Any linearized joint regression equation system of different observation series can be written out in the Canonical Block-Angular (CBA) form as follows:

where vectors **y**_{t,1},** y**_{t,2},…, **y**_{t,K}
and **e**_{t,1},** e**_{t,2},…, **e**_{t,K}
represent the measurements and their errors, respectively, from the different
observation series (k = 1, 2,..., K) during an observing period t (t = 1, 2,...).
Vectors **b**_{t,1},** b**_{t,2},…, **b**_{t,K} represent
entirely unknown regression parameters.
Those regression parameters that are common to several observation series
are represented here by a common vector **c**_{t} whose prior estimates at time t-1
are given by .
Matrices X_{t,1}, X_{t,2},…, X_{t,K} and G_{t,1}, G_{t,2},…, G_{t,K}
are the Jacobians that are related to the unknown parameters and to the common parameters,
respectively.
Vector **a**_{t,c} represents random errors of the common parameters
that typically stem from various calibration drifts and/or modeling errors.

Moreover, it is possible to determine an additional set of common parameters
for transforming the joint covariance matrix of the residuals
**e**_{t,1},** e**_{t,2},…, **e**_{t,K}
into a strictly **block-diagonal** form.
This can be done by making use of those **CANONICAL COMMON FACTORS**
(e.g. Empirical Orthogonal Functions G_{t,1}, G_{t,2},…, G_{t,K})
that are found by performing a Principal Component Analysis on
all the residuals after they have first been orthonormalized separately within every
block k (k = 1, 2,..., K).
This computational procedure for finding the conventional Canonical Correlations
between only two blocks (K = 2) of data was described by Lange already in
1969.
These tedious preparatory measures are required for speeding up the optimal
FKF computations in realtime.
Reference is also made to the discussion on Multivariate Statistical Methods in
Statistical Calibration of Observing Systems
(Lange, 1999), see page 16,
and PCT/FI96/00621
(Lange, 1997), see pages 11-12.