
SQUARES INTO FUZZY
REGRESSION
Murat Alper Basaran
Nigde University, TURKEY
Biagio Simonetti
University of Sannio, ITALY
Luigi D’Ambra
University Federico II of Naples, ITALY
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Outline
• Brief Information on Partial Least Squares (PLS).
• Brief Information on Fuzzy Set Theory (FST)
• Fuzzy Linear Regression (FLR).
• Illustrating the necessity of FLR with data as an
alternative modeling.
• Illustration of PLS based FLR using data set.
• Conclusion
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Partial Least Squares (PLS)
• Applied to multiple and multivariate cases.
• The main objective is to reduce the dimension of data.
• The components are constructed which are the linear combination of
original independent variables.
• Based on the constructed components, regression equation is
developed.
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Fuzzy Set Theory (FST)
• Introduced by Zadeh in 1965 to model impression in natural language.
Since then, it is the fundamental part of Soft Computing and is used in many
disciplines.
• A fuzzy set is a function from real numbers to [0,1].
• As a mathematical tool, the function generally called membership function is
used to represent an impression given by either a word in natural language
such as young and rich or a measurement which can not be gauged
exactly.
• Fuzzy number is a membership function which satisfies some conditions
such as normality (at least one element has 1 membership grade),
boundedness (bounded from below and above)
• The most used fuzzy numbers are symmetric and asymmetric triangular
ones.
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Fuzzy Linear Regression (FLR)
• When data set consisting of a few number of observations
• Assumptions of classic regression is not met.
• When data set consisting of verbal statements.
Alternative modeling technique called FLR can be an option which is given as follows:
Y X A
Reel Reel Fuzzy
Fuzzy Reel Fuzzy
Fuzzy Fuzzy Fuzzy
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Fuzzy Linear Regression (FLR)
Two research directions related to FLR are :
1. Improving parameter estimation techniques
2. Dealing with the problems occured in the phase of appyling it to real data
sets.
When literature is examined, it is observed that application part of FLR is
relatively not robust. Therefore, several problems possibly emerging from
the application such as missing observations, outliers, influential cases,
and other types of issues need to be overcome.
In this paper, the second part of the research is dealt with when data set
consisting of many independent vaiables.
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Illustration of FLR
• 5 independent variables and one dependent variable
• X1:quality of the construction material
• X2: area of he first floor
• X3: area of the second floor
• X4: total number of room
• X5:total number of room, number of Japanese room
• Y: Price
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Illustration of FLR
• 6 independent and one dependent variables (Chocolates Data )
• X1:Size
• X2:Energy
• X3:Protein
• X4:Fat
• X5:carbohydrate
• X6:Sodium
• Y: Price
Running PLS, then one component is constructed.
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Conclusion
• PLS should be extended for fuzzy independent and dependent variables in
order to be more applicable.
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Bibliography
• Tanaka, H., Uejima, S., Asai, K., (1982). regression
analysis with fuzzy model, IEEE Transactions on
Systems, Man, and Cybernetics SMC-12, 903-907.
• Tanaka,H., Guo, P., (1999). Possibilistic Data Aanalysisi
for Operation Reserach, Springer-Verlag, New York.
• Diamond, P., (1988), Fuzzy least squares, Information
Science ,46, 141-157.
• Zadeh, L.A., (1978), Fuzzy sets as a basis for a theoryof
possibility, Fuzzy Sets and Systems, 1(1), 3-28.
• Garthwaite., P.H., (1994)., An Interpretation of Partial
Least Squares, Journal of the American Statistical
Association, Vol. 89, No. 425., pp. 122-127.
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