Environmental Engineering Reference
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in which n is the number of observations in the original time series, and S is the
length of the season in the series. A good choice of K is 2
n p . These statistics both
have
˅ ¼ K m p and m p are the
number of parameters that must be estimated in the model under consideration.
Large values of Q or Q* signify a lack of
ˇ 2 distribution with
˅
degrees of freedom, where
fitting the model, whose sources can be
traced by the following speci
c tests.
Independence of residuals could be checked by various routine statistical
methods. However, this is best carried out by sample autocorrelation function of the
residuals (RSAC). If independence holds, the RSAC would re
ect it by having no
signi
cant correlations.
Normality could also be checked by several methods but the easiest one is the
p-p plot (normal probability plot) of the residuals. In this method, a plot is drawn in
which the x-axis indicates the percentiles of the standard normal distribution and
y-axis represents the observed percentiles of the standardized residuals. If residuals
follow a normal distribution, the p-p plot would resemble a straight line from (0, 0)
to (1, 1). Wild departures from this line are a sign of non-normality and may need
some sort of transformation for the data to make them normal.
To ascertain the absence of any systematic pattern in the residuals, especially of
the periodic nature, the integrated paradigm can be used. The integrated (cumulative)
periodogram for a random series having no periodic feature is a straight line of 45
°
passing through (0, 0) and (0.5, 1). A signi
cant departure from this line suggests
some periodic aspects left in the residuals, which could be retrieved and incorpo-
rated, in the model.
In general a
flow diagram of the iterative approach for the Box-Jenkins model
building and stages to reach an adequate model is illustrated in Fig. 4.2 , and
Table 4.1 .
Postulate General Class of Models
Identify Model to be Tentatively Entertained
Estimate Parameters in Tentatively Entertained Model
No
Diagnostic Checking
(is the model adequate ?)
Yes
Use Model for Forecasting or Control
Fig. 4.2 Stage in the iterative approach to model building (Box and Jenkins 1976 )
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