Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 52

4.11 Guidelines for Identi cation of Seasonal Models

As in case of non-seasonal identi cation of the seasonal model (Eq. ( 4.23 )),

requires whether to include ʴ in the model. This task is accomplished by performing

cant test using Z.
The behavior of the SACF and PSACF of the Z b ; Z b þ 1 ; ...; Z n values at the non-
seasonal level is used to decide which (if any) of the non-seasonal moving average
operators of order q, and

a signi

h q ðÞ ¼1 h 1 ð Þh q B q and the non-seasonal

autoregressive operator of order p, and / p ðÞ ¼1 / 1 B / p B p should be

employed.
The behavior of the SACF and the SPACF of Z b ; Z b þ 1 ; ...; Z n values at the
seasonal level are used to determine which (if any) of the seasonal moving average
operators of order Q, and

2s H Q ; s B Qs and the
seasonal autoregressive operator of order P, and F p B ðÞ ¼1 F 1 ; s B s F 2 ; s B

H Q B ðÞ ¼1 H 1 ; s B s H 2 ; s B

2s
F P ; s B Ps should be utilised. In this regard, some guidelines are available which
are used to specify a tentative model, see Bowerman and O ' Connell ( 1987 ).

ed the appropriate operators, and inserting the appropriate sta-
tionarity transformation, one obtains the model.

Having identi

/ p ð B Þ/ p ð B s Þr S r d y t ¼ d þ h q ð B Þh Q ð B s Þ a t

ð 4 : 24 Þ

whose parameters should be estimated and its accuracy should be checked. These
steps are similar to those of non-seasonal models.

4.12 Diagnostic Checking

Diagnostic checking is used to see whether or not the identi ed and estimated

model is adequate. If the model proved to be inadequate, it must be modi ed and

improved. The diagnostic methods employed will help to decide how the model can
be improved.
A good way of finding the adequacy of an overall model is to analyze the
residuals obtained from the model. According to the assumptions, which the model
is based upon, the residuals should be independent and identically distributed. They
should follow a normal distribution. Thus, the diagnostic examination methods
consist of checking for independence, normality, and the absence of any systematic
behavior in the residuals.
Before embarking on a speci

c test, an overall check, which utilizes the sample
autocorrelation function of the residuals, is employed. This is called a portmanteau
lack of t test. The test statistic for the portmanteau test is either the Box-Pierce (Box

and Jenkins 1976 ) Statistic Q ¼ n 0 K

i ð a Þ or the preferred Ljung-Box (Box and
Jenkins 1976 ) statisticQ ¼ n 0 ð n 0 þ 2 Þ K

i¼1 r

2

2

i ð a Þ=ð n 0 i Þ .where

n 0 ¼ n ð d SD Þ

i¼1 r