Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 58

4.16 Adaptive Control Procedures

One cannot expect a forecasting system to produce perfect forecasts of a time series.
Therefore, it is required to determine whether the forecast errors are tolerable or are
beyond the reasonable bounds. To this end, assume a history of T one-step ahead
forecast errors, e 1 ðcÞ; e 2 ðcÞ; ...; e T ðcÞ is computed. Here,

ʳ

denotes the fact that the

ʳ

forecast errors are dependent on the smoothing constant , used to compute a one-step

ned by C ðc; T Þ ¼ T

ahead forecast. The cumulative forecast error can be de

t¼1 e t ðcÞ

until time period T is reached. Then C( ʳ ,T) will be equal to C ðc; T 1 Þþ eT ðcÞ .

Now, the mean absolute deviation is de ned by:

T
t¼1 e t ðcÞ

j j

D ðc; T Þ = ð 4 : 42 Þ

T

Then the tracking signal TS( ʳ ,t)isde ned as:

TS ðc; T Þ = C ðc; T Þ
D ðc; T Þ

ð 4 : 43 Þ

If TS( ʳ ,T)is “ large ” , it means that C( ʳ ,T) is also large relative to D( ʳ ,T).
Equivalently this re

ects that absolute errors are large and some measures should be
taken. These measures can be taken as:

1. Change the smoothing constant,
2. Change the model,
3. Use different values of smoothing constant at different times.

Some procedures have been developed to implement these measures on the
computer (Chow 1965 ).
Considering the de cit of water in Iran, protection of water resources against

pollution is vital. In this regard, water quality monitoring is a tool which produces
up to date information. Having a great amount of raw data without interpretation is
not suf

cient, and it is necessary to analysis data and predicts the variation of water
quality in the future for any decision making on water quality management.
Recently, more researchers have become interested in the application of time series
models for the prediction of water quality. Time series approach for analyzing water
resources were

first applied by Thomann ( 1967 ) who studied variation of temper-
ature by the time and dissolved oxygen level for the Delaware Estuary. The data
was obtained by continuous recording by monitoring stations, operated jointly by
the U.S. Geological Survey Department and the city of Philadelphia. Carlson et al.
( 1970 ) and McMichael and Hunter ( 1972 ) reported the successful use of the
Box-Jenkins method for time series analysis.
The Box-Jenkins method for the time series analysis was applied to model the
hourly water quality data recorded in the St. Clair River near Corunna, Ontario, for