The Application of Fuzzy Interval Correlation Evaluating the Relationship Between Transportation Engineering and Air Pollution - Innovative Management in Information and Production - page 268

Fig. 1 Fuzzy correlation with interval data

2.1 Definition 1

Let ( X i =[ a i , b i , c i , d i ] , Y i =[ e i , f i , g i , h i ] ; i = 1 , 2 ,..., n ) be a sequence of paired

trapezoid fuzzy sample on population Ω with its pair of centroid ( cx i , cy i ) and pair

of area x i = area ( x i ) , y i = area ( y i ) .

i

Σ ( cx i − cx )( cy i − cy ) ln ( 1 + | ar xy | )

= 1

Σ

2 Σ

r xy =

2 , λ

ar xy = 1 − ,

i i

| ar xy |

( cx i − cx ) ( cy i − cy )

= 1 = 1

where

n

Σ

i = 1 (

x i − x i )( y i − y i )

ar xy = Σ

2 Σ 2 .

(2)

n n

i = 1 (

x i − x i )

i = 1 (

y i − y i )

Then fuzzy correlation is defined as:

1. W hen cr xy ≥ 0 , λ ar xy ≥ 0 , fuzzy correlation =( cr xy , min ( 1 , cr xy + λ ar xy ))

2. W hen cr xy ≥ 0 , λ ar xy < 0 , fuzzy correlation =( cr xy − λ ar xy , cr xy )

3. W hen cr xy < 0 , λ ar xy ≥ 0 , fuzzy correlation =( cr xy , cr xy + λ ar xy )

4. W hen cr xy < 0 , λ ar xy < 0 , fuzzy correlation =( max ( − 1 , cr xy − λ ar xy ) , cr xy )

2.2 Definition 2

Let X ji =[ a 1 i , a 2 i ] and Y ji =[ b 1 i , b 2 i ] be a sequence of paired fuzzy sample on

population Ω .Let

b k )

n
i

Σ ( a ji − a j )( b ki −

= 1

r jk =

2 2 ,

= , , = , .

( j 1 2 k 1 2

b k )

a ji − a j ) ( b ki −