15.4 Ensemble Averages, Mean and Autocorrelation
535
If X 1 and X 2 have zero means, the covariance is identical to the autocorrelation.
From the above statistical averages, we obtain a gross description of the probability
distribution. Just as in rigid body dynamics, the total mass, the mass moment of
inertia, the product moment of inertia and centroid give only gross information,
which do not enable us to make more detailed study about certain quantities, such
as the internal force; similarly, mean, variance and so on, only provide a somewhat
limited information. In order to make a more refined study of the random processes,
detailed information about probability distributions are required.
15.5 Stationary Process, Ergodic Process and Temporal
Averages
We have so far considered the properties of random process at a fixed time instant or
at two fixed time instants. The variation of these properties, when time instants are
assumed to vary is discussed next.
15.5.1 Stationary Process
A random process is said to be stationary, if its probability distributions are invariant
under a shift of time scale, that is, independent of the origin. This suggests that
the probability density at any time instant is valid for later times as well. This is
somewhat analogous to the assumptions made in the case of steady-state vibration,
where the resulting vibration is independent of the initial conditions.
Wave and wind generation are random processes, which can be treated as
stationary processes without introducing much error in the analysis. Earthquake
ground motions, however, cannot be treated as stationary processes, as the probability distributions vary with time. For a process to be strictly stationary, it can have
no beginning and no end and each sample should extend from −∞ to +∞.
For stationary processes, the first-order probability distribution p(x) is independent
of time. As such E [X ] and σ
2 are also independent of time.
In case of second-order probability density function, p(x 1 , x 2 ) is a function of
time lag between t 1 and t 2 and not a function of t 1 and t 2 individually. Time lag
τ = t 2 − t 1 . Then the autocorrelation function is
E[X 1 X 2 ] = E[X (t)X (t + τ )] = R(τ )
(15.20)
Since by definition of stationary process, the probability density function p(t, t +
τ ) is independent of t, the autocorrelation function is related to time lag only.R(τ )
will be used to denote autocorrelation function of a stationary random process.
535
If X 1 and X 2 have zero means, the covariance is identical to the autocorrelation.
From the above statistical averages, we obtain a gross description of the probability
distribution. Just as in rigid body dynamics, the total mass, the mass moment of
inertia, the product moment of inertia and centroid give only gross information,
which do not enable us to make more detailed study about certain quantities, such
as the internal force; similarly, mean, variance and so on, only provide a somewhat
limited information. In order to make a more refined study of the random processes,
detailed information about probability distributions are required.
15.5 Stationary Process, Ergodic Process and Temporal
Averages
We have so far considered the properties of random process at a fixed time instant or
at two fixed time instants. The variation of these properties, when time instants are
assumed to vary is discussed next.
15.5.1 Stationary Process
A random process is said to be stationary, if its probability distributions are invariant
under a shift of time scale, that is, independent of the origin. This suggests that
the probability density at any time instant is valid for later times as well. This is
somewhat analogous to the assumptions made in the case of steady-state vibration,
where the resulting vibration is independent of the initial conditions.
Wave and wind generation are random processes, which can be treated as
stationary processes without introducing much error in the analysis. Earthquake
ground motions, however, cannot be treated as stationary processes, as the probability distributions vary with time. For a process to be strictly stationary, it can have
no beginning and no end and each sample should extend from −∞ to +∞.
For stationary processes, the first-order probability distribution p(x) is independent
of time. As such E [X ] and σ
2 are also independent of time.
In case of second-order probability density function, p(x 1 , x 2 ) is a function of
time lag between t 1 and t 2 and not a function of t 1 and t 2 individually. Time lag
τ = t 2 − t 1 . Then the autocorrelation function is
E[X 1 X 2 ] = E[X (t)X (t + τ )] = R(τ )
(15.20)
Since by definition of stationary process, the probability density function p(t, t +
τ ) is independent of t, the autocorrelation function is related to time lag only.R(τ )
will be used to denote autocorrelation function of a stationary random process.