Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 49

4.8 Classi cation of Non-seasonal Time Series Models

For a time series consisting of Z b ; Z b þ 1 ; ...; Z n , where Z t is the original or trans-
formed values of a time series, an autoregressive model of order p ; AR ð p Þ ,is

de ned as:

Z t = / 1 Z t 1 þ / 2 Z t 2 þþ/ r Z t p þ a t ð 4 : 12 Þ

where ϕ 1

··· ϕ r are

fixed coef cients and a t ,t=1,2, … , n are independent random

2
t

variables with zero mean and constant variance . They are usually assumed as

normally distributed. Using the backward shift operator B, Eq. ( 4.12 ) can be written
as:

r

/ p ð B Þ Z t = a t ð 4 : 13 Þ

/ p ð B Þ =1 / 1 B / p B p and BZ t = Z t 1 ; ...; B p Z t = Z t p .
A moving average model of order q, MA (q), is represented as:

where

Z t = a t h 1 a t 1 h q a t q ð 4 : 14 Þ

Or employing the backward shift operator B,

Z t = h q ð B Þ a t ð 4 : 15 Þ

With

h q ð B Þ =1 h 1 B h q B q ð 4 : 16 Þ

The general nonseasonal autoregressive moving average model of order (p, q) is

Z t = d þ / 1 Z t 1 þþ/ p Z t p þ a t h 1 a t 1 h q a t q ð 4 : 17 Þ

ʴ

. It has an autoregressive part which expresses
the current value Z t as a function of past values Z t 1 ; Z t 2 ; ...; Z t p with unknown
coef

This model utilizes a constant term

ϕ p . In addition, it has a moving average part which is
represented by a t ; a t 1 ; ...; a t q , with unknown

cients (parameters) ϕ 1 , … ,

q 1 ; ...; q q . The
variable Z t is also considered as a function of a random variable, as, a t 1 ; ...; a t q .
In Eq. ( 4.17 ), the constant term

xed parameters

ʴ can be shown as equal to equal μ˕ p (B), where

μ is the mean of the stationary time series Z t . In concise notation, Eq. ( 4.17 )is

presented as:

/ p ð B Þ Z t = d þ h q ð B Þ a t ð 4 : 18 Þ

There are statistical tests, which can be used to decide whether to include ʴ in the

model.