Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 51

where S is the length of season, for instance S = 4 for quarterly data and S = 12 for
monthly data. Let y t denote an appropriate pre-differencing transformation. This can
be shown as, y t ¼ ln y t if a logarithmic transformation is needed. Another example is
the normalizing transformation of Box and Cox ( 1964 ) which is de ned as:

(

y k l 1
y k l

k 6 ¼ 0

y t ¼ ð 4 : 20 Þ

lny ; k ¼ 0

where y k is the geometric mean of y k values.
Then the general stationery transformation is given by:

Z t ¼ r S r d y t ¼ ð 1 B s Þ D ð 1 B Þ d y t ð 4 : 21 Þ

where d is the degree of non-seasonal differencing and D is the degree of seasonal
differencing used to reach stationary. Either of D and d can be taken as 0, 1, 2 or at
most 3.
The SACF within each season behaves as it was described for non-seasonal
models. Ignoring the behavior of the SACF within each season and only consid-
ering it at lag s multiples of S can describe the seasonal behavior of the time series.

Similarly, the SPACF of seasonal models can be studied within and between
seasons. Once the stationarity transformation is performed, there is:

'

Z t ¼ ð 1 B s Þ D ð 1 B Þ d y t ð 4 : 22 Þ

which provides Z b ; Z b þ 1 ; ...; Z n model describing these values. This model con-
sists of two components determined by their respective operators. One set of
operator

s models the seasonal pattern while the other set does the non-seasonal
pattern, of the data. The general model of order (p, P, q, Q) is written as:

'

/ p ð B ÞU p ð B s Þ Z t ¼ d þ h q ð B ÞH Q ð B s Þ a t

ð 4 : 23 Þ

where / p ðÞ ¼1 / 1 B / p B p is the non-seasonal autoregressive operator

2s U P ; s B Ps is called the seasonal
autoregressive operator of order P ; h q ðÞ ¼1 h 1 ð Þh q B q is the non-
seasonal moving average operator of order q, H Q B ðÞ ¼1 H 1 ; s B s H 2 ; s B

U p B ðÞ ¼1 U 1 ; s B s U 2 ; s B of order p,

2s

H Q ; s B Qs is called the seasonal moving average operator of order Q, and ʴ is a

constant term. / 1 ; / 2 ; ...; / p ; U 1 ; s ; U 2 ; s ; ...; U P ; s ; h 1 ; h 2 ; ...; h q ; h 1 ; s ; h 2 ; s ; ...; h Q ; s

and are unknown parameters which can be estimated from sample data.

a t ; a t 1 ; ...

ʴ

are random shocks which are assumed to be statistically independent of
each other, and identically distributed as normal with zero mean, and a constant
variance. This is true for each and every time period t.