Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 46

final model is determined, a model is used to compute the future values.
These values are expressed in two forms: point estimates and interval estimates.
The future values along with their respective probability limits are the products of
time series modelling.
A point to be considered in forecasting future values of a positive variable is that
negative lower limit for the forecast is obtained. In this situation, zero is used in
place of a negative lower limit.
The Box-Jenkins approach is based on the notion of stationary time series brie

When a

y

explained in the following section.

4.6 Stationary and Non-stationary Time Series

Classical Box-Jenkins models are used for stationary time series. Thus, to tenta-
tively identify a Box-Jenkins model, it is necessary to verify that the time series
used in forecasting is stationary. If it is not, the time series should be transformed
into a series of stationary time series values.
Intuitively, a time series is called stationary if their statistical property, such as
mean, variance remains essentially constant through time. If n values of y 1 ,y 2 ,

…

,y n
of a time series are observed, by examining their plot against time, their stationery
can be checked. If n values seem to

fluctuate with constant variation around a
constant level, then it is reasonable to believe that the time series is stationary.
In practice, this is done with the help of sample autocorrelation function (SACF) and
sample partial autocorrelation function (SPACF). If n values do not uctuate around

a constant mean or do not fluctuate with constant variation, then it is reasonable to

believe that the time series is nonstationary. In this case, one can sometimes trans-
form the nonstationary time series values into stationary time series values, by taking
the first, second or higher differences of the nonstationary time series values (Bendat

and Piersol 1966 ).
The

first difference of the time series values y 1 ,y 2 , … ,y n are de ned as:

Z t = y t y t 1 = r y t
t =2

ð 4 : 3 Þ

; ...; n

where, the difference operator is related to backward shift operator B, i

∇

e :r =1 B ð 4 : 4 Þ

where By = y t − 1 and consequently, B j y t =y t − j.