gery, 
rector 
an be 
egory 
| each 
(0) 
ressed 
@) 
and |. 
in Eq. 
(3) 
ien the 
ector I 
PL: Pel XB -SU-ABYZ "Q-ÀB) 
(21)? det(Z^) 
VAR... CON M 
Z! = : : : *diag[o^,,..,0? 4] (4) 
cov, ^ VAR, 
Ay e Aw 
dz 
Aui Ayn 
The category proportion B has following constraints, 
0<B;<1, g=1,...N), 
N 
Y 5-1 5 
7=1 
From the concept of maximum likelihood estimation 
(Rodgers, 1976), the proportion which maximize P(I; B) 
under the above constraints is the solution. This problem 
can be solved by constrained non-linear optimization 
technique (Konno and Yamashita 1978). 
3. VERIFICATION 
To verify the maximum likelihood estimation, the 
comparison between the result from the proposed method 
and conventional method (generalized inversion method) 
based on the simulated Mixel data. 
3.1 Simulated Mixel Data 
3.1.1 Classified Categories and the Supervised Data In 
this study, the classified categories are selected from sub- 
scene (380 pixels and 310 lines) of LANDSAT-5 TM data 
of HAKONE area of Japan which acquired at 6. June '87. 
As the classified categories, 5 categories of residential area, 
bare soil, grass land, broad leaf tree and needle leaf tree are 
selected from vegetation map and land use map (Amano, 
Furuya and Ishikawa, 1990). Average and variance of each 
category are calculated and shown in Table 1 and 2, 
respectively. These data are use as the supervised data for 
the estimation. The calculation of P(I; B) of Eq. (4) is 
numerically sensitive to the correlation matrix R of each 
band, Table 3 shows the correlation matrix R calculated 
from whole image data except band 6 (thermal IR band). In 
this study, because of the high correlation of each band, the 
first and second principal components (stretched into 8 bits) 
of the imagery are used as the multispectral data. Table 4 
and 5 show that eigenvalue and eigenvector of each 
principal component and the average and variance of the 
supervised data in the principal components, respectively. 
3.1.2 Generation of Simulated Mixel Data The 100 
simulated Mixel data are generated in the following manner. 
(1) Category Proportion 
By using uniform random number from 0 to 1, proportion 
of each category is generated as in Eq. (6), 
= N EI . (6) 
where r, is uniform random number from 0 to 1. 
(2) Supervised data 
In this method, the supervised data of each category (each 
component of matrix A of Eq. (1) ) is allowed to fluctuate, 
so by using random number in multivariate normal 
distribution (Takane, 1980) which average and variance are 
those of each category and band. 
(3) Observation Error 
Also the observation error of each band is generated by 
normal distribution which average is 0 and variance is 1/12 
[count?]. This variance correspond to quantitization error. 
(4) Mixel data 
After above simulated data from (1—3) and Eq. (1), the 100 
Mixel data is generated. 
3.2 Generalized Inversion Method 
As a conventional category proportion method, the 
generalized inversion method (Inamura, 1987) is used in this 
study to compare with the proposed method. In this case the 
number of category (5 categories) is greater than the number 
of band (2 bands), so this problem is non-determine. And 
this method, the non-negative constraint is not considered. 
4. RESULT 
Eq. (4) is solved by the grid search method with 1/64 of 
step width (Kanno and Yamashita, 1978). From the results 
of two methods, two kinds of root means square error 
(RMSE) are calculated, one is total RMSE: RMSE, and the 
other is maximum RMSE: RMSE,, 
$13 j=1 (7) 
where BS, B® and S are estimated proportion of j-th 
category, actual proportion of j-th category and number of 
the Mixel data (100), respectively. Table 6 shows these 
RMSE's from the two methods. the result from maximum 
likelihood estimation shows the better result correspond to 
that from generalized inversion method. This is because that 
in maximum likelihood estimated, supervised data is 
represented by 2 parameters (average and variance), while 
in generalized inversion method it is represented by only 
one parameter (average). 
5. MAXIMUM ERROR OCCURRENCE PROBABILITY 
To predict the degree of estimation error in each category, 
the index named maximum error occurrence probability 
between category k an category | : P.(E k, I) is proposed as 
follows.