Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 50

If the stationary time series Z b ; Z b þ 1 ; ...; Z n is the original series, then assuming

ʼ is equal to zero, this implies that these original time series values are fluctuating

around a zero mean, whereas uctu-

ating around a non-zero mean. In such a case one can use Z t Z in place of Z t .
Then

l 6 ¼ 0 implies that these original values are

can be removed from the model. If the stationary time series
Z b ; Z b þ 1 ; ...; Z n are different from those of the original time series values, where

ʴ

ʼ
is not assumed to be zero, it can be assumed that there is a deterministic trend in
those original values. Here the deterministic trend refers to a tendency to the
original values to move persistently upward (if < 0). If a

time series does not exhibit a deterministic trend, then any trend (or failure of the
series to

ʴ > 0) or downward (if ʴ

fluctuate around a central value) is stochastic. The stochastic trend is more
realistic in practical situations since it does not dictate a certain path to be taken by
the future values.

4.9 Guidelines for Choosing a Non-seasonal Models

ARMA (p, q) models of Eq. ( 4.17 ) are speci ed by choosing suitable orders for AR

operator ʸ q (B). This boils down to specifying p and q as

positive integers. It is illustrated by some guidelines for choosing such numbers.
See Bowerman and O

ϕ p (B) and MA operator

' Connell ( 1987 ).

4.10 Seasonal Box-Jenkins Models

Seasonality may be de ned as the common feature of most time series data being

the periodic pattern of fluctuations in time series values. For example the meteo-

rological time series recorded in a location such as temperature, rainfall, and
radiation, exhibit a marked periodic behavior of 12 months. River discharges have
periodic nature too. This feature can be accompanied by any one or more of a trend,
cyclical and irregular fluctuations. Seasonal change is an example of non-sta-

tionarity, which can be removed by a seasonal differentiation. Seasonal differencing
takes the differences of two similar observations, one from each period. For
example in the case of monthly average temperature, the difference of the values of
the same months from consecutive years removes seasonality. For seasonal dif-
ferencing, the seasonal operator sisde ned as:

∇

r s ¼ 1 B s ð 4 : 19 Þ