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2: Raghuveer M. Rao, Manoj K. Arora
The higher the correlation coefficient better is the classification accuracy of
a class.
Thus, it can be seen that a number of measures may be adopted to evaluate
the accuracy of a fuzzy classification. However, as pointed out by Binaghi et
al. (1999), though each measure has something to offer, none is universally
applicable. In their opinion, a common constraint with them is that they do
not preserve the location information and therefore propose a fuzzy error
matrix based approach.
The layout of a fuzzy error matrix is similar to an error matrix (see
Table 2.1), to evaluate crisp classifications, with the exception that elements
of a fuzzy error matrix can be any non-negative real numbers instead of nonnegative integer numbers. The elements of the fuzzy error matrix represent
class proportions corresponding to reference data (i. e. fuzzy reference data)
and classified outputs (i. e. fuzzy classified image) respectively.
Let Rn and Cm be the sets of reference and classification data assigned to
class nand m, respectively where the values of nand m are bounded by one
and the number of classes L. Note here that Rn and Cm are fuzzy sets and {R Il }
and {Cm } form two fuzzy partitions of the sample data set Xwhere x denotes
a sample element in X. The membership functions of R'1 and Cm are given by
JlRn : X ~ [0,1]
and
Jlc m : X ~ [0, 1] ,
(2.31)
(2.32)
where [0, 1] denotes the interval of real numbers from 0 to 1 inclusive. Here,
JlRn (x) and Jlc m (x) is the gradual membership of the testing sample x in Rn and
Cm , respectively. Since, in the context of fuzzy classification, these membership
functions also represent the proportion of a class in the testing sample, the
orthogonality or sum-normalization is often required, i. e.
",C JlRt(x) = 1 .
~l=l
(2.33)
The procedure used to construct the fuzzy error matrixM employs fuzzy min
operator to determine the element M(m,n) in which the degree of membership
in the fuzzy intersection Cm n Rn is computed as
M(m, n) = ICm n Rnl = L min (Jle m , JlR II ) •
(2.34)
XEX
Once the fuzzy error matrix is generated, conventional error matrix based
measures such as OA, PA and UA may be computed in the similar fashion to
indicate the accuracy of a fuzzy classification and of individual class. Thus,
the use of fuzzy error matrix based measures to evaluate fuzzy classification
have conformity with the measures based on conventional error matrix based
measures for crisp classification, and therefore may be more appropriate than
the distance and entropy based measures. However, further research needs to
be conducted to operationalize these measures in remote sensing community.
2: Raghuveer M. Rao, Manoj K. Arora
The higher the correlation coefficient better is the classification accuracy of
a class.
Thus, it can be seen that a number of measures may be adopted to evaluate
the accuracy of a fuzzy classification. However, as pointed out by Binaghi et
al. (1999), though each measure has something to offer, none is universally
applicable. In their opinion, a common constraint with them is that they do
not preserve the location information and therefore propose a fuzzy error
matrix based approach.
The layout of a fuzzy error matrix is similar to an error matrix (see
Table 2.1), to evaluate crisp classifications, with the exception that elements
of a fuzzy error matrix can be any non-negative real numbers instead of nonnegative integer numbers. The elements of the fuzzy error matrix represent
class proportions corresponding to reference data (i. e. fuzzy reference data)
and classified outputs (i. e. fuzzy classified image) respectively.
Let Rn and Cm be the sets of reference and classification data assigned to
class nand m, respectively where the values of nand m are bounded by one
and the number of classes L. Note here that Rn and Cm are fuzzy sets and {R Il }
and {Cm } form two fuzzy partitions of the sample data set Xwhere x denotes
a sample element in X. The membership functions of R'1 and Cm are given by
JlRn : X ~ [0,1]
and
Jlc m : X ~ [0, 1] ,
(2.31)
(2.32)
where [0, 1] denotes the interval of real numbers from 0 to 1 inclusive. Here,
JlRn (x) and Jlc m (x) is the gradual membership of the testing sample x in Rn and
Cm , respectively. Since, in the context of fuzzy classification, these membership
functions also represent the proportion of a class in the testing sample, the
orthogonality or sum-normalization is often required, i. e.
",C JlRt(x) = 1 .
~l=l
(2.33)
The procedure used to construct the fuzzy error matrixM employs fuzzy min
operator to determine the element M(m,n) in which the degree of membership
in the fuzzy intersection Cm n Rn is computed as
M(m, n) = ICm n Rnl = L min (Jle m , JlR II ) •
(2.34)
XEX
Once the fuzzy error matrix is generated, conventional error matrix based
measures such as OA, PA and UA may be computed in the similar fashion to
indicate the accuracy of a fuzzy classification and of individual class. Thus,
the use of fuzzy error matrix based measures to evaluate fuzzy classification
have conformity with the measures based on conventional error matrix based
measures for crisp classification, and therefore may be more appropriate than
the distance and entropy based measures. However, further research needs to
be conducted to operationalize these measures in remote sensing community.