Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 42

data well. Lohani and Wang ( 1987 ) also reported to have used this model to study
the monthly water quality data in the Chung Kang River located at the northern part
of Miao-Li County in the middle of Taiwan. Jayawardena and Lai ( 1991 ) applied an
adaptive Auto Regressive Moving Average (ARMA) model approach for water
quality forecasting. MacLeod and Whit

eld ( 1996 ) analyzed water quality data
using Box-Jenkins time series analysis of the Columbia River at Revelstoke.
Caissie et al. ( 1998 ) studied water temperature in the Catamaran Brook stream. The
short-term residual temperatures were modeled using different air to water relations,
namely a multiple regression analysis, a second-order Marcov for process, and a
Box-Jenkins time series model. Asadollahfardi ( 2002 ) applied Box Jenkins and
Exponential smoothing models to monthly surface water quality data in Tehran for
3 years. Most of the models indicated seasonality. Kurunc et al. ( 2004 ) applied
Auto Regressive Integrated Moving Average (ARIMA) and Thomas

ring tech-
niques for 13 years of monthly data about the Duruacasu station at Yesilirmark
River. Asadollahfardi et al. ( 2012 ) also worked about water quality of Jaj-Rud
River and applied ARMA time series models.

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4.3 Time Series

A time series is a chronological sequence of observations on a particular variable,
such as daily, monthly or annual air or water quality, daily mean temperature, and
so on (Box and Jenkins 1976 ).
Time series data are often examined in the hope of discovering a historical
pattern that can be exploited in the preparation of a forecast. To identify this pattern,
it is often convenient to think of a time series as consisting of several components.
The components of a time series are trend, cycle, seasonal variations, and irregular

fluctuations (Haan 1977 ).
Trend refers to the upward or downward movement, which characterize a time
series over a period of time. Thus, the trend re

ects the long-run growth or decline
in time series. Its movements can represent a variety of factors. For example, long-
run downward movements in DO might be due to gradual pollution (Bowerman and
O ' Connell 1987 ; Haan 1977 ).

Cycle refers to recurring up and down movements around trend levels. These

10 years or even longer, measured from peak
to peak or trough to trough. For instance, in arid zones, a cycle of an 11 year length
is expected, for dry and wet years.
Seasonal variations are periodic patterns in a time series which repeat themselves
in a certain length of time, (e.g. in a calendar year). Seasonal variations are usually
caused by factors such as weather and customs (Cruz and Yevjevich 1972 ). The
obvious example is the average monthly temperature, which is clearly seasonal in
nature. Other examples are those of water quality parameter related to temperature.
Average DO is high in winter and lower in the summer months, exhibiting a
seasonal pattern.

fluctuations can have a duration of 2

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