Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 47

4.7 The Sample Autocorrelation and Partial
Autocorrelation Functions

The behavior of the SACF and the SPACF are important in tentative identi cation

of stationary time series models. For the values of a stationary time series Z b ,Z b+1 ,
… ,Z n which may be the original time series values or the transformed time series

values, SACF is de

ned as follows. The sample autocorrelation at lag k denoted by

r k is:

n k
t = b ð z t z Þð z t þ k z c Þ
n
t = b ð z t z c Þ

r k =

2

ð 4 : 5 Þ

P n
t = b z t
n b þ 1

z =

, K, it calls the sample
autocorrelation function (SACF). This quantity measures the linear relationship
between time series observations separated by a lag of k time units. The r k is a
coef

Considering r k a function of lag k, for k = 1, 2, …

cient of correlation and it is always between 1 and +1. The standard error of

−

r k is given by:

2 3

1 þ 2 k 1
j¼1

2
j

r

4 5

n b þ 1

ð 4 : 6 Þ

s r k =

2

k =1 ; 2 ; ...

The t r k statistic is then computed as:

t r k = r k

ð 4 : 7 Þ

s r k

which is used to test the signi …

Plotting r k against k provides the SACF. The behavior of this function is a key
tool for identi

cance of r k , for k = 1, 2,

cation of the stationary of a time series and its order.
To employ the Box-Jenkins approach, one must examine and try to classify the
behavior of the SACF. The SACF for a non-seasonal time series can display a
variety of behaviors (Bowerman and O

' Connell 1987 ). These are explained as

follows: