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thode der
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ssification
he results
ncy table:
.e. winter
e number
se fallible
eloped by
in (1972)
; the pro-
(1)
where p is the true proportion and 0 and $ are probabi-
lities of misclassification and q is 1 — p. This equation
applies to the binomial case with only one land use
category.
In case of k mutually exclusive land use categories,
(multinomial case) the result for the category j is
k
mi =) pili (2)
i=1
where 0;; is the probability that a unit, which belongs
to the category ? is classified to the category j. The
bias in the estimates 7; — p; is corrected ba applying
the following equation
P—AxII (3)
where P is & x 1 column vector of the estimates p;, À is
kxk Matrix with the terms nj; — aij/a.; and II is the
kx 1 column vector with the results of the maximum
likelihood classification. To correct the misclassified
results of the ML classifier given by the vector II the
error matrix À must be known.
Let us consider the results of the ML classificati-
on with multitemporal ERS-SAR GTC data for five
major crops. In the study area following contingency
table is available from which A can be derived:
fallible ML classifier
Crop 1 2 3 4 5 6 3
T 1. 2751. 794. 907. 419. 120 2019... 7010
r à 186 ..511...166. .211...41.;-515...1630
u 3 681 352 824 263 107 868 3095
e 4 211 915 420 2368 460 2256 6630
5 55 184 94 368 111 464 1276
GIS 6 305 412 242 341 65 589 1954
> 4189 3168 2653 3970 904 6111 21595
Where: 1 - winter wheat, 2 - winter barley, 3 - summer
barley, 4 - sugar beets, 5 - maize and 6 - others
The disadvantage of the above procedure is obvious.
The true classifications have to be provided for the
entire area. But this data is not available. This is the
am of the image classification.
Some information can, however, be gained using
sampling techniques. Suppose a small sample can be
provided with true classifications. The size of that
sample is denoted n. The entire area consisting of N
pixels is classified using only the ML- classifier. Thus
the n members of the sample are classified twice.
The results give following contingency table:
fallible maximum
likelihood
classifier
Crop s3da 2 k
1 311 9122 ‘+ ay aj.
True 2 ag; 822 Ak a2.
classifier 3 a31 aa23 -:- ask as.
(GIS)
k' “ayy Vago ev. Cag | ar.
an An par n
X1 Xo X3 Xk N-n
where a;; is the number of pixels in the sample whose
true category is ¢ and whose fallible category is j. X =
(X1, X2, ---, Xx) is a vector of frequencies, where X;
is the number of pixels in the image with N units
whose fallible category is 7.
After Tenebein (1972), estimates of p and the
misclassification probabilities 0 can be derived as fol-
lows
k
pi =) aij(X; + a;)/(a;N) (4)
j=1
ó; = (X; + a j)aij/ (a.5.N pi) (5)
The results of the maximum likelihood classificati-
on of the N - members of the image (X; + a ;) are
corrected by multiplying with the ratios a;;/a.; and
summing over j. In the case of k land use categories,
the estimates are derived as given in equation 3.
In addition a coefficient of reliability K is defined. It
measures the strength of the relationship between the
true measurements and maximum likelihood classifier
for each category.
: k
K: = pi(Y 68/73 — 1)/gi (6)
j=1
The variance of p; is
: Didk 7 Piqk
iz 1 —E; K;
vig) =a ky + BK)
If K = 0, the true classification is the only way to
obtain reliable estimates for the fraction of considered
crop area. If K = 1, the ML classifier is not fallible,
thus the true classifications are not required. In samp-
ling optical data K is within the range of 0.3 - 0.8 (
Smiatek, 1993).
It is necessary, however, to accept the sampling er-
ror as the error probabilities are estimated from a sam-
ple. But, the required sample size can be estimated
according to the desired accuracy criteria. This is the
great advantage of the sampling procedure proposed
here.
625
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996
