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
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