5.4
=. OO
and have obtained an expression for the expectation of X(t), em e 6.
which is much more convenient to be studied than the infinite
individual expectations X(t,) for the different values ti.
In a similar way we can write down the moments of the second
order (variances and co-variances) of the function X(t), which
are given by | T.
v(X(t)) =E[(x(s) - x0)
K(t4,*5) - E[(X(,2-. X4) (X(t,) - X(t,))]
The latter magnitude K is frequently called correlation function
and the branch of the theory which only deals with X, V and
K, is known as correlation theory. Of very great practical
importance is the case in which the magnitude V(X(t)) and
K(t,>t,) do not depend on the origin of measurements of t,
but only on the difference (+, - t) between t and t. Then
V(X(t)) is a constant and K(t,>t,) is a function of only one
variable Tz t - i. nandom functions of this type are caliea
stationary, to distinguish them from the general n e
functions.
Based on this theory of random functions, the individual parts
of the processes of flight mission, film development and
measurement can be described rather completely. The parameters
X(t), V and K(7) of the random functions, which occur in these
processes, can be determined by experiments, or assumptions about
their size can be made for the time being. Then the stochastic
properties of the image coordinates can be defined by a
relatively small number of parameters and proper tests and
estimation methods can be developed to cope with the image
errors which so far we have called random and systematic. It
is also possible to estimate some of the parameters X, V ana K
simultaneously with the other unknowns, as.was done for the
conventional method of least squares by Kubik (1971) and others.
It is believed that the development of this theory will provide
the tools for a very sophisticated error theory, with which we e | ©
can even study such unconventional factors as large local |
atmospheric turbulences, unproper treatment of individual
photos and others.
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