341 
3 THE AUTOREGRESSIVE 
MODEL OF CLASS 
FREQUENCES 
Several different stochastic models can be used for the 
matic map class frequences modeling. However the com 
putational complexity is serious limiting factor, therefore 
the autoregressive model (7) was chosen. 
N 
Yt = X] AiYt-\ + A (7) 
>=i 
Where Y t E t are , A'—dimensional vectors, /1, are KxK ma 
trices of unknown model parameters and N is the order of 
model. E t is the white noise vector with following proper 
ties for t > N : 
7 (t — 1) = *y(N) + t — 1 — N 
Ul = Yy(t-1) — Ky(t-l)K(t-l)^y(i-l) 
Yy(t-1) Ky(t-l) 
Vl_i 
Ky(t-i) Yz(t-1) 
T 
H«-.)= E YkYi 
k=N+1 
E Z t Yl 
k=N+l 
k (1 -,i= E 
k=N+1 
V/v is a positive definite matrix and 
l(N) > N(l + K) - 2 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
£[£<] = 0 
E[Ê t ÊJ_ x ] = 0 i J. 0 i < t (8) 
4 NUMERICAL 
REALIZATION 
= 0 0 < i < t 
We assume, that probability density of E has multidimen 
sional normal distribution independent of previous data 
and is the same for every time t. 
E\È,ÈJ) = n (9) 
fi is a constant covariance A'—dimensional matrix. The 
task consist of finding the estimation Y (3) in dépendance 
of known process history. 
= (10) 
To construct estimator (3), we need to derive the condi 
tional probability density 
(ii) 
Using Bayesian estimation theory (Peterka,1981), we 
can express (11) in the form of Student’s distribution 
PWIU'-») = ir K > 2 T((i(t) -P+K + l)/2)/{r(( 7 (() 
-/» + 1)/2)(1 + ZjV^Z,)^ 2 |A,_,|'' 2 [1 + (Y, - Pl lZi f 
K-i(Y, -PLz,)H 1 + zfvr f !_ n z,)}"w-s+K + m ]{12) 
The predictor (13) can be evaluated using matrix V t 
(17) updating and its following inversion. Another pos 
sibility is direct updating of P t . According to work (Pe- 
terka,1981), to ensure the numerical stability of solution , 
it is advantageous to calculate (15) by the means of the 
square-root filter REFIL (Peterka,1981), which guarantees 
the positive definiteness of matrix (17). The filter REFIL 
updates directly the Cholesky square root of the matrix 
vr l • 
The numerical complexity of proposed classifier is larger 
than the conventional per-point Bayesian one. If we denote 
the number of arithmetic operations necessary to classify 
one pixel then the Bayesian classifier in its most effi 
cient version needs: 
h(*) = I<d(d + 3)/2 h{+) = I<(d - 1 )(d + 2)/2 + 2K 
The contextual Bayesian classifier is computationally 
more demanding: 
h(*) = Kd(d + 3)/2 + 3K 2 N + 2K 2 N 2 + SEN + 16A 
h{+) = K(d— l)(d+2)/2+K+3K 2 N+1.5K 2 N 2 +'2.5K N+n 
with conditional mean value ( 3) 
y; = PJ_ x z t 
Where n is the number of pixels in thematic map win 
dow. To avoid overflow problems the smallest possible sin- 
(13) gle class predictor value (13) was chosen to be 0.001. 
where P t -1 is estimation (15) of the Kx/3 matrix (14) 
P T = [Au- 
.., Ayv], 
(14) 
A-1 = v~i 
-l)Ezy(t—1) 
(15) 
Zt = [Y t T _ 1 ,. 
V’T ] T 
■ ■ ? U-aJ 
(16) 
is the flxl data vector (/? = KN). The following notation 
was used in (12): 
vu = K_! + V N (17) 
5 EXPERIMENTAL RESULTS 
The contextual classification algorithm was applied to 
agricultural type of Thematic Mapper subscene from North 
Moravia . The comparison was made by the Bayesian per- 
point classifier. The area studied is large cooperative farm 
situated in Vizovice Hills. The objective of the study was 
to determine its land-use, land-cover conditions. Ground 
areas of homogeneous landforms and land cover conditions