Continuous Random Processes
A random process X()t is as mentioned a random function of time meaning that for any point
in time the value of X
t is a random variable. A realisation of a random process (e.g. water
level variation) is illustrated in Figure 2.14.
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
1.50
2.00
2.50
3.00
3.50
Water level [m]
Time [days]
Figure 2.14: Realization of the water level variation as function of time.
In accordance with the definition of the mean value of a random variable the mean value of all
the possible realisation of the stochastic process at time t is given by:
XX()txfxt( , )
dx (2.58)
The correlation between all possible realisations at two points in time and is described
through the so-called autocorrelation function
t 1 t 2
RXX(, )tt 12. The autocorrelation function is
defined by:
RXX(, )tt 12 EXt Xt() ( ) 1 2 xxf xx tt dxdx 12 XX( , ; , )1 212 1 2
(2.59)
The auto-covariance function is defined as:
12 1 1 2 2
112 2 12121
(, ) ( () ())( ( ) ( ))
( ( )) ( ( )) ( , ; , )
XX X X
XXXX
Ctt EXt t Xt t
x t x t f x x t t dx dx 2
(2.60)
For tt 12 t the auto-covariance function becomes the covariance function:
(^2) () (, ) (, ) (^2) ()
XXXtCttRttXX Xt (2.61)
where ( )X t is the standard deviation function.
The above definitions for the scalar processX( )t
t))T
may be extended to cover also vector valued
processes having covariance functions
. For i these become the auto-covariance functions and when
these are termed the cross-covariance functions. Finally the correlation function may be
defined as:
X( ) (tXtXt X 12 ( ), ( ),..., n(
CXXXijcovij(), ( )tXt 12 j
ij/
12
12
12
cov ( ), ( )
(), ( )
ij() ( )
ij
ij
XX
X tXt
Xt X t
tt