Nonparametric Statistic and the Runs Test for Stationarity - Signal Processing of Random Physiological Signals - page 68

The test failed to reject H 0 , since the calculated test value of D is less than the
table value (0 . 2 < 0 . 410). It is concluded that the data are stationary.

8.2.2 One-Sample Runs Test
For the One-Sample Runs test, the sample record must be illustrated by symbols; gener-
ally, by taking the median of the sample data and assigning all values above the median
a symbol, n 1 , and all values below the median another symbol, n 2 . Once the values are
represented by symbols, the number of runs can be determined. A run is defined as a
succession of identical symbols.
For small samples, n 1 and n 2

20, the number of runs, r , is then compared to the
value from the Runs table. If r falls within the critical values, then the null hypothesis is
accepted, which infers failure to reject the null hypothesis, H 0 .
If n 1 or n 2 is larger than 20, the table value is not used. For large samples, an
approximation of the sampling distribution of r is the normal distribution. Therefore, a
Z score is computed to test H 0 , as shown in (8.9):

<

1

2 n 1 n 2
n 1

r −

n 2 +

r − μ r

+

Z = = 2 n 1 n 2 (2 n 1 n 2 (8.9)

σ r

− n 1 − n 2 )

( n 1 + n 2 ) 2 ( n 1 + n 2 − 1)

The calculated Z value from (8.9) is then compared to values in the Normal Distribution
Table, which shows the one-tailed probability. For a two-tailed test, double the probability
value ( p ) given in the table. If the p associated with the observed value of r is less than
or equal to the significant level ( ), then the null hypothesis H 0 is rejected and it is

concluded that the data are nonstationarity.

α

8.2.3 Testing for Ergodicity
8.2.3.1 Wilcoxon-Matched Pairs Signed-Ranks Test
The Wilcoxon-matched pairs test is a powerful test because it considers relative magni-
tude and direction of differences. The process steps are as follows: