With these new expressions, the law of variance 
propagation permits the expression of the 
corresponding covariance matrices in equally 
convenient forms. The covariance matrix of the 
estimated signal and the residual noise become 
respectively: 
1 
C=C > 6° (BR PB) La 
ss ss n 
+ ot (B‘PBC B'PB « o? B'pp)! (A.8) 
n ss n 
C= oF IP = BB ERY 5] 
nn n 
+ o* B(B'PBC B'PB + o B'PB) 'B (A.9) 
n ss n 
Moreover, taking into account (A.1), the estimated 
value of the observables can be written as: 
(A. 10) 
Applying the law of variance propagation, its 
covariance matrix becomes: 
t 
C** = BC -B (A. 11) 
ox ss 
Finally, indicating with the symbol e the error in 
the estimate of the signal, i.e. the difference 
between its theoretic value and its estimate: e = 
s - s, the covariance matrix C becomes, taking 
into account (A.8), and applying the law of 
covariance propagation, 
C -C -C^- 
ee ss ss 
c? (B'PB)'!« o* (B*PBC_ B*PB + o% B*PB)-* (A.12) 
n n ss n 
Consequently one has: 
C~=2C -C (A. 13) 
ss ss ee 
and: 
Qm eg pl-n Bim Pho (A. 14) 
nn n ee ££ 
where the last matrix in expression (A.14) is the 
covariance matrix of the estimate for the expected 
value of the observables: 
BC B 
ee 
C = (A.15) 
££ 
having indicated with the error of the estimate 
of the expected value of the observables, i.e. the 
difference between its theoretical value and its 
estimate: 
388 
€ =o - & 7 B(s - s) = Be 
The least squares criterion, 
formulation of the  collocation method, can 
provide, besides an estimate for a filtered 
signal, also an estimate for a predicted signal 
s=t: the stochastic parameters can also be 
eBtimated in every point. One has to keep in mind 
however, that only the properly called stochastic 
parameters can be estimated. 
Consequently the covariance matrix C only consists 
of the properly called stochastic parámeters, and 
the crosscovariance matrix between the filtered 
expressed in the 
and the predicted signal C ,(C ,- ct )is divided 
in two parts: one contáinihg the covariance 
between the predicted signal and the properly 
called stochastic parameters in the filtered 
signal, and one identically zero. This null matrix 
is exactly the reason of the impossibility to 
predict the parameters, which are strictly non 
stochastic. 
Given the functional: 
C 0 S 
ss st 
1 m ta lic C gl | 
2 ts tt 
0 à Prn |n 
+ A (Be = Ot + n - €) = min 
A being a vector of Lagrange multipliers, and 
taking into account expression {A.7), one has: 
t zc ppc B P')'a = 
ts ss n 
- c B'PB (B'PBC B'PB « o? B'PB) !B'P a^ (4.16) 
ts ss n 
or. 
t = z 
with z a service vector: 
z = B'PB (B'PBC .B'PB + o? B*PB) !p'P a* 
s 
which is to be computed once at the end of the 
filtering. 
Applying the law of covariance propagation, the 
covariance matrix of the predicted signal becomes: 
C^ - C B'PB (B'PBC B'PB « o? B'PB) !B'PBC | (A. 17) 
tt ts ss n st