Nonparametric Statistic and the Runs Test for Stationarity - Signal Processing of Random Physiological Signals - page 60

Unfortunately, it is not often possible to describe the properties of a stationary ran-
dom process by computing time averages over specific sample functions in the ensemble,
since many times an ensemble cannot be collected, but rather a single long recording of
a signal is acquired. In this case, the long recording is divided into smaller segments and
the data are tested for the Ergodic properties.
For example , the mean value μ

x ( k ) and the autocorrelation function R x (

λ, k )of

the k th sample function can be obtained with (8.1) and (8.2).

T

1
T

μ x ( k ) = Lim

T

x k ( t ) dt (8.1)

→∞

0

T

1
T

R x (

λ, k ) = Lim

T

x k ( t ) x k ( t + τ ) dt (8.2)

→∞

0

The random process { x ( t ) } is Ergodic and Stationary if the mean value μ

x ( k )

and the autocorrelation function R x ( , k ) of the short-time segments do not differ

when computed over different sample functions. It should be noted that only stationary
random processes can be Ergodic . Another way of thinking is “all Ergodic Random
Processes are Stationary.” For many applications, verification of weak stationarity justifies
an assumption of strong stationary (Bendat and Piersol).

τ

8.1.2 Alternative Methods to Test for Stationarity
So far, the methods described require integral functions and parametric statistical testing
or moments describing the properties of the random process. A disadvantage in the use
of parametric statistics is that the random data must be tested for independence and
normality. By normality of the distribution, we mean that the data must be “Gaussian
distributed.” By far the most common method used is to test whether the data are Ergodic.
In general, three assumptions are made, when random data are tested for Ergodicity. The
basic assumptions are as follows.

1. That any given sample record will properly reflect the nonstationary character

of the random process.