The Application of Fuzzy Interval Correlation Evaluating the Relationship Between Transportation Engineering and Air Pollution - Innovative Management in Information and Production - page 266

stock markets was developed ( Wu and Hsu 2002 ). In traditional statistical theory,
the observations should be observated under probability distribution. In practice, the
observations are sometimes explained by linguistic terms such as “Very important,”
“Important,” “Normal,” “Unimportant,” “Very unimportant,” or “Maximum value
and Minimum value,” which are only approximately known, rather than equating
with randomness. Measuring the correlation coefficient between two variables
including fuzziness is a challenge to the classical statistical theory. A lot of studies
investigate the topic of the fuzzy correlation analysis and its application in the
social or economic science fields ( Bustince and Burillo 1995 ; Yu 1993 ; Liu and
Kao 2002 ; Hong 2006 ). For example, Hong and Hwang ( 1995 )and Yu ( 1993 )
define a correlation formula to measure the interrelation of intuitionist fuzzy sets.
However, the range of their defined correlation is from 0 to 1, which contradicts
with the conventional awareness of correlation which should range from 1to1.

In order to overcome this issue, Chiang and Lin ( 1999 ) take random sample from the
fuzzy sets and treat the membership grades as the crisp observations. Their derived
coefficient is between

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1 and 1; however, the sense is that the fuzziness is gone.
Liu and Kao ( 2002 ) calculated the fuzzy correlation coefficient based on Zadeh's
extension principles. They used a mathematical programming approach to derive
fuzzy measures based on the classical definition of the correlation coefficient. Their
derivation is very probable; however, in order to use this scheme, the mathematical
programming should be required.
In addition, formulas in these studies are quite complicated or required some
mathematical programming which really limited the access of some researchers with
no strong mathematical background. In this thesis, we propose a simple solution
of a fuzzy correlation coefficient without programming. In addition, the provided
solutions are based on the classical definition of Pearson correlation which is quite
easy and straightforward. The definitions provided in this study can also be used for
interval-valued fuzzy data.
The remainder of the paper proceeds as follows. The fuzzy interval correlation
is introduced in Sect. 2 . Section 3 presents its results of the relationship of the
simulation. Section 4 presents its empirical results. Finally, the conclusions are
drawn in Sect. 5 .

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2 Fuzzy Interval Correlation

In general, we need to study the relationship between the variables x and y ;the
most direct and simple way is to draw a scatter plot, which can approximately
illustrate the relationship between these variables such as positive correlation,
negative correlation, or noncorrelation. Pearson's correlation coefficient is often
considered to evaluate that presents a measure of how two random variables are