Efficacy of Time Series Tests: A Critical Assessment - Hydrologic Time Series Analysis: Theory and Practice - page 157

7.8 Investigating Periodicity
The harmonic analysis was performed to examine the periodic/cyclic nature
of the annual rainfall series for past 46 years (1957-2002) and maximum
rainfall (one-day maximum, consecutive 2-, 3-, 4-, 5- and 6-day maximum)
series for past 47 years (1956-2002) of Kharagpur. The harmonic analysis was
performed up to 23 maximum number of harmonics in all these rainfall series.
It was found that the periodicity effect is not apparent in the seven rainfall
series and about 75% variation in all the rainfall series is caused by 15
harmonics cumulatively. Furthermore, since the periodicity in a time series is
generally introduced due to the earth's rotation around the sun (Kite, 1989), it
is expected to be inherent in the time series having a period of less than one
year (e.g., monthly and/or seasonal) rather than annual or annual maximum
rainfall series. In this context, the average monthly rainfall time series of
Kharagpur were also subjected to the harmonic analysis, which revealed six-
month and one-year periodicities in the rainfall series.
7.9 Investigating Persistence
The persistence in the total annual and the selected consecutive days' maximum
rainfall series was tested by using the autocorrelation technique. Separate
autocorrelograms were prepared for all the rainfall time series as shown in
Figs 7.4(a-g). According to the time series lengths of 46 and 47 years, the
autocorrelation functions/coefficients were determined up to a maximum order
of 12 years. The upper and lower bounds for defining the non-critical
(acceptable) range were computed by the Anderson's test (Anderson, 1942)
for large samples. It is apparent from these figures that for all the rainfall time
series, the autocorrelograms have a zigzag path closer to zero, and the
autocorrelation coefficients are more or less same for a given time lag. Further,
all the autocorrelograms have almost the similar profile over the time lag
scale.
Moreover, for the annual rainfall time series, the autocorrelation function
at nine years time lag falls outside the acceptable region [Fig. 7.4(a)]. The
appearance of autocorrelation coefficient ( r k ) beyond the upper critical limit
indicates persistency in the annual rainfall time series of Kharagpur with nine
years time lag. For the remaining time lags, the autocorrelation coefficients
were found within the acceptable range. Thus, it can be safely inferred that the
annual rainfall time series of Kharagpur is slightly persistent in nature with a
nine-year time lag. This means that the annual rainfall of a t th year has some
relationship with the annual rainfall of ( t + 9) th year, though it is not significant
in this study as the deviation of r k from the critical limit (i.e., 0.14) may not
be considered very significant for practical purposes. However, persistency
from the annual rainfall series was removed through transformation of the
series by removing autocorrelation for nine years time lag prior to the